U.S. patent application number 14/946849 was filed with the patent office on 2016-03-24 for method and apparatus for deriving benchmarks for trading instruments.
The applicant listed for this patent is EBS Group Limited. Invention is credited to Edward R. Howorka, Neena Jain, David Jifeng Liu, Jeffrey Edward Power, Nasir Ahmed Zubairi.
Application Number | 20160086276 14/946849 |
Document ID | / |
Family ID | 32713266 |
Filed Date | 2016-03-24 |
United States Patent
Application |
20160086276 |
Kind Code |
A1 |
Howorka; Edward R. ; et
al. |
March 24, 2016 |
METHOD AND APPARATUS FOR DERIVING BENCHMARKS FOR TRADING
INSTRUMENTS
Abstract
Benchmarks for the price of a financial instrument such as FX
spot rate for a currency pair are calculated by an algorithm based
on a previous benchmark and a market price. The market price is
derived from a deal price and a quote price. The deal price is
based on deals conducted since the last benchmark and the quote
price is based on bids and offers entered since the last benchmark.
For each of the deal and quote prices, a price, weight and scatter
is calculated which is used to calculate a benchmark price, weight
and scatter and a benchmark error.
Inventors: |
Howorka; Edward R.; (Morris
Plains, NJ) ; Power; Jeffrey Edward; (Rockaway,
NJ) ; Zubairi; Nasir Ahmed; (New York, NY) ;
Liu; David Jifeng; (East Rutherford, NJ) ; Jain;
Neena; (South Plainfield, NJ) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
EBS Group Limited |
London |
|
GB |
|
|
Family ID: |
32713266 |
Appl. No.: |
14/946849 |
Filed: |
November 20, 2015 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
12365339 |
Feb 4, 2009 |
|
|
|
14946849 |
|
|
|
|
10983170 |
Nov 5, 2004 |
8548892 |
|
|
12365339 |
|
|
|
|
10745694 |
Dec 29, 2003 |
|
|
|
10983170 |
|
|
|
|
60438025 |
Jan 2, 2003 |
|
|
|
Current U.S.
Class: |
705/37 |
Current CPC
Class: |
G06Q 40/00 20130101;
G06Q 40/04 20130101; G06Q 30/0283 20130101 |
International
Class: |
G06Q 40/04 20060101
G06Q040/04; G06Q 30/02 20060101 G06Q030/02 |
Claims
1. A method for calculating benchmark prices for trades in an
instrument, the method being carried out by one or more programmed
computers, the method comprising: receiving, by the one or more
programmed computers, information indicating actual prices which
occurred on one or more electronic trading systems, the actual
prices being quote and/or deal prices; calculating, by the one or
more programmed computers, a new benchmark price at each of a
plurality of successive time instants, each new benchmark price
being calculated as a function of (a) a previously calculated
benchmark price, (b) a market price determined as a function of the
actual prices which occurred since the previous benchmark price was
calculated, and (c) a phantom price, wherein the phantom price is:
equal to the previous benchmark price if no actual prices occurred
since the previous benchmark price was calculated; and calculated
as a function of the actual prices which occurred since the last
benchmark price was calculated, if such actual prices occurred; and
delivering, by the one or more programmed computers, the calculated
new benchmark price to at least one recipient.
2. A method according to claim 1, wherein the prices include both
quote prices and deal prices.
3. A method according to claim 1, wherein a weight is assigned to
each of the prior benchmark price, the market price and the phantom
price when calculating the new benchmark price, the weight assigned
to the phantom price being sufficiently less than the weight
assigned to the previous benchmark price and the market price such
that the phantom price has substantially no effect on the newly
calculated benchmark price if actual prices occurred since the last
benchmark was calculated.
4. A method according to claim 3, wherein the weight of the phantom
price is greater than zero but not more than 0.5*10.sup.-6.
5. A method according to claim 3, wherein a scatter is assigned to
each of the prior benchmark price, the market price and the phantom
price when calculating the new benchmark price, the scatter
assigned to the phantom price being sufficiently less than the
scatter assigned to the previous benchmark price and the market
price such that the phantom price has substantially no effect on
the newly calculated benchmark price if actual prices occurred
since the last benchmark was calculated.
6. A method according to claim 5, wherein the scatter of the
phantom price is between zero and 1% of the big figure of the
instrument being traded.
7. A method according to claim 1, wherein the recipient is a
trading entity.
8. A method according to claim 1, wherein the recipient is a
trader.
9. A method according to claim 1 wherein the actual price is the
same as the quote price when there are no deal prices.
10. A method according to claim 1 wherein the one or more
programmed computers form a market rate feeder server.
11. The method of claim 1, wherein the actual price is the same as
the quote price when there arc no read prices.
12. The method of claim 1, wherein the one or more programmed
computers form a market rate fee server.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application is a Continuation of U.S. application Ser.
No. 12/365,339 filed Feb. 4, 2009, which is a divisional of U.S.
application Ser. No. 10/983,170, filed Nov. 5, 2004, now U.S. Pat.
No. 8,548,892 issued Oct. 1, 2013, which is a continuation of U.S.
application Ser. No. 10/745,694, filed Dec. 29, 2003, now
abandoned, which claims benefit of U.S. Provisional Application No.
60/438,025, filed Jan. 2, 2003, each of which are hereby
incorporated by reference in their entirety.
FIELD OF THE INVENTION
[0002] This invention relates to methods, apparatus and systems for
deriving benchmarks for use in trading instruments, particularly
but not exclusively, financial instruments such as foreign exchange
(FX), including FX spot, FX forwards and other foreign exchange
products.
BACKGROUND TO THE INVENTION
[0003] Benchmarks are used in the trading of instruments such as FX
spot to provide a reference point which indicates the actual state
of the market and which is neither biased to the buy side or sell
side of the market. As markets can move very rapidly it is
important to recalculate the benchmarks frequently and to
distribute them to participating traders and other interested
parties so that the benchmarks remain a reliable indicator.
[0004] Benchmarks may be derived in a number of ways. However, all
are based on the market history, and access to good information as
to the current state of the market, and trades conducted in the
market is essential. Typically, a financial institution will trade
a variety of currency pairs and benchmarks need to be established
in each of those pairs.
SUMMARY OF THE INVENTION
[0005] The present invention aims to provide an improved algorithm
for calculation of benchmarks. The algorithm calculates benchmarks
on the basis of the prices of deals conducted since a previous
benchmark was established, quotes at present in the market and the
previous benchmark price. The invention further aims to provide a
computerized system which can calculate those benchmarks according
to the algorithm and then deliver them to traders.
[0006] More specifically there is provided a method of establishing
a benchmark price for trades in a instrument at a given time,
comprising: acquiring prices at which deals in the instrument have
been conducted since a previous benchmark price was established;
acquiring the prices of quotes at present in the market; and
calculating a benchmark price from the previous benchmark price and
a measure of deal prices and quote prices occurring since the
previous benchmark price.
[0007] Preferably a market price is calculated from the measure of
deal prices and quote prices; and the benchmark is calculated from
the market price and the previous benchmark price. Preferably the
deal prices and the quote prices are acquired from a trading
system.
[0008] Preferably, the benchmark is established periodically every
t seconds. This enables the benchmark price to remain an accurate
reflection of the true state of the market.
[0009] Preferably, the calculation of market prices includes the
weighting of deal prices according to size and/or age. Weighting
may be performed in terms of multiples of a minimum deal size,
preferably according to the formula W(d.sub.i)=V.sub.1*2.sup.-ct
where V.sub.1 is the volume of the deal, t is the elapsed time
since the previous benchmark, and c is a constant determining the
speed at which deals are marked down over time.
[0010] Preferably, the calculation of market prices includes the
summing of the weights of all deal prices occurring since the
previous benchmark.
[0011] The market price may be calculated from a deal price
obtained from the arithmetic average of individual deal prices for
deals conducted since the previous benchmark was established,
weighted by their deal weight. Deal price calculation may further
include calculation of a scatter for deals conducted since the
previous benchmark price was established. This deal price scatter
may be derived from the standard deviation of deals measured with
respect to deal price and weight.
[0012] The calculation of market prices may also comprise weighting
quote prices according to size and age. Weighting may be in terms
of a minimum quote size and may be according to their distance from
the best quote in the market. Weighting may be performed according
to the formula: W(Q.sub.i)=V.sub.1*2.sup.-P(Q1)-B where V.sub.1 is
the volume of a quote, P(Q.sub.i) is the quote price and B is the
best bid or offer depending on whether Q.sub.i is a bid or an
offer.
[0013] All quote prices occurring since the previous benchmark may
also be summed and adjusted by a constant representing demand as
part of the process of obtaining the market price.
[0014] Preferably, a quote scatter price is also calculated for all
quotes entered into the market since the previous benchmark price
was established. This scatter may be derived fro the standard
derivation of quote prices measured with respect to quote price and
weight.
[0015] Preferably, the market price is derived from an average of
the quote and deal prices each sealed by its own weight. A market
price weight is calculated from the sum of the deal and quote price
weights. Preferably a market price scatter is calculated, for
example from the standard derivation of weighted market prices.
[0016] Preferably, a phantom price is also calculated. If no prices
have been entered since the last benchmark, this phantom price is
equal to the previous benchmark. If prices have been entered, this
price is zero.
[0017] The phantom price may have a weight which is equal to a very
small constant, for example 0.5*10.sup.-6. The phantom price may
also have a scatter which is equal to a small constant, such as 1%
of the base figure of the instrument being traded.
[0018] Preferably, the benchmark price is derived from the market
price, a market price weight, the previous benchmark price, and a
previous benchmark price weight. The latter weight may be
calculated from the sum of the market price weight and the last
benchmark price weight modified by a time markdown.
[0019] Preferably, a benchmark price error is also calculated. This
error may be a standard error derived from the benchmark scatter
and the benchmark price weight.
[0020] A further aspect of the invention provides a method of
establishing a benchmark price for trades in a instrument at a
given time, comprising: acquiring from at least one trading system
deal price information relating to prices at which deals in the
instrument have been conducted since a previous benchmark price was
established; calculating from the deal price information, a deal
price weight, a deal price and a deal scatter; acquiring from the
at least one trading system price quote information relating to the
prices of quotes at present in the trading system; calculating from
the quote price information a quote price weight, a quote price and
a quote price scatter; calculating a market price weight, a market
price and a market price scatter from the deal price weight, the
deal price and the deal price scatter, the quote price weight, the
quote price and the quote price scatter; and calculating a
benchmark price from the previous benchmark price and the market
price.
[0021] The invention further provides a method of establishing a
benchmark price for trades in an instrument at a given time,
comprising the step of: periodically calculating the benchmark
price from a previous benchmark price and a measure of deal prices
and quote prices occurring since the previous benchmark price.
[0022] The invention also provides a method of establishing a
benchmark price for trades in an instrument at a given time,
comprising the step of: periodically calculating the benchmark
price from a previous benchmark price and a measure of deal prices
occurring since the previous benchmark price, wherein the deal
prices are weighted in accordance with their age.
[0023] The invention also provides a method of establishing a
benchmark price for trades in an instrument at a given time,
comprising the step of: periodically calculating the benchmark
price from a previous benchmark price and a measure of deal prices
occurring since the previous benchmark price, wherein the deal
prices are weighted in accordance with their volume.
[0024] The invention further provides A method of establishing a
benchmark price for trades in an instrument at a given time,
comprising the step of: periodically calculating the benchmark
price from a previous benchmark price and a measure of deal prices
occurring since the previous benchmark price, wherein the deal
prices are weighted in accordance with their age and volume.
BRIEF DESCRIPTION OF THE DRAWINGS
[0025] Embodiments of the invention will now be described, by way
of example only, and with reference to the accompanying drawings,
in which:
[0026] FIG. 1 illustrates how deal prices may be scattered or
condensed during any time interval;
[0027] FIG. 2 is an example of a benchmark established in a quote
only market;
[0028] FIG. 3 is a further example of a benchmark established in a
quote only market;
[0029] FIG. 4 is a still further example of a benchmark established
in a quote only market;
[0030] FIG. 5 is an example of a benchmark established in a deal
only market;
[0031] FIG. 6 is a further example of a benchmark established in a
deal only market;
[0032] FIG. 7 is a still further example of a benchmark established
in a deal only market;
[0033] FIG. 8 is a schematic block diagram of a system for
implementing calculating and delivering benchmarks to traders;
and
[0034] FIG. 9 is a schematic block diagram of a further embodiment
of a system for implementing calculating and delivering benchmarks
to traders in which benchmarks are also exchanged with a third
party.
DESCRIPTION OF BEST MODE
[0035] The algorithm used to calculate benchmark prices will first
be described. A computerized system which can calculate those
benchmarks according to the algorithm and then deliver them to
traders will then be described.
[0036] The following description considers how benchmark prices are
derived from the various trading data that is available, for
example from an anonymous trading system, or another trading system
or systems. The Benchmark price is derived from the market price
and the previous Benchmark price. The market price is itself
derived from the deal price and the quote price. The deal price is
calculated from actual deals that have been done on the system and
the quote price is calculated from quotes (bids and offers) that
have been entered into the system by traders.
[0037] In the foreign exchange market, for example, the price of a
currency pair moves up and down. The Benchmark Price attempts to
describe the price best by utilizing market information acquired
from one or more sources, such as an anonymous trading system. The
Benchmark Price is calculated at a given frequency, for example,
every t seconds. At each tick, a Benchmark is published along with
a margin of error called the Standard Error.
[0038] Using the market information from the trading system is like
sampling the price population at the tick time. This sample
includes deal and quote transactions. These transactions act as
traders' votes on where the market is.
[0039] The currency pair price is a population (P) at any given
moment. Suppose its mean is MN(P) (the "true price") and its
Standard Deviation is SD(P). The Benchmark Price approximates MN(P)
by sampling this price population. In the following description, P
is used to represent the "price population" as well as the price;
that is, the Mean price of the population, just as MN(P).
[0040] A general overview will first be given, showing how
Benchmark prices are calculated. The derivation of each element
will then be discussed in more detail.
Prices
[0041] At time n, Benchmark Price P.sub.n is derived from Market
Price at the time, and P.sub.n-1, the Benchmark Price at the last
tick n-1. Market price consists of Deal Price calculated from deals
that occurred in the past time interval, and Quote Price calculated
from quotes at the moment.
[0042] These four elements may be expressed as follows:
[0043] At time n: DealPrice=function (deal price and volume in the
past time interval of n-1 to n);
QuotePrice=function (quotes price and volume at n);
MarketPrice=function (QuotePrice, DealPrice); and
BenchmarkPrice=function (MarketPrice, BenchmarkPrice at n-1).
Weight and Scatter
[0044] For each of these prices, an associated weight W and a value
of Scatter, or an approximated Standard Deviation, .sigma. are also
calculated. That is, a triplet of (P, W, .sigma.) is associated
with each data point.
[0045] In the following description, the manner in which each of
these three P, W, .sigma. will be discussed for DealPrice,
QuotePrice, MarketPrice and BenchmarkPrice.
[0046] The weight serves as the sample size in statistical terms.
In this example, in common with most anonymous trading systems,
traders can trade at a minimum trading size. The parameter, which
is defined by the trading system, is the minimum amount of a bid or
offer order. In an interbank trading system trading spot FX this is
typically in the order of $US 1 million ($1M). Every minimum
trading size in the trades should be considered as ONE "vote" for a
price. All "votes" should be taken into consideration. The larger
the volume of a transaction, the more the "votes" (more confidence
about the price) carried by the trade.
[0047] In other words, a trading amount is converted to units of
"minimum trading size" when it is being weighted. Where the minimum
is $1M, a bid of $10M would count as 10 votes when being
weighed.
[0048] The weight of the price W and Scatter a are used internally.
The weight W is used when the price is combined with other prices
to form a new one.
Standard Error
[0049] From the weight W and Scatter .sigma., a margin of error or
a Standard Error is calculated. It provides a measure of confidence
in the Benchmark Price derived from the market data.
[0050] Consider a Benchmark B as a sample of the price population P
using market information (prices/volumes of deals and quotes). Let
MN(B), WT(B), SD(B), and SE(B) be the mean (Benchmark Price),
Weight (sample size), Standard Deviation, and Standard Error,
respectively.
[0051] If Standard Error SE(B)=SE (pips), then we can say that the
sample mean MN(B) has a 95% chance of lying within two SE of the
mean of the population of MN(P) (the "true price"), or, the
Benchmark Price has a 95% chance satisfying:
MN(P)-2*SE<Benchmark Price<MN(P)+2*SE
[0052] That is, the Benchmark Price is almost certain (a 95%
chance) in the 2*SE-neighborhood of the "true price". Apparently,
the smaller the SE, the more precise is the Benchmark Price.
[0053] For example, if MN(S)=0.9779 and SE(S)=0.00002, and P is the
"true price" of the currency pair, then the Benchmark Price of
0.9779 has a 95% chance lying within (P-0.00004, P+0.00004) or,
conversely, P has a 95% chance of lying within the Benchmark Price
interval of (0.97786, 0.97794).
[0054] The Standard Error and its formula is discussed in greater
detail later on.
[0055] Thus, it can be seen that the Deal Price and the Quote Price
are the fundamental building blocks of the Benchmark Price. The
manner in which these are obtained will now be discussed in
detail.
Deal Price
[0056] The Benchmark is calculated at a certain frequency. At each
tick t when the Benchmark is to be calculated, there may be deals
which have occurred since calculation of the last Benchmark at time
(t-1). Deal Price is derived from the deals in the past time
interval. It is an average price of all deal prices weighted by the
volume and age of deals.
Deal Price Weight (W.sub.d)
[0057] Deals of larger amounts count more than smaller amounts.
Recent deals count more than aged deals.
"Minimum Trading Size" Adjustment
[0058] As mentioned above, each deal is weighted by the "minimum
trading size". That is, the Volume V.sub.i of a deal d.sub.i is the
deal's amount in the units of minimum trading size. In the $10M
example above, the volume would be 10.
Time Markdown
[0059] In the past time interval since the last benchmark price,
many deals may have occurred. Deals that happened in the same
second (or some other defined time interval) are considered the
same "age". Going backwards, the deals that occurred in the second
away from the tick-time are discounted by an exponential function
2.sup.-ct, where t is the number of seconds away from the tick-time
and c is a constant.
[0060] For example, if c=0.5 and t=1 (one second earlier than
tick-time), 2.sup.-ct=2.sup.-0.5=0.707. That is, the deal that
happened one second ago is worth 70.7% of the deal that happened
now (at tick-second). If c=1 and t=1, 2.sup.-ct=2.sup.-1=0.5. That
is, the deal that happened one second ago is worth one-half of the
deal that happened now (at tick-second).
Weight of a Deal
[0061] To apply the minimum trading size adjustment and time
markdown to deals that occurred in the past time interval, the
weight of a deal d.sub.i must be calculated as follows:
W(d.sub.i)=V.sub.i*2.sup.-ct, Equation 1
where V.sub.i is the Volume (deal amount in minimum trading size
units) of deal d.sub.i; t is time elapsed in seconds since d.sub.i
occurred (to the tick-time); c is a constant coefficient of
adjustment defining the speed of deal price markdown.
Deal Price Weight
[0062] The Deal Price Weight consists of all weights of all deals
that occurred in the past time interval. It is a summation of all
deal weights:
W.sub.d=.SIGMA..sub.iW(d.sub.i)=.SIGMA..sub.iV.sub.i*2.sup.-ct
Equation 2
where .SIGMA..sub.i runs over all deals occurring in the past time
interval; V.sub.i is the Volume (deal amount in "minimum trading
size" units) of deal d.sub.i; t is time elapsed in second since
d.sub.i occurred (to the tick-time); c is a constant coefficient of
adjustment defining the speed of deal price markdown.
[0063] Table 1 lists the time-markdown-values of 2.sup.-ct
according to time elapsed (t) for C=0.5 and C=1.0:
TABLE-US-00001 TABLE 1 c = 0.5 c = 1.0 t -ct 2.sub.-ct -ct
2.sub.-ct 0 0.0 1.0000 0.0 1.0000 1 -0.5 0.7071 -1.0 0.5000 2 -1.0
0.5000 -2.0 0.2500 3 -1.5 0.3536 -3.0 0.1250 4 -2.0 0.2500 -4.0
0.0625 5 -2.5 0.1768 -5.0 0.0312 6 -3.0 0.1250 -6.0 0.0156 7 -3.5
0.0884 -7.0 0.0078 8 -4.0 0.0625 -8.0 0.0039 9 -4.5 0.0442 -9.0
0.0020 10 -5.0 0.0312 -10.0 0.0010
[0064] It will be seen that the larger the value of C, the faster
the markdown.
[0065] In the special case of a Benchmark frequency of one second,
the Benchmark is calculated every second and only deals that
occurred in the past second are used in the calculation. Then
Equation 2 becomes
W.sub.d=.SIGMA..sub.iW(d.sub.i)=.SIGMA..sub.iV.sub.i*2.sup.-ct=.SIGMA..s-
ub.iV.sub.i*2.sup.-cx0=.SIGMA..sub.iV.sub.i*1=.SIGMA..sub.iV.sub.i.
[0066] This suggests that the parameter c that controls the
markdown speed looses its effect, and the time markdown disappears,
as will been seen below, the time markdown is used in the form of a
historical price markdown, where c plays the same role.
Deal Price (P.sub.d)
[0067] If P(d.sub.i) is the price of a deal d.sub.i, then the
combined deal price of all deals that occurred in the past time
interval is simply the arithmetic average of all deals weighted by
their Weights of Deal:
P d = i P ( d i ) * W ( d i ) i W ( d i ) Equation 3 )
##EQU00001##
where .SIGMA..sub.i runs through all deals d.sub.i occurred in the
past time interval.
[0068] In the case of a one-second Benchmark, each deal's weight is
its Volume. Table 2 lists the Deal Price of five deals that
occurred in the past time interval. The Deal Price is a real
number, not restricted by whole pips 0.9779882353.
TABLE-US-00002 TABLE 2 Deals Price (P) Weight (W) P*W 1 0.9781 2
1.956200 2 0.9780 2 1.956000 3 0.9781 1 0.978100 4 0.9780 7
6.846000 5 0.9779 5 4.889500 Total: 17 16.625800 Deal Price (Pd):
0.9779882353
Deal Price Scatter (.sigma..sub.d)
[0069] In the past time interval, deal prices may scatter widely,
reflecting a market in which traders disagree with each other on
where the market is and the votes are widespread. Alternatively,
deal prices may concentrate at one spot, where everyone agrees
where the market is. This is illustrated in FIG. 1.
[0070] The Deal Price Scatter can be measured by the Standard
Deviation of deals:
.sigma. d = .SIGMA. i W ( d i ) ( P ( d i ) - P d ) 2 .SIGMA. i W (
d i ) Equation 4 ##EQU00002##
where P.sub.d is the Deal Price calculated from equation 3 above;
W(d.sub.i) is the individual Deal Weight derived from equation 1
above; .SIGMA..sub.i runs through all deals d.sub.i that occurred
in the past time interval.
[0071] The Standard Deviation (SD) of a population is a measurement
of the spread of the population distribution. It is like an average
distance of the population to the mean of the population. Large SD
is a good indication of volatility showing that prices are widely
dispersed from the average.
[0072] Using the example above, table 3 shows the Scatter:
Scatter=sqrt (0.0000000776/17)=0.0000675831.
TABLE-US-00003 TABLE 3 Deals Price (P) Weight (W) P*W W*(P - Pd)2 1
0.9781 2 1.956200 0.0000000250 2 0.9780 2 1.956000 0.0000000003 3
0.9781 1 0.978100 0.0000000125 4 0.9780 7 6.846000 0.0000000010 5
0.9779 5 4.889500 0.0000000389 Total: 17 16.625800 0.0000000776
Deal Price (Pd): 0.9779882353 Scatter (.sigma..sub.d):
0.0000675831
[0073] It will be appreciated from the above discussion that
Equations 1, 2, and 3 define, respectively, the three elements
(P.sub.d, W.sub.d, .sigma..sub.d) of Deal Price.
Quote Price
[0074] Quote Price is derived from all live quotes in the anonymous
trading system or other data source at the tick-time.
[0075] When there are more offers in the market, more people are
trying to sell. The liquidity on the offer/selling side results in
sells at the prices of the "front line" of bids, the Best Bid.
[0076] Conversely, if there are more bids, more people are buying.
The liquidity on the bid/buying side results in buys at the prices
of the "front line" of offers, the Best Offer.
[0077] Quote Price maps market liquidity to "Buying Power" at the
Best Bid and Best Offer prices at the time.
Quote Price Weight (W.sub.q)
[0078] As with Deal Price, a quote with a small amount is worth
less than a quote with a large amount. A far-away-from-the-market
quote is not worth as much as a close-to-the-market quote. To
describe the weight of a quote, we need to take into consideration
both the amount and the positioning of the quote with respect to
the best Bid/Offer.
"Minimum trading size" Adjustment
[0079] A quote can be at the minimum trading size. Therefore, each
quote is weighted by the minimum trading size. That is, the Volume
of a quote Q.sub.i is the quote's amount in the units of minimum
trading size.
Distance Markdown
[0080] The front line of a market is drawn by the Best Bid
(P.sub.b) and Best Offer (P.sub.o). The distance of a quote price
from the Best, measured in the number of pips, shows "how good the
quote price is". The term "pip" is well understood in the financial
trading art and refers to the least significant digit of the price
that is quoted.
[0081] At any moment, the Best Bid is the highest bid, and the Best
Offer is the lowest offer of the Market from which data is
obtained, regardless of credit situation. This is not always the
best price that is displayed to traders, as many trading systems do
not display to traders prices which they cannot deal, for example
submitted by a counterparty with whom they have no or insufficient
credit.
[0082] If P(Q.sub.i) is the price of a quote Q.sub.i, and B is the
price of the Best of the same side, then the distance of this quote
to the Best is |P(Q.sub.i)-B|. The position markdown is defined by
the exponential function:
2.sup.-|P(Q1)-B|
[0083] When the distance is 0, 1, 2, 3, 4, . . . , (pips),
2.sup.-|P(Qi)-B|=1, 1/2, 1/4, 1/8, 1/16, . . . .
Weight of a Quote
[0084] To apply minimum trading size and the distance markdown, the
weight of a quote Q.sub.i is:
W(Q.sub.i)=V.sub.i*2.sup.-P(Qi)-B|, Equation 5
where V.sub.i is the Volume (quote amount in "minimum trading size"
unit) of quote Q.sub.i; P(Q.sub.i) is the price of quote Q.sub.i;
B=P.sub.b (Best Bid) if Q.sub.i is a bid, B=P.sub.o (Best Offer) if
Q.sub.i is an offer, and |P(Qi)-B| is in the number of pips, the
distance to the Best Price.
[0085] It can be seen that a quote is weighted by its Volume and
marked down by its distance to the Best Price.
[0086] Table 4 shows an example of weight of quotes. All quotes
have the same Volume, but the distances to the Best vary.
TABLE-US-00004 TABLE 4 Offers Price Volume Distance Weight 1 0.9780
10 0 10.0000 2 0.9781 10 1 5.0000 3 0.9782 10 2 2.5000 4 0.9783 10
3 1.2500 5 0.9784 10 4 0.6250
[0087] The weight of the quotes goes down 1/2 each time the quote
moves one more pip away from the market.
[0088] In the example of Table 5 below, the quotes' distances to
the Best vary, and their Volumes get larger when moving away from
the market.
TABLE-US-00005 TABLE 5 Bids Price Volume Distance Weight 1 0.9779
10 0 10.0000 2 0.9778 20 1 10.0000 3 0.9777 40 2 10.0000 4 0.9776
80 3 10.0000 5 0.9775 160 4 10.0000
[0089] As can be seen from Table 5, a volume of twenty at one-pip
away is worth (is equivalent to) a volume of 10 at the zero-pip; a
volume of 160 at 4-pips away is worth (is equivalent to) the volume
of 10 at zero-pip.
Weight of all Quotes
[0090] The weight of a group of quotes is merely the summation of
the weight of each individual quote.
[0091] The Weight of Bids is
V.sub.b=.SIGMA..sub.i.sup.bW(Q.sub.i)=.SIGMA..sub.i.sup.bV.sub.i/2.sup.(-
Pb-Pi), Equation 6
and the Weight of Offers is
[0092]
V.sub.o=.SIGMA..sub.i.sup.oW(Q.sub.i)=.SIGMA..sub.i.sup.oV.sub.i/2-
.sup.(Pi-Po), Equation 7
where W(Q.sub.i) is the weight of quote Q.sub.i;
.SIGMA..sub.i.sup.b sums up all bids, .SIGMA..sub.i.sup.o sums up
all offers; V.sub.i is the Volume of quote Q.sub.i; P.sub.b is the
Best Bid, P.sub.o is the Best Offer, and (P.sub.b-P.sub.i),
(P.sub.i-P.sub.o) are in pips.
[0093] Note the (P.sub.b-P.sub.i).gtoreq.0 since P.sub.b is the
highest bid, and (P.sub.i-P.sub.o).gtoreq.0 since P.sub.o is the
lowest offer.
[0094] Thus, the Weight of Bids V.sub.b is the volume-equivalent of
the bids at the Best Bid price, and the Weight of Offers V.sub.o is
the volume-equivalent of the offers at Best Offer price.
[0095] The Quote Price Weight consists of a scaled summation of Bid
Weight and Offer Weight. The Quote Price Weight is:
W.sub.q=.lamda..SIGMA..sub.iW(Q.sub.i)=.lamda.(V.sub.b+V.sub.o)
Equation 8
where .lamda. is a constant coefficient which controls the share of
quote's weight in Market Price, as will be discussed;
V.sub.b is the Weight of Bids; and
V.sub.o is the Weight of Offers.
[0096] Where the market is one-sided, that is there are only bids
or offers in the market, W.sub.q=0. That is, one-sided quotes do
not carry any weight. In the next section to be described, the
quotes are converted to "deals" in the calculation of a quote's
price. An "empty" side causes the "deal price" to be zero.
Therefore, the weight of non-deal is zero.
[0097] Table 6 shows an example of Weight of Offers. Although there
are five offers in the market with a total volume of 50, the offers
are worth 19.375 at the Best Offer price.
TABLE-US-00006 TABLE 6 Offers Price Volume Distance Weight 1 0.9780
10 0 10.0000 2 0.9781 10 1 5.0000 3 0.9782 10 2 2.5000 4 0.9783 10
3 1.2500 5 0.9784 10 4 0.6250 Vo: 19.3750
[0098] Table 7 shows an example of Weight of Bids. Although there
are five bids in the market with a total volume of 310, the bids
are worth 50 at the Best Bid price.
TABLE-US-00007 TABLE 7 Bids Price Volume Distance Weight 1 0.9779
10 0 10.0000 2 0.9778 20 1 10.0000 3 0.9777 40 2 10.0000 4 0.9776
80 3 10.0000 5 0.9775 160 4 10.0000 Vb: 50.0000
[0099] Table 8 shows an example of the Quote Price Weight.
(V.sub.b+V.sub.o)=69.375 volume of quote, but the Quote Price
Weight is reduced to 6.940 due to the .lamda. scale-down.
TABLE-US-00008 TABLE 8 Price Volume Distance Weight Offers 1 0.9780
10 0 10.0000 2 0.9781 10 1 5.0000 3 0.9782 10 2 2.5000 4 0.9783 10
3 1.2500 5 0.9784 10 4 0.6250 Vo: 19.3750 Bids 1 0.9779 10 0
10.0000 2 0.9778 20 1 10.0000 3 0.9777 40 2 10.0000 4 0.9776 80 3
10.0000 5 0.9775 160 4 10.0000 Vb: 50.0000
[0100] From Table 8, if A=0.1, then:
W.sub.q=.lamda.(V.sub.b+V.sub.o)=0.1*(19.375+50.0)=6.9375.
Quote Price (P.sub.q)
[0101] The weight of bids V.sub.b and offers V.sub.o defines
"buying and selling demand" at the time. The Buys/Sells resulting
from the demand is a function of the prices (P.sub.b, P.sub.o) and
the liquidity (V.sub.b, V.sub.o) at the time. That is,
Buys=Function.sub.b (V.sub.b, V.sub.o, P.sub.b, P.sub.o).
Similarly, Sells=Function.sub.s (V.sub.b, V.sub.o, P.sub.b,
P.sub.o).
[0102] The buy volume is proportional to V.sub.b (buying demand)
and the sell volume is directly proportional to V.sub.o (selling
demand). Therefore, Function.sub.b and Function.sub.s are
linear:
deal-volume=.lamda.*demand,
or more specifically,
Buy-volume=.lamda.*(buying demand)=.lamda.*V.sub.b,
Sell-volume=.lamda.*(selling demand)=.lamda.*V.sub.o;
where .lamda. is a constant coefficient (a percentage) that
converts quote volume into deal volume.
[0103] Deals almost always occur only at or near the "front line"
of the market. That is, buys resulting from the buying demand are
at the Best Offer price of P.sub.o, and sells resulting from the
selling demand are at the Best Bid price of P.sub.b. There will be
some deviation from this, particularly where a party does not have
credit with the owner or owners of the best offer or bid and so
trades below the front line. Even so the trade will almost always
be very close to that front line.
[0104] The bids, and the resulting "buying power", are now
converted into deals of buy at the price of P.sub.o and volume of
.lamda.*V.sub.b. The offers, and the resulting "selling power", are
now converted into deals of sell at price P.sub.b and volume of
.lamda.*V.sub.o.
[0105] As quotes are mapped into "deals" of buys of .lamda.V.sub.b
volume at price P.sub.o and sells of .lamda.V.sub.o volume at price
P.sub.b, the deal price formula of Equation 3 can be used to derive
the Quote Price P.sub.q:
P q = P b * .lamda. V o + P o * .lamda. V b .lamda. V o + .lamda. V
b = P b * V o + P o * V b V o + V b Equation 9 ##EQU00003##
where P.sub.b and P.sub.o are Best Bid and Best Offer,
respectively; V.sub.b and V.sub.o are Weight of Bids and Weight of
Offers, respectively. P.sub.q is the average price of buys at
P.sub.o with volume .lamda.V.sub.b and sells at P.sub.b with volume
.lamda.V.sub.o. Note that in Equation 8, the .lamda.s are cancelled
out. That is, .lamda. does not influence the outcome of Quote
Price. Quote Price is solely determined by the Best Prices and the
liquidity of the market.
[0106] Mapping quotes to "deals" requires both sides of the market
to work. In the cases of no quote or a one-sided-market, P.sub.q=0.
As stated above, in these cases, the quote's weight is also
zero.
[0107] By way of example, and continuing from the results of Table
8 above, where P.sub.b=0.9779, P.sub.o=0.9780, V.sub.b=50.0, and
V.sub.o=19.375, we have
P q = P b * V o + P o * V b V o + V b = 0.9779 * 19.375 + 0.9780 *
50.0 19.375 + 50.0 = 0.9779720720719 ##EQU00004##
[0108] As the buying demand is larger
(V.sub.b=50.0>V.sub.o=19.375), the Quote Price P.sub.q is pushed
up, at almost the Best Offer price.
Quote Price Scatter (.sigma..sub.q)
[0109] Since the quotes are mapped into deals, we can apply the
formula of Deal Scatter on the quotes'. According to Equation 4,
the quote "scatterness" the Standard Deviation of quotes is:
.sigma. q = .lamda. V o ( P b - P q ) 2 + .lamda. V b ( P o - P q )
2 .lamda. V o + .lamda. V b = V o ( P b - P q ) 2 + V b ( P o - P q
) 2 V o + V b Equation 10 ##EQU00005##
where P.sub.q is the Quote Price derived from Equation 8 above;
P.sub.b and P.sub.o are EBS Best Bid and Best Offer, respectively;
V.sub.b and V.sub.o are Weight of Bids and Weight of Offers,
respectively.
[0110] It can now be seen that the three elements of the triplet
(P.sub.q, W.sub.q, .sigma..sub.q) of Quote Price are defined by
Equations 9, 8 and 10 respectively.
[0111] Where the market is one sided, .sigma..sub.q=0.
[0112] Table 9 shows how the quote scatter is obtained, using the
results of Table 8 and the value of P.sub.q obtained above.
TABLE-US-00009 TABLE 9 Pb Po Pq Vb Vo Vo(Pb - Pq)2 Vb(Po - Pq)2 Vb
+ Vo Scatter (.sigma..sub.q) 0.9779 0.9780 0.9779720721 50.000
19.375 0.0000001006 0.0000000390 69.375 0.0000448645
[0113] The Deal Price triplet (P.sub.d, W.sub.d, .sigma..sub.q) and
the Quote Price triplet (P.sub.q, W.sub.q and .sigma..sub.q) have
now been defined. These values need to be consolidated in the
Market Price to give a good indication of where the market actually
is.
Market Price
[0114] The Market Price Triplet (P.sub.m, W.sub.m, .sigma..sub.m)
is constructed by simply combining Deal Price information (P.sub.d,
W.sub.d, .sigma..sub.d) and Quote Price information (P.sub.q,
W.sub.q, .sigma..sub.q). Without using historical information,
Market Price is calculated from current live quotes and the latest
deals that occurred in the past time interval.
Market Price (P.sub.m)
[0115] The Market Price PM is derived by averaging the Quote Price
and Deal Price, and each is scaled by its Weight:
P m = P q * W q + P d * W d W q + W d Equation 11 ##EQU00006##
where P.sub.q is the Quote Price from Equation 8 above; P.sub.d is
the Deal Price from Equation 3; W.sub.q is the Quote Weight from
Equation 7; and W.sub.d is the Deal Weight from Equation 2.
[0116] Equation 8 stated that W.sub.q=.lamda.(V.sub.b+V.sub.o), and
the constant coefficient .lamda. is used to control the share of
quotes. Equation 11 shows that since Quote Price is weighted by its
weight of W.sub.q, and W.sub.q=.lamda.(V.sub.b+V.sub.o), .lamda.
plays an important role of scaling quote weight in respect to Deal
Weight. Thus, if .lamda.=0.1, it means a 10 volume of quotes is
worth one volume of deals.
[0117] Following on from the Example of Table 10, Table 11 shows
how Equation 10 can be used to obtain the market price.
TABLE-US-00010 TABLE 11 Pq 0.9779720721 0.9779882353 Pd Wq 6.9375
17.0000 Wd .sigma..sub.q 0.0000448645 0.0000675831 .sigma..sub.d
Market Price: 0.9779835509
Market Price Weight (W.sub.m)
[0118] The Market Price Weight is summation of Deal Price Weight
and Quote Price Weight:
W.sub.m=W.sub.d+W.sub.q Equation 12
[0119] W.sub.d and W.sub.q calculations are stated by Equations 2
and 7 respectively.
[0120] Thus, continuing with the example, of Table 10, Table 12
shows the Market Price weigh W.sub.m
TABLE-US-00011 TABLE 12 Wq Wd Wm 6.9375 17.0000 23.9375
Market Price Scatter (.sigma..sub.m)
[0121] Applying the definition of Standard Deviation, we get the
Market Price Scatter as follow:
Equation 13 ##EQU00007## .sigma. m .lamda. V o ( P b - P m ) 2 +
.lamda. V b ( P o - P m ) 2 + .SIGMA. i W ( d i ) ( P ( d i ) - P m
) 2 .lamda. V o + .lamda. V b + .SIGMA. i W ( d i )
##EQU00007.2##
[0122] If P.sub.q=0 (no quotes, one-sided-market), then quotes are
no longer part of the consideration and Equation 13 becomes:
.sigma. m .SIGMA. i W ( d i ) ( P ( d i ) - P m ) 2 .SIGMA. i W ( d
i ) Equation 13 ( b ) ##EQU00008##
[0123] Table 13 continues the example of table 10 and shows
calculations of Market Price Scatter:
TABLE-US-00012 TABLE 13 Price Weight W*(P - Pm Deals (P) (W)
.lamda. Pm)2 0.9779836 1 0.9781 2 0.0000000271 2 0.9780 2
0.0000000005 3 0.9781 1 0.0000000136 4 0.9780 7 0.0000000019 5
0.9779 5 0.0000000349 Bid-> 0.9780 50.000 0.1 0.0000000014 Buy
Offer-> 0.9779 19.375 0.1 0.0000000135 Sell Total: 23.937500
0.0000000929 Scatter 0.0000622966 (.sigma..sub.m ):
[0124] At each tick t, the Market Price uses the current, latest
information of the market. But in some off-peak hours or in some
less-active currency pairs, complete current information may not be
available. In those situations, Market Price (P.sub.m, W.sub.m,
.sigma..sub.m) must be combined with the previous Benchmark Price
to derive the current price.
[0125] Having now obtained the Market Price, the Benchmark Price
may now be derived.
Benchmark Price
[0126] Suppose the current tick is n, then current Benchmark Price
triplet (P.sub.n, W.sub.n, .sigma..sub.n) is derived from the
current Market Price (P.sub.m, W.sub.m, .sigma..sub.m), the last
Benchmark Price (P.sub.n-1, W.sub.n-1, .sigma..sub.n-1) calculated
at the last tick, and a Phantom Price (P.sub.ph, W.sub.ph,
.sigma..sub.ph). As the last benchmark was calculated one
time-interval ago, it is discounted accordingly.
[0127] The weight of the last Benchmark Price is marked down by a
faction of 2.sup.-cT, where T is the time interval from the last
Benchmark to the current tick.
[0128] Phantom Price is an imaginary price introduced to balance
the no-price situation. The Phantom price has a very tiny weight
that injects little influence on the calculations when prices are
present, but which prevents the Benchmark weight from becoming zero
where there are no prices in the Benchmark period. This, in turn,
prevents the Standard Error from becoming infinite. It follows that
in an active market, the phantom price is not essential to the
calculation of the benchmark.
[0129] When prices exist in the market, Phantom Price is equal to
Market Price. When there is no price in the market, Phantom Price
equals the previous Benchmark Price. That is:
P.sub.ph=P.sub.m when P.sub.m.noteq.0; P.sub.ph=P.sub.n-1 when
P.sub.m=0.
[0130] The weight of the Phantom Price W.sub.ph is a tiny constant,
for example a half of one-millionth (0.5*10.sup.-6). In the case of
the minimum trading amount of one million (of base currency) for
most currencies, the weight is unit of the base currency. For
example, while the minimum-trading amount for EUR/USD is one
million (1,000,000) Euro, its Phantom Price's weight is 0.5 Euro.
It will be seen from later discussion that this modest amount can
ensure that the Standard Error does not become infinite.
[0131] The scatter of Phantom Price .sigma..sub.ph is a tiny
constant as well. It is set at 1% of a Big Figure of a currency.
The purpose of having .sigma..sub.ph is to prevent the scatter from
becoming zero (therefore zero Standard Error) in quiet market.
Benchmark Price (P.sub.n)
[0132] The Benchmark price is the average on the Market Price and
(n-1) Benchmark Price, weighted by their Weights. The (n-1)
Benchmark Price Weight has an extra time-markdown): Thus
P n = P m * W m + P n - 1 * W n - 1 * 2 cT + P ph * W ph W m + W n
- 1 * 2 - cT + W ph Equation 14 ##EQU00009##
where P.sub.m is the Market Price from Equation 10 above; W.sub.m
is the Market Price Weight from Equation 11; P.sub.n-1 is the
previous Benchmark Price; P.sub.ph is Phantom Price.
P.sub.ph=P.sub.m if P.sub.m.noteq.0; P.sub.ph=P.sub.n-1 if
P.sub.m=0; W.sub.ph is the weight of Phantom Price, a small
constant, set at 0.5*10.sup.-6; W.sub.n-1 is the previous Benchmark
Price Weight; T is the tick time interval in seconds; and c is a
constant coefficient of adjustment defining the speed of
deal/Benchmark price markdown.
[0133] The coefficient c is the same one used in the deal price
elapsed-time markdown.
[0134] In the case of a still-market (P.sub.m=0, W.sub.m=0),
P.sub.n=P.sub.n-1. That is, Benchmark Price does not change.
[0135] Referring back to the example as discussed in Table 11,
suppose c=1.0 and T=1, and suppose P.sub.n-1=0.9778096128,
W.sub.n-1=19.0458, the Benchmark Price P.sub.n by using Equation 13
is shown in the Table below.
TABLE-US-00013 TABLE 14 Wm + Wn - 1 * Pm Wm Pn - 1 Wn - 1 Pph Wph
2-ct + Wph Pn 0.9779835509 23.9375 0.9778095128 19.0458
0.9778095128 0.000001 33.4604009156 0.9779340193
Benchmark Price Weight (W.sub.n)
[0136] The Benchmark Price Weight is a summation of Market Price
Weight and the last Benchmark Price Weight after applying the time
markdown of 2.sup.-cT:
W.sub.n=W.sub.m+W.sub.n-1*2.sup.-cT+W.sub.ph Equation 15
[0137] Where the Phantom Price Weight W.sub.ph is a small
constant.
[0138] Again, the Phantom Price is introduced solely to prevent the
weight of the Benchmark from going down to zero in a quiet market
when there is neither any price nor deal. Since its weight is tiny,
it does not effect the price in a regular or busy market.
[0139] In the case of a still-market (P.sub.m=0, W.sub.m=0),
suppose W.sub.1 is the last Benchmark Price before an interval with
no new active prices, then the Benchmark Price Weights are:
W 1 , W 2 = W 1 * 2 - cT + W ph , W 3 = W 2 * 2 - cT + W ph = W 1 *
2 - 2 cT + W ph * 2 - cT + W ph , , W n = W n - 1 * 2 - cT + W ph =
W 1 * 2 - ( n - 1 ) cT + W ph * 2 - ( n - 2 ) cT + + W ph * 2 - cT
+ W ph = W 1 * 2 - ( n - 1 ) cT + W ph ( 1 - 2 - ( n - 1 ) cT ) / (
1 - 2 - cT ) = W 1 * 2 - ( n - 1 ) cT + W ph / ( 1 - 2 - cT ) - W
ph * 2 - ( n - 1 ) cT / ( 1 - 2 - cT ) ##EQU00010##
[0140] When n.fwdarw..infin. (c=1 and Benchmark is calculated every
second: T=1), then
W.sub.n.fwdarw.W.sub.ph/(1-2.sup.-cT)=2*W.sub.ph. That is, in a
quiet market, the weight converges to 2*W.sub.ph. If Phantom Price
is 0.5*10.sup.-6 (0.0000005), then the weight converges to
10.sup.-6 (one-millionth).
[0141] Continuing with the example of Table 14, suppose c=1.0, T=1,
Equation 14 provides a Benchmark Price Weight
W.sub.n=W.sub.m+W.sub.n-1*2.sup.-cT=33.4603999156.
Benchmark Price Scatter (.sigma..sub.n)
[0142] Before calculating the Benchmark Price scatter a, the
combined values of the Market Price and Phantom Price must be
obtained.
[0143] In general, with two samples of the same population, there
are two sets of means, sample sizes (weights) and Standard
Deviations (Scatters). The two samples can be combined and the
combined samples' Standard Deviation SD can be derived from the SDs
of the two separate samples. Thus:
( 4.3 .0 ) .sigma. 1 + 2 = ( N 1 - 1 ) .sigma. 1 2 N 1 + N 2 - 1 +
( N 2 - 1 ) .sigma. 2 2 N 1 + N 2 - 1 + N 1 N 2 ( M 1 - M 2 ) 2 ( N
1 + N 2 ) ( N 1 + N 2 - 1 ) Equation 16 ##EQU00011##
[0144] Where M.sub.1, M.sub.2, .sigma., .sigma..sub.2 and N.sub.1,
N.sub.2 are the means, SDs and sample sizes of the two samples. The
combined price, weight and scatter triplet of
(P.sub.c/W.sub.c/.sigma..sub.c) are then obtained from the Market
Price (P.sub.m, W.sup.m, .sigma..sub.m) and the Phantom Price
(P.sub.ph, W.sub.ph, .sigma..sub.ph).
[0145] As the Phantom Price equal the Market Price when a Market
Price exists, and equals the previous Benchmark Price when no
Market Price exists, P.sub.c=P.sub.m if P.sub.m.noteq.0;
P.sub.c=P.sub.n-1 if P.sub.m=0.
[0146] The combined weight W.sub.c=W.sub.m+W.sub.ph.
[0147] Finally, by applying the combined standard deviation of
Equation 16 on (P.sub.m, W.sub.m, .sigma..sub.m), and (P.sub.ph,
W.sub.ph, .sigma..sub.ph):
.sigma. c = ( W m - 1 ) .sigma. m 2 W m + W ph - 1 + ( W ph - 1 )
.sigma. ph 2 W m + W ph - 1 + W m W ph ( P m - P ph ) 2 ( W m + W
ph ) ( W m + W ph - 1 ) ##EQU00012##
[0148] From this formula, it can be concluded that when
P.sub.m.noteq.0, .sigma..sub.c.apprxeq..sigma..sub.m. That is, when
prices exist, .sigma..sub.c is just the Market Price's scatter
.sigma..sub.m. When P.sub.m=0, .sigma..sub.c.apprxeq.
.sigma..sub.m.sup.2+.sigma..sub.ph.sup.2 That is, when there is no
prices, .sigma..sub.c is bounded by at least one .sigma..sub.ph,
since
.sigma..sub.m.sup.2+.sigma..sub.ph.sup.2.gtoreq..sigma..sub.ph.
This effectively prevents .sigma..sub.c from going down to
zero.
[0149] With (P.sub.c, W.sub.c, .sigma..sub.c) having been derived,
the Benchmark Price scatter can be calculated.
[0150] Applying Equation 16 on (P.sub.c, W.sub.c, .sigma..sub.c)
and (P.sub.n-1, W.sub.n-1*, 2.sup.-cT, .sigma..sub.n-1) gives:
.sigma. n = ( W c - 1 ) .sigma. c 2 W c + W n - 1 * 2 - cT - 1 + (
W n - 1 * 2 - cT - 1 ) .sigma. n - 1 2 W c + W n - 1 * 2 - cT - 1 +
W c W n - 1 * 2 - cT ( P c - P n - 1 ) 2 ( W c + W n - 1 * 2 - cT )
( W c + W n - 1 * 2 - cT - 1 ) Equation 17 ##EQU00013##
[0151] Equation 17 functions like the combined scatter over
(P.sub.m, W.sub.m, .sigma..sub.m) and (P.sub.n-1, W.sub.n-1*,
2.sup.-cT, .sigma..sub.n-1), since when P.sub.m.noteq.0,
P.sub.c=P.sub.n, W.sub.c.apprxeq.W.sub.m and
.sigma..sub.c.apprxeq..sigma..sub.m
[0152] In the case of a still-market (P.sub.m=0, W.sub.m=0),
.sigma..sub.n.apprxeq. .sigma..sub.n-1.sup.2+.sigma..sub.ph.sup.2.
That is, the scatter is bounded by at least one .sigma..sub.ph. It
makes sense that, when there is no price in market, the scatter is
at least one .sigma..sub.ph (it is set at 1% of a Big Figure or one
pip).
[0153] Following the example of Table 14, Table 15 shows the
Benchmark Price Scatter:
TABLE-US-00014 TABLE 15 Pm Wm .sigma..sub.m Pn - 1 Wn - 1
.sigma..sub.n-1 Wm + Wn - 1 * 2.sub.-ct .sigma..sub.n 0.9779835509
23.9375 0.0000622966 0.9778095128 19.0458 0.0000761207
33.4603999156 0.0001029197
Benchmark Price Error (E.sub.n)
[0154] The Standard Error is a commonly used index of the error
entailed in estimating a population mean based on the information
in a random sample of size N:
E n = .sigma. 2 N ##EQU00014##
[0155] In this case, a is the Benchmark Price Scatter
.sigma..sub.n, and
N=W.sub.n=W.sub.m+W.sub.n-1*2.sup.-cT+W.sub.ph:
E n = .sigma. n 2 W m + W n - 1 * 2 - cT + W ph Equation 18
##EQU00015##
[0156] It can be seen that E.sub.n is always smaller than the
Benchmark Price Scatter .sigma..sub.n. The larger the weights,
corresponding to more sample points, the smaller the error.
[0157] In the case of a still-market (P.sub.m=0, W.sub.m=0):
E n = .sigma. n W n - 1 * 2 - cT + W ph ##EQU00016##
when n.fwdarw..infin., the Error E.sub.n.fwdarw..sigma..sub.n/
2*W.sub.ph. When W.sub.ph=0.5*10.sup.-6,
E.sub.n.fwdarw..sigma..sub.n/ 0.000001=1000*.sigma..sub.n.
[0158] In other words, if the scatter .sigma..sub.n is at one pip,
then the Standard Error is at 10 big figures.
[0159] The Standard Error (E.sub.n) has its special meaning, as
explained below. 2*E.sub.n defines the "95%-bracket" around
Benchmark Price P.sub.n. That is, in a still-market, if the scatter
is one pip, then the "950-bracket" is bounded by 20 big figures on
each side of the Benchmark Price; if the scatter is two pips, then
the "95%-bracket" is bounded by 40 big figures on each side of the
Benchmark Price.
[0160] Continuing with the example of Table 15, the Benchmark Price
Error can now be calculated.
E.sub.n=sqrt(.sigma..sub.n.sup.2/(W.sub.m+W.sub.n-1*2.sup.-cT+W.sub.ph))-
=sqrt(0.0001029197.sup.2/33.46040092)=0.0000177923.
[0161] Therefore, twice the Standard
Error=2*0.0000177923=0.0000355847 (pips).
[0162] That is, at tick n, the Benchmark Price of 0.9779340193 has
a 95% chance lying within 0.0000355847 pips of the true price P,
or, the Benchmark Price of 0.9779340193 has a 95% chance
satisfying:
P-0.0000355847<0.9779340193<P+0.0000355847
[0163] The Standard Error of the mean may be calculated as
follows:
Central Limit Theorem
[0164] If P is a population of size m and S is a sample of P with
size n (n<m), let MN(S) be the mean of S, and SD(S) be the
Standard Deviation of S, then Central Limit Theorem states that:
[0165] (1) When n is large, the distribution of the means M.sub.i
of ALL possible [0166] samples of size n is approximately a Normal
Distribution (that is, M.sub.1 M.sub.2, . . . , M.sub.k form a
normal distribution); [0167] (2) MN (M.sub.i)=MN (P) That is, the
mean of the distribution of M.sub.i equals to the mean of the
population. [0168] (3) SD(M.sub.i)=SD(P)/sqrt(n) Standard Error
(SE) That is, the SD of the distribution of M.sub.i equals the SD
of the population divided by the square root of n.
[0169] Since M.sub.i is a Normal Distribution, then the mean of a
random sample S has a 68% chance of lying within one SE of the
population mean, or a 95% change lying within two SE of the
population mean:
MN(P)-2*SE<MN(S)<MN(P)+2*SE.
[0170] (Conversely, the population mean has a 68% chance of lying
within one SE of the mean of a single randomly chosen sample S, and
a 95% chance of lying within two SE of the mean of a single
randomly chosen sample S).
SE Approximation
[0171] Since SD (P) is usually not available, it is a common
practice when n is large to use the Standard Deviation of a sample
S instead:
approximated by
SE=SD(M.sub.i)=SD(P)/sqrt(n)SD(S)/sqrt(n)
[0172] Therefore, for any sample of size n the SE of the Mean can
be calculated (approximated).
[0173] The following section gives some examples of Benchmark
calculations following the Equations derived in the foregoing
sections.
Example 1
[0174] This example is a summary of the examples used above. Deal
Price=(Pd, Wd, .sigma..sub.d), Quote Price=(Pq, Wq, .sigma..sub.q),
Market Price=(Pm, Wm, .sigma..sub.m), and Benchmark Price at n=(Pn,
Wn, .sigma..sub.n).
TABLE-US-00015 ##STR00001## ##STR00002## ##STR00003##
##STR00004##
Example 2.1
[0175] Quote only market: There are only two quotes in market. They
are 1 pip apart, 1 million on each side. We want to find out
Benchmark's (Pn, Wn, .sigma..sub.n) and SE.
[0176] Note that when there is no deal, Market Price is the same as
the Quote Price. By applying Benchmark formulas, we have:
TABLE-US-00016 Quote Prices Benchmarks Bid Vb Offer Vo Pq Wq
.sigma..sub.q Pn Wn .sigma..sub.n SE 0 0 0 0.9800 1 0.9801 1
0.98005 0.2 0.000050 0.98005 0.2 0.000050 0.00011180 0.9800 1
0.9801 1 0.98005 0.2 0.000050 0.98005 0.3 0.000050 0.00009129
0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.35 0.000050
0.00008452 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.375
0.000050 0.00008165 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005
0.3875 0.000050 0.00008032 0.9800 1 0.9801 1 0.98005 0.2 0.000050
0.98005 0.39375 0.000050 0.00007968 0.9800 1 0.9801 1 0.98005 0.2
0.000050 0.98005 0.396875 0.000050 0.00007937 0.9800 1 0.9801 1
0.98005 0.2 0.000050 0.98005 0.3984375 0.000050 0.00007921 0.9800 1
0.9801 1 0.98005 0.2 0.000050 0.98005 0.3992188 0.000050 0.00007913
0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.3996094 0.000050
0.00007910 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.3998047
0.000050 0.00007908 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005
0.3999023 0.000050 0.00007907 0.9800 1 0.9801 1 0.98005 0.2
0.000050 0.98005 0.3999512 0.000050 0.00007906 0.9800 1 0.9801 1
0.98005 0.2 0.000050 0.98005 0.3999756 0.000050 0.00007906 0.9800 1
0.9801 1 0.98005 0.2 0.000050 0.98005 0.3999878 0.000050 0.00007906
0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.3999939 0.000050
0.00007906 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.3999969
0.000050 0.00007906 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005
0.3999985 0.000050 0.00007906 0.9800 1 0.9801 1 0.98005 0.2
0.000050 0.98005 0.3999992 0.000050 0.00007906 0.9800 1 0.9801 1
0.98005 0.2 0.000050 0.98005 0.3999996 0.000050 0.00007906 0.9800 1
0.9801 1 0.98005 0.2 0.000050 0.98005 0.3999998 0.000050 0.00007906
0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.3999999 0.000050
0.00007906 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005 0.4
0.000050 0.00007906 0.9800 1 0.9801 1 0.98005 0.2 0.000050 0.98005
0.4 0.000050 0.00007906 0.9800 1 0.9801 1 0.98005 0.2 0.000050
0.98005 0.4 0.000050 0.00007906 SE = 0.00007906, 2*SE = 0.0001581,
4*SE = 0.000316
[0177] The relationship between the Benchmark and the Bid and offer
prices is shown in FIG. 2. A number of conclusions can be drawn as
follows:
(1) The Benchmark Price P.sub.n is in the middle of the bid and
offer, (P.sub.b+P.sub.o)/2, as expected. (2) The Benchmark Price
Weight W.sub.n converges to (2.times.total volume.times..lamda.).
Namely, W.sub.n l->2.times.2.times.0.1=0.4 after 16 rounds (16
seconds). (3) The Benchmark Price Scatter .sigma..sub.n is 1/2 of
the quote spread, as expected. SE is larger the D.sub.n because
W.sub.n is small (0.4)
Example 2.2
[0178] Quote only market: There are only two quotes in market. They
are 2 pips apart, 1 million on each side. We want to find out
(P.sub.n, W.sub.n, .sigma..sub.n) and SE. This example is the same
as 2.1 above, except for the larger spread.
[0179] Note that when there is no deal, Market Price is the same as
the Quote Price. By applying Benchmark formulas, we have:
TABLE-US-00017 Quote Prices Benchmarks Bid Vb Offer Vo Pq Wq
.sigma..sub.q Pn Wn .sigma..sub.n SE 0 0 0 0.9800 1 0.9802 1 0.9801
0.2 0.000100 0.9801 0.2 0.000100 0.00022361 0.9800 1 0.9802 1
0.9801 0.2 0.000100 0.9801 0.3 0.000100 0.00018257 0.9800 1 0.9802
1 0.9801 0.2 0.000100 0.9801 0.35 0.000100 0.00016903 0.9800 1
0.9802 1 0.9801 0.2 0.000100 0.9801 0.375 0.000100 0.00016330
0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.3875 0.000100
0.00016064 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.39375
0.000100 0.00015936 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801
0.396875 0.000100 0.00015874 0.9800 1 0.9802 1 0.9801 0.2 0.000100
0.9801 0.3984375 0.000100 0.00015842 0.9800 1 0.9802 1 0.9801 0.2
0.000100 0.9801 0.3992188 0.000100 0.00015827 0.9800 1 0.9802 1
0.9801 0.2 0.000100 0.9801 0.3996094 0.000100 0.00015819 0.9800 1
0.9802 1 0.9801 0.2 0.000100 0.9801 0.3998047 0.000100 0.00015815
0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.3999023 0.000100
0.00015813 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.3999512
0.000100 0.00015812 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801
0.3999756 0.000100 0.00015812 0.9800 1 0.9802 1 0.9801 0.2 0.000100
0.9801 0.3999878 0.000100 0.00015812 0.9800 1 0.9802 1 0.9801 0.2
0.000100 0.9801 0.3999939 0.000100 0.00015812 0.9800 1 0.9802 1
0.9801 0.2 0.000100 0.9801 0.3999969 0.000100 0.00015811 0.9800 1
0.9802 1 0.9801 0.2 0.000100 0.9801 0.3999985 0.000100 0.00015811
0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.3999992 0.000100
0.00015811 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801 0.3999996
0.000100 0.00015811 0.9800 1 0.9802 1 0.9801 0.2 0.000100 0.9801
0.3999998 0.000100 0.00015811 0.9800 1 0.9802 1 0.9801 0.2 0.000100
0.9801 0.3999999 0.000100 0.00015811 0.9800 1 0.9802 1 0.9801 0.2
0.000100 0.9801 0.4 0.000100 0.00015811 0.9800 1 0.9802 1 0.9801
0.2 0.000100 0.9801 0.4 0.000100 0.00015811 0.9800 1 0.9802 1
0.9801 0.2 0.000100 0.9801 0.4 0.000100 0.00015811 SE = 0.00015811,
2*SE = 0.0003162, 4*SE = 0.00063247
[0180] The relationship between the Benchmark and the Bid and offer
prices is shown in FIG. 3. A number of conclusions can be drawn as
follows:
[0181] There is no change for (1) and (2). But the wider spread
changes .sigma..sub.n and SE:
(1) P.sub.n is at the middle of the bid and offer,
(P.sub.b+P.sub.o)/2, as expected. (2) No weight change, W.sub.n
converges to (2.times.total
volume.times..lamda.)=2.times.2.times.0.1=0.4 after 16 rounds. (3)
.sigma..sub.n is 1/2 of the quote spread, as expected. But since
the spread is twice as large as in Example 2.1, .sigma..sub.n and
SE both double their values. It makes sense: wider spread makes the
market less certain.
Example 2.3
[0182] Quote only market: There are only two quotes in market. They
are 1 pip apart, 2 million on each side. We want to find out
(P.sub.n, W.sub.n, .sigma..sub.n) and SE. (This example is the same
as 2.1, only with larger volumes).
[0183] Note that when there is no deal, Market Price is the same as
the Quote Price. By applying Benchmark formulas, we have:
TABLE-US-00018 Quote Prices Benchmarks Bid Vb Offer Vo Pq Wq
.sigma..sub.q Pn Wn .sigma..sub.n SE 0 0 0 0.9800 2 0.9801 2
0.98005 0.4 0.000050 0.98005 0.4 0.000050 0.00007906 0.9800 2
0.9801 2 0.98005 0.4 0.000050 0.98005 0.6 0.000050 0.00006455
0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7 0.000050
0.00005976 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.75
0.000050 0.00005774 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005
0.775 0.000050 0.00005680 0.9800 2 0.9801 2 0.98005 0.4 0.000050
0.98005 0.7875 0.000050 0.00005634 0.9800 2 0.9801 2 0.98005 0.4
0.000050 0.98005 0.79375 0.000050 0.00005612 0.9800 2 0.9801 2
0.98005 0.4 0.000050 0.98005 0.796875 0.000050 0.00005601 0.9800 2
0.9801 2 0.98005 0.4 0.000050 0.98005 0.7984375 0.000050 0.00005596
0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7992188 0.000050
0.00005593 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7996094
0.000050 0.00005592 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005
0.7998047 0.000050 0.00005591 0.9800 2 0.9801 2 0.98005 0.4
0.000050 0.98005 0.7999023 0.000050 0.00005591 0.9800 2 0.9801 2
0.98005 0.4 0.000050 0.98005 0.7999512 0.000050 0.00005590 0.9800 2
0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999756 0.000050 0.00005590
0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999878 0.000050
0.00005590 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999939
0.000050 0.00005590 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005
0.7999969 0.000050 0.00005590 0.9800 2 0.9801 2 0.98005 0.4
0.000050 0.98005 0.7999985 0.000050 0.00005590 0.9800 2 0.9801 2
0.98005 0.4 0.000050 0.98005 0.7999992 0.000050 0.00005590 0.9800 2
0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999996 0.000050 0.00005590
0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999998 0.000050
0.00005590 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005 0.7999999
0.000050 0.00005590 0.9800 2 0.9801 2 0.98005 0.4 0.000050 0.98005
0.8 0.000050 0.00005590 0.9800 2 0.9801 2 0.98005 0.4 0.000050
0.98005 0.8 0.000050 0.00005590 SE = 0.00005590, 2*SE = 0.0001118,
4*SE = 0.00022361
[0184] The relationship between the Benchmark and the Bid and offer
prices is shown in FIG. 4. A number of conclusions can be drawn as
follows:
[0185] There is no change for (1) and .sigma..sub.n; Weight
doubles; The doubling of the Weight makes SE smaller:
1) P.sub.n is at the middle of the bid and offer,
(P.sub.b+P.sub.o)/2, as expected. (2) Weight doubles. W.sub.n
converges to (2.times.total
volume.times..lamda.)=2.times.4.times.0.1=0.8 after 16 rounds. (3)
.sigma..sub.n is 1/2 of the quote spread, as expected. But since
the heavier Weight (more certainty), SE decreases almost 30%
comparing with Example 2.1 (it is inversely proportional to
sqrt(Weight)).
Example 3.1
[0186] Deal only market: There are only one 1 million deals
occurring every second at the same price. We want to find out
(P.sub.n, W.sub.n, .sigma..sub.n) and SE. Note that when there is
no quote, Market Price is the same as the Deal Price. By applying
Benchmark formulas, we have:
TABLE-US-00019 Deal Prices Benchmarks Deal V Pd Wd .sigma..sub.d Pn
Wn .sigma..sub.n SE 0.00000 0 0.000000 0.9800 1 0.98000 1
0.00000000 0.98000 1 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.5 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.75 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.875 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9375 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.96875 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.984375 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9921875 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9960938 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9980469 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9990234 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9995117 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9997559 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9998779 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.999939 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999695 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999847 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999924 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999962 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999981 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.999999 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999995 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999998 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999999 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 1.9999999 0.000000 0.00000000 0.9800 1 0.98000 1
0.00000000 0.98000 2 0.000000 0.00000000
[0187] FIG. 5 shows the relationship between Deal Price and
Benchmark price.
[0188] The following conclusions can be drawn:
(1) The Benchmark Price P.sub.n is at the deal price, as expected.
(2) The Benchmark Price Weight W.sub.n converges to (2.times.total
volume). Namely, W.sub.n->2.times.1=2 after 22 rounds (22
seconds). (3) The Benchmark Price Scatter .sigma..sub.n and SE are
zeros, as expected. Since there is only one deal in market.
Example 3.2
[0189] Deal only market: There is only one 1 million deal occurring
every second alternately at two different prices one pip apart. We
want to find out (P.sub.n, W.sub.n, .sigma..sub.n) and SE. This
example is like Example 3.1, only deal prices are not at a steady
price but "vibrate" one-pip apart. By applying the formulas, we
have:
TABLE-US-00020 Deal Prices Benchmarks Deal V Deal V Pd Wd
.sigma..sub.d Pn Wn .sigma..sub.n SE 0.00000 0 0.000000 0.9800 1
0.98000 1 0.00000000 0.98000 1 0.000000 0.00000000 0.9801 1 0.98010
1 0.00000000 0.98007 1.5 0.000047 0.00003849 0.9800 1 0.98000 1
0.00000000 0.98003 1.75 0.000045 0.00003415 0.9801 1 0.98010 1
0.00000000 0.98007 1.875 0.000047 0.00003443 0.9800 1 0.98000 1
0.00000000 0.98003 1.9375 0.000047 0.00003358 0.9801 1 0.98010 1
0.00000000 0.98007 1.96875 0.000047 0.00003360 0.9800 1 0.98000 1
0.00000000 0.98003 1.984375 0.000047 0.00003340 0.9801 1 0.98010 1
0.00000000 0.98007 1.9921875 0.000047 0.00003340 0.9800 1 0.98000 1
0.00000000 0.98003 1.9960938 0.000047 0.00003335 0.9801 1 0.98010 1
0.00000000 0.98007 1.9980469 0.000047 0.00003335 0.9800 1 0.98000 1
0.00000000 0.98003 1.9990234 0.000047 0.00003334 0.9801 1 0.98010 1
0.00000000 0.98007 1.9995117 0.000047 0.00003334 0.9800 1 0.98000 1
0.00000000 0.98003 1.9997559 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9998779 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.999939 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9999695 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.9999847 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9999924 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.9999962 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9999981 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.999999 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9999995 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.9999998 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 1.9999999 0.000047 0.00003333 0.9800 1 0.98000 1
0.00000000 0.98003 1.9999999 0.000047 0.00003333 0.9801 1 0.98010 1
0.00000000 0.98007 2 0.000047 0.00003333 SE = 0.00003333, 2*SE =
0.000067, 4*SE = 0.00013
[0190] FIG. 6 shows the relationship between Deal Prices and
Benchmarks over time.
[0191] From FIG. 6, we can conclude:
[0192] There is no change for Weight, but everything else
changes:
(1) The Benchmark Price P.sub.n also "vibrates" in between deal
prices, but with only a 40% "amplitude" of the deals. This is
because of the Benchmark at tick n is smoothed by the price at the
previous tick. (2) The Benchmark Price Weight W.sub.n converges to
(2.times.total volume)=2.times.1=2. (3) The Benchmark Scatter is at
0.47 pips (47% of the deal price spread).
Example 3.3
[0193] Deal only market: There are only two 1 million deals
occurring every second at two different prices one-pip apart. We
want to find out (P.sub.n, W.sub.n, .sigma..sub.n) and SE. By
applying Benchmark formulas, we have:
TABLE-US-00021 Deal Prices Benchmarks Deal V Deal V Pd Wd
.sigma..sub.d Pn Wn .sigma..sub.n SE 0.00000 0 0.000000 0.9800 1
0.9801 1 0.98005 2 0.00005000 0.98005 2 0.000050 0.00003536 0.9800
1 0.9801 1 0.98005 2 0.00005000 0.98005 3 0.000050 0.00002887
0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.5 0.000050
0.00002673 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.75
0.000050 0.00002582 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005
3.875 0.000050 0.00002540 0.9800 1 0.9801 1 0.98005 2 0.00005000
0.98005 3.9375 0.000050 0.00002520 0.9800 1 0.9801 1 0.98005 2
0.00005000 0.98005 3.96875 0.000050 0.00002510 0.9800 1 0.9801 1
0.98005 2 0.00005000 0.98005 3.984375 0.000050 0.00002505 0.9800 1
0.9801 1 0.98005 2 0.00005000 0.98005 3.9921875 0.000050 0.00002502
0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.9960938 0.000050
0.00002501 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.9980469
0.000050 0.00002501 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005
3.9990234 0.000050 0.00002500 0.9800 1 0.9801 1 0.98005 2
0.00005000 0.98005 3.9995117 0.000050 0.00002500 0.9800 1 0.9801 1
0.98005 2 0.00005000 0.98005 3.9997559 0.000050 0.00002500 0.9800 1
0.9801 1 0.98005 2 0.00005000 0.98005 3.9998779 0.000050 0.00002500
0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.999939 0.000050
0.00002500 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.9999695
0.000050 0.00002500 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005
3.9999847 0.000050 0.00002500 0.9800 1 0.9801 1 0.98005 2
0.00005000 0.98005 3.9999924 0.000050 0.00002500 0.9800 1 0.9801 1
0.98005 2 0.00005000 0.98005 3.9999962 0.000050 0.00002500 0.9800 1
0.9801 1 0.98005 2 0.00005000 0.98005 3.9999981 0.000050 0.00002500
0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.999999 0.000050
0.00002500 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005 3.9999995
0.000050 0.00002500 0.9800 1 0.9801 1 0.98005 2 0.00005000 0.98005
3.9999998 0.000050 0.00002500 0.9800 1 0.9801 1 0.98005 2
0.00005000 0.98005 3.9999999 0.000050 0.00002500 0.9800 1 0.9801 1
0.98005 2 0.00005000 0.98005 3.9999999 0.000050 0.00002500 SE =
0.00002500, 2*SE = 0.00005000, 4*SE = 0.0001
[0194] FIG. 7 shows the relationship between deal prices and
Benchmark prices from which one may conclude:
[0195] There is no change for Weight (1) but everything else
changes:
(1) The Benchmark Price P.sub.n is at the middle of deal prices, as
expected. (2) The Benchmark Price Weight W.sub.n converges to
(2.times.total volume)=2.times.2=4. (3) The Benchmark Scatter is at
0.5 pips (50% of the deal price spread). SE decreases compared to
Example 3.2 above, as the Weight has doubled, and SE is inversely
proportional to sqrt(Weight).
[0196] Having discussed how the Benchmarks are derived, the
following description details how a system may be set up to
calculate benchmarks and distribute them to traders. FIGS. 8 and 9
show two examples of systems for deriving and providing benchmarks
to traders. In the system of FIG. 8, benchmarks are derived from
price and deal data obtained from a single source. In that of FIG.
9 they are derived from priced and deal data supplied from more
than one source.
[0197] From the discussion above, it is clear that for a benchmark
to be calculated for a given currency pair, the previous benchmark
must be available together with bids and offer prices entered into
the system since the last benchmark together with the prices of
deals done in the time since the last benchmark was calculated.
[0198] These bid and offer prices and deals are conveniently
supplied from a trading system, for example an anonymous trading
system such as that described in U.S. Pat. No. 6,014,627 of Togher
et al. However, the information could be supplied from any suitable
trading system, trading in the instrument to which the benchmark
relates. Such a system must provide price and deal information and
may or may not be anonymous. Bid/offer prices and deal prices may
even be supplied from a non-computerized source such as an open out
cry market or a voice broker.
[0199] Most trading systems enable traders to trade a variety of
currency pairs. For example, USD: EUR, USA:JPY, USD:CAD; EUR:JPY;
EUR:CAD and CAD:JPY and other GB currency pairs. In a preferred
embodiment the system will establish a benchmark every t seconds
for each of these currency pairs as well as other currency pairs
that may be traded. The benchmarks will then be communicated to
traders trading on the system and to traders trading on other
systems, for example on conversational systems or direct dealing
systems. The benchmarks may also be released more generally into
the financial community.
[0200] In the embodiment of FIG. 8, the anonymous trading system is
shown at 10. The price and deal information is passed via a gateway
12 to a market rate feed server 14 where the benchmark calculation
is performed according to the algorithm described above. Market
Rate Feeders (MRF) are, in themselves, a well-known tool for
distributing price information from a source to a number of
subscribers, however it is not conventional for them to calculate
benchmarks.
[0201] The system of U.S. Pat. No. 6,014,627 operates by using a
series of arbitrators, based on geographical location, to keep a
trading book of orders (bids and offers) entered into the system by
traders. Traders receive market views showing the best bids and
offers which are filtered by a Market Distributor according to
their ability to deal prices. The arbitrator holds a log of all
deals conducted on the system and provides deal information, since
the last benchmark, together with bids and offers entered and
active in the system, via a city node, which is a local node via
which trading banks are connected to the system, to the MRF server
14. This market data is delivered to traders 20 on other systems in
XML or other formats over a TCP/IP link via a wide area network 22
and a Conversational Direct Dealing (CDD) server 24 and a Secure
Communication Server (SCS) 26. The market data is enhanced with the
benchmarks calculated by the MRF 14. The benchmarks are also
communicated back to the anonymous trading system for distribution,
via the market distributor, to traders trading on the anonymous
trading system 10. The benchmarks may be used to calculate the big
figure (most significant digits) of the currency rate at the
anonymous trading system.
[0202] Thus, the market rate feed unit 14 differs from prior art
units in that the benchmark calculation algorithm described above
is implemented. The MRF 14 calculates benchmarks at predetermined
equal intervals over a trading day, for example every second. Thus,
the MRF receives the order book from the trading system together
with the deals done since the last benchmark, calculates the new
benchmark and distributes the benchmark.
[0203] In the variant of FIG. 9, benchmarks for some currencies may
be calculated from trading information provided from a different
source than the anonymous trading system 10. This source 28 may be
a different trading system, which, for example has greater
liquidity in some currency pairs than trading system 10. The
benchmarks for some currency pairs will be calculated from the
price and deal data provided from the third party trading system 30
via the third party server 28.
[0204] As an alternative, the market rate feed server 14 could be
configured to receive bid/offer and deal price information for a
given currency pair from both the anonymous trading system 10 and
the third party trading system 30.
* * * * *