CATEGORÍA DE INVESTIGACIÓN Mención especial:

“Can correlation risk be hedged?”

Wilfrido Castillo Miranda Olea

Contents

1 Introduction 7

2 Fundamentals of index and constituent variance and correlation 8

2.1 Basic definitions of variances and correlations . . . . . . . . . . . . . . . . 8

2.2 Definitions of realized and implied quantities . . . . . . . . . . . . . . . . . 10

2.2.1 Realized and expected variances . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Realized and expected correlations . . . . . . . . . . . . . . . . . . 12

3 Variance and correlation swaps 14

3.1 Variance swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Correlation swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Dispersion trade 17

4.1 Defining dispersion trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Implementing dispersion trade by variance swaps . . . . . . . . . . . . . . 18

4.3 Connection between dispersion trade and hedging correlation risks . . . . . 19

5 Wishart affine stochastic correlation model (WASC) 22

5.1 Affine diffusion process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Model Assumptions and Specifications . . . . . . . . . . . . . . . . . . . . 24

5.3 Reasons for choosing WASC . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.4 Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4.1 Parameter specifications . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4.2 Parameter reduction techniques . . . . . . . . . . . . . . . . . . . . 30

6 Historical backtesting 33

2

7 Simulation results 37

7.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.1.1 Toy Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.2 Stress Test: Misestimated Future Realized Variances . . . . . . . . . . . . 51

7.2.1 Toy Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8 Conclusions and future research directions 62

9 References 64

3

Executive Summary

This paper summarizes our team’s effort in researching the topic of hedging correlation

risk among constituent stocks in an equity index. Since the average pairwise correlation

is difficult to replicate, the idea that it can be approximated closely by canonical proxy

correlation introduced by Bossu (2007) has been utilized throughout our paper. This idea

was tested robustly using market data from Dow Jones 65 Composite Average and it is

shown that the spread between these two correlation measures diminishes significantly as

the number of stocks increases from 3 to 50. This gives rise to the central idea of our

paper that with a sufficient number of stocks in the index, the problem of hedging average

pairwise correlation essentially becomes hedging proxy correlation.

Traditionally, dispersion trade is defined as taking opposite positions in index and its

constituents’ variance swap contracts. In our paper, we will show that correlation swaps

can be hedged by dispersion trade by appropriately buying weights of variance swaps.

The appropriate trading weights in dispersion trade have been derived in our report. By

doing so, the dispersion trade can be implemented in our simulation process.

To capture the properties of stochastic variances and covariances among the constituent

stocks, we need a stock price model like multi-dimensional Heston stochastic volatility

model to achieve this goal. In our paper, a model called Wishart affine stochastic correla-

tion (WASC) model proposed by Bru (1991) is introduced to generate stock price paths.

To simplify the parameter estimation process for the WASC model, we have assumed

that the pairwise correlations among all constituent stocks are identical for initial and ex-

pected long-term variance-covariance matrix for asset returns in our simulation process.

The hedging efficiency of correlation swaps by dispersion trades can then be investigated

5

in two settings. First, it is tested in a toy model setting with the assumption that future

realized variances can be estimated correctly. To allow readers a better understanding

of the feasibility of our dispersion trade strategy in real-world situations, stress testings

with the assumption that the future realized variances are estimated wrongly have been

performed in our model. Our results show that the dispersion trade with our proposed

trading weights performs reasonably well in both cases. As a consequence, this paper

provides readers with a practical framework in both modeling correlations among stocks

and hedging correlation risk among them using dispersion trade.

6

1 Introduction

Hedging correlation risk within constituent stocks in an equity index is never an easy task.

Despite the fact that a lot of practitioners apply dispersion trade for hedging purposes,

the trading weights of constituent stocks in the dispersion trade have not converged both

in the academic field and the financial industry. This gives the motivation to undergo fur-

ther research to understand dispersion trade more deeply in order to analyze the hedging

efficiency of correlation swaps by dispersion trade.

This paper is organized by first addressing the fundamental concepts of index and con-

stituent variance and correlation. The idea of hedging canonical proxy correlation instead

of the average pairwise correlation is also discussed is Section 2. Then, properties and

payoff structures of variance swap and correlation swap are introduced. Section 4 defines

the dispersion and the connection between dispersion trade and hedging correlation risks

using variance swaps is also discussed in this section. Section 5 focuses on the theoretical

model called Wishart affine stochastic correlation model whom we choose to simulate the

stock price paths. All parameters and mathematical concepts are explained in detail in

this section to help readers understand that it is a good model to capture correlation

components among stocks. After that, we proceed to incorporate this model into the dis-

persion trade and implement the whole dispersion trade process. Implementation details

and simulation results are discussed in Section 7. Finally, the relationship between dis-

persion trading and hedging correlation risks among constituent stocks can be understood

more rigorously.

7

2 Fundamentals of index and constituent variance

and correlation

2.1 Basic definitions of variances and correlations

Assume that a time period τ = [t, T ], where T ≥ t ≥ 0. In addition, to simplify the no-

tations throughout the report, we will further assume that the interest rate is always zero.

Let St = (S1,t, S2,t, ..., Sn,t) be a vector of stock prices at time t and Si,t be the price

of stock i at time t. Therefore, all elements in the vector St must be positive. Suppose

there are n stocks in a particular index and its index price at time t is It. To simplify

our calculations of index returns from our simulation model, we further assume the index

weight of each stock wi to stay constant throughout [0, T ] and it implies that no rebal-

ancing occurrs in the whole time period. Then, the index price at time t (It) as follows

from its constituent stocks:

It =

n∏

i=1

(Si,t)wi (1)

where∑n

i=1 wi = 1

This means that the index price is basically the geometric weighted-average of the prices

of constituent stocks. According to Bossu (2007), this assumption is a simplified system

for the computation of equity indices because it allows us to take logarithms of these

quantities more easily in later derivation of formulas. In practice, most equity indices

are computed based on arithmetic weighted-average of market values. With a view to

match the real-world practice, we have calibrated our simulation model using the arith-

metic weighted-average approach by using real-world data in Section 7 and the results are

8

similar to the toy model setting with the geometric weighted average assumption.

Realized variance is calculated based on a prespecified set of sampling stock prices over a

certain period of time. In the market convention, the computations of the realized vari-

ance assumes a zero mean return. Denote σ2i (τ) as the realized variance of the returns on

stock Si and m as the number of time steps during the time period τ . Hence, σ2i (τ) can

be computed as:

σ2i (τ) ≈ 1

m

m−1∑

k=0

[ln

(Stk+1

Stk

)]2

(2)

where t = t0 < t1 < ... < tn = T

Likewise, the realized variance for index price, σ2I (τ), can be computed by replacing Stk+1

and Stk in the above equation.

Let ρij(τ) denote the realized correlation between returns on asset Si and returns on

asset Sj during time period τ . Similar to the realized variance, the computations of the

realized correlation assumes zero mean returns in market convention. Therefore, it is

defined as:

ρij(τ) ≈

m−1∑k=0

[ln

(Si,tk+1

Si,tk

)ln

(Sj,tk+1

Sj,tk

)]

√m−1∑k=0

[ln

(Si,tk+1

Si,tk

)]2√

m−1∑k=0

[ln

(Sj,tk+1

Sj,tk

)]2(3)

where t = t0 < t1 < ... < tn = T

9

2.2 Definitions of realized and implied quantities

After introducing the basic terminology about how to compute variances and correlations

for stocks and indices, the definitions on realized quantities which are used in dispersion

trade are discussed in this section.

2.2.1 Realized and expected variances

The following are definitions for realized variances which will be used in later sections to

define other quantities.

By multiplying the realized variance of each stock by their respective weight in the index,

the index-weighted average realized variance of constituent stocks in period τ under the

index:

σ2avg(τ) =

n∑

i=1

wiσ2i (τ) (4)

In addition, by squaring the sum of the products between the index weights and realized

volatility of each stock, the sum of the index-weighted realized variance of constituent

stocks in period τ under the index:

σ2sum(τ) =

(n∑

i=1

wiσi(τ)

)2

(5)

Moreover, by summing the products between the square of index weights and realized

10

variances of each stock, the residual term can be defined as:

ǫ(τ) =

n∑

i=1

w2i σ

2i (τ) (6)

In accordance with the classical theories in stochastic calculus, the price of a derivative

contract at any time before the maturity can be computed by taking the conditional expec-

tation of discounted payoff of the derivative contract at its maturity. This can be applied

to derivative contracts based on variance quantities. Following the initial assumption

of zero interest rate for all periods and under the P -equivalent risk-neutral (martingale)

measure P at time tk ≤ T , we can define the price of a derivative contract based on

expected variance quantities under risk neutral measure as:

The price of expected variance on stock i in the period τ at time tk:

σ2i,tk

(τ) = E[σ2i (τ)|Ftk ] (7)

The price of expected index-weighted average variance of constituent stocks in the period

τ at time tk:

σ2avg,tk

(τ) =

n∑

i=1

wiσ2i,tk

(τ) = E[σ2avg(τ)|Ftk ] (8)

Note that for tk ∈ [t, T ], equation (7) can be rewritten as:

σ2i,tk

(τ) =tk − t

τσ2

i ([t, tk]) +T − tk

τσ2

i,tk([tk, T ]) (9)

11

2.2.2 Realized and expected correlations

In this part, the definitions of both realized and expected correlations are addressed and

are used again in the implementation of dispersion trade. The following are definitions

for realized correlations:

The average pairwise correlation among constituent stocks in an index is:

ρavgPair(τ) =

∑i<j wiwjρij(τ)∑

i<j wiwj

(10)

The clean correlation among constituent stocks in an index is:

ρclean(τ) =σ2

I (τ)− ǫ(τ)

σ2sum(τ)− ǫ(τ)

(11)

The canonical correlation among constituent stocks in an index is:

ρcanonical(τ) =σ2

I (τ)− ǫ(τ)

σ2avg(τ)− ǫ(τ)

(12)

According to Bossu (2007), if the following assumptions hold:

1. The number of stocks, n, becomes very large.

2. All realized volatilities of each stock, σi(τ), never become zero or go to infinity

3. All pairwise correlations between each pair of stock, ρij(τ), never go to zero

the following condition can be obtained:

Max(wi)

Min(wi)= O(

√n) (13)

12

where O(n) is the function with the order n and i = 1, 2, ..., n

With the above conditions, Bossu (2007) proves that the residual term ǫ(τ) goes to zero

and the proxy for canonical correlation can be defined as follows:

ρproxy(τ) =σ2

I (τ)

σ2avg(τ)

≈ ρcanonical(τ) (14)

In Section 6, we have performed the historical backtesting to study the relationship be-

tween ρproxy(τ) and ρavgPair(τ) by using market data of Dow Jones 65 Composite Average

dataset over the last 10 years and we found that ρproxy(τ) is a close approximation to

ρavgPair(τ) when the number of stocks in the index, n, becomes reasonably large. This

becomes one of the major building blocks for our hedging strategy in later sections.

Again, similar to the price of a derivative contract based on expected variance quantities,

under the P -equivalent risk-neutral measure P , we can compute the price of derivative

contracts based on correlation quantities by taking conditional expectation of the payoff

of these contracts at their maturities. Hence, with the initial assumption of zero interest

rate, the price of average pairwise correlation and proxy correlation can be defined as

follows:

ρavgPair,tk(τ) = E[ρavgPair(τ)|Ftk ] (15)

ρproxy(τ) = E

[σ2

I (τ)

σ2avg(τ)

|Ftk

](16)

13

3 Variance and correlation swaps

Two major kinds of products are involved heavily in dispersion trade and they are variance

and correlation swaps. In this section, their definitions and payoff structures are discussed

in detail. For simplicity and without loss of generality, we assume the following for contract

specifications:

1. All swaps contracts exchange cash flow only once at maturity time T. In other words,

they are forward contracts.

2. τ = [t, T ] equals one year.

3. The notional value equals one.

4. t0 denotes the time the trader enters a contract.

14

3.1 Variance swap

A variance swap on asset Si allows two parties to exchange cash flows based on the realized

variance of an asset’s returns over time. The underlying of the variance swap is σ2i (τ).

When two parties enter a variance swap contract, the fair strike of the variance swap at

time t0 should be the price that makes the contract have value zero and it is denoted as

σ2i,t0

(τ). This is obtained by equation (7) and it is the risk-neutral conditional expectation

of σ2i (τ) given the information up to time t0. At maturity, one of the parties has to pay

another party to settle the contract. The profit and loss profile (P&L) of a long position

in a variance swap on asset Si at maturity time T is expressed as:

σ2i,T (τ)− σ2

i,t0(τ) (17)

For the value of a short position, the vise versa holds.

15

3.2 Correlation swap

A correlation swap on index I is a swap contract whose underlying is ρavgPair(τ). Similar

to variance swaps, to make sure there is no exchange of cash flows at the beginning of the

contract, the fair strike of a correlation swap at time t0 can be denoted as ρavgPair,t0(τ).

This is obtained by taking the risk-neutral conditional expectation of ρavgPair(τ) given

the information up to time t0 and it has been shown in equation (15). At maturity, one

of the parties has to pay the other party to settle the contract. The profit and loss profile

(P&L) of a long position in a correlation swap on index I at maturity time T is expressed

as:

ρavgPair,T (τ)− ρavgPair,t0(τ) (18)

For the value of a short position, the vise versa holds.

16

4 Dispersion trade

4.1 Defining dispersion trade

The variance of the index, σ2I , is not equal to the sum of the weighted variances of

its components, ǫ(τ), due to the correlation between the constituents. In other words,

both the expected volatility as well as the historical volatility of the basket of stocks are

different from that of the index. Though the index is comprised of all the constituent

stocks which have correlations among themselves, it still trades as one asset and it only

has one unique value for expected volatility and realized volatility. On the other hand,

the basket of stocks is created synthetically and all stocks trade independently and have

their own values of expected and realized volatilities. For the sake of calculating both

the expected and realized volatility of the basket of stocks, the correlations among stocks

have to be considered.

Thus, there is always a spread between trading the basket of stocks and index and trading

on this spread is called dispersion trading. By assuming bwi to be the buying weights of

each stock in the dispersion trade in time period [t,T] and the number of stocks in the

index is n,

DispersionSpread =

n∑

i=1

bwiσ2i (τ)− σ2

I (τ) (19)

17

4.2 Implementing dispersion trade by variance swaps

From the equation of dispersion spread, it is obvious that by taking positions in the index

variance and its constituents’ variances, one can have exposure to the correlations among

underlying stocks. This strategy of dispersion trade can be achieved through variance

swaps. For example, selling a variance-swap on the index and buying variance swaps on

the individual constituents of the index constructs a short exposure in correlations among

stocks and gives a long exposure to the dispersion spread. After expanding σ2I in the

equation for dispersion spread, it can be easily interpreted that this trading strategy is

betting that the correlation among stocks is going to decrease.

LongDispersionSpread =

n∑

i=1

bwiσ2i (τ)−

n∑

i=1

w2i σ

2i (τ)−

n∑

i6=j

wiwjρij(τ)σi(τ)σj(τ) (20)

where bwi and wi are buying weight and index weight of the stock i respectively

18

4.3 Connection between dispersion trade and hedging correla-

tion risks

Our objective in this paper is to produce an approximate hedge for the payoff of a cor-

relation swap based on average pairwise correlation through replicating ρproxy, a proxy

measure for ρavgPair, by using dispersion trade. In this section, we will establish and jus-

tify the trading weights of each constituent stock in the dispersion trade. This forms the

major building blocks for our later sections which we will illustrate through Monte Carlo

simulations the effectiveness of this hedging strategy under both theoretical and real-world

settings. Our idea of offsetting the variation of ρproxy through dispersion trades is largely

due to Bossu(2007). However, while Bossu(2007) is able to price a forward contract with

ρproxy as the underlying under his toy-model setting, this is not our objective here. Al-

though our dispersion trade strategy may be able to offset the variation of ρproxy, it is not

designed to be an exact replication of a forward contract on ρproxy. Our eventual goal is

to come up with a good hedge for ρavgPair.

Before specifying the weights of stocks in dispersion trade, we will introduce a new nota-

tion. Let ρproxy,t(τ) 1denote the ratio at time t between the strike price of variance swap

on σ2I (τ) and the strike price of variance swap on σ2

avg(τ) :

ρproxy,t(τ) ≡σ2

I,t(τ)

σ2avg,t(τ)

(21)

1We use the notation ρproxy,t(τ) instead of ρproxy,t(τ) to highlight the fact that it is not a price. In

other words, ρproxy,t(τ) is not the conditional expectation ofσ2

I (τ)σ2

avg(τ) at time t. It is merely a ratio of two

prices.

19

A dispersion trade can be viewed as buying β units of variance swap on σ2avg(τ) and selling

1 unit of variance swap on σ2I (τ) :

D(β, τ) ≡ βσ2avg(τ)− σ2

I (τ) (22)

More generally speaking, a dispersion trade is composed of β long positions in σ2avg(τ) for

each short position in σ2I (τ) . The reader may probably argue that there exists no liquid

traded variance swaps on σ2avg(τ) . However, for the discussion in this paper, we assume

that we could synthetically construct a variance swap on σ2avg(τ) by trading variance swaps

on the component assets with appropriate weightings (see equation (4)) Now, suppose we

set β as ρproxy,0(τ), the payoff of the dispersion trade becomes:

PNL(DispersionTrade) = β × PNL(σ2avg(τ))− PNL(σ2

I (τ))

=σ2

I,0(τ)

σ2avg,0(τ)

(σ2avg(τ)− σ2

avg,0(τ))− (σ2I (τ)− σ2

I,0(τ))

= −σ2avg(τ)(ρproxy(τ)− ρproxy,0(τ))

(23)

From the above equation, the payoff of our dispersion trade exactly offsets the differences

between ρproxy(τ) and ρproxy,0(τ) multiplied by the realized variance of the weighted av-

erage of constituent stock returns. Next, we make an estimate of σ2avg(τ) and denote it

by σ2avg(τ) 2. We will now redefine dispersion trade by scaling equation (23) by

1

σ2avg(τ)

to obtain:

∆avg =ρproxy,0(τ)

σ2avg(τ)

∆I = − 1

σ2avg(τ)

(24)

2Again, we use the notation σ2avg(τ) instead of σ

2avg(τ) to highlight the fact that it is not a price. The

market price σ2avg(τ) may be the best estimator of σ

2avg(τ) , but it is not the only estimator.

20

PNL(DispersionTrade) = ∆avgPNL(σ2avg(τ))−∆IPNL(σ2

I (τ))

= −σ2avg(τ)

σ2avg(τ)

(ρproxy(τ)− ρproxy,0(τ))(25)

The above equations tell us that as long as σ2avg is an accurate estimate for σ2

avg , the

variation of ρproxy can be mostly, if not completely, offset by dispersion trade. Through

this strategy, we expect at least a partial hedge of ρavgPair and we will now move on to

describe our methodology of testing the hedge’s effectiveness under various simulation

scenarios.

21

5 Wishart affine stochastic correlation model (WASC)

With a view undergoing the Monte Carlo simulation for stock price paths, a stock price

model which is flexible enough to capture the stochastic correlations between different

constituent stock returns in the index is needed. Moreover, the model should have the

ability to reproduce many financial stylized facts like volatility skews and mean-reverting

feature of volatility of asset prices. In this section, we will first address the properties

and advantages of using affine model to model correlations among stocks. After this, the

pros and cons of Wishart affine stochastic correlation model (WASC hereafter) proposed

by Fonseca, Grasselli, and Tebaldi (FGT hereafter) in 2006 and robust WASC model

specifications will be explained.

5.1 Affine diffusion process

Affine diffusion processes can be generalized as follows in a stochastic differential equation

form:

dx = [a + Ax]dt + CT diag(√

v(x))dW (26)

v(x) = b + Bx (27)

where W is a n-vector of independent standard Weiner processes, a, A, b, B,C are a set

of parameters, v(x) is the n-vector volatility process and the instantaneous covariance of

x, CT diag(v(x))C is well-defined (positive semi-definite) for all states in which v(x) ≥ 0

The normal affine process is defined as a class of time-homogeneous Markov processes.

The major property of the affine process is that the logarithm of the characteristic func-

tion of the transition distribution of such process is affine related to the initial state. By

22

solving a system of ordinary differential equations within the affine diffusion processes,

the coefficients of the affine relationship can be obtained. However, the parameters of an

affine diffusion cannot be chosen randomly and they cannot violate certain restrictions

to be admissible. These imposed restrictions ensure the existence of a solution to the

stochastic differential equation.

According to Duffie (2002), the advantages of affine process can be summarized as follows:

1. In general, solving multi-dimensional partial differential equations (PDEs) is needed

to price financial assets which depend on multiple factors. By turning the under-

lying factors to be described by affine diffusion processes, pricing derivatives only

requires solving the systems of ordinary differential equations (ODEs) which are

less complicated than PDEs. This property of the affine diffusion process provides

a practical framework for the implementation of these models by alleviating the

problem of the curse of dimensionality.

2. Another useful feature of the affine diffusion process is the computational tractability

and flexibility in the interpretation of the factors. This property allows the models

to capture lots of empirical evidence of the financial time series data ranging from

jumps to stochastic volatility in different forms.

3. The parameters in an affine diffusion can be verified to ensure admissibility easily

through the implementation of the checking procedures in software like Matlab.

23

5.2 Model Assumptions and Specifications

The quoted model assumptions which were explicitly written in the original article (FGT,

2006 b, p.3-4) are:

Assumption 1

The continuous time diffusive Factor Model is considered to be affine in the terminology

of Duffie and Kan (1996).

Assumption 2

The evolution of asset returns is conditionally Gaussian while the stochastic covariance

matrix follows a Wishart process.

Assumption 3

The Brownian motions of the assets’ returns and those driving the covariance matrix are

linearly correlated.

In mathematical terms, Assumption 2 declares that a n-dimensional risky asset St fol-

lows the risk-neutral dynamics given by:

dSt = diag[St](r1dt +

√∑

t

dZt) (28)

where 1=(1, ..., 1)T and Zt ∈ Rn is a vector of independent Brownian motions

As for Zt, each component inside is defined as follows:

dZkt =

√1− Tr[RkRT

k ]dBkt + Tr[RkdW T

t ], k = 1, ..., n (29)

24

where B is a n-by-1 Brownian motion vector independent of W, Rk is a correlation matrix

in which the elements Rij

k represents the correlation between the scalar Brownian motion

Zk and the scalar Brownian motion W ij.

The quadratic variation of the risky assets is the symmetric matrix∑

t whose risk-neutral

dynamics follows the Wishart process:

d∑

t

= (ΩΩT + M∑

t

+∑

t

MT )dt +

√∑

t

dWtQ + QT (dWt)T

√∑

t

(30)

where Ω, M, Q ∈ Mn, Ω invertible, and Wt ∈ Mn is a matrix of independent Brownian

motion

Mn in the equation above denotes square matrices and M is a negative semi-definite

matrix that makes the term variance-covariance at time t (i.e.∑

t) in the above equation

mean-reverting to its expected long-term variance-covariance (i.e.∑

∞). By assuming

further that Q is the volatility of volatility matrix accounting for fluctuations in the

variance-covariance matrix∑

t, the following relationships involving Q,M,∑

∞ and Ω

can be defined:

−ΩΩT = M∑

∞

+∑

∞

MT (31)

ΩΩT = βQT Q, β > n− 1 (32)

25

FGT proposed the correlation matrix Rk should be given by:

0 0 0

ρ1 ... ρn

0 0 0

←− kth row (33)

where ρi ∈ [−1, 1], i=1,...,n.

26

5.3 Reasons for choosing WASC

1. The WASC model can be viewed as a multi-factor extension of the Heston stochastic

volatility model. In the model, the evolution of asset prices are shaped by the joint

diffusion of multiple Brownian motions, whose coefficients also evolve stochastically

according to the Wishart process matrix as first introduced by Bru (1991). In our

implementation part later in this report, we will simulate the price paths of three

stocks using WASC. Each stock return is determined by the same set of Brownian

motions, but with each coefficient to these Brownian motions being a component

of the square-root matrix of the stochastically evolving variance-covariance matrix.

This enables us to create random variance and covariance of asset return movements

that variance swaps and correlation swaps are based upon.

2. In addition to having the advantages associated of being an affine model, WASC

also has the property of being solvable, i.e. the solution of option pricing can be

derived by applying a Fast Fourier Transform (FGT, 2006). Furthermore, not only

can the WASC model undo the ”volatility leverage effect” (a negative correlation

between noise driving stock returns and noise that shocks volatility), but it also

has the flexibility to accommodate the ”correlation leverage effect” observed first

by Roll (1988) and also by Ang and Chen (2002), which represents ”the asymmetric

response of correlation to positive and negative shocks on asset returns: a decrease in

correlation will increase the dispersion of individual assets and thus the probability

that any asset may reach high values at maturity” (FGT, 2006 ).

3. Despite the existence of a large amount of parameters needed to be estimated, it

suffices to make intelligent assumptions about the parameters to make use of the

WASC model in simulating stock price paths incorporating the correlation structures

27

among them. The standard constant volatility Black-Scholes model is a special case

of the WASC model by appropriately assigning the zero value to certain parameters.

Therefore, for the sake of making WASC more practicable, parameters reduction

techniques are illustrated in the later section and it is demonstrated that the number

of input parameters decreases largely.

28

5.4 Model implementation

5.4.1 Parameter specifications

To implement the WASC model, we need to specify clearly the restrictions, dimensions

and economic meaning of the input parameters. All these input parameters in WASC

model are described in detail in Table 1.

Table 1: Input Parameters for the WASC Model

Symbol Description Dimension Restriction

r Risk-free rate Scalar Positive∑∞ Expected long-term variance-covariance n× n Symmetric

matrix for asset returns Positive definite∑0 Initial variance-covariance matrix n× n Symmetric

Positive definite

QT Q Variance-covariance matrix for∑

t n× n Symmetric

β Gindikin coefficient (see Bru 1991) Scalar Greater than n-1

ρ Correlation between noise of asset return n× 1 ρi ∈ [−1, 1]

and noise of√∑

t

M Mean-reverting coefficients for∑

t n× n Negative

Need not be estimated Semi-definite

29

5.4.2 Parameter reduction techniques

The WASC model provides a flexible framework to accommodate many stylized facts

observed in the equity market. The flip side of such flexibility is its (potentially) large

number of parameters. The number of parameters as a function of model dimension is

(n3 + 6n2 + 3n)/2. Thus, even for a small n (i.e. 3 and 4), the number of parameters

can rise significantly (45 and 86 respectively). Some of these parameters, moreover, can

be difficult to calibrate. For instance, the model contains a variance-covariance matrix

for the variance-covariance matrix of the stock returns - the so called ”vol vol” matrix.

Since the variance and covariance of the variance and covariance among stocks cannot

be observed directly from the market, estimation of the vol vol matrix is a potentially

troublesome task. In order to make WASC model easier to implement, we have made

reasonable assumptions so that major advantages of WASC model can be preserved and

the number of input parameters needed to be estimated can be reduced to a large extent.

Firstly, estimation processes of∑

∞ and∑

0 can be simplified by restricting these matrices

to be the following form:

∑

∞

=

ρijσiσj

σ2i

ρijσiσj

(34)

This is why we first assign values to the QT Q matrix and then compute Q from eigenvalue

decomposition method. Now, given both the QT Q matrix and the∑

∞ matrix, we only

need the Gindikin coefficient β to compute M from (31) and (32) through the eigenvalue

30

decomposition technique again but with a twist. By doing so, we no longer need to es-

timate M . For a given QT Q and∑

∞, β can adjust how quickly the∑

t matrix reverts

back to∑

∞. The larger the β, the less time it takes for the∑

t matrix to revert back to

∑∞, and vice versa.

By adopting the parameter estimation approach just described, we have greatly reduced

the number of parameters needed to be estimated relative to the original numbers sug-

gested at the beginning of this section. If we make the assumption that∑

∞ is a single-

valued diagonal matrix and applies the same treatment to QT Q, the only significant

parameter estimation task left is calibration of∑

0 to the market data. It is obvious that

∑0 can be easily obtained through historical stock prices data. The effect of parameters

reduction following the number of stocks is shown in Table 2 below.

Table 2: Parameters reduction for the WASC Model

Number of stocks

Original Simplified

Three Four n n

r 1 1 1 1∑∞ 6 10 n(n+1)/2 1∑0 6 10 n(n+1)/2 n(n+1)/2

QT Q 6 10 n(n+1)/2 1

β 1 1 1 1

ρ 3 4 n 1

M 0 0 0 0

Total 23 36 1.5n2 + 2.5n + 2 0.5n2 + 0.5n + 5

31

The figure below shows the simulation results of WASC of five stock price paths with

different volatilities after parameter reduction process. We have assumed initial and long-

term pairwise correlations between each pair of stock prices to be 0.3 and 0.8 respectively.

From the bottom plot in the figure, we can see all pairwise correlations start with 0.3 and

converge to 0.8 over time. As can be seen in the middle plot, the volatilities of all stocks

are stochastic in nature. Also, the stock with the highest volatility in middle plot has

the greatest fluctuations in the upper plot of stock prices. This figure verifies that our

simulation model can still retain important properties in WASC after parameter reduction

process.

Figure 1: Simulation of WASC after parameter reduction

32

6 Historical backtesting

Bossu (2007) proposed the idea that the canonical correlation measure, ρproxy can be a

good estimate for the average pairwise correlation ρavgPair. From equation (14), it has

been stated by Bossu (2007) that when the number of stocks becomes sufficiently large,

the canonical proxy correlation will be a good approximation for the canonical correlation.

The aim of this section is to test if the canonical proxy correlation is a good approximation

for average pairwise correlation. We gathered our data from Datastream databases from

the Johnson School of Cornell University. For the historical backtesting, the constituents

and index data from Dow Jones 65 composite average index of last 10 years (12/1996 -

12/2006) were tested. The key computational procedures is as follows:

1. The data was cleaned to eliminate those stocks which did not have traded continu-

ously between 12/1996 and 12/2006 to avoid any distortions in computing correla-

tion measures.

2. The index weights were computed according to arithmetic average methodology.

More precisely, the weight of stock i at time t can be computed from the following

formula:

Market Cap of Stock i at time t

Total Market Cap of all stocks in Index at time t(35)

where Market Cap at time t is computed as Price(t)× Shares(t)

3. All correlation measures are computed from prices data using 2-year moving window

period.

We ran 13 cases for different number of stocks and created a table as shown below to

demonstrate the effect of the number of stocks on the spread (in terms of correlation

units), ρproxy − ρavgPair, between the canonical proxy and average pairwise correlations.

33

To visualize the effect more clearly, the graphs showing clean, canonical proxy and average

pairwise correlations for the case of 5, 10, 35 and 50 stocks have also been included (Figure

2 - 3).

It can be observed from the table and the graphs that as the number of the stocks grows,

the spread between average pairwise and canonical proxy correlations decreases. This

confirms Bossu’s (2007) theory that the approximation by canonical proxy correlation

gets more accurate as the number of stocks increases.

Table 3: The effect of number of stocks on correlation spreads

Number of Stocks Spread3 0.5330

5 0.3147

8 0.1531

10 0.1393

12 0.1258

15 0.0719

20 0.0584

25 0.0573

30 0.0428

35 0.0262

40 0.0165

45 0.0146

50 0.0113

34

Figure 2: The graphs of correlations with 5 and 10 stocks

35

Figure 3: The graphs of correlations with 20 and 50 stocks

36

7 Simulation results

As discussed earlier in Section 4.3, dispersion trade is implemented as buyingρproxy,0

σ2avg

units

of component variance swaps (swaps on σ2avg ) and selling

1

σ2avg

units of index variance

swaps (swaps on σ2I ). As long as the estimate σ2

avg closely approximates σ2avg , dispersion

trade will be able to offset the variations due to ρproxy. This provides at least a partial

hedge against a correlation swap on ρavgPair . The goal of this section is to examine the

effectiveness of our dispersion trade hedging strategy via Monte Carlo simulations. We

will generate stock prices through the WASC model framework and use those paths to

produce several scenarios to test the hedging strategy’s robustness. We implement our

hedging strategy in two settings:

1. Toy model setting with our choice of input parameters, where the Index is calculated

as the geometric average of the component stocks

2. Calibration to the prices data of Dow Jones 65 Composite Average , where the

Index is computed as the market-cap weighted average of the component stocks.

We also stress test our implementation models in Section 7.2 when σ2avg is a not a

close approximation of σ2avg.

37

7.1 Implementation

In our implementation, σ2avg is obtained by taking the expectation of future realized vari-

ance from multiple Monte Carlo simulations. This method assumes that the future re-

alized covariance matrix will be accurately specified by the WASC model and the input

parameters. We compute the expected values of future realized variances ( σ2avg ) by

averaging over 30 simulated price paths for each time-step. Since the future realized

variance-covariances are produced by the WASC model, this approach will produce fairly

accurate estimates of σ2avg . 1500 simulation runs along with 800 time-steps for each run

will be generated.

38

7.1.1 Toy Model Setting

To conform with our theoretical set-up, we will first perform simulations under a toy model

setting, where the following assumptions hold: 1) index is a geometric weighted average of

its constituents, 2) constituents’ weights are equal and held constant for the whole period,

and 3) the correlation parameters in the∑

∞ and∑

0 matrices of the WASC model are

identical for each pair of stocks. Then, we will proceed to test our hedging strategy

under a calibrated parameter setting where most parameters in the WASC simulation

framework are calibrated from the Dow Jones 65 Composite Average dataset3 and that

this three assumptions cease to be valid. To simplify the simulation process, we take the

notional amount of correlation swap to be 1 and all profit and loss (P&L) are in terms of

correlation points.

I. Static Hedge Under static hedge, the hedge for dispersion trade is performed only

once over the entire contract period of correlation swap. We first explore a simple index

consisting of three stocks and another composing of ten stocks. Please refer to Appendix A

for the input parameters. Below are the results and the discussion for those two scenarios.

3Daily closing stock prices in the composite from Dec. 1996 to Dec. 2006 from Datastream.

39

(i) Three Stock Index Scenario The figure on the left can be interpreted as the

payoff histogram of a correlation swap on ρproxy4 hedged with dispersion trade over 1500

simulation runs. The P&L is very close to zero on average with a low standard deviation

(0.01 correlation points). The figure on the right represents the payoff histogram of a

correlation swap on ρavgPair hedged with dispersion trade over the same 1500 simulation

runs. The dispersion trade strategy seems to hedge correlation swap on ρavgPair with

reasonably good results. The mean P&L of this hedged portfolio is -0.03 correlation

points and the standard deviation is 0.05 correlation points.

Figure 4: Three-stock index scenario

4Technically, a correlation swap on ρproxy has the payoff ρproxy−ρproxy,0 , where ρproxy,0 is the forwardprice of ρproxy at contract inception. Here, we calculate the payoff as ρproxy − ρproxy,0 . This minortechnical difference, however, interferes very little, if any, with our objective of testing the effectivenessof dispersion trade hedging strategy applied to correlation swap on ρavgPair .

40

(ii) Ten Stock Index Scenario The histograms below are plotted with the identical

mechanism as the last one scenario, except the fact that the number of stocks now increases

to ten. Interestingly, the figure on the right shows that the hedge for correlation swap

on ρavgPair has improved considerably. The mean and the standard deviation have been

reduced 25 times and 5 times respectively. This result reflects that our dispersion trade

hedging strategy works significantly better with more stocks in the index.

Figure 5: Ten-stock index scenario

41

II. Dynamic Hedge In dynamic hedge, rebalancing of the hedging portfolio are imple-

mented with updated dispersion trade weights during the contract period of the correlation

swap. We will rebalance the hedged portfolios four times throughout the contract period

with equally spaced time interval in between. The table below compares hedging results

for static hedge and dynamic hedge. Being consistent with our theoretical discussion,

dynamic hedging improves the hedge for ρproxy considerably. As for ρavgPair , dynamic

hedge only works slightly better that static hedge.

Table 4:

Mean of Correlation Swap Portfolio ρproxy ρavgPair

Static hedge (1 Step) -0.0012 -0.0297

Dynamic hedge (4 steps) -0.0006 -0.0291

Std Dev of Correlation Swap Portfolio ρproxy ρavgPair

No hedge 4.03% 4.66%

Static hedge (1 Step) 0.31% 1.21%

Dynamic hedge (4 steps) 0.15% 1.19%

42

7.1.2 Calibration

In this sub-section, we calibrate the WASC model to the price data from Dow Jones 65

Composite Average 5 and relax some key assumptions in our toy model setting. The main

differences between the Toy Model Setting and the Calibration are listed below:

1. The index is now a dynamically adjusted, market-cap weighted arithmetic average

of its constituents instead of a constant weighted geometric average.

2. Since most parameters are now calibrated, instead of being chosen arbitrarily, all

correlation parameters in the∑

∞ and∑

0 matrices now become unique and reflect

historical tendency.

3. Like the∑

∞ and∑

0 matrices, the volatility of volatility matrix in the WASC

model, or the QQ matrix, is also calibrated to reflect past history.

We take the closing prices from December 1996 as our model’s initial stock prices. We

also take the number of shares data from that date and hold them constant throughout

our simulation. Thus, in our simulation, stocks’ weights will adjust through prices only.

To reduce computation, we only randomly select ten stocks to form an index. The rest of

the parameters can be found in Appendix.

In order to come up with test-case scenarios, we also compute three covariance matrices

from the Dow-Jones data. Their characteristics are tabled below:

5A more detailed description of our calibration process can be found in the Appendix A.

43

Table 5:

Covariance Matrix Name Description∑small The covariance matrix with the lowest combined

sum of individual variance-covariance compo-

nents during a 252-day period. Intuitively, this

matrix represents the quietest period with high

diversification benefit.∑overall The overall covariance matrix during the entire

10-year period∑big The covariance matrix with the highest com-

bined sum of individual variance-covariance

components during a 252-day period. Intu-

itively, this matrix represents the most volatile

period with low diversification benefit.

For the following analysis, we made the same assumption by taking the notional

amount of correlation swap to be 1 and all profit and loss (P&L) are in terms of cor-

relation points.

44

I. Static Hedge In this strategy, we keep the buying weights in the dispersion trade

constant over the period of the contract. We explore the strategy where the WASC

process starts with the most and least covariance matrices as discussed above. Below are

the results and discussion for those two scenarios, each with 3000 simulation over 800

time-steps for each run.

Scenario One: Low Vol Converges to Normal In this scenario, we set the initial

instantaneous covariance parameters (the∑

0 matrix) to be∑

small. The∑

0 matrix will

evolve towards∑

inf , which is set to be∑

overall, and will fluctuate around there.

In the figures below, we plot the payoff histogram of a hedged portfolio consisting of

dispersion trade and correlation swap on ρproxy (left figure) or ρavgPair (right figure). The

histogram on the left uses the proxy measure of correlation. With our hedging strategy, the

empirical result shows that the difference between the P&Ls is close to zero on average.

The standard deviation is only 0.5 correlation point. The histogram on the right uses

the actual pairwise measure of correlation for the same hedging strategy. The strategy

seems to approximately hedge the swap with reasonably good results. The mean of the

difference between the P&Ls is 1.8 correlation points and the standard deviation is 4

correlation points. Hence, this shows that our strategy could be successfully applied to

the real data with market capital weighted Index to hedge the correlation swaps very well,

demonstrating that dispersion trade can still perform well when the index is constructed

as an arithmetically market-cap weighted index composite.

45

Figure 6: Low Vol Returns to Normal

46

Scenario Two: High Vol Converges to Normal In this scenario, we set the initial

instantaneous covariance parameter (the∑

0 matrix) as∑

big. The∑

0 matrix will evolve

towards∑

∞, which is set to be∑

overall, and will fluctuate around there.

The histogram on the left uses the proxy measure of correlation. With our hedging

strategy, the empirical result shows that the difference between the P&Ls is close to zero

on average. The standard deviation is only 0.4 correlation points. The histogram on

the right uses the actual pairwise measure of correlation for the same hedging strategy.

The strategy seems to approximately hedge the swap with reasonably good results. The

mean of the difference between the P&Ls is close to zero and the standard deviation is 4

correlation points.

These results are consistent with the above conclusion in Scenario 1. The figures show

that withdrawing those abovementioned assumptions do not dramatically alter our results

derived from the toy model setting.

47

Figure 7: High Vol Returns to Normal

48

II. Dynamic Hedge Similar to in the toy-model setting, rebalancing is performed four

times during the contract period of the correlation swap with equally spaced time interval

in between. For each scenario, 1200 simulation runs along with 800 time-steps for each

run will be generated.

Scenario One: Low Vol Converges to Normal The table below compares the

results between static and dynamic hedging when the initial covariance matrix is∑

small

and∑

inf is∑

overall. In this scenario, dynamic hedging only improves hedging performance

slightly.

Table 6: Low Vol Converges to Normal

Mean of Correlation Swap Portfolio ρproxy ρavgPair

Static hedge (1 Step) 0.000812 0.0283

Dynamic hedge (4 steps) 0.000677 0.0281

Std Dev of Correlation Swap Portfolio ρproxy ρavgPair

No hedge 4.56% 4.74%

Static hedge (1 Step) 0.46% 4.27%

Dynamic hedge (4 steps) 0.42% 4.13%

49

Scenario Two: High Vol Converges to Normal The table below compares the

results between static and dynamic hedging when the initial covariance matrix is∑

big

and∑

∞ is∑

overall. In this scenario, dynamic hedging marginally improves hedging

performance.

Table 7: High Vol Converges to Normal

Mean of Correlation Swap Portfolio ρproxy ρavgPair

Static hedge (1 Step) -0.0053 0.016

Dynamic hedge (4 steps) 0.0071 0.0175

Std Dev of Correlation Swap Portfolio ρproxy ρavgPair

No hedge 6.41% 5.13%

Static hedge (1 Step) 0.8% 3.37%

Dynamic hedge (4 steps) 0.71% 3.2%

50

7.2 Stress Test: Misestimated Future Realized Variances

This section tests the effectiveness of Dispersion Trade hedge when the estimation of

future realized variances turns out to be a bad estimate (i.e., when σ2avg does not closely

match σ2avg ). We will first return to the original toy model setup: 1) index is a geometric

weighted average of its constituents, 2) constituents’ weights are equal and held constant,

and 3) the correlation parameters in the∑

∞ and∑

0 matrices of the WASC model are

identical for each pair of stocks. Similar to section 7.1, we will then remove these setups

later to test the dispersion trade hedging strategy’s robustness.

7.2.1 Toy Model Setting

Now, we briefly illustrate two scenarios where the initial σavg , assumed to be 30%, fails

to be a good approximation of σavg , which turns out to be mostly around 38%. From

equation (24), an underestimation of σ2avg should result an over-hedge 6. For each scenario,

1000 simulation runs along with 800 time-steps for each run will be generated. The rest

of the parameters are documented in the Appendix.

6By over-hedge, we mean that if ρproxy − ρproxy,0 has a positive (negative) payoff, then the hedgedportfolio will have a negative (positive) payoff.

51

Scenario One: Correlation Drops Low We assume that initially the correlation

swap is mispriced at 0.5 correlation points. The actual realized average pairwise corre-

lation is close to 0. The top diagram depicts the payoff of unhedged correlation swap

on ρavgPair. The middle diagram depicts the payoff of correlation swap hedged with dis-

persion trade. The bottom diagram depicts the payoff of ρproxy7hedged with dispersion

trade.

7Again, technically speaking, the payoff is computed as ρproxy − ρproxy,0 .

52

Figure 8: Correlation Drops Low

Here, as expected, the unhedged correlation swap looses about 0.5 correlation points

(top subplot), but the dispersion-trade-hedged correlation swap position has much less

payoff magnitude in absolute terms (middle subplot). Moreover, dispersion trade does

indeed over-hedge ρproxy is fairly close to ρproxy in this scenario and correlation swap is

also over-hedged by dispersion trade.

53

Scenario Two: Correlation Shoots High We assume that initially the correlation

swap is mis-priced at 0 correlation point and the actual realized average pairwise correla-

tion is close to 0.5.

Figure 9: Correlation Shoots High

54

Here, as expected, the un-hedged correlation swap gains about 0.5 correlation points

(top subplot), but the Dispersion Trade hedged correlation swap position has much less

payoff magnitude in absolute terms (middle subplot). Moreover, dispersion trade over-

hedges ρproxy , as shown in the bottom subplot. Unlike scenario one, nevertheless, since

the payoff of ρproxy is lower than that of ρavgPair , the over-hedging of ρproxy now becomes

a fairly accurate hedge for the portfolio including dispersion trade and correlation swap.

55

7.2.2 Calibration

The set-up of this setting is identical to the Calibration from section 7.1.2. The reader

can refer back for a more detailed specification. For each scenario, 1000 simulation runs

along with 800 time-steps for each run will be generated.

Scenario One: Market Becomes Quiet In this scenario, we set the initial instanta-

neous covariance parameter (the∑

0 matrix) as∑

overall .∑

0 will evolve towards∑

∞,

which is set to be∑

small , and will fluctuate around there. σavg takes the value of 30.76%.

While the value of σavg varies among simulation runs, it has an average value of 23.01%8.

Equation (25) predicts that ρproxy will be mostly under-hedged 9. Figure 10 depicts port-

folios on correlation swap and figure 11 depicts portfolios on (ρproxy − ρproxy,0) . For

both figures, the top row depicts naked swap positions, and the bottom three rows depict

hedged positions.

8Technically, it is the square root of the mean value of σ2avg realized over 1,000 simulation runs.

9By under-hedge, we mean that if ρproxy − ρproxy,0 has a positive (negative) payoff, then the hedgedportfolio will have a positive (negative) payoff, but in a smaller magnitude.

56

Figure 10: Market Becomes Quiet (Correlation Swap)

57

Figure 11: Market Becomes Quiet (Portfolios of (ρproxy − ρproxy,0))

58

Figure 11 shows that ρproxy is indeed mostly under-hedged, although the magnitude is

very small. Because ρproxy exhibits deviation from ρavgPair ((first row of both figures)), the

reader can find in figure 10 that dispersion trade fails to be a good hedge for correlation

swap in this setting. Furthermore, as hedging frequency increases, there is an improvement

of hedging performance from the (ρproxy − ρproxy,0) + dispersion trade portfolio (figure

11), but a deterioration of hedging performance from the portfolio of dispersion trade and

correlation swap (figure 10). The reason is due to the fact that though dispersion trade

hedges ρproxy more efficiently, it will not hedge ρavgPair more efficiently unless ρproxy is

sufficiently close to ρpair .

59

Scenario Two: Market Filled with Fear and Greed Under this scenario, we set

the initial∑

0 matrix as∑

overall . The covariance matrix will evolve towards∑

big and

fluctuate around there. σavg remains 30.76%; σavg has a “mean” value of 34.47% with the

same interpretation as scenario one.

Figure 12: Market Filled with Fear and Greed (Correlation Swap)

60

Figure 13: Market Filled with Fear and Greed (Portfolios of (ρproxy − ρproxy,0))

Figure 13 confirms our expectation that dispersion trade would over-hedge ρproxy (due

to σavg underestimation). Unlike scenario one, ρproxy is now a good estimation of ρavgPair,

which causes dispersion trade to produce decent hedging performance over correlation

swap (figure 12).

61

8 Conclusions and future research directions

We have tested the Dispersion Trade hedging strategy under various scenarios. Our

results show that with the proposed weights in the Dispersion Trade, we could hedge the

proxy correlation swap very well and the pairwise correlation swap reasonably well. Our

results also suggest that as number of stock increases, the hedge is better. The Dispersion

Trade’s hedging performance is highly dependent on our estimate of future realized average

variance (σ2avg). The two most important factors that influence Dispersion Trade’s hedging

performance are 1) the spread between ρproxy and ρavgPair and 2) the accuracy of σ2avg.

Correctly Estimated Fu-ture Realized Variances

Stress Test: MisestimatedFuture Realized Variances

Toy Model Setting Dispersion Trade is a good

hedge for correlation swap.

Both correct estimate of σ2avg

and difference between ρproxy

and ρavgPair are important fac-

tors of hedging performance.

Calibration Dispersion Trade is still a good

approximate hedge for correla-

tion swap.

Both correct estimate of σ2avg

and difference between ρproxy

and ρavgPair are important fac-

tors of hedging performance.

As shown from our Historical Backtesting results with the dataset of Dow Jones 65

Composite Average Index, the spread between ρproxy and ρavgPair diminishes when the

number of stocks in the index increases. According to Table 3, the spread between ρproxy

and ρavgPair is about 0.14 correlation points for 10 stocks. The spread drops below 0.05

correlation points when the number of stocks in the index is greater than 30. This

shows that in order for ρproxy to closely approximate ρavgPair, the number of stocks in the

index only needs to be reasonably large. Obviously, further research has to be done to

determine the optimal number of stocks for different equity indices for this approximation

relationship to hold. This is particularly important for financial institutions from the

62

point of view of risk management and structuring product sales. By doing so, they can

understand which equity indices they can hedge the correlation risks among constituents

more effectively through dispersion trade and decide which indices products they would

like to offer to their clients.

63

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66