We develop a liquidity aggregation algorithm from a mathematical model based around volume-weighted average price (VWAP) calculations taking into consideration market depth.

1. Exploring Liquidity Aggregation Algorithm to
Calculate Volume for a Given VWAP Spread
A winning liquidity aggregation strategy requires the ability to both
tabulate potential profitability on bid/ask spreads and to accumulate
maximum liquidity (volume) available as part of market depth.
Executive Summary
There are special purpose trading tools available
in the market that can be used for liquidity aggre-
gation, especially where voluminous positions
must be hedged as a result of adverse moves in
security prices.
The latest trading tools have developed several
sophisticated features that can be leveraged when
a trader hedges positions within a very truncated
time span that creates minimal price movement
in the ladder. One of those fundamental features
is to acquire a position based on the user-entered
spread using volume-weighted average price
(VWAP) calculation.
This white paper covers the algorithmic aspects
used to calculate the available VWAP volume for
a given spread, including:
• What is VWAP? The elaborated problem
statement of the algorithm.
• Simplified case study to find the solution.
• Formation of generic mathematical models.
• Deriving the algorithm from mathematical
models.
What Is VWAP?
In finance, VWAP is the ratio of the value traded
of a particular security to the total volume traded
over a particular time horizon. In our problem
statement, however, we will consider the market
depth of a pricing ladder to constitute the volume
weighted average. It is a measure of the average
price a security traded at over the trading
time span.
Let’s understand the problem statement with
reference to the example in Figure 1 (next page)
of a ladder of EURUSD bid/ask prices.
Simplified Case Study to Find Solution
Figure 1 provides a problem statement based on
the market depth of the ask side of a ladder —
where the prices are sorted in ascending order
(i.e., best ask being the least price a trader can
afford for buying the security). If the trader
agrees to pay $1.274730 and expects to hedge as
much volume as possible, then without VWAP the
available volume in the illustrated ladder would be
23M (Price: $1.273957) + 16M (Price: $1.274730) =
39M.
However, our target is to maximize the volume
that can be hedged in this scenario. Hence, the
idea here is to afford some volume from an
expensive lot at the cost of the price that is saved
when acquiring cheaper lots in the ladder.
So, if we take the best two prices in the ladder
(price: $1.273957, volume: 23M and price:
• Cognizant 20-20 Insights
cognizant 20-20 insights | march 2014

2. 2
$1.274730, volume: 16M), then the VWAP becomes
(1.273957*23 + 1.274730*16)/(23+16) = $1.274274.
This price is less than what the trader can afford
($1.274957). Hence, we take the next best prices
in the ladder until we reach the VWAP price
mentioned by the trader.
After taking the third-best price in the ladder
(price: $1.273957, volume: 23M, price: $1.274730,
volume: 16M, price: $1.275502, volume: 41M), our
VWAP becomes = (1.273957*23 + 1.274730*16 +
1.275502*41) / (23+16+41) = $1.274903.
Interestingly, we have afforded a price in the
ladder that is individually greater than the target
price set by the trader (ladder price: $1.275502,
trader quote: $1.274957), but the effective price
(VWAP) is less, so we can afford the next price in
the ladder.
While taking the fourth price in the ladder, we
see that if we afford 3M of $1.277048, then our
VWAP becomes = (1.273957*23 + 1.274730*16
+ 1.275502*41 + 1.277048*3) / (23+16+41+3) =
$1.274981, which is greater than the trader quote,
whereas if we take 2M of $1.277048, then our
VWAP becomes = (1.273957*23 + 1.274730*16
+ 1.275502*41 + 1.277048*3) / (23+16+41+3) =
$1.274956, which is just less than the trader quote.
Hence, if we go by VWAP, we can afford a volume
of 82M with a VWAP of $1.274956.
In the next section, we will derive an algebraic
model to identify the way to determine the VWAP
volume.
Formation of Generic Mathematical
Model
The following steps are critical for deriving the
algorithmic model used above:
• Consider generic values in market depth (V1:P1,
V2:P2,…Vn:Pn). Here, V1 is the volume at the 1st
position in the ladder, and P1 is the correspond-
ing price.
• There can be Max 1 lot (Min 0 lot) in market
depth where volume will split to support
a given VWAP. Let’s say it is Vx — Px. So the
market depth becomes
V1 — P1
V2 — P2
...
Vx — Px
...
Vn — Pn
• Let’s assume that the required volume from
the Vx — Px lot is Vr. Then total money spent
will become :
V1*P1+V2*P2+….+Vr*Px =
Sum(Vi*Pi) + Px*Vr where i
varies from 1 to (x-1).
• Let’s assume that the VWAP given is Pv (89.2
in the above case). Then, total money spent will
be = (V1+V2+…Vx-1+Vr)*Pv = (Sum(Vi)+Vr)*Pv
where i varies from 1 to (x-1).
• Equating both sides gives us:
Sum(Vi*Pi) + Px*Vr where i varies
from 1 to (x-1) = (V1+V2+…Vx-
1+Vr)*Pv=(Sum(Vi)+Vr)*Pv where i
varies from 1 to (x-1)
Or,
cognizant 20-20 insights
Figure 1
As the ladder above shows, Best Ask is 1.273957
with a volume of 23M. Market depth is 1.27473
with volume 16M, 1.275502 with volume 41M,
1.277048 with volume 8M, and so on. Let’s as-
sume that a trader wishes to hedge his position,
and he agrees to afford a spread of 10 pips from
the Best Ask. Hence, he can afford a price of
1.274957 and he is expecting to hedge an infinite
volume (or as much as possible). We will try to
find out technically how much will he be able to
hedge (assuming that no other trader is present
in the market at this point in time to acquire any
similar position in the same ladder).

3. 3cognizant 20-20 insights
Vr*Px — Vr*Pv = Sum(Vi)*Pv -
Sum(Vi*Pi)
Or,
Vr*(Px-Pv)= Sum(Vi)*Pv -
Sum(Vi*Pi)
Or,
Vr = {Sum(Vi)*Pv - Sum(Vi*Pi)}/
(Px-Pv)
So, the total volume that can be technically availed
is — Sum(Vi) + Vr where i varies from 1 to (x-1),
and value of Vr is -: { Sum(Vi)*Pv - Sum(Vi*Pi)}/
(Px-Pv)
Deriving the Algorithm from a
Mathematical Model
The basic steps of the algorithm include:
• Assumptions:
For the ask side of the ladder, prices are sorted
in ascending order.
>> Every price with its volume will be called a
lot in the algorithm.
>> Algorithm for ASK is shown below; algorithm
for BID will be exactly same as that of ASK;
only the comparator signs will be reversed
(all greater than will become less than and
vice-versa).
>> Input to the algorithm is a set of lots and the
VWAP price is quoted by the trader (Pv).
>> Output to the algorithm will be the volume
and effective VWAP achieved.
• Algorithm:
Initialize below variables:
>> totalPriceMultipliedByVol = 0
>> totalVol = 0
>> For every lot in the ladder execute the below
steps (start a for loop):
»» Take the price of the lot, multiply with the
volume available (let’s say Vi). Store it in
a variable (e.g., X).
»» If (totalPriceMultipliedByVol + X<Pv*(
totalVol*Vi)), then totalVol = totalVol + Vi
and totalPriceMultipliedByVol = totalPri-
ceMultipliedByVol + X.
»» Then this is the lot that will be split. Store
this lot (Px:Vx); break the loop.
>> Beyond the loop, check on whether Px:Vx
is found. If it is not found, then that means
we can consume the entire ladder. Hence,
in that case the effective VWAP will be to-
talPriceMultipliedByVol/totalVol, and VWAP
volume will be totalVol.
>> If Px:Vx is found, then:
»» VWAP volume (totalVol + VV), where
VV= (totalVol*Pv — totalPriceMultiplied-
ByVol)/(Px — Pv).
»» Effective VWAP = (totalPriceMultiplied-
ByVol + VV*Px)/(VV + totalVol).
Benefits of Volume Calculation Based
on VWAP
If we hedge our position with a given price, we
will end up with the cumulative volume of those
lots where the individual lot prices are either less
than or equal to the given quote. Whereas, if we
go with volume acquisition based on the VWAP
algorithm, we can afford a few lots that have indi-
vidual prices above the given quote, thus generat-
ing more volume.
In our example, for example, we could get a cumu-
lative volume of 39M (23M with price 1.273957,
16M with price 1.274730), whereas with appli-
cation of VWAP algorithm we could hedge our
position with 82M of volume.
One noteworthy point here is that the VWAP
algorithm is useful only when we wish to acquire
more volume within a given spread of price.
Hence, specifically for liquidity aggregation
purposes, volume calculation based on VWAP
prices is certainly effective.
Within the currency trading (FX) domain (FX
ladder), the VWAP algorithm is more popular
and frequently used, because of the existence of
numerous exchange venues (like EBS, Reuters,
HotSpot, 360T, etc.) that provide huge liquidity for
many currencies. Hence, even usage of a virtual
order book (VOB) along with the VWAP algorithm
is very effective in liquidity aggregation.

4. About Cognizant
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About the Author
Nishit Kumar Ghosh is a Project Manager who leads Cognizant’s Wealth Management Team at one of the
company’s major banking clients. He has a decade of experience in techno-functional roles in market
risk domain, FX front-office trading algorithm development, wealth management and low-latency pro-
gramming. Nishit has worked in Europe and APAC regions delivering transformational projects for large
financial institutions. He can be reached at Nishitk.Ghosh@cognizant.com.