U.S. patent application number 14/967803 was filed with the patent office on 2016-11-17 for device, method and system of pricing financial instruments.
The applicant listed for this patent is SUPER DERIVATIVES, INC.. Invention is credited to David Gershon.
Application Number | 20160335720 14/967803 |
Document ID | / |
Family ID | 44225309 |
Filed Date | 2016-11-17 |
United States Patent
Application |
20160335720 |
Kind Code |
A1 |
Gershon; David |
November 17, 2016 |
DEVICE, METHOD AND SYSTEM OF PRICING FINANCIAL INSTRUMENTS
Abstract
Some demonstrative embodiments include methods, devices and
systems of pricing financial instruments. In one embodiment, a
pricing module may be configured to receive first input data
corresponding to at least one parameter defining a first option on
an underlying asset and second input data corresponding to at least
one current market condition relating to said underlying asset,
and, based on said first and second input data, to determine a
price of the first option according to a volatility smile
satisfying a first criterion relating to a sum of a first
correction corresponding to the first option and a second
correction corresponding to a second option representing a position
opposite to a position of a the first option and having
substantially a same absolute delta value as the first option,
wherein the first correction relates to a difference between a
theoretical price of the first option and the price of the first
option according to the volatility smile, and wherein the second
correction relates to a difference between a theoretical price of
the second option and the price of the second option according to
the volatility smile. Other embodiments are described and
claimed.
Inventors: |
Gershon; David; (Tel Aviv,
IL) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SUPER DERIVATIVES, INC. |
NEW YORK |
NY |
US |
|
|
Family ID: |
44225309 |
Appl. No.: |
14/967803 |
Filed: |
December 14, 2015 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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14091602 |
Nov 27, 2013 |
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14967803 |
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13713200 |
Dec 13, 2012 |
8620792 |
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14091602 |
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12983992 |
Jan 4, 2011 |
8626630 |
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13713200 |
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61291942 |
Jan 4, 2010 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06Q 40/06 20130101;
G06Q 40/04 20130101 |
International
Class: |
G06Q 40/04 20060101
G06Q040/04; G06Q 40/06 20060101 G06Q040/06 |
Claims
1. (canceled)
2. A system comprising: at least one interface to interface over a
communication network with at least a trading system and a market
data system; and a processor configured to determine at least one
price of a first option on an underlying asset, based on a model
price of the first option according to a pricing model, the at
least one price of the first option comprising at least one price
selected from the group consisting of a bid price of the first
option, and an offer price of said first option, the processor
configured to trigger transmission of a trade of the first option
to be submitted to the trading system via the communication
network, the trade comprising the at least one price of the first
option, the processor configured to process real time market data
from the market data system to detect a change in one or more
parameters corresponding to the underlying asset, the processor
configured to, based at least on the detected change in the one or
more parameters, automatically recalculate at least one updated
price of the first option, and automatically trigger transmission
of an updated trade of the first option to be submitted to the
trading system via the communication network, the updated trade
comprising the at least one updated price of the first option,
wherein the model price of the first option satisfies a first
criterion and a second criterion, the first criterion relating to a
sum of a first correction corresponding to the first option and a
second correction corresponding to a second option, and the second
criterion relating to a difference between the first correction
corresponding to the first option and the second correction
corresponding to the second option, wherein the second option
represents a position opposite to a position of the first option
and has a same absolute delta value as the first option, wherein
the first correction relates to a difference between the model
price of the first option and a price of the first option according
to a Black-Scholes model with an At-The-Money (ATM) volatility, and
wherein the second correction relates to a difference between a
model price of the second option according to the pricing model and
a price of the second option according to the Black-Scholes model
with the ATM volatility.
3. The system of claim 2, wherein said processor is to
automatically recalculate the at least one updated price of the
first option based at least on a change in the model price of said
first option.
4. The system of claim 2, wherein said processor is to
automatically recalculate the at least one updated price of the
first option based at least on a change in a price of said
underlying asset.
5. The system of claim 2, wherein the processor is to determine the
model price of said first option based on first data corresponding
to at least one parameter defining the first option, and second
data corresponding to at least one current market condition
relating to said underlying asset.
6. The system of claim 5, wherein said first data comprises an
indication of at least one parameter selected from the group
consisting of a type of said first option, an expiration date of
said first option, a trigger for said first option, and a strike of
said first option.
7. The system of claim 5, wherein said second data comprises an
indication of at least one parameter selected from the group
consisting of a spot value, a forward rate, an interest rate, a
volatility, an at-the-money volatility, a delta risk reversal, a
delta butterfly, a delta strangle, a 10 delta risk reversal, a 10
delta butterfly, a 10 delta strangle, a 25 delta risk reversal, a
25 delta butterfly, a 25 delta strangle, a caplet, a floorlet, a
swap rate, a security lending rate, and an exchange price.
8. The system of claim 2, wherein the first criterion requires that
the sum of the first and second corrections is proportional to a
sum of first and second volatility convexities corresponding to the
first and second options, and wherein the second criterion requires
that a difference between the first and second corrections is
proportional to a difference between first and second delta
convexities corresponding to the first and second options.
9. The system of claim 8, wherein the first criterion requires that
the sum of the first and second corrections is proportional to the
sum of the first and second volatility convexities according to a
first proportionality function, which is based on said delta, and
wherein the second criterion requires that the difference between
the first and second corrections is proportional to the difference
between the first and second delta convexities according to a
second proportionality function, which is based on said delta.
10. The system of claim 2, wherein the first and second criteria
require satisfying the following equations: .zeta. C .DELTA. +
.zeta. P .DELTA. = A ( .DELTA. ) Vega .DELTA. d 1 2 ( 1 .sigma. K
Call + 1 .sigma. K Put ) .zeta. C .DELTA. - .zeta. P .DELTA. = B (
.DELTA. ) Vega .DELTA. d 1 S t ( 1 .sigma. K Call + 1 .sigma. K Put
) ##EQU00027## wherein .zeta..sub.C.sup..DELTA. and
.zeta..sub.P.sup..DELTA. denote said first and second corrections,
wherein .DELTA. denotes said delta, wherein A(.DELTA.) and
B(.DELTA.) denote first and second functions of A, respectively,
wherein Vega.sup..DELTA. denotes a vega of the first and second
options, wherein t denotes a time to expiration of said first
option, wherein d.sub.1 denotes a predefined function of the time
to expiration of said first option, wherein S denotes a price of
said underlying asset, and wherein .sigma..sub.K.sub.Call and
.sigma..sub.K.sub.Put denote a volatility of the first option and a
volatility of the second option, respectively.
11. The system of claim 2, wherein said first option includes a
Vanilla option.
12. The system of claim 2, wherein said underlying asset comprises
a financial asset.
13. The system of claim 2, wherein said underlying asset is related
to at least one asset type selected from the group consisting of a
commodity, a stock, a bond, a currency, an interest rate, and the
weather.
14. A product including a non-transitory storage medium having
stored thereon instructions that, when executed by a machine,
result in: determining at least one price of a first option on an
underlying asset, based on a model price of the first option
according to a pricing model, the at least one price of the first
option comprising at least one price selected from the group
consisting of a bid price of the first option, and an offer price
of said first option; triggering submission of a trade of the first
option to a trading system via a communication network, the trade
comprising the at least one price of the first option; processing
real time market data from a market data system to detect a change
in one or more parameters corresponding to the underlying asset;
and based at least on the detected change in the one or more
parameters, automatically recalculating at least one updated price
of the first option, and automatically triggering submission of an
updated trade of the first option to the trading system via the
communication network, the updated trade comprising the at least
one updated price of the first option, wherein the model price of
the first option satisfies a first criterion and a second
criterion, the first criterion relating to a sum of a first
correction corresponding to the first option and a second
correction corresponding to a second option, and the second
criterion relating to a difference between the first correction
corresponding to the first option and the second correction
corresponding to the second option, wherein the second option
represents a position opposite to a position of the first option
and has a same absolute delta value as the first option, wherein
the first correction relates to a difference between the model
price of the first option and a price of the first option according
to a Black-Scholes model with an At-The-Money (ATM) volatility, and
wherein the second correction relates to a difference between a
model price of the second option according to the pricing model and
a price of the second option according to the Black-Scholes model
with the ATM volatility.
15. The product of claim 14, wherein the instructions result in
automatically recalculating the at least one updated price of the
first option based at least on a change in the model price of said
first option.
16. The product of claim 14, wherein the instructions result in
determining the model price of said first option based on first
data corresponding to at least one parameter defining the first
option, and second data corresponding to at least one current
market condition relating to said underlying asset.
17. The product of claim 14, wherein the first criterion requires
that the sum of the first and second corrections is proportional to
a sum of first and second volatility convexities corresponding to
the first and second options, and wherein the second criterion
requires that a difference between the first and second corrections
is proportional to a difference between first and second delta
convexities corresponding to the first and second options.
18. The product of claim 14, wherein the first and second criteria
require satisfying the following equations: .zeta. C .DELTA. +
.zeta. P .DELTA. = A ( .DELTA. ) Vega .DELTA. d 1 2 ( 1 .sigma. K
Call + 1 .sigma. K Put ) .zeta. C .DELTA. - .zeta. P .DELTA. = B (
.DELTA. ) Vega .DELTA. d 1 S t ( 1 .sigma. K Call + 1 .sigma. K Put
) ##EQU00028## wherein .zeta..sub.C.sup..DELTA. and
.zeta..sub.P.sup..DELTA. denote said first and second corrections,
wherein .DELTA. denotes said delta, wherein A(.DELTA.) and
B(.DELTA.) denote first and second functions of .DELTA.,
respectively, wherein Vega.sup..DELTA. denotes a vega of the first
and second options, wherein t denotes a time to expiration of said
first option, wherein d.sub.1 denotes a predefined function of the
time to expiration of said first option, wherein S denotes a price
of said underlying asset, and wherein .sigma..sub.K.sub.Call and
.sigma..sub.K.sub.Put denote a volatility of the first option and a
volatility of the second option, respectively.
Description
CROSS REFERENCE
[0001] The present application is a Continuation application of
U.S. patent application Ser. No. 14/091,602, filed on Nov. 27,
2013, which is a Continuation application of U.S. patent
application Ser. No. 13/713,200, filed Dec. 13, 2012, which is a
Continuation application of U.S. patent application Ser. No.
12/983,992, filed Jan. 4, 2011, which claims the benefit of and
priority from U.S. Provisional Patent application 61/291,942,
entitled "Method and system of pricing financial instruments",
filed Jan. 4, 2010, the entire disclosures of all of which are
incorporated herein by reference.
FIELD
[0002] The disclosure relates generally to financial instruments
and, more specifically, to methods and systems for pricing, e.g.,
real-time pricing, of options and/or for providing automatic
trading capabilities.
BACKGROUND
[0003] Pricing financial instruments is a complex art requiring
substantial expertise and experience. Trading financial
instruments, such as options, involves a sophisticated process of
pricing typically performed by a trader.
[0004] The term "option" in the context of the present application
is broadly defined as any financial instrument having option-like
properties, e.g., any financial derivative including an option or
an option-like component. This category of financial instruments
may include any type of option or option-like financial instrument,
relating to some underlying asset. Assets as used in this
application include anything of value; tangible or non-tangible,
financial or non-financial, for example, stocks; currencies;
commodities, e.g., oil, metals, or sugar; interest rates;
forward-rate agreements (FRA); swaps; futures; bonds; weather,
e.g., the temperature at a certain area; electricity; gas emission;
credit; mortgages; indices; and the like. For example, as used
herein, options range from a simple Vanilla option on a single
stock and up to complex convertible bonds whose convertibility
depends on some key, e.g., the weather.
[0005] The term "Exchange" in the context of the present
application relates to any one or more exchanges throughout the
world, and includes all assets/securities, which may be traded in
these exchanges. The terms "submit a price to the exchange",
"submit a quote to the exchange", and the like generally refer to
actions that a trader may perform to submit a bid and/or offer
prices for trading in the exchange. The price may be transferred
from the trader to the exchange, for example, by a broker, by
online trading, on a special communication network, through a
clearing house system, and/or using in any other desired system
and/or method.
[0006] The price of an asset for immediate, e.g., 1 or 2 business
days, delivery is called the spot price. For an asset sold in an
option contract, the strike price is the agreed upon price at which
the deal is executed if the option is exercised. For example, a
stock option involves buying or selling a stock. The spot price is
the current stock price on the exchange in which is the stock is
traded. The strike price is the agreed upon price to buy/sell the
stock if the option is exercised.
[0007] To facilitate trading of options and other financial
instruments, a market maker suggests a bid price and offer price
(also called ask price) for a certain option. The bid price is the
price at which the market maker is willing to purchase the option
and the offer price is the price at which the market maker is
willing to sell the option. As a market practice, a first trader
interested in a certain option may ask a second trader for a quote,
e.g., without indicating whether the first trader is interested to
buy or to sell the option. The second trader quotes both the bid
and offer prices, not knowing whether the first trader is
interested in selling or buying the option. The market maker may
earn a margin by buying options at a first price and selling them
at a second price, e.g., higher than the first price. The
difference between the offer and bid prices is referred to as
bid-offer spread.
[0008] A call option is the right to buy an asset at a certain
price ("the strike") at a certain time, e.g., on a certain date. A
put option is the right to sell an asset at a strike price at a
certain time, e.g., on a certain date. Every option has an
expiration time in which the option ceases to exist. Prior to the
option expiration time, the holder of the option may determine
whether or not to exercise the option, depending on the prevailing
spot price for the underlying asset. If the spot price at
expiration is lower than the strike price, the holder will choose
not to exercise the call option and lose only the cost of the
option itself. However, if the strike is lower than the spot, the
holder of the call option will exercise the right to buy the
underlying asset at the strike price making a profit equal to the
difference between the spot and the strike prices. The cost of the
option is also referred to as the premium.
[0009] A forward rate is defined as the predetermined rate or price
of an asset, at which an agreed upon future transaction will take
place. The forward rate may be calculated based on a current rate
of the asset, a current interest rate prevailing in the market,
expected dividends (for stocks), cost of carry (for commodities),
and/or other parameters depending on the underlying asset of the
option.
[0010] An at-the-money forward option (ATM) is an option whose
strike is equal to the forward rate of the asset. In some fields,
the at-the-money forward options are generically referred to as
at-the-money options, as is the common terminology in the
commodities and interest rates options. The at the money equity
options are actually the at the money spot, i.e. where the strike
is the current spot rate or price.
[0011] An in-the-money call option is a call option whose strike is
below the forward rate of the underlying asset, and an in the-money
put option is a put option whose strike is above the forward rate
of the underlying asset. An out-of-the-money call option is a call
option whose strike is above the forward rate of the underlying
asset, and an out-of-the-money put option is a put option whose
strike is below the forward rate of the underlying asset.
[0012] An exotic option, in the context of this application, is a
generic name referring to any type of option other than a standard
Vanilla option. While certain types of exotic options have been
extensively and frequently traded over the years, and are still
traded today, other types of exotic options had been used in the
past but are no longer in use today. Currently, the most common
exotic options include "barrier" options, "digital" options,
"binary" options, "partial barrier" options (also known as "window"
options), "average" options, "compound" options and "quanto"
options. Some exotic options can be described as a complex version
of the standard (Vanilla) option. For example, barrier options are
exotic options where the payoff depends on whether the underlying
asset's price reaches a certain level, hereinafter referred to as
"trigger", during a certain period of time. The "pay off" of an
option is defined as the cash realized by the holder of the option
upon its expiration. There are generally two types of barrier
options, namely, a knock-out option and a knock-in option. A
knock-out option is an option that terminates if and when the spot
reaches the trigger. A knock-in option comes into existence only
when the underlying asset's price reaches the trigger. It is noted
that the combined effect of a knock-out option with strike K and
trigger B and a knock-in option with strike K and trigger B, both
having the same expiration, is equivalent to a corresponding
Vanilla option with strike K. Thus, knock-in options can be priced
by pricing corresponding knock-out and vanilla options. Similarly,
a one-touch option can be decomposed into two knock-in call options
and two knock-in put options, a double no-touch option can be
decomposed into two double knock-out options, and so on. It is
appreciated that there are many other types of exotic options known
in the art.
[0013] Certain types of options, e.g., Vanilla options, are
commonly categorized as either European or American. A European
option can be exercised only upon its expiration. An American
option can be exercised at any time after purchase and before
expiration. For example, an American Vanilla option has all the
properties of the Vanilla option type described above, with the
additional property that the owner can exercise the option at any
time up to and including the option's expiration date. As is known
in the art, the right to exercise an American option prior to
expiration makes American options more expensive than corresponding
European options.
[0014] Generally in this application, the term "Vanilla" refers to
a European style Vanilla option. European Vanilla options are the
most commonly traded options; they are typically traded over the
counter (OTC). American Vanilla options are more popular in the
exchanges and, in general, are more difficult to price.
[0015] U.S. Pat. No. 5,557,517 ("the '517 patent") describes a
method of pricing American Vanilla options for trading in a certain
exchange. This patent describes a method of pricing Call and Put
American Vanilla options, where the price of the option depends on
a constant margin or commission required by the market maker.
[0016] The method of the '517 patent ignores data that may affect
the price of the option, except for the current price of the
underlying asset and, thus, this method can lead to serious errors,
for example, an absurd result of a negative option price. Clearly,
this method does not emulate the way American style Vanilla options
are priced in real markets.
[0017] The Black-Scholes (BS) model (developed in 1973) is a widely
accepted method for valuing options. This model calculates a
theoretical value (TV) for options based on the probability of the
payout, which is commonly used as a starting point for
approximating option prices. This model is based on a presumption
that the change in the spot price of the asset generally follows a
Brownian motion, as is known in the art. Using such Brownian motion
model, known also as a stochastic process, one may calculate the
theoretical price of any type of financial derivative, either
analytically or numerically. For example, it is common to calculate
the theoretical price of complicated financial derivatives through
simulation techniques, such as the Monte-Carlo method, introduced
by Boyle in 1977. Such techniques may be useful in calculating the
theoretical value of an option, provided that a computer being used
is sufficiently powerful to handle all the calculations involved.
In the simulation method, the computer generates many propagation
paths for the underlying asset, starting at the trade time and
ending at the time of the option expiry. Each path is discrete and
generally follows the Brownian motion probability, but may be
generated as densely as necessary by reducing the time lapse
between each move of the underlying asset. Thus, if the option is
path-dependant, each path is followed and only the paths that
satisfy the conditions of the option are taken into account. The
end results of each such path are summarized and lead to the
theoretical price of the derivative.
[0018] The original Black-Scholes model was derived for calculating
theoretical prices of European Vanilla options, where the price of
the option is described by a relatively simple formula. However, it
should be understood that any reference in this application to the
Black-Scholes model refers to use of the Black-Scholes model or any
other suitable model for evaluating the behavior of the underlying
asset, e.g., assuming a stochastic process (Brownian motion),
and/or for evaluating the price of any type of option, including
exotic options. Furthermore, this application is general and
independent of the way in which the theoretical value of the option
is obtained. It can be derived analytically, numerically, using any
kind of simulation method or any other technique available.
[0019] For example, U.S. Pat. No. 6,061,662 ("the '662 patent")
describes a method of evaluating the theoretical price of an option
using a Monte-Carlo method based on historical data. The simulation
method of the '662 patent uses stochastic historical data with a
predetermined distribution function in order to evaluate the
theoretical price of options. Examples is the '662 patent are used
to illustrate that this method generates results which are very
similar to those obtained by applying the Black-Scholes model to
Vanilla options. Unfortunately, methods based on historical data
alone are not relevant for simulating financial markets, even for
the purpose of theoretical valuation. For example, one of the most
important parameters used for valuation of options is the
volatility of the underlying asset, which is a measure for how the
price and/or rate of the underlying asset may fluctuate. It is well
known that the financial markets use a predicted, or an expected,
value for the volatility of the underlying assets, which often
deviates dramatically from the historical data. In market terms,
expected volatility is often referred to as "implied volatility",
and is differentiated from "historical volatility". For example,
the implied volatility tends to be much higher than the historical
volatility of the underlying asset before a major event, such as
risk of war, and in anticipation of or during a financial
crisis.
[0020] It is appreciated by persons skilled in the art that the
Black-Scholes model is a limited approximation that may yield
results very far from real market prices and, thus, corrections to
the Black-Scholes model must generally be added by traders. For
example, in the Foreign Exchange (FX) Vanilla market, and in
commodities, the market trades in volatility terms and the
translation to option price is performed through use of the
Black-Scholes formula. In fact, traders commonly refer to using the
Black-Scholes model as "using the wrong volatility with the wrong
model to get the right price".
[0021] In order to adjust the BS price, in the Vanilla market,
traders use different volatilities for different strikes, i.e.,
instead of using one volatility per asset per expiration date, as
is required by the BS model, a trader may use different volatility
values for a given asset depending on the strike price. This
adjustment is known as volatility "smile" adjustment. The origin of
the term "smile", in this context, is the typical shape of the
volatility vs. strike, which is similar to a flat "U" shape
(smile).
[0022] The phrase "market price of an option" is used herein to
distinguish between the single value produced by some known models,
such as the Black-Scholes model, and the actual bid and offer
prices traded in the real market. For example, for some options,
the market bid side may be twice the Black-Scholes model price and
the offer side may be three times the Black-Scholes model
price.
[0023] Many exotic options are characterized by discontinuity of
the payout and, therefore, a discontinuity in some of the risk
parameters near the trigger(s). This discontinuity prevents an
oversimplified model such as the Black-Scholes model from taking
into account the difficulty in risk-managing the option.
Furthermore, due to the peculiar profile of some exotic options,
there may be significant transaction costs associated with
re-hedging some of the risk factors. Existing models, such as the
Black-Scholes model, completely ignore such risk factors.
[0024] Several options pricing models were introduced since 1973,
but none of these models was able to replicate the market prices
universally and/or consistently. The most famous pricing models
include, the stochastic volatility model, which assumes that the
volatility itself is another stochastic process correlated with the
underlying process; the local volatility model, where the
volatility is a function of time and the underlying asset; and
Libor based models, such as BGM, which generate the swaption prices
from the Libor rates which are correlated stochastic processes.
[0025] Many factors may be taken into account in calculating option
prices and corrections. The term "Factor" is used herein broadly as
any quantifiable or computable value relating to the subject
option. Some of the notable factors are defined as follows.
[0026] Volatility ("Vol") is a measure of the fluctuation of the
return realized on an asset, e.g., a daily return. An indication of
the order of magnitude the volatility can be obtained by historical
volatility, i.e., the standard deviation of the daily return of the
assets for a certain past period.
[0027] However, the markets trade based on a volatility that
reflects the market expectations of the standard deviation in the
future. The volatility reflecting market expectations is called
implied volatility. In order to buy/sell volatility one commonly
trades Vanilla options. For example, in the foreign exchange
market, the implied volatilities of ATM Vanilla options for
frequently used option dates and currency pairs are available to
users in real-time, e.g., via screens such as REUTERS, Bloomberg or
directly from FX option brokers.
[0028] Volatility smile, as discussed above, relates to the
behavior of the implied volatility with respect to the strike,
i.e., the implied volatility as a function of the strike, where the
implied volatility for the ATM strike is the given ATM volatility
in the market. Typically the plot of the implied volatility as a
function of the strike shows a minimum that looks like a smile. For
example, usually in equity options the minimum volatility is below
the ATM strike.
[0029] Delta is the rate of change in the price of an option in
response to changes in the price of the underlying asset; in other
words, it is a partial derivative of the option price with respect
to the spot. For example, a 25 delta call option is defined as
follows: if against buying the option on one unit of the underlying
asset, 0.25 units of the underlying asset are sold, then for small
changes in the underlying asset price, assuming all other factors
are unchanged, the total change in the price of the option and the
profit or loss generated by holding 0.25 units of the asset are
null.
[0030] Vega is the rate of change in the price of an option or
other derivative in response to changes in volatility, i.e., the
partial derivative of the option price with respect to the
volatility.
[0031] Volatility Convexity is the second partial derivative of the
price with respect to the volatility, i.e. the derivative of the
Vega with respect to the volatility, denoted dVega/dVol.
[0032] Straddle is a strategy, which includes buying Vanilla call
and put options having the same strike price and the same
expiration.
[0033] At-the-money Delta neutral straddle is a straddle wherein
the Delta of the call option and the Delta of the put option have
the same value with opposite sign. The buyer of the at-the-money
Delta neutral straddle strategy is automatically Delta-hedged
(protected from small changes in the price of the underlying
asset).
[0034] Risk Reversal (RR) is a strategy, which includes buying a
Vanilla call option and selling a Vanilla put option with the same
expiration sand the same Delta with opposite sign. In some markets,
the RR corresponds to the difference between the implied volatility
of a call option and a put option with the same delta (in opposite
directions). Traders in the currency and/or commodity option
markets generally use 25delta RR, which is the difference between
the implied volatility of a 25delta call option and a 25delta put
option. Thus, 25delta RR may be calculated as follows:
25delta RR=implied Vol(25delta call)-implied Vol(25delta put)
[0035] The 25delta RR may correspond to a combination of buying a
25 delta call option and selling a 25 delta put option.
Accordingly, the 25delta RR may be characterized by a slope of Vega
of such combination with respect to spot. Thus, the price of the
25delta RR may characterize the price of the Vega slope, since
practically the convexity of 25delta RR at the current spot is
close to zero. Therefore, the 25delta RR as defined above may be
used to price the slope dVega/dspot.
[0036] Strangle is a strategy of buying call and put options with
the same expiration. In some applications, the call and put options
may have the same Delta with opposite signs. The strangle price can
be presented as the average of the implied volatility of the call
and put options. For example:
25delta strangle=0.5(implied Vol(25delta call)+implied Vol(25delta
put))
[0037] The 25delta strangle may be characterized by practically no
slope of Vega with respect to spot at the current spot, but a lot
of convexity, i.e., a change of Vega when the volatility changes.
Therefore, it is used to price convexity.
[0038] Since the at-the-money Vol may be known, it is more common
to quote the butterfly strategy, in which one buys one unit of the
strangle and sells 2 units of the ATM 25 option. In some assets,
the strangle/butterfly is quoted in terms of volatility. For
example:
25delta butterfly=0.5*(implied Vol(25delta call)+implied
Vol(25delta put))-ATM Vol
[0039] The reason it is more common to quote the butterfly rather
than the strangles is that butterfly provides a strategy with
almost no Vega but significant convexity. Since butterfly and
strangle are related through the ATM volatility, which may be
known, they may be used interchangeably. The 25delta put and the
25delta call can be determined based on the 25delta RR and the
25delta strangle. The ATM volatility, 25 delta risk reversal and/or
the 25 delta butterfly may be referred to, for example, as the
"Volatility Parameters". The Volatility Parameters may include any
additional and/or alternative parameters and/or factors.
[0040] Bid/offer spread is the difference between the bid price and
the offer price of a financial derivative. In the case of options,
the bid/offer spread may be expressed, for example, either in terms
of volatility or in terms of the price of the option. For example,
the bid/ask spread of exchange traded options is quoted in price
terms (e.g., cents, etc). The bid/offer spread of a given option
depends on the specific parameters of the option. In general, the
more difficult it is to manage the risk of an option, the wider is
the bid/offer spread for that option.
[0041] In order to quote a price, traders typically try to
calculate the price at which they would like to buy an option
(i.e., the bid side) and the price at which they would like to sell
the option (i.e., the offer side). Many traders have no
computational methods for calculating the bid and offer prices, and
so traders typically rely on intuition, experiments involving
changing the factors of an option to see how they affect the market
price, and past experience, which is considered to be the most
important tool of traders.
[0042] One dilemma commonly faced by traders is how wide the
bid/offer spread should be. Providing too wide a spread reduces the
ability to compete in the options market and is considered
unprofessional, yet too narrow a spread may result in losses to the
trader. In determining what prices to provide, traders need to
ensure that the bid/offer spread is appropriate. This is part of
the pricing process, i.e., after the trader decides where to place
the bid and offer prices, he/she needs to consider whether the
resultant spread is appropriate. If the spread is not appropriate,
the trader needs to change either or both of the bid and offer
prices in order to show the appropriate spread.
[0043] Option prices that are quoted in exchanges typically have a
relatively wide spread compared to their bid/ask spread in the OTC
market, where traders of banks typically trade with each other
through brokers. In addition the exchange price typically
corresponds to small notional amounts of options (lots). A trader
may sometimes change the exchange price of an option by suggesting
a bid price or an offer price with a relatively small amount of
options. This may result in the exchange prices being distorted in
a biased way.
[0044] In contrast to the exchanges, the OTC option market has a
greater "depth" in terms of liquidity. Furthermore, the options
traded in the OTC market are not restricted to the specific strikes
and expiration dates of the options traded in the exchanges. In
addition, there are many market makers, which quote bid/offer
prices, which are totally different from the bid/offer prices in
the exchange.
[0045] One of the reasons that exchange prices of options are
quoted with a wide spread is that the prices of options
corresponding to many different strikes, and many different dates
may change very frequently, e.g., in response to each change in the
price of the underlying assets. As a result, the people that
provide the bid and ask prices to the exchange have to constantly
update a large number of bid and ask prices simultaneously, e.g.,
each time the price of the underlying assets changes. In order to
avoid this tedious activity, it is mostly preferred to use "safe"
bid and ask prices, which will not need to be frequently
updated.
SUMMARY
[0046] Some demonstrative embodiments include devices, systems
and/or methods of pricing financial instruments.
[0047] Although some embodiments are described herein with
reference to pricing a Vanilla option, other embodiments may be
implemented for pricing any other suitable exotic option, e.g.,
based on the pricing of a corresponding vanilla option.
[0048] Some demonstrative embodiments include methods, devices
and/or systems implementing a pricing model for pricing, e.g., in
real time, options in substantially all asset classes, for example,
in a way that truly replicates the traded prices of the options,
e.g., as traded in the interbank market.
[0049] In some demonstrative embodiments, a pricing module may
receive first input data corresponding to at least one parameter
defining an option on an underlying asset and second input data
corresponding to at least one current market condition relating to
the underlying asset.
[0050] In some demonstrative embodiments, the first input data may
include an indication of at least one of a type of the option, an
expiration date of the option, a trigger for the option, and a
strike of the option.
[0051] In some demonstrative embodiments, the second input data may
include an indication of at least one of a spot value, a forward
rate, an interest rate, a volatility, an at-the-money volatility, a
delta risk reversal, a delta butterfly, a delta strangle, a 10
delta risk reversal, a 10 delta butterfly, a 10 delta strangle, a
25 delta risk reversal, a 25 delta butterfly, a 25 delta strangle,
a caplet, a floorlet, a swap rate, a security lending rate, and an
exchange price.
[0052] In some demonstrative embodiments, the pricing module may
price the option based on the first and second input data.
[0053] The Black Scholes (BS) model assumes that there is a single
volatility for any maturity regardless of the strike and, that this
single volatility, which reflects the rate of fluctuation of the
price of the underlying asset, is constant throughout the life of
the option. Therefore the BS model assumes that a trader only has
to constantly re-hedge the price of the underlying asset, e.g., by
always keeping the Delta amount of the underlying asset, in order
to eliminate the price risk of the option. In reality, this
assumption is not true. Typically, the volatility changes when the
price of the underlying asset changes. Therefore, there is a
different "volatility value" for different strikes. The BS model
ignores the cost of rehedging the volatility changes.
[0054] In some demonstrative embodiments, the pricing module may
consider the re-hedging of two "axes", e.g., which may be almost
orthogonal to one another. A first "axis" may result from the fact
that there is the volatility "smile", wherein the volatility may be
affected by changes in the price of the underlying asset price. The
first axis may be re-hedged, for example, using a risk reversal
strategy. The second axis may result from a Vega hedged book
becoming un-hedged, e.g., when the volatility changes. The second
axis may be re-hedged, for example, using the strangle
strategy.
[0055] In some demonstrative embodiments, the pricing module may
price the option according to a volatility smile, which may satisfy
one or more predefined criterions.
[0056] The term "volatility smile" as used herein relates to the
behavior of the implied volatility with respect to the strike,
i.e., the implied volatility as a function of the strike, where the
implied volatility for the ATM strike is the given ATM volatility
in the market. The plot of the implied volatility as a function of
the strike may typically show a minimum that looks like a smile,
e.g., usually in equity options the minimum volatility is below the
ATM strike. However, the plot of the implied volatility as a
function of the strike may have any other suitable behavior and/or
shape, e.g., different from a "U" or "smile" shape.
[0057] In some demonstrative embodiments, the volatility smile may
satisfy the criterions, for example, with respect to a pair of
options forming a Delta neutral strategy, e.g., a first option
including the option to be priced and a second option representing
a position opposite to a position of a the first option and having
substantially the same absolute delta value as the first
option.
[0058] In some demonstrative embodiments, the volatility smile may
satisfy the criterions, for example, with respect to each pair of
options forming a Delta neutral strategy, e.g., including a first
option and a second option representing a position opposite to a
position of a the first option and having substantially the same
absolute delta value as the first option.
[0059] In some demonstrative embodiments, the volatility smile may
satisfy a first criterion relating to a sum of a first correction
corresponding to the first option and a second correction
corresponding to the second option.
[0060] In some demonstrative embodiments, the first correction
relates to a difference between a theoretical price of the first
option and the price of the first option according to the
volatility smile, and the second correction relates to a difference
between a theoretical price of the second option and the price of
the second option according to the volatility smile. The
theoretical value may be determined according to any suitable
model, e.g., the BS model or any other model.
[0061] In some demonstrative embodiments, the first criterion may
require that the sum of the first and second corrections is
proportional to a sum of first and second volatility convexities
corresponding to the first and second options, respectively.
[0062] In some demonstrative embodiments, the sum of the first and
second volatility convexities is a predefined function of a
volatility of the first option according to the volatility smile
and a volatility of the second option according to the volatility
smile.
[0063] In some demonstrative embodiments, the second criterion may
require that the difference between the first and second
corrections is proportional to a difference between first and
second delta convexities corresponding to the first and second
options, respectively.
[0064] In some demonstrative embodiments, the difference between
the first and second delta convexities is a second predefined
function of the volatility of the first option according to the
volatility smile and the volatility of the second option according
to the volatility smile.
[0065] In some demonstrative embodiments, the first criterion
requires that the sum of the first and second corrections is
proportional to the sum of first and second volatility convexities
according to a first proportionality function, which is based on
the delta.
[0066] In some demonstrative embodiments, the second criterion
requires that the difference between the first and second
corrections is proportional to the difference of the first and
second delta convexities according to a second proportionality
function, which is based on the delta.
[0067] In some demonstrative embodiments, at least one of the first
and second proportionality functions includes a predefined
combination of the delta and one or more market-based
parameters.
[0068] In some demonstrative embodiments, the pricing module may
determine the market-based parameters based on the second input
data.
[0069] In some demonstrative embodiments, the first and second
proportionality functions are decreasing functions of delta.
[0070] In some demonstrative embodiments, the first and second
criterion require satisfying the following equations,
respectively:
.zeta. C .DELTA. + .zeta. P .DELTA. = A ( .DELTA. ) Vega .DELTA. d
1 2 ( 1 .sigma. K Call + 1 .sigma. K Put ) ##EQU00001## .zeta. C
.DELTA. - .zeta. P .DELTA. = B ( .DELTA. ) Vega .DELTA. d 1 S t ( 1
.sigma. K Call + 1 .sigma. K Put ) ##EQU00001.2##
wherein .zeta..sub.C.sup..DELTA. and .zeta..sub.P.sup..DELTA.
denote the first and second corrections, wherein .DELTA. denotes
the delta, wherein A(.DELTA.) and B(.DELTA.) denote first and
second functions of A, respectively, wherein Vega.sup..DELTA.
denotes a Vega of the first and second options, wherein t denotes a
time to expiration of the first option, wherein d.sub.1 denotes a
predefined function of the time to expiration of the first option,
wherein S denotes a price of the underlying asset, and wherein
.sigma..sub.K.sub.Call and .sigma..sub.K.sub.Put denote a
volatility of the first option according to the volatility smile
and a volatility of the second option according to the volatility
smile, respectively.
[0071] In some demonstrative embodiments, the pricing module may
determine a volatility of the first option based on the first and
second criterions. For example, the pricing module may determine
the volatility of the first option according to the volatility
smile.
[0072] In some demonstrative embodiments, the pricing module may
determine the first correction corresponding to the first option
based on the volatility of the first option.
[0073] In some demonstrative embodiments, the pricing module may
determine the price of the first option based on the first
correction and the theoretical price of the first option, e.g.,
based on a sum of the theoretical price and the first
correction.
[0074] In some demonstrative embodiments, the pricing module may
determine a price of an exotic option on the underlying asset based
on the volatility smile. For example, the pricing module may
determine a price of a vanilla option on the underlying asset,
e.g., according to the volatility smile described above, and
determine the price of the exotic option based on the price of the
vanilla option.
[0075] In some demonstrative embodiments, the pricing module may
provide an output based on the price of the first option, e.g., in
real time.
[0076] In some demonstrative embodiments, the pricing module may
communicate the output via a communication network.
[0077] In some embodiments, a system may implement the pricing
module to provide price information for substantially any suitable
option on substantially any suitable asset based on input market
data. The market data may be easily obtained, e.g., on a real time
basis. Thus, a real-time price of any desired option may be
determined, e.g., based on real time prices received from the
exchanges and/or OTC market.
[0078] In some demonstrative embodiments, the price may be updated,
e.g., substantially immediately and/or automatically, for example,
in response to a change in spot prices and/or option prices. This
may enable a user to automatically update prices for trading with
the exchanges.
[0079] The trader may want, for example, to submit a plurality of
bid and/or offer (hereinafter "bid/offer") prices for a plurality
of options, e.g., ten bid/offer prices for ten options,
respectively. When entering the bids/offers to a quoting system,
the trader may check the price, e.g., in relation to the current
spot prices, and may then submit the bids/offers to the exchange.
Some time later, e.g., a second later, the spot price of the stock,
which is the underlying asset of one or more of the options, may
change. A change in the spot prices may be accompanied, for
example, by changes in the volatility parameters, or may include
just a small spot change while the volatility parameters have not
changed. In response to the change in the spot price, the trader
may want to update one or more of the submitted bid/offer prices.
The desire to update the bid/offer prices may occur, e.g.,
frequently, during trade time.
[0080] The system implementing the pricing module according to some
demonstrative embodiments, may automatically update the bid/offer
prices entered by the trader, e.g., based on any desired criteria.
For example, the system may evaluate the trader's bids/offers
versus bid and offer prices of the options, which may be estimated
by the system, e.g., when the trader submits the bid/offer prices.
The system may then automatically recalculate the bid and/or offer
prices, e.g., whenever the spot changes, and may automatically
update the trader's bid/offer prices. The system may, for example,
update one or more of the trader's bid/offer prices such that a
price difference between the bid/offer price calculated by the
pricing module and the trader's bid/offer price is kept
substantially constant. According to another example, the system
may update one or more of the trader's bid/offer prices based on a
difference between the trader's bid/offer prices and an average of
bid and offer prices calculated by the pricing module. The system
may update one or more of the trader's bid/offer prices based on
any other desired criteria.
[0081] It is noted, that a change of the spot price, e.g., of a few
pips, may result in a change in one or more of the volatility
parameters of options corresponding to the spot price. It will be
appreciated that the system according to some embodiments, may
enable automatically updating one or more option prices submitted
by a trader, e.g., while taking into account the change in the spot
price, in one or more of the volatility parameters, and/or in any
other desired parameters, as described above.
[0082] In some demonstrative embodiments, the system may enable the
trader to submit one or more quotes in the exchange in a form of
relative prices vs. prices determined by the pricing module. For
example, the trader may submit quotes for one or more desired
strikes and/or expiry dates. The quotes submitted by the trader may
be in any desired form, e.g., relating to one or more corresponding
prices determined by the pricing module. For example, the quotes
submitted by the trader may be in the terms of the bid/offer prices
determined by the pricing module plus two basis points; in the
terms of the mid market price determined by the pricing module
minus four basis points, and the like. The system may determine the
desired prices, for example, in real time, e.g., whenever a price
change in the exchange is recorded. Alternatively, the system may
determine the desired prices, according to any other desired timing
scheme, for example, every predefined time interval, e.g., every
half a second.
[0083] A change in a spot price of a stock may result in changes in
the prices of a large number of options related to the stock. For
example there could be over 200 active options relating to a single
stock and having different strikes and expiration dates.
Accordingly, a massive bandwidth may be required by traders for
updating the exchange prices of the options in accordance with the
spot price changes, e.g., in real time. This may lead the traders
to submit to the exchange prices which may be "non-competitive",
e.g., prices including a "safety-margin", since the traders may not
be able to update the submitted prices according to the rate at
which the spot prices, the volatility, the dividend, and/or the
carry rate may change.
[0084] Some demonstrative embodiments, may allow automatically
updating of one or more bid and/or offer prices submitted by a
trader, e.g., as described above. This may encourage the traders to
submit with the exchange more aggressive bid and/or offer prices,
since the traders may no longer need to add the "safety margin"
their prices for protecting the traders against the frequent
changes in the spot prices. Accordingly, the trading in the
exchange may be more effective, resulting in a larger number of
transactions. For example, a trader may provide the system with one
or more desired volatility parameter and/or rates. The trader may
request the system to automatically submit and/or update bid and/or
offer prices on desired amounts of options, e.g., whenever there is
a significant change in the spot price and/or in the volatility of
the market. The trader may also update some or all of the
volatility parameters. The system may be linked, for example, to an
automatic decision making system, which may be able to decide when
to buy and/or sell options using the pricing module.
BRIEF DESCRIPTION OF THE DRAWINGS
[0085] For simplicity and clarity of illustration, elements shown
in the figures have not necessarily been drawn to scale. For
example, the dimensions of some of the elements may be exaggerated
relative to other elements for clarity of presentation.
Furthermore, reference numerals may be repeated among the figures
to indicate corresponding or analogous elements. The figures are
listed below.
[0086] FIG. 1 is a schematic illustration of a system, in
accordance with some demonstrative embodiments.
[0087] FIG. 2 is a schematic flow-chart illustration of a method,
in accordance with some demonstrative embodiments.
[0088] FIGS. 3A-3D are schematic illustrations of graphs depicting
volatility smiles, in accordance with some demonstrative
embodiments.
[0089] FIG. 4 is schematic illustration of an article of
manufacture, in accordance with some demonstrative embodiments.
DETAILED DESCRIPTION
[0090] In the following detailed description, numerous specific
details are set forth in order to provide a thorough understanding
of some embodiments. However, it will be understood by persons of
ordinary skill in the art that some embodiments may be practiced
without these specific details. In other instances, well-known
methods, procedures, components, units and/or circuits have not
been described in detail so as not to obscure the discussion.
[0091] Some portions of the following detailed description are
presented in terms of algorithms and symbolic representations of
operations on data bits or binary digital signals within a computer
memory. These algorithmic descriptions and representations may be
the techniques used by those skilled in the data processing arts to
convey the substance of their work to others skilled in the
art.
[0092] An algorithm is here, and generally, considered to be a
self-consistent sequence of acts or operations leading to a desired
result. These include physical manipulations of physical
quantities. Usually, though not necessarily, these quantities take
the form of electrical or magnetic signals capable of being stored,
transferred, combined, compared, and otherwise manipulated. It has
proven convenient at times, principally for reasons of common
usage, to refer to these signals as bits, values, elements,
symbols, characters, terms, numbers or the like. It should be
understood, however, that all of these and similar terms are to be
associated with the appropriate physical quantities and are merely
convenient labels applied to these quantities.
[0093] Discussions herein utilizing terms such as, for example,
"processing", "computing", "calculating", "determining",
"establishing", "analyzing", "checking", or the like, may refer to
operation(s) and/or process(es) of a computer, a computing
platform, a computing system, or other electronic computing device,
that manipulate and/or transform data represented as physical
(e.g., electronic) quantities within the computer's registers
and/or memories into other data similarly represented as physical
quantities within the computer's registers and/or memories or other
information storage medium that may store instructions to perform
operations and/or processes.
[0094] The terms "plurality" and "a plurality" as used herein
includes, for example, "multiple" or "two or more". For example, "a
plurality of items" includes two or more items.
[0095] Some embodiments may include one or more wired or wireless
links, may utilize one or more components of wireless
communication, may utilize one or more methods or protocols of
wireless communication, or the like. Some embodiments may utilize
wired communication and/or wireless communication.
[0096] Some embodiments may be used in conjunction with various
devices and systems, for example, a Personal Computer (PC), a
desktop computer, a mobile computer, a laptop computer, a notebook
computer, a tablet computer, a server computer, a handheld
computer, a handheld device, a Personal Digital Assistant (PDA)
device, a handheld PDA device, an on-board device, an off-board
device, a hybrid device, a vehicular device, a non-vehicular
device, a mobile or portable device, a non-mobile or non-portable
device, a wireless communication station, a wireless communication
device, a cellular telephone, a wireless telephone, a Personal
Communication Systems (PCS) device, a PDA device which incorporates
a wireless communication device, a device having one or more
internal antennas and/or external antennas, a wired or wireless
handheld device (e.g., BlackBerry, Palm Treo), a Wireless
Application Protocol (WAP) device, or the like.
[0097] Some demonstrative embodiments of the present invention are
described herein in the context of a model for calculating a value,
e.g., the market value, of a financial instrument, e.g., a stock
option. It should be appreciated, however, that models in
accordance with the invention may be applied to other options,
financial instruments and/or markets, and the embodiments are not
limited to stock options. One skilled in the art may apply the
embodiments to other options and/or option-like financial
instruments, e.g., options on interest rate futures, options on
commodities, and/or options on non-asset instruments, such as
options on the weather and/or the temperature, and the like, with
variation as may be necessary to adapt for factors unique to a
given financial instrument.
[0098] The term "financial instrument" may refer to any suitable
"asset class", e.g., Foreign Exchange (FX), Interest Rate, Equity,
Commodities, Credit, weather, energy, real estate, mortgages, and
the like; and/or may involve more than one asset class, e.g.,
cross-asset, multi asset, and the like. The term "financial
instrument" may also refer to any suitable combination of one or
more financial instruments.
[0099] The term "derivative financial instrument" or "option" may
refer to any suitable derivative instruments, e.g., forwards,
swaps, futures, exchange options and OTC options, which derive
their value from the value and characteristics of one or more
underlying assets.
[0100] Reference is now made to FIG. 1, which schematically
illustrates a block diagram of a system 100, in accordance with
some demonstrative embodiments.
[0101] In some demonstrative embodiments, system 100 may include a
pricing module ("pricing application") 160 to price one or more
derivative financial instruments, e.g., as described below.
[0102] In some demonstrative embodiments, system 100 includes one
or more user stations or devices 102, for example, a PC, a laptop
computer, a PDA device, and/or a terminal, to allow one or more
users to price the one or more financial assets using pricing
module 160, e.g., as described herein.
[0103] In some demonstrative embodiments, devices 102 may be
implemented using suitable hardware components and/or software
components, for example, processors, controllers, memory units,
storage units, input units, output units, communication units,
operating systems, applications, or the like.
[0104] The user of device 102 may include, for example, a trader, a
business analyst, a corporate structuring manager, a salesperson, a
risk manager, a front office manager, a back office, a middle
office, a system administrator, and the like.
[0105] In some demonstrative embodiments, system 100 may also
include an interface 110 to interface between users 102 and one or
more elements of system 100, e.g., pricing module 160. Interface
110 may optionally interface between users 102 and one or more
Financial-Instrument (FI) systems and/or services 140. Services 140
may include, for example, one or more market data services 149, one
or more trading systems 147, one or more exchange connectivity
systems 148, one or more analysis services 146 and/or one or more
other suitable FI-related services, systems and/or platforms.
[0106] In some demonstrative embodiments, pricing module 160 may be
capable of communicating, directly or indirectly, e.g., via
interface 110 and/or any other interface, with one or more suitable
modules of system 100, for example, one or more of FI systems 140,
a database, a storage, an archive, an HTTP service, an FTP service,
an application, and/or any suitable module capable of providing,
e.g., automatically, input to pricing module 160 and/or receiving
output generated by pricing module 160, e.g., as described
herein.
[0107] In some demonstrative embodiments, pricing module 160 may be
implemented as part of FI systems/services 140, as part of device
102 and/or as part of any other suitable system or module, e.g., as
part of any suitable server, or as a dedicated server.
[0108] In some demonstrative embodiments, pricing module 160 may
include a local or remote application executed by any suitable
computing system 183. For example, computing system 183 may include
a suitable memory 187 having stored-thereon pricing-application
instructions 189; and a suitable processor 185 to execute
instructions 189 resulting in pricing module 160.
[0109] In some demonstrative embodiments, computing system 183 may
include or may be part of a server to provide the functionality of
pricing module 160 to users 102. In other embodiments, computing
system 183 may be implemented as part of user station 102. For
example, instructions 189 may be downloaded and/or received by
users 102 from another computing system, such that pricing module
160 may be locally-executed by users 102. For example, instructions
189 may be received and stored, e.g., temporarily, in a memory or
any suitable short-term memory or buffer of user device 102, e.g.,
prior to being executed by a processor of user device 102. In other
embodiments, computing system 183 may include any other suitable
computing arrangement, server and/or scheme.
[0110] In some demonstrative embodiments, computing system 183 may
also execute one or more of FI systems/services 140. In other
embodiments, pricing application 160 may be implemented separately
from one or more of FI systems/services 140.
[0111] In some demonstrative embodiments, interface 110 may be
implemented as part of pricing module 160, FI systems/services 140
and/or as part of any other suitable system or module, e.g., as
part of any suitable server.
[0112] In some demonstrative embodiments, interface 110 may be
associated with and/or included as part of devices 102. In one
example, interface 110 may be implemented, for example, as
middleware, as part of any suitable application, and/or as part of
a server. Interface 110 may be implemented using any suitable
hardware components and/or software components, for example,
processors, controllers, memory units, storage units, input units,
output units, communication units, operating systems, applications.
In some demonstrative embodiments, interface 110 may include, or
may be part of a Web-based pricing application interface, a
web-site, a web-page, a stand-alone application, a plug-in, an
ActiveX control, a rich content component (e.g., a Flash or
Shockwave component), or the like.
[0113] In some demonstrative embodiments, interface 110 may also
interface between users 102 and one or more of FI systems and/or
services 140.
[0114] In some demonstrative embodiments, interface 110 may be
configured to allow users 102 to enter commands; to define a
derivative financial instrument to be priced by pricing module 160;
to define and/or structure a trade corresponding to the derivative
financial instrument; to receive a pricing of the derivative
financial instrument from pricing module 160; to analyze the trade;
to transact the trade; and/or to perform any other suitable
operation.
[0115] In some demonstrative embodiments, pricing module 160 may be
capable of pricing, e.g., accurately and/or in real-time, an
option, e.g., any suitable Vanilla option, on any suitable
underlying asset, e.g. options on currencies, interest rates,
commodities, equity, energy, credit, weather, and the like.
[0116] In some demonstrative embodiments, given the price of
European Vanilla options, one can obtain the probability function,
denoted P(S.sub.T), which represents the probability that the price
of underlying asset at time T to be S.sub.T, e.g., regardless of
the pricing model. For example, since:
Price Call = df R .intg. K .infin. S T ( S T - K ) P ( S T ) then:
( 1 ) P ( S T ) = .differential. 2 Price Call .differential. K 2 (
2 ) ##EQU00002##
wherein Price.sub.call denotes the price of a call option, df.sub.R
denotes a factor for time T calculated using a term currency annual
interest rate R, and K denotes the strike price.
[0117] Accordingly, in some demonstrative embodiments, pricing
module 160 may use the probability function obtained from the
vanilla model to calculate the price of any other suitable, e.g.,
exotic, option via, for example, a suitable Monte Carlo
simulation.
[0118] Hence, although some embodiments are described herein with
reference to pricing a Vanilla option, it will be appreciated that
other embodiments may be implemented for pricing any other suitable
exotic option, e.g., based on the pricing of a corresponding
vanilla option.
[0119] In some demonstrative embodiments, pricing module 160 may
implement the pricing model described below for pricing, in real
time, options in substantially all asset classes in a way that
truly replicates the traded prices of the options, e.g., as traded
in the interbank market.
[0120] In some demonstrative embodiments, pricing module 160 may
calculate one or more values of the volatility parameter, denoted
.sigma.=.sigma.(K), for one or more strikes K, e.g., for each
strike K; and determine the price of the option based on the
calculated volatility parameters, for example, by applying the
Black-Scholes (BS) model, or any other suitable model, to the
determined volatility parameters, e.g., as described in detail
below.
[0121] In some demonstrative embodiments, pricing module 160 may
determine a correction to be applied to a theoretical value of the
option. The theoretical value may be determined according to any
suitable model, e.g., the BS model or any other model.
[0122] In some demonstrative embodiments, pricing module 160 may
price the option according to a volatility smile, which may satisfy
one or more predefined criterions.
[0123] In some demonstrative embodiments, the volatility smile may
satisfy the criterions, for example, with respect to a pair of
options forming a Delta neutral strategy. For example, the pair of
options may include, for example, a first option, e.g., the option
to be priced, and a second option representing a position opposite
to a position of a the first option and substantially the same
absolute delta value as the first option. The term "absolute delta
value" as used herein relates to an absolute of the delta. For
example, first and second delta values may be the same of they have
substantially the same absolute value, regardless of the sign.
[0124] In some demonstrative embodiments, the volatility smile may
satisfy the criterions, for example, with respect to each pair of
options including a first option and a second option representing a
position opposite to a position of a the first option and having
substantially the same absolute delta value as the first
option.
[0125] In some demonstrative embodiments, the volatility smile may
satisfy a first criterion relating to a sum of a first correction
corresponding to the first option and a second correction
corresponding to the second option.
[0126] In some demonstrative embodiments, the first correction
relates to a difference between a theoretical price of the first
option and the price of the first option according to the
volatility smile, and the second correction relates to a difference
between a theoretical price of the second option and the price of
the second option according to the volatility smile, e.g., as
described in detail below.
[0127] In some demonstrative embodiments, the notation d.sub.1 may
be defined as follows:
d 1 = log ( F / K ) .sigma. t + 1 2 .sigma. t ( 3 )
##EQU00003##
wherein F denotes the forward rate, and t denotes the time to
expiration of the option.
[0128] The BS model for Vanilla call and put options may be
represented using the notation d.sub.1, e.g., as follows:
BS.sup.Call=df.sub.R(FN(d.sub.1)-KN(d.sub.1-.sigma. {square root
over (t)})) (4)
BS.sup.Put=df.sub.R(-FN(-d.sub.1)+KN(-d.sub.1+.sigma. {square root
over (t)}) (5)
wherein BS.sup.call denotes the BS value of the call option,
BS.sup.Put denotes the BS value of the put option, and wherein N(x)
denotes the cumulative normal distribution function of x, e.g., as
follows:
N ( x ) = .intg. - .infin. x - t 2 / 2 2 .pi. t ( 6 )
##EQU00004##
[0129] The BS values BS.sup.Call and BS.sup.Put according to
Equations 4 and 5 may represent the respective prices of a call
option to buy and a put option to sell one unit of asset at the
predetermined strike price K at a predetermined expiration date
t.
[0130] The delta of a call option and a put option, denoted
.DELTA..sub.Call and .DELTA..sub.Put, respectively, i.e., the rate
of change in the price of the call option and the put option,
respectively, in response to changes in the price of the underlying
asset, may be determined as follows:
.DELTA..sub.Call=df.sub.LN(d.sub.1) (7)
.DELTA..sub.Put=-df.sub.LN(-d.sub.1) (8)
wherein df.sub.L is a discount factor, which is calculated using a
base annual interest like rate L. For example, in stocks L is the
dividend rate, in commodities L is the carry or convenience rate,
and in currencies L is the base currency interest rate. The
discount factors df.sub.L and df.sub.R may be related by the
formula F=Sdf.sub.L/df.sub.R, wherein S is the current price (rate)
of the asset.
[0131] Accordingly, a call option and a put option having the same
delta satisfy the following condition:
d.sub.1(K.sub.Call)=d.sub.1(K.sub.Put) (9)
wherein K.sub.Call denotes the strike of the call option, and
K.sub.Put denotes the strike of the put option.
[0132] In some demonstrative embodiments, the rate of change,
denoted Vega, in the price of an option in response to changes in
the volatility may be defined as follows:
Vega=df.sub.LS {square root over (t)}n(d.sub.1) (10)
wherein n(t) denotes the normal probability density function of t,
e.g., as follows:
n ( t ) = - t 2 / 2 2 .pi. ( 11 ) ##EQU00005##
[0133] In some demonstrative embodiments, a first strategy ("same
delta Risk Reversal") may be defined to include buying a call
option and selling a put option having a delta of the same value
and an opposite sign of the delta of the call option; and a second
strategy ("same delta strangle") may be defined to include buying a
call option and buying a put option having a delta of the same
value and an opposite sign of the delta of the call option.
According to Equations 9 and 10, a put option and a call option
having the same delta with opposite signs may also have the same
Vega (hereinafter referred to as "having the same delta").
Accordingly, the derivatives of the Vega of the first and second
strategies may satisfy the following conditions:
.differential. Vega Call .DELTA. .differential. S - .differential.
Vega Put .DELTA. .differential. S = - df L n ( d 1 ) d 1 ( 1
.sigma. K Call + 1 .sigma. K Put ) == - Vega .DELTA. d 1 S t ( 1
.sigma. K Call + 1 .sigma. K Put ) and: ( 12 ) .differential. Vega
Call .DELTA. .differential. .sigma. + .differential. Vega Put
.DELTA. .differential. .sigma. = df L S t N ( d 1 ) d 1 2 ( 1
.sigma. K Call + 1 .sigma. K Put ) == Vega .DELTA. d 1 2 ( 1
.sigma. K Call + 1 .sigma. K Put ) ( 13 ) ##EQU00006##
wherein Vega.sub.Call.sup..DELTA. and Vega.sub.Put.sup..DELTA.
denote the value of Vega for the call and put options,
respectively, on the same underlying asset and having the same
Delta.
[0134] The BS model assumes that there is a single volatility for
any maturity regardless of the strike and, that this single
volatility, which reflects the rate of fluctuation of the price of
the underlying asset, is constant throughout the life of the
option. Therefore the BS model assumes that a trader only has to
constantly re-hedge the price of the underlying asset (by always
keeping the Delta amount of the underlying asset) in order to
eliminate the price risk of the option. It is well known that in
reality this assumption is not true. Typically the volatility
changes when the price of the underlying asset changes. Therefore,
there is a different "volatility value" for different strikes. The
BS model ignores the cost of rehedging the volatility changes.
[0135] In some demonstrative embodiments, pricing module 160 may
implement a pricing model ("the Gershon model"), which may at least
partially fix this flaw of the BS model, e.g., as described
herein.
[0136] In some demonstrative embodiments, the Gershon model may
consider the re-hedging of two "axes", e.g., which may be almost
orthogonal to one another. A first "axis" may result from the fact
that there is the volatility "smile", wherein the volatility may be
affected by changes in the price of the underlying asset price. The
first axis may be re-hedged using the risk reversal. The second
axis may result from a Vega hedged book becoming un-hedged, e.g.,
when the volatility changes. The second axis may be re-hedged using
the strangle.
[0137] In some demonstrative embodiments, the Delta neutral
straddle strategy may be defined to include call and put options
with the same strike, denoted K.sub.0, at which:
.DELTA..sub.Call.sup.(K.sup.0.sup.)=-.DELTA..sub.Put.sup.(K.sup.0.sup.)
(14)
[0138] Therefore, d.sub.1=0 and
K.sub.0=Fe.sup.1/2.sigma..sup.2.sup.t. According to this
definition, the Delta, denoted .DELTA..sub.0, of the call or the
put of the Delta neutral straddle strategy is:
.DELTA..sub.0=df.sub.L/2 (15)
[0139] The volatility, denoted .sigma..sub.0, may be defined as the
volatility, which, if substituted in the BS model for the strike
K.sub.0, yields the market price of the option with the strike
K.sub.0.
[0140] In some demonstrative embodiments, the Gershon model may
implement a correction ("Zeta"), denoted .zeta., to be added to the
value of an option determined according to the BS model ("the BS
value"), e.g., a difference between the value of the option
according to the Gershon model ("the market price") at the strike
K, and the BS value with the volatility .sigma..sub.0 that is used
in the BS model for the strike of the at-the-money Delta neutral
straddle. The correction .zeta., may be defined, for example, as
follows:
.zeta.=Market price(K)-BS(K) (16)
[0141] In some demonstrative embodiments, the Gershon model may
assume that .sigma..sub.0 is the BS volatility such that
BS.sup.C(.sigma..sub.0,K.sub.0) generates the correct market price
for the strike K.sub.0.
[0142] In some demonstrative embodiments, the correction, denoted
.zeta..sub.C, via the function .sigma.(K) to the call option may be
represented as follows:
.zeta..sub.C(K)=BS.sup.Call(.sigma..sub.K,K)-BS.sup.Call).sigma..sub.0,K-
) (17)
[0143] In some demonstrative embodiments, the correction, denoted
.zeta..sub.p, via the function .sigma.(K) to the put option may be
represented as follows:
.zeta..sub.P(K)=BS.sup.Put(.sigma..sub.K,K)-BS.sup.Put(.sigma..sub.0,K)
(18)
[0144] By definition of the corrections .zeta..sub.C and
.zeta..sub.p, the corrections .zeta..sub.C and .zeta..sub.p at
K.sub.0 satisfy
.zeta..sub.C.sup..DELTA..sup.0=.zeta..sub.P.sup..DELTA..sup.0=0. It
is noted, that since buying a call option together with selling a
put option with the same strike is equivalent to entering a forward
deal at a forward rate equal the strike, the value of the
correction .zeta..sub.C for the call option is identical to the
value of the correction .zeta..sub.P for the put option with the
same strike, e.g., regardless of the pricing model.
[0145] In some demonstrative embodiments, the sum of the first and
second corrections may be proportional to the sum of first and
second volatility convexities corresponding to the first and second
options, respectively, according to a first proportionality
function, which is based on the delta.
[0146] For example, the correction of the strangle strategy, which
is the sum of the corrections .zeta. corresponding to the call and
put options on the same underlying asset and having the same Delta,
may be proportional to the sum of the derivatives of Vega with
respect to the volatility, in accordance with Equation 13, e.g., as
follows:
.zeta. C .DELTA. + .zeta. P .DELTA. = A ( .DELTA. ) Vega .DELTA. d
1 2 ( 1 .sigma. K Call + 1 .sigma. K Put ) ( 19 ) ##EQU00007##
wherein A(.DELTA.) denotes a first proportionality function of
.DELTA., e.g., as described below.
[0147] In some demonstrative embodiments, the difference between
the first and second corrections may be proportional to the
difference of first and second delta convexities corresponding to
the first and second options, respectively, according to a second
proportionality function, which is based on the delta.
[0148] The term "Delta convexity" as used herein may relate to a
derivative of Vega with respect to the spot S.
[0149] For example, the correction of the risk-reversal strategy,
which is the difference between the corrections corresponding to
the call and put options on the same underlying asset and having
the same Delta, may be proportional to the difference of the
derivatives of Vega with respect to S, for example, in accordance
with Equation 12, e.g., as follows:
.zeta. C .DELTA. - .zeta. P .DELTA. = B ( .DELTA. ) Vega .DELTA. d
1 S t ( 1 .sigma. K Call + 1 .sigma. K Put ) ( 20 )
##EQU00008##
wherein B(.DELTA.) denotes a second proportionality function of
.DELTA., e.g., as described below.
[0150] In some demonstrative embodiments, the functions A(.DELTA.)
and B(.DELTA.) are decreasing functions of A and have to satisfy
market conditions, e.g., as described in detail below. The
functions A(.DELTA.) and B(.DELTA.) may depend on any suitable
parameters and/or factors, e.g., the time to expiration t, and the
like.
[0151] In some demonstrative embodiments, the proportionality
functions A(.DELTA.) and/or B(.DELTA.) may include a predefined
combination of the delta and one or more market-based
parameters.
[0152] In some demonstrative embodiments, module 160 may determine
the market-based parameters based on the second input data.
[0153] In some demonstrative embodiments, the market-based
parameters of the proportionality functions A(.DELTA.) and/or
B(.DELTA.) may depend on the maturity f the option and/or any other
suitable factor, e.g., except for the strike, d or the volatility.
In other embodiments, the market-based parameters may depend on any
other suitable factor.
[0154] In some demonstrative embodiments, the proportionality
functions A(.DELTA.) and/or B(.DELTA.) may be decreasing functions
of delta.
[0155] In some demonstrative embodiments, the market date may
relate to a plurality of option prices may be obtained from the
market. The market-based parameters of the proportionality
functions A(.DELTA.) and/or B(.DELTA.) may be determined by fitting
the proportionality functions A(.DELTA.) and/or B(.DELTA.) to the
market data. Equations 19 and 20 may then be used with the
determined proportionality functions A(.DELTA.) and/or B(.DELTA.)
to price an option of any suitable strike, e.g., as described
herein.
[0156] In one embodiment, in the market of currency options (FX),
the 25.DELTA. risk reversal and 25.DELTA. butterfly may be traded.
Hence, the functions A(.DELTA.) and/or B(.DELTA.) may be determined
such that Equations 19 and 20 satisfy the traded 25.DELTA. risk
reversal and 25.DELTA. butterfly of the market. Optionally, a
number of free parameters in the functions A(.DELTA.) and/or
B(.DELTA.) may be selected to satisfy additional conditions. For
example, in some currency pairs, additional delta values may be
traded in the market, e.g., the 10.DELTA. risk reversal and/or the
10.DELTA. butterfly. Accordingly, the functions A(.DELTA.) and/or
B(.DELTA.) may be determined such that Equations 19 and 20 satisfy
the 10.DELTA. risk reversal and/or the 10.DELTA. butterfly of the
market.
[0157] In another embodiment, in the market of Equity, the
functions A(.DELTA.) and/or B(.DELTA.) may be determined depending
on a plurality of strike prices of options traded in the market.
For example, the functions A(.DELTA.) and/or B(.DELTA.) may be
determined by requiring a best fit between the prices according to
Equations 19 and 20 and between the exchange prices of the
plurality of strikes and/or depending on suitable fixed strikes
that are more liquid.
[0158] In another embodiment, in the interest rates caps and floors
market, the functions A(.DELTA.) and/or B(.DELTA.) may be
determined based on the caplets/floorlets to generate the correct
market prices for the caps and floors.
[0159] In another embodiment, in the Swaptions market the functions
A(.DELTA.) and/or B(.DELTA.) may be determined based on a best fit
for swaption prices of the same swap length and the same expiration
with different strikes (fixed rate of the swap), which are
typically denoted by a difference, in basis points, from the
at-the-money forward strike.
[0160] In one example, the functions A(.DELTA.) and B(.DELTA.) may
defined as follows, e.g., for .DELTA..sub.0>.DELTA.:
A(.DELTA.)=.alpha..sub.1e.sup.-.beta..sup.1.sup.(.DELTA..sup.0.sup.-.DEL-
TA.) (21)
B(.DELTA.)=.alpha..sub.2e.sup.-.beta..sup.2.sup.(.DELTA..sup.0.sup.-.DEL-
TA.) (22)
wherein .alpha..sub.1, .alpha..sub.2, .beta..sub.1, .beta..sub.2
denote four respective market parameters to be determined, e.g.,
based on the traded market data.
[0161] It is noted, that there may be no need to handle the
situation of .DELTA..sub.0<.DELTA., e.g., since Equations 19 and
20 are simultaneously solved for both call and put options, e.g.,
as described below, and, therefore, .DELTA..sub.0 is the maximal
Delta to be handled.
[0162] In some demonstrative embodiments, pricing module 160 may
implement the Gershon model to determine the volatility
.sigma..sub.K.sub.Call corresponding to a given a strike
K.sub.Call>K.sub.0, for example, by solving the following
equation:
.zeta. P ( .sigma. K Put , K Put ) = .zeta. C ( .sigma. K Call , K
Call ) 1 - B ( .DELTA. ) / A ( .DELTA. ) S t d 1 1 + B ( .DELTA. )
/ A ( .DELTA. ) S t d 1 ( 23 ) ##EQU00009##
wherein .sigma..sub.K.sub.Put denotes the volatility of a put
option at the strike K.sub.Put<K.sub.0, which may be determined,
e.g., based on Equations 19 and 20, for example, as follows:
.sigma. K Put = ( 2 .zeta. C ( K Call , .sigma. K Call ) / ( Vega
.DELTA. d 1 2 ( A ( .DELTA. ) + B ( .DELTA. ) S t d 1 ) ) - 1
.sigma. K Call ) - 1 ( 24 ) ##EQU00010##
wherein the strike K.sub.Put may be determined, e.g., based on
Equations 3 and 9, for example, as follows:
K Put = F ( d 1 .sigma. K Put t + 1 / 2 .sigma. K Put 2 t ) ( 25 )
##EQU00011##
and wherein, as mentioned above:
.DELTA. = .DELTA. ( d 1 ) ; d 1 = log ( F / K Call ) .sigma. K Call
t + 1 2 .sigma. K Call t ( 26 ) ##EQU00012##
[0163] In some demonstrative embodiments, Equation 23 may be
solved, e.g., using any suitable numerical method or algorithm, to
determine the value of .sigma..sub.K.sub.Call.
[0164] Additionally or alternatively, pricing module 160 may
implement the Gershon model to determine the volatility
.sigma.K.sub.Put corresponding to the given strike
K.sub.Put<K.sub.0, since Equations 19 and 20 are symmetric with
respect to the call and put options. For example, the volatility
.sigma..sub.K.sub.Put may be determined explicitly by solving the
following equation:
.zeta. C ( .sigma. K Call , K Call ) = .zeta. P ( .sigma. K Put , K
Put ) 1 - B ( .DELTA. ) / A ( .DELTA. ) S t d 1 1 + B ( .DELTA. ) /
A ( .DELTA. ) S t d 1 ( 27 ) ##EQU00013##
wherein, e.g., based on Equations 19 and 20:
.sigma. K Call = ( 2 .zeta. P ( K Put , .sigma. K Put ) / ( Vega
.DELTA. d 1 2 ( A ( .DELTA. ) - B ( .DELTA. ) S t d 1 ) ) - 1
.sigma. K Put ) - 1 ( 28 ) ##EQU00014##
wherein, e.g., based on Equation 3:
K Call = F ( d 1 .sigma. K Call t + 1 / 2 .sigma. K Call 2 t ) ( 29
) ##EQU00015##
and wherein, as mentioned above:
- d 1 = log ( F / K Put ) .sigma. K Put t + 1 2 .sigma. K Put t (
30 ) ##EQU00016##
[0165] In some demonstrative embodiments, Equation 27 may be
solved, e.g., using any suitable numerical method or algorithm, to
determine the value of .sigma..sub.K.sub.Put.
[0166] Following is an example, in accordance with one embodiment,
of determining the functions A(.DELTA.) and B(.DELTA.) using the 25
delta strikes. However, it will be appreciated that in other
embodiments the functions A(.DELTA.) and B(.DELTA.) may be
determined in any other suitable manner, e.g., using any suitable
data and/or parameters.
[0167] In some markets, e.g., the currencies and/or commodities
markets, the 25 delta strikes may be traded. Accordingly, the
values of .sigma..sub.25.DELTA.C and .sigma..sub.25.DELTA.P for the
call and put options, respectively, may be received from the
market. The values of the functions A(.DELTA.) and B(.DELTA.) at
the 25 delta strikes may be determined, for example, since at the
25 delta 0.25=df.sub.LN(d.sub.1), then:
d 1 = N - 1 ( 0.25 / df L ) ( 31 ) Vega 25 .DELTA. = df L S t n ( N
- 1 ( 0.25 / df L ) ) ( 32 ) A ( .DELTA. = 25 ) = ( .zeta. C 25
.DELTA. + .zeta. P 25 .DELTA. ) df L S t n ( N - 1 ( 0.25 / df L )
) ( N - 1 ( 0.25 / df L ) ) 2 ( 1 .sigma. 25 .DELTA. C + 1 .sigma.
25 .DELTA. P ) = = .alpha. 1 - .beta. 1 ( 0.5 df L - 0.25 ) ( 33 )
B ( .DELTA. = 25 ) = ( .zeta. C 25 .DELTA. - .zeta. P 25 .DELTA. )
df L n ( N - 1 ( 0.25 / df L ) ) ( N - 1 ( 0.25 / df L ) ) ( 1
.sigma. 25 .DELTA. C + 1 .sigma. 25 .DELTA. P ) = = .alpha. 2 -
.beta. 2 ( 0.5 df L - 0.25 ) ( 34 ) ##EQU00017##
[0168] The values, denoted BS(25.DELTA.call) and BS(25.DELTA.put),
of the respective 25 delta call and put options according to the BS
model may be determined, for example, by substituting d1 of
Equation 31 into Equations 4 and 5, as follows:
BS(25.DELTA.call)=0.25S-K.sub.25.DELTA.Cdf.sub.RN(N.sup.-1(0.25/df.sub.L-
)-.sigma..sub.25.DELTA.c {square root over (t)}) (35)
BS(25.DELTA.call)=0.25S-K.sub.25.DELTA.Pdf.sub.RN(N.sup.-1(0.25/df.sub.L-
)-.sigma..sub.25.DELTA.p {square root over (t)}) (36)
wherein:
K.sub.put.sup.25.DELTA.=Fe.sup.(N.sup.-1.sup.(0.25/df.sup.L.sup.).sigma.-
.sup.25.DELTA.p.sup. {square root over
(t)}+1/2.sigma..sup.25.DELTA.p.sup.2.sup.t) (37)
K.sub.put.sup.25.DELTA.=Fe.sup.(N.sup.-1.sup.(0.25/df.sup.L.sup.).sigma.-
.sup.25.DELTA.c.sup. {square root over
(t)}+1/2.sigma..sup.25.DELTA.c.sup.2.sup.t) (38)
Accordingly:
BS(25.DELTA.call)=0.25S-df.sub.RFe.sup.-(N.sup.-1.sup.(0.25/df.sup.L.sup-
.).sigma..sup.25.DELTA.c.sup. {square root over
(t)}-1/2.sigma..sup.25.DELTA.c.sup.2.sup.t)N(N.sup.-1(0.25/df.sub.L)-.sig-
ma..sub.25.DELTA.c.sup.2t) (39)
BS(25.DELTA.call)=0.25S-df.sub.RFe.sup.-(N.sup.-1.sup.(0.25/df.sup.L.sup-
.).sigma..sup.25.DELTA.p.sup. {square root over
(t)}-1/2.sigma..sup.25.DELTA.p.sup.2.sup.t)N(N.sup.-1(0.25/df.sub.L)-.sig-
ma..sub.25.DELTA.P.sup.2t) (40)
[0169] The corrections .zeta..sub.25.DELTA.C and
.zeta..sub.25.DELTA.p corresponding to the call and put options at
the 25 delta strike may be determined, e.g., based on the
above-listed Equations, for example, as follows:
The values of d.sub.1 corresponding to the call and put options at
the 25 delta strike may be determined, for example, as follows:
d 1 25 .DELTA. C 0 = log ( F / K ) + 1 2 .sigma. 0 2 t .sigma. 0 t
= = N - 1 ( 0.25 / df L ) .sigma. 25 .DELTA. C .sigma. 0 + 1 2 (
.sigma. 0 - .sigma. 25 .DELTA. C 2 .sigma. 0 ) t ( 41 ) d 1 25
.DELTA. P 0 = - N - 1 ( 0.25 / df L ) .sigma. 25 .DELTA. P .sigma.
0 + 1 2 ( .sigma. 0 - .sigma. 25 .DELTA. P 2 .sigma. 0 ) t ( 42 )
##EQU00018##
and the corrections .zeta..sub.25.DELTA.C and .zeta..sub.25.DELTA.p
may be determined by subtracting the BS value for .sigma..sub.0
from the values BS(25.DELTA.call) and BS(25.DELTA.put),
respectively, for example, as follows:
.zeta. 25 .DELTA. C = df R F [ 0.25 / df L - N ( N - 1 ( 0.25 / df
L ) .sigma. 25 .DELTA. c .sigma. 0 + 1 2 ( .sigma. 0 - .sigma. 25
.DELTA. c 2 / .sigma. 0 ) t ) -- - ( N - 1 ( 0.25 / df L ) .sigma.
25 .DELTA. c t - 1 2 .sigma. 25 .DELTA. c 2 t ) ( N ( N - 1 ( 0.25
/ df L ) - .sigma. 25 .DELTA. C 2 t ) - - N ( N - 1 ( 0.25 / df L )
.sigma. 25 .DELTA. C .sigma. 0 - 1 2 ( .sigma. 0 + .sigma. 25
.DELTA. c 2 / .sigma. 0 ) t ) ) ] ( 43 ) .zeta. 25 .DELTA. P = df R
F [ - 0.25 / df L + N ( N - 1 ( 0.25 / df L ) .sigma. 25 .DELTA. P
.sigma. 0 - 1 2 ( .sigma. 0 - .sigma. 25 .DELTA. p 2 / .sigma. 0 )
t ) -- ( N - 1 ( 0.25 / df L ) .sigma. 25 .DELTA. P t + 1 2 .sigma.
25 .DELTA. p 2 t ) ( N ( N - 1 ( 0.25 / df L ) + .sigma. 25 .DELTA.
P 2 t ) - - N ( N - 1 ( 0.25 / df L ) .sigma. 25 .DELTA. P .sigma.
0 - 1 2 ( .sigma. 0 + .sigma. 25 .DELTA. p 2 / .sigma. 0 ) t ) ) ]
( 44 ) ##EQU00019##
[0170] The relationships of .alpha..sub.1(.beta..sub.1) and
.alpha..sub.2(.beta..sub.2) may be determined, for example, based
on Equations 19, 20, 21, 22, 43 and 44. One or more additional
parameters of the functions A(.DELTA.) and B(.DELTA.) may be
determined based on one or more additional parameters, e.g., the
.sigma..sub.10.DELTA.call and/or .zeta..sub.10.DELTA.put
parameters.
[0171] Following is an example, in accordance with some
demonstrative embodiments, of a method of solving Equations 19 and
20, for example, by representing elements of equations 19 and 20 in
terms of the notation d.sub.1. However, it will be appreciated that
in other embodiments Equations 19 and 20 may be solved in any other
suitable manner, e.g., using any suitable representation, notation
and/or any other solving method and/or algorithm.
[0172] In some demonstrative embodiments, Equations 19 and 20 may
be rewritten as follows, for example, using the definition of Vega,
e.g., according to Equation 10:
.zeta. C .DELTA. + .zeta. P .DELTA. = A ( d 1 ) df R F t n ( d 1 )
d 1 2 ( 1 .sigma. K Call + 1 .sigma. K Put ) ( 45 ) .zeta. C
.DELTA. - .zeta. P .DELTA. = B ( d 1 ) df L n ( d 1 ) d 1 ( 1
.sigma. K Call + 1 .sigma. K Put ) ( 46 ) ##EQU00020##
wherein A(d.sub.1) and B(d.sub.1) denote first and second
proportionality functions of d.sub.1. The functions A(d.sub.1) and
B(d.sub.1) may include one or more market-based parameters, which
may be determined based on the market data, e.g., as described
above.
[0173] In some demonstrative embodiments, a first combination of
Equations 45 and 46, for example, a sum of Equations 45 and 46 may
yield a first combined Equation, e.g., as follows:
.zeta. C .DELTA. = 1 2 ( A ( d 1 ) df R F t d 1 + B ( d 1 ) df L )
n ( d 1 ) d 1 ( 1 .sigma. K Call + 1 .sigma. K Put ) ( 47 )
##EQU00021##
[0174] In some demonstrative embodiments, the volatility
.sigma..sub.K.sub.Call may be represented as a function of the
notation d.sub.1. For example, the following representation of the
volatility .sigma..sub.K.sub.Call may be achieved, for example, by
rearranging Equation 26, e.g., since for a call option
K.sub.C>K.sub.0=Fe.sup..sigma..sup.0.sup.2.sup.t:
t .sigma. K Call = 2 log K Call F + d 1 2 + d 1 ( 48 ) 1 t .sigma.
K Call = d 1 - 2 log K Call F + d 1 2 2 log F K Call ( 49 )
##EQU00022##
[0175] In some demonstrative embodiments, the volatility
.sigma..sub.K.sub.Put may be represented as a function of
.zeta..sub.C and d.sub.1. For example, the volatility
.sigma..sub.K.sub.Put may be represented as follows, e.g., by
rearranging Equation 47 using Equations 48 and 49:
1 .sigma. K Put = 2 .zeta. C .DELTA. ( A ( d 1 ) df R F t d 1 + B (
d 1 ) df L ) n ( d 1 ) d 1 - t ( d 1 - 2 log K Call F + d 1 2 ) 2
log F K call ( 50 ) .sigma. K Put = 2 log F K call ( A ( d 1 ) df R
F t d 1 + B ( d 1 ) df L ) n ( d 1 ) d 1 4 log F K call .zeta. C
.DELTA. - ( A ( d 1 ) df R F t d 1 + B ( d 1 ) df L ) n ( d 1 ) d 1
t ( d 1 - 2 log K Call F + d 1 2 ) ( 51 ) ##EQU00023##
[0176] In some demonstrative embodiments, a second combination of
Equations 45 and 46 may yield a second combined Equation. For
example, Equation 46 may be subtracted from Equation 45 and
rearranged, e.g., as follows:
.zeta. P .DELTA. = ( A ( d 1 ) df R F t d 1 - B ( d 1 ) df L ) n (
d 1 ) d 1 ( 1 .sigma. K Call + 1 .sigma. K Put ) ( 52 )
##EQU00024##
[0177] In some demonstrative embodiments, the corrections
.zeta..sub.C.sup..DELTA. and/or .zeta..sub.P.sup..DELTA. may be
represented as a function of d.sub.1, K and .sigma..sub.0. For
example, the corrections .zeta..sub.C.sup..DELTA. and/or
.zeta..sub.P.sup..DELTA. may be represented as follows, e.g., by
combining and rearranging Equations 4, 5, 17, 18, 25, 48 and/or
49:
.zeta. C .DELTA. = .zeta. C .DELTA. ( K Call , d 1 , .sigma. 0 ) =
df R N ( d 1 ) - df R K Call N ( - 2 log K Call F + d 1 2 ) -- df R
N ( log F K Call .sigma. 0 t + 1 2 .sigma. 0 2 t ) + df R K Call N
( log F K Call .sigma. 0 t - 1 2 .sigma. 0 t ) ( 53 ) .zeta. P
.DELTA. = .zeta. P .DELTA. ( K Call , d 1 , .sigma. 0 ) = df R F (
( d 1 t .sigma. K Putt + 1 2 .sigma. K Putt 2 t ) N ( d 1 + .sigma.
K Putt t ) - N ( d 1 ) ) -- df R F ( ( d 1 t .sigma. K Putt + 1 2
.sigma. K Putt 2 t ) N ( 2 d 1 t .sigma. K Putt + .sigma. K Putt 2
t 2 .sigma. 0 t + .sigma. 0 t 2 ) - N ( 2 d 1 t .sigma. K Putt +
.sigma. K Putt 2 t 2 .sigma. 0 t - .sigma. 0 t 2 ) ) ( 54 )
##EQU00025##
wherein, for example, the volatility .sigma..sub.K.sub.Put may be
replaced according to Equation 51.
[0178] In some demonstrative embodiments, the value of d.sub.1 may
be determined, e.g., using any suitable numeric method, for
example, by requiring that Equation 52 is equal to Equation 54,
e.g., as described below.
[0179] In some demonstrative embodiments, a method of determining
the value of d.sub.1 may include selecting an initial value for
d.sub.1.
[0180] In some demonstrative embodiments, the method of determining
the value of d.sub.1 may include determining the value of the
correction .zeta..sub.C.sup..DELTA. using the value of d.sub.1,
e.g., according to Equation 53.
[0181] In some demonstrative embodiments, the method of determining
the value of d.sub.1 may include determining the value of the
volatility .sigma..sub.K.sub.Put may using the value of d.sub.1 and
the determined correction .zeta..sub.C.sup..DELTA., e.g., according
to Equation 51.
[0182] In some demonstrative embodiments, the method of determining
the value of d.sub.1 may include determining the value of the
correction .zeta..sub.P.sup..DELTA. using the value of d.sub.1 and
the determined volatility .sigma..sub.K.sub.Put, e.g., according to
Equation 54.
[0183] In some demonstrative embodiments, the method of determining
the value of d.sub.1 may include substituting the determined value
of the correction .zeta..sub.P.sup..DELTA. into Equation 52 and
determining whether or not the determined value of the correction
.zeta..sub.P.sup..DELTA. satisfies Equation 52.
[0184] In some demonstrative embodiments, if, for example, the
determined value of the correction .zeta..sub.P.sup..DELTA. does
not satisfy Equation 52, then another value of d.sub.1 may be
selected and the determining of the value of the correction
.zeta..sub.C.sup..DELTA., determining the value of the volatility
.zeta..sub.K.sub.Put, determining the value of the correction
.zeta..sub.C.sup..DELTA. and determining whether or not the
determined value of the correction .zeta..sub.P.sup..DELTA.
satisfies Equation 52 may be repeated iteratively, e.g., until
Equation 52 is satisfied. The value of d.sub.1 may be selected
according to any suitable solver algorithm.
[0185] In some demonstrative embodiments, the method of determining
the value of d.sub.1 may be performed using any suitable solver,
for example, a solver including bisection for convergence and/or
stability. In one embodiment, the solver may include a
Newton-Raphson solver. In other embodiments, the solver may include
any other suitable solver type, e.g., a Brent solver and the
like.
[0186] In some demonstrative embodiments, Equations 17 and/or 18
may be simplified using any suitable approximation, e.g., in order
to allow solving of Equations 23 and/or 24 in a more efficient
and/or quicker manner. In one example, Equations 17 and/or 18 may
be rewritten using the format of a Taylor-series approximation,
e.g., as follows:
.zeta. C ( .sigma. K Call , K Call ) = ( .sigma. K Call - .sigma. 0
) df L S t N ( d 1 ) ( 1 + ( .sigma. K Call - .sigma. 0 ) d 1 2 ( d
1 .sigma. K Call - t ) ) ( 55 ) .zeta. P ( .sigma. K Put , K Put )
= ( .sigma. K Put - .sigma. 0 ) df L S t N ( d 1 ) ( 1 + ( .sigma.
K Put - .sigma. 0 ) d 1 2 ( d 1 .sigma. K Put + t ) ) ( 56 )
##EQU00026##
[0187] In some demonstrative embodiments, pricing module 160 may
receive from user 102, e.g., via interface 110, first input data
including one or more parameters defining an option to be priced
("the requested option").
[0188] In some demonstrative embodiments, pricing module 160 may
receive, e.g., from market data service 149, second input data
corresponding to at least one current market condition relating to
an underlying asset of the option, e.g., including real time market
data corresponding to an asset class of the requested option.
[0189] For example, for FX instruments, pricing module 160 may
receive from market data service 149 market data including one or
more of spot rates, forward rates, interest rates, at the money
volatility for different maturities, 25 delta risk reversals for
different maturities, 25 delta butterflies for different maturities
and, optionally, other delta risk reversals and/or butterflies,
e.g., the 10 delta risk reversal and/or the 10 delta butterfly.
[0190] For interest-rate instruments, pricing module 160 may
receive from market data service 149 market data including one or
more of Libor rates, e.g., all Libor rates in all available
countries, swap rates for all maturities, interest-rates future
prices in currencies, where available, cap floor volatilities or
prices for several strikes, swaption at the money volatilities and
other strikes such as 100 or 200 basis points over and under the at
the money forward strike.
[0191] For equity options pricing module 160 may receive from
market data service 149 market data including exchange prices for
stocks and indices, exchange prices for options on stocks and
indices, forward prices for several maturities, and/or security
lending rates and interest rates, and the like.
[0192] In some demonstrative embodiments, pricing module 160 may
determine the functions A(.DELTA.) and/or B(.DELTA.), for example,
based on the received market data, e.g., using Equations 31-34, as
described above.
[0193] In some demonstrative embodiments, pricing module 160 may
determine the volatility smile corresponding to the option. For
example, pricing module may determine one or more volatilities
.sigma..sub.k corresponding to a Vanilla option having one or more
respective strikes K, for example, based on Equations 23 and/or 27,
e.g., depending on the whether the option is a call option or a put
option.
[0194] In some demonstrative embodiments, pricing module 160 may
perform any suitable extrapolation and/or interpolation operations
to determine a volatility surface and/or the volatility
corresponding to the strike and expiration time of the requested
option, e.g., based on the determined volatilities
.sigma..sub.k.
[0195] In some demonstrative embodiments, pricing module 160 may
determine the correction .zeta. to be added to the BS value of the
Vanilla option in accordance with the volatility smile, for
example, according to Equations 23 and/or 27, e.g., depending on
the whether the option is a call option or a put option.
[0196] In some demonstrative embodiments, pricing module 160 may
determine the price of the Vanilla option based on the correction
.zeta. and the BS value of the Vanilla option, e.g., according to
Equation 16.
[0197] In some demonstrative embodiments, pricing module 160 may
determine the price of the requested option, e.g., based on the
determined price of the corresponding Vanilla option.
[0198] In some demonstrative embodiments, interface 110 and pricing
module 160 may be implemented as part of an application or
application server to process user information, e.g., including
details of a defined option to be priced, received from user 102,
as well as real time trade information, received, for example, from
market data service 149. System 100 may also include storage 161,
e.g., a database, for storing the user information and/or the trade
information.
[0199] The user information may be received from user 102, for
example, via a communication network, for example, the Internet,
e.g., using a direct telephone connection or a Secure Socket Layer
(SSL) connection, a Local Area Network (LAN), or via any other
communication network known in the art. Pricing module 160 may
communicate a determined price corresponding to the defined option
to user 102 via interface 110, e.g., in a format convenient for
presentation to user 102.
[0200] A system, e.g., system 100, for pricing financial
derivatives according to some embodiments, may provide price
information for substantially any suitable option on substantially
any suitable asset based on input market data. The market data may
be easily obtained, e.g., by pricing module 160, on a real time
basis. Thus, pricing module 160 may provide user 102 with a
real-time price of any desired option, e.g., based on real time
prices received from the exchanges and/or OTC market. Pricing
module 160 may update the price, e.g., substantially immediately
and/or automatically, for example, in response to a change in spot
prices and/or option prices. This may enable user 102 to
automatically update prices for trading with the exchanges.
[0201] A trader may want, for example, to submit a plurality of bid
and/or offer (hereinafter "bid/offer") prices for a plurality of
options, e.g., ten bid/offer prices for ten options, respectively.
When entering the bids/offers to a quoting system, the trader may
check the price, e.g., in relation to the current spot prices, and
may then submit the bids/offers to the exchange. Some time later,
e.g., a second later, the spot price of the stock which is the
underlying asset of one or more of the options may change. A change
in the spot prices may be accompanied, for example, by changes in
the volatility parameters, or may include just a small spot change
while the volatility parameters have not changed. In response to
the change in the spot price, the trader may want to update one or
more of the submitted bid/offer prices. The desire to update the
bid/offer prices may occur, e.g., frequently, during trade
time.
[0202] A system according to some demonstrative embodiments, e.g.,
system 100, may automatically update the bid/offer prices entered
by the trader, e.g., based on any desired criteria. For example,
pricing module 160 may evaluate the trader's bids/offers versus bid
and offer prices of the options, which may be estimated by pricing
module 160, e.g., when the trader submits the bid/offer prices.
Pricing module 160 may then automatically recalculate the bid
and/or offer prices, e.g., whenever the spot changes, and may
automatically update the trader's bid/offer prices. Pricing module
160 may, for example, update one or more of the trader's bid/offer
prices such that a price difference between the bid/offer price
calculated by pricing module 160 and the trader's bid/offer price
is kept substantially constant. According to another example,
pricing module 160 may update one or more of the trader's bid/offer
prices based on a difference between the trader's bid/offer prices
and an average of bid and offer prices calculated by pricing module
160. Pricing module 160 may update one or more of the trader's
bid/offer prices based on any other desired criteria.
[0203] It is noted, that a change of the spot price, e.g., of a few
pips, may result in a change in one or more of the volatility
parameters of options corresponding to the spot price. It will be
appreciated that a pricing module according to some embodiments,
e.g., pricing module 160, may enable automatically updating one or
more option prices submitted by a trader, e.g., while taking into
account the change in the spot price, in one or more of the
volatility parameters, and/or in any other desired parameters, as
described above.
[0204] According to some demonstrative embodiments, pricing module
160 may enable the trader to submit one or more quotes in the
exchange in a form of relative prices vs. prices determined by the
pricing module 160. For example, the trader may submit quotes for
one or more desired strikes and/or expiry dates. The quotes
submitted by the trader may be in any desired form, e.g., relating
to one or more corresponding prices determined by pricing module
160. For example, the quotes submitted by the trader may be in the
terms of the bid/offer prices determined by pricing module 160 plus
two basis points; in the terms of the mid market price determined
by pricing module 160 minus four basis points, and/or in any other
suitable format and/or terms. Pricing module 160 may determine the
desired prices, for example, in real time, e.g., whenever a price
change in the exchange is recorded. Alternatively, pricing module
160 may determine the desired prices, according to any other
desired timing scheme, for example, every predefined time interval,
e.g., every half a second.
[0205] A change in a spot price of a stock may result in changes in
the prices of a large number of options related to the stock. For
example there could be over 200 active options relating to a single
stock and having different strikes and expiration dates.
Accordingly, a massive bandwidth may be required by traders for
updating the exchange prices of the options in accordance with the
spot price changes, e.g., in real time. This may lead the traders
to submit to the exchange prices which may be "non-competitive",
e.g., prices including a "safety-margin", since the traders may not
be able to update the submitted prices according to the rate at
which the spot prices, the volatility, the dividend, and/or the
carry rate may change.
[0206] According to some demonstrative embodiments, pricing module
160 may be implemented, e.g., by the exchange or by traders, for
example, to automatically update one or more bid and/or offer
prices submitted by a trader, e.g., as described above. This may
encourage the traders to submit with the exchange more aggressive
bid and/or offer prices, since the traders may no longer need to
add the "safety margin" their prices for protecting the traders
against the frequent changes in the spot prices. Accordingly, the
trading in the exchange may be more effective, resulting in a
larger number of transactions. For example, a trader may provide
pricing system 100 with one or more desired volatility parameter
and/or rates. The trader may request system 100 to automatically
submit and/or update bid and/or offer prices on desired amounts of
options, e.g., whenever there is a significant change in the spot
price and/or in the volatility of the market. The trader may also
update some or all of the volatility parameters. In addition,
system 100 may be linked, for example, to an automatic decision
making system, which may be able to decide when to buy and/or sell
options using pricing module 160.
[0207] Reference is made to FIG. 2, which schematically illustrates
a method of pricing an option in accordance with some demonstrative
embodiments. In some demonstrative embodiments, one or more of the
operations of the method of FIG. 2 may be performed and/or
implemented by any suitable device and/or system, for example,
suitable computing device and/or system, e.g., system 100 (FIG. 1)
and/or pricing module 160 (FIG. 1).
[0208] As indicated at block 202, the method may include receiving
first input data corresponding to at least one parameter defining a
first option on an underlying asset. For example, module 160 (FIG.
1) may receive e.g., from user 102 (FIG. 1), the first input data
defining an option to be priced, e.g., as described above.
[0209] As indicated at block 204, the method may include receiving
second input data corresponding to at least one current market
condition relating to the underlying asset. For example, module 160
(FIG. 1) may receive e.g., from services 149 (FIG. 1), the second
input data corresponding to the underlying asset, e.g., as
described above.
[0210] As indicated at block 206, the method may include
determining a price of the first option based on the first and
second input data, according to a volatility smile satisfying one
or more predefined criterions.
[0211] As indicated at block 208, determining the price of the
first option may include determining the price of the first option
according to a volatility smile satisfying a first criterion
relating to a sum of a first correction corresponding to the first
option and a second correction corresponding to a second option
representing a position opposite to a position of a the first
option and having a same delta as the first option.
[0212] In some demonstrative embodiments, the first correction may
relate to a difference between a theoretical price of the first
option and the price of the first option according to the
volatility smile, and/or the second correction may relate to a
difference between a theoretical price of the second option and the
price of the second option according to the volatility smile. For
example, module 160 (FIG. 1) may determine the price of the first
option according to a volatility smile satisfying Equations 19 and
20, e.g., as described above.
[0213] As indicated at block 210, determining the price of the
first option according to the volatility smile may include
determining market-based parameters of first and second
proportionality functions based on the second input data. For
example, module 160 (FIG. 1) may determine the market-based
parameters of the proportionality functions A(.DELTA.) and
B(.DELTA.) based on the market data, e.g., as described above.
[0214] As indicated at block 212, determining the price of the
first option according to the volatility smile may include
determining the first correction based on the first and second
criterions. For example, module 160 (FIG. 1) may determine the
correction .zeta. corresponding to the first option according to
Equations 23 and/or 27, e.g., as described above.
[0215] As indicated at block 214 determining the first correction
may include determining a volatility of the first option based on
the first and second criterions, and determining the first
correction based on the volatility of the first option. For
example, module 160 (FIG. 1) may determine the volatility .sigma.
corresponding to the first option, and the correction .zeta.
corresponding to the volatility .sigma., e.g., as described
above.
[0216] Following are examples of volatility smiles determined with
respect to options on various asset classes, using the volatility
smile mode as described herein in accordance with some
demonstrative embodiments. It should be noted that the trade
information used in these examples have been randomly selected from
the market for demonstrative purposes only and is not intended to
limit the scope of the embodiments described herein to any
particular choice of the trade information.
[0217] The volatility smiles were determined using the following
proportionality functions:
A(.DELTA.)=c.sub.1e.sup.-C.sup.2.sup.(.DELTA..sup.0.sup.-.DELTA.)
(57)
B(.DELTA.)=c.sub.1'e.sup.-C.sup.2.sup.'(.DELTA..sup.0.sup.-.DELTA.)
(58)
wherein c.sub.1, c.sub.1', c.sub.2, c.sub.2' denote four respective
market parameters to be determined, e.g., based on the traded
market data.
[0218] The following examples demonstrate the results of the
volatility smile model with respect to different asset classes,
e.g., at the same time. The following examples relate to options on
currencies, e.g., options on the exchange rate of EURO (EUR) to US
dollar (USD) (EUR/USD), which are traded in the OTC market; options
on Interest Rates, e.g., swaptions on EUR swap rates, which are
traded in the OTC market; options on Commodities, e.g., options on
West Texas intermediate (WTI) crude oil, which is exchange traded;
and options on Equities, e.g., options on the DAX index, which is
exchange traded. All of the examples relate to assets, which are
very liquid and commonly traded, therefore the market data may be
assumed to be accurate. The examples relate to different
maturities. The following examples are based on market data on Dec.
27, 2010.
[0219] A first example relates to FX options on EUR/USD with an
expiration of one year. The FX options market trades ATM delta
neutral volatility as well as delta strikes. The inputs received
from the market are summarized in Table 1:
[0220] Delta neutral ATM vol .sigma..sub.0=14.45; Forward
rate=1.31408
TABLE-US-00001 TABLE 1 Delta 5 .DELTA. Put 10 .DELTA. Put 25
.DELTA. Put ATM 25 .DELTA. Call 10 .DELTA. Call 5 .DELTA. Call
Strike 0.951 1.053 1.1956 1.3279 1.4541 1.5933 1.7016 Market Vol
21.19 18.775 16.225 14.45 13.825 14.325 15.08
[0221] Based on the above market data, the market-based parameters
may be determined as follows, e.g., using the model described
above: c1=0.002, c2=0.5, c1'=0.0042, c2'=1.6.
[0222] A volatility smile ("the model volatility smile")
corresponding to the FX options may be determined according to
Equations 19 and 20, e.g., as described above. Table 2 includes
seven volatilities corresponding to seven respective strikes
determined according to the volatility smile:
TABLE-US-00002 TABLE 2 Strike 0.9423 1.0562 1.1941 1.3279 1.454
1.5933 1.7016 Model 21.638 18.533 16.288 14.450 13.723 14.343
15.023 Vol
[0223] FIG. 3A schematically illustrates a first graph 302
depicting the model volatility smile based on Table 2, and a second
graph 304 depicting the market volatilities of Table 1. As shown in
FIG. 3A, the differences between the model volatility smile and the
market volatilities are generally negligible.
[0224] A second example relates to options on EUR swaps rate with a
maturity of ten years and an expiration of one year. The
interest-rates market trades ATM forward strikes (ATMF, where the
strike is the forward rate) and other strikes may be measured with
respect to a difference in basis points from the forward rate. The
inputs received from the market are summarized in Table 3:
Forward=3.671
TABLE-US-00003 [0225] TABLE 3 Market Data -100 -50 -25 ATMF +25 +50
+100 +200 Strike 2.671 3.171 3.421 3.671 3.921 4.171 4.671 5.671
Market Vol 31.3 27.7 26.3 25.1 24 23.2 22.2 21.9
[0226] Based on the above market data, the market-based parameters
may be determined as follows, e.g., using the model described
above: .sigma..sub.0=24.5, c1=0.0045, c2=1.5, c1'=0.0095,
c2'=0.1.
[0227] The model volatility smile corresponding to the IR options
may be determined according to Equations 19 and 20, e.g., as
described above. Table 4 includes volatilities corresponding to
respective strikes determined according to the volatility
smile:
TABLE-US-00004 TABLE 4 Strike 2.644177 2.874236 3.12931 3.387316
3.643633 3.921 Model 31.21694 29.41119 27.83829 26.42833 25.13876
23.94235 Vol Strike 4.171 4.421 4.671 4.9 5.671 Model 23.12829
22.52502 22.11125 21.88758 21.9418 Vol
[0228] FIG. 3B schematically illustrates a first graph 306
depicting the model volatility smile based on Table 4 and a second
graph 308 depicting the market volatilities of Table 3. As shown in
FIG. 3B, the differences between the model volatility smile and the
market volatilities are generally negligible.
[0229] A third example relates to options on WTI crude oil with
expiration on Nov. 15, 2012 (687 days). The underlying asset of
these options is the WTI future contract of December 12 (December
2012). The market data is taken from the Nymex exchange (CME), and
includes about 20 strikes with their corresponding volatility
implied from the exchange price for option premium. The inputs
received from the market are summarized in Table 5:
Forward=92.61
TABLE-US-00005 [0230] TABLE 5 Strike 65 70 75 80 85 90 95 100 105
110 Market 29.75 28.97 28.06 27.04 26.32 25.68 25.07 24.47 23.89
23.8 Vol Strike 115 120 125 130 135 140 145 150 160 Market 23.65
23.65 23.74 23.83 24.3 24.16 24.4 24.6 25.02 Vol
[0231] Based on the above market data, the market-based parameters
may be determined as follows, e.g., using the model described
above: .sigma..sub.0=24.491, c1=0.0105, c2=0.015, c1'=0.0165,
c2'=0.65.
[0232] The model volatility smile corresponding to the WTI options
may be determined according to Equations 19 and 20, e.g., as
described above. Table 6 includes volatilities corresponding to
respective strikes determined according to the volatility
smile:
TABLE-US-00006 TABLE 6 Strike 64.52 70.87 78.17 82.23 86.58 91.21
96.05 100.00 105.00 110.00 Model 30.20 28.76 27.38 26.71 26.05
25.40 24.79 24.35 23.94 23.69 Vol Strike 115 120 125 130 135 140
145 150 160 Model 23.58 23.58 23.66 23.80 23.98 24.19 24.41 24.65
25.15 Vol
[0233] FIG. 3C schematically illustrates a first graph 310
depicting the model volatility smile based on Table 6 and a second
graph 312 depicting the market volatilities of Table 5. As shown in
FIG. 3C, the differences between the model volatility smile and the
market volatilities are generally negligible.
[0234] A fourth example relates to options on the DAX index with
expiration on Dec. 21, 2012 (725 calendar days). The market
volatilities are taken from the exchange settlement prices for the
expiry date of Dec. 21, 2012. The inputs received from the market
are summarized in Table 7:
Forward=7187.635
TABLE-US-00007 [0235] TABLE 7 Strike 4200 4600 5000 5400 5800 6200
6600 7000 7400 Market 33.16 31.62 30.14 28.72 27.33 25.95 24.60
23.30 22.11 Vol Strike 7600 8000 8400 8800 9200 9600 10000 10400
11000 Market 21.57 20.61 19.82 19.19 18.65 18.16 17.72 17.32 17.02
Vol
[0236] Based on the above market data, the market-based parameters
may be determined as follows, e.g., using the model described
above: .sigma..sub.0=22.00, c1=0.005, c2=0.2, c1'=0.025,
c2'=0.1.
[0237] The model volatility smile corresponding to the DAX options
may be determined according to Equations 19 and 20, e.g., as
described above. Table 8 includes volatilities corresponding to
respective strikes determined according to the volatility
smile:
TABLE-US-00008 TABLE 8 Strike 4238.85 4693.35 5159.1 5639.83
6127.86 6606.64 7060.75 7482.51 7600 Model 33.79 31.31 29.26 27.50
25.94 24.54 23.29 22.15 21.84 Vol Strike 8000 8400 8800 9200 9600
10000 10400 11000 Model 20.81 19.87 19.06 18.45 18.05 17.85 17.80
17.89 Vol
[0238] FIG. 3D schematically illustrates a first graph 314
depicting the model volatility smile based on Table 8 and a second
graph 316 depicting the market volatilities of Table 7. As shown in
FIG. 3D, the differences between the model volatility smile and the
market volatilities are generally negligible.
[0239] Reference is made to FIG. 4, which schematically illustrates
an article of manufacture 400, in accordance with some
demonstrative embodiments. Article 400 may include a
machine-readable storage medium 402 to store logic 404, which may
be used, for example, to perform at least part of the functionality
of pricing module 160 (FIG. 1); and/or to perform one or more
operations described herein.
[0240] In some demonstrative embodiments, article 400 and/or
machine-readable storage medium 402 may include one or more types
of computer-readable storage media capable of storing data,
including volatile memory, non-volatile memory, removable or
non-removable memory, erasable or non-erasable memory, writeable or
re-writeable memory, and the like. For example, machine-readable
storage medium 402 may include, RAM, DRAM, Double-Data-Rate DRAM
(DDR-DRAM), SDRAM, static RAM (SRAM), ROM, programmable ROM (PROM),
erasable programmable ROM (EPROM), electrically erasable
programmable ROM (EEPROM), Compact Disk ROM (CD-ROM), Compact Disk
Recordable (CD-R), Compact Disk Rewriteable (CD-RW), flash memory
(e.g., NOR or NAND flash memory), content addressable memory (CAM),
polymer memory, phase-change memory, ferroelectric memory,
silicon-oxide-nitride-oxide-silicon (SONOS) memory, a disk, a
floppy disk, a hard drive, an optical disk, a magnetic disk, a
card, a magnetic card, an optical card, a tape, a cassette, and the
like. The computer-readable storage media may include any suitable
media involved with downloading or transferring a computer program
from a remote computer to a requesting computer carried by data
signals embodied in a carrier wave or other propagation medium
through a communication link, e.g., a modem, radio or network
connection.
[0241] In some demonstrative embodiments, logic 404 may include
instructions, data, and/or code, which, if executed by a machine,
may cause the machine to perform a method, process and/or
operations as described herein. The machine may include, for
example, any suitable processing platform, computing platform,
computing device, processing device, computing system, processing
system, computer, processor, or the like, and may be implemented
using any suitable combination of hardware, software, firmware, and
the like.
[0242] In some demonstrative embodiments, logic 404 may include, or
may be implemented as, software, a software module, an application,
a program, a subroutine, instructions, an instruction set,
computing code, words, values, symbols, and the like. The
instructions may include any suitable type of code, such as source
code, compiled code, interpreted code, executable code, static
code, dynamic code, and the like. The instructions may be
implemented according to a predefined computer language, manner or
syntax, for instructing a processor to perform a certain function.
The instructions may be implemented using any suitable high-level,
low-level, object-oriented, visual, compiled and/or interpreted
programming language, such as C, C++, Java, BASIC, Matlab, Pascal,
Visual BASIC, assembly language, machine code, and the like.
[0243] The processes and displays presented herein are not
inherently related to any particular computer or other apparatus.
Various general-purpose systems may be used with programs in
accordance with the teachings herein, or it may prove convenient to
construct a more specialized apparatus to perform the desired
method. The desired structure for a variety of these systems will
appear from the description below. In addition, some embodiments
are not described with reference to any particular programming
language. It will be appreciated that a variety of programming
languages may be used to implement the teachings of the invention
as described herein.
[0244] Functions, operations, components and/or features described
herein with reference to one or more embodiments, may be combined
with, or may be utilized in combination with, one or more other
functions, operations, components and/or features described herein
with reference to one or more other embodiments, or vice versa.
[0245] While certain features of the invention have been
illustrated and described herein, many modifications,
substitutions, changes, and equivalents may occur to those skilled
in the art. It is, therefore, to be understood that the appended
claims are intended to cover all such modifications and changes as
fall within the true spirit of the invention.
* * * * *