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University of Amsterdam

Faculty of Economics and Business Master of Science in Econometrics Thesis

Dispersion Trading

A signal-based trading approach

Author: Daan Olivier Rotsteege

Student number: 10259384

Date: August 14, 2015

Specialisation: Financial Econometrics

Supervisor: Prof. dr. C. G. H. Diks

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Abstract

The aim of this thesis is to investigate the characteristics and trading opportunities of the implied volatility spread between CAC 40 index options and its corresponding portfolio of single stock options. Dispersion trading is a trading strategy based on monetising this implied volatility dispersion, by creating a hedging portfolio with options or third generation volatility

products. The focus in this thesis is on signal trading strategies which make use of combinations of options and weighting schemes created by principal component analysis (PCA) and differential evolution and combinatorial search (DECS), where the latter weighting

scheme is optimised with market impact constraints. Herein, a specific dispersion trade is entered into based on market signals about a collection of volatility smiles and (forecasted)

implied correlations. It is found that the profitability of a highly active naive dispersion trading strategy is very sensitive to extreme market events; signal trading can reduce this

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Acknowledgement

My gratitude and appreciation goes out to my internal supervisor and co-director of the Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Prof. dr. C.G.H. Diks, for his invaluable advice, selflessness and pleasant conversations. Furthermore, I would like to thank Prof. dr. H.P. Boswijk, for his questions and comments in the final period of my study.

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Contents

Abstract . . . i Acknowledgement . . . ii List of Figures . . . v List of Tables . . . vi 1 Introduction 1 2 Theory 4 2.1 Dispersion trading . . . 4 2.1.1 The concept . . . 4 2.1.2 Optimal dispersion . . . 6 2.1.3 Market neutrality . . . 7 2.1.4 Tracking P&L . . . 7

2.2 Options as hedging strategy . . . 8

2.2.1 Why options? . . . 8

2.2.2 Price and value . . . 9

2.2.3 Combinations . . . 13

2.2.4 The volatility surface . . . 15

2.3 Swaps as hedging strategy . . . 17

2.3.1 Why swaps? . . . 17

2.3.2 Price and value . . . 17

2.3.3 Volatility dispersion trading and correlation trading . . . 19

2.4 Volatility and correlation . . . 20

2.4.1 Portfolio variance . . . 20 2.4.2 Implied correlation . . . 21 2.5 Tracking portfolio . . . 22 3 Methodology 24 3.1 Overview . . . 24 3.2 Tracking portfolio . . . 25 3.2.1 PCA analysis . . . 25

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3.3 Implied volatility . . . 28 3.3.1 Index options . . . 28 3.3.2 Stock options . . . 29 3.4 Strategies . . . 30 3.4.1 Naive strategy . . . 30 3.4.2 Position forecasting . . . 32 3.4.3 Combination forecasting . . . 34 3.4.4 Remarks . . . 35 3.5 Tracking P&L . . . 36 3.5.1 Evaluation . . . 36 4 Data 39 5 Evaluation 41 5.1 Preliminary results and analysis . . . 41

5.2 Naive dispersion trading . . . 46

5.3 Position signal dispersion trading . . . 50

5.4 Combination signal dispersion trading . . . 54

5.5 Mixing signals . . . 55

5.6 Robustness checks . . . 57

5.6.1 Do the strategy returns have finite variance? . . . 57

5.6.2 Is dispersion trading profitable under a transaction costs scenario? . . . . 57

5.6.3 Remarks . . . 58

6 Conclusion 60

Appendix A 65

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List of Figures

2.1 Example of S&P 500 and random generated tracking portfolio implied volatility. 6

2.2 Evolution of the Greeks for an ATM straddle and an OTM strangle. . . 16

2.3 Evolution of the Greeks for the Variance Swap. . . 18

4.1 The market conditions of the CAC 40 index during the trading period. . . 40

5.1 PCA method tracking portfolio characteristics. . . 42

5.2 DECS method tracking portfolio characteristics. . . 43

5.3 The historical volatility surface of the CAC 40 index over the period 01-01-2010 to 31-05-2010, using both put and call options. . . 45

5.4 The cumulative returns of the delta-hedged naive trading strategies against the CAC 40 index. . . 50

5.5 Straddle implied correlation with DECS. . . 53

A.1 Implied correlation DECS tracking portfolio (strangle). . . 65

A.2 Implied correlations PCA tracking portfolio. . . 66

A.3 Implied volatilities for the straddle combination. . . 67

A.4 Cumulative return DECS signal trading (delta-hedged). . . 68

A.5 Cumulative return PCA signal trading (delta-hedged). . . 69

A.6 Daily returns of the delta-hedged naive and combination strategies. . . 70

A.7 Daily returns of the delta-hedged EGARCH strategies. . . 71

A.8 Daily returns of the delta-hedged Bollinger Bands strategies. . . 72

A.9 Daily historical returns of the CAC 40 index and some constituents. . . 73

A.10 Implied volatility CAC 40 for different strike prices over the period 01-01-2010 to 31-05-2010. . . 74

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List of Tables

5.1 Garch-type modelling. . . 44

5.2 Non delta-hedged strategies. . . 47

5.3 Delta-hedged strategies. . . 48

5.4 Straddle implied volatility spread. . . 52

5.5 Delta-hedged strategies under transaction costs. . . 59

A.1 Augmented Dickey-Fuller test return series. . . 75

A.2 Characteristics of the implied volatilities. . . 76

A.3 Characteristics of the implied volatility spread. . . 77

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Chapter 1

Introduction

In the period after the Global Financial crisis, the correlations between stocks increased to new historical records (Kolanovic, 2010). As a result, the trading strategy called Dispersion Trading gained renewed interest by sophisticated hedge funds and proprietary trading desks (Marshall, 2009), but remained limited in the academic world. Prior to this period, the strategy has been discussed in some business papers and reports, especially because of implied correlation spikes, that occurred on account of global events (e.g. London terrorist bombings and the 9/11 terrorist attacks), in which some hedge funds unwound a short position on the high correlation observed in these indecisive financial markets. At the same time, the studies of Bakshi et al. (2003), Bakshi and Kapadia (2003) and Bollen and Whaley (2004) contributed to empirical evidence that generally index options are traded against a premium compared to their theoretical Black-Scholes prices, while individual stock options do not appear to be overpriced.

Dispersion trading is a trading strategy which aims to profit from ostensible risk premiums in implied volatilities and is closely related to correlation trading. Because the value of an index is equal to a weighted average of the underlying stocks prices, by the Law of One Price the implied volatility derived from index options should also be equal to the implied volatility derived from options on the corresponding portfolio of stocks. Thus the mispricing suggests that index volatility is more rich and the volatility of the constituents is cheaper. Several papers have investigated this implication (e.g., Deng, 2008; Bakshi and Kapadia, 2003) and as a result two main hypotheses are made. The first argument is a risk-based hypothesis, which states that index options are more expensive because the market volatility risk premium is smaller for stock options compared to index options and that index options hedge a certain correlation risk (Driessen et al., 2005). This is confirmed by Bakshi et al. (2003), who address the differential pricing of index and single stock options to the different skewness of the risk-neutral distribution of the underlying asset.

On the other side of the academic literature one assigns the expensive index options to market inefficiencies. Bollen and Whaley (2004) state that the net buying pressure drives the index option prices out of parity. Herein it is suggested that as a market maker builds up a larger position in a given option, the volatility risk exposure of his portfolio, i.e. vega, also

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increases. As a result, hedging costs will increase and the market maker is forced to demand a higher price for the option, which leads to an increase in implied volatility. This hypothesis is

complemented by Gˆarleanu et al. (2006) and Lakonishok et al. (2007), who both argue that the

demand pattern of stock options is different from index options.

Some major changes in the US options market around the beginning of 2000, such as the launch of the International Securities Exchange (ISE) and an overall market reduction in the bid-ask spreads made way for a natural experiment to investigate both hypotheses. By the launch of the ISE as the first electronic options exchange in the US, costs for taking advantage of any differential pricing of indices and associating stocks reduced. Therefore following the demand and supply-based argument, it would be expected that the market became more efficient after this change and hence the profitability of dispersion trading reduced. Deng (2008) shows that dispersion trading was profitable in the five year period prior these structural changes but that in the same timespan after the 2000 break point average monthly returns decreased significantly, from 24% to -0.03%.

Marshall (2009) evaluates the efficiency of US options in pricing volatility in the period of 2005-2007. Using a modification of the Markowitz variance equation to estimate the volatility of the portfolio of stocks underlying the index, she was able to show the existence of a volatility

premium implicit in index options on the S&P 500 index. Even when a transaction costs

scenario was taken into account, there were a significant number of days with potential volatility dispersion trading opportunities. The results of Marshall are of great importance because it proves the existence of volatility dispersion in the US option market in this specific period, however the results do not imply a trading strategy and can merely be used as a signal of potential arbitrage.

Identification of dispersion trading opportunities can be done in various ways, but the most elegant way is by looking at the implied correlation of the index, which is an average correlation measure derived from the implied volatilities of index options and individual options. Another measure is the volatility dispersion statistic. Although the identification methods give approx-imately the same signal, a more vital choice of the strategy is how the volatility discrepancy is monetised. Typically, a dispersion trade can be entered by taking positions in plain vanilla options or variance/volatility swaps. Hereby, one takes a short position in the overpriced el-ement and a long position on the cheaper elel-ement of the strategy. The advantage of using swaps is that delta-hedging is not labour intensive and that they give direct exposure to the variance/volatility of the underlying, however since the financial crisis of 2008 the liquidity of these swaps on individual stocks has decreased (Martin, 2013). Using plain vanilla options for dispersion trading usually involves taking positions in straddles and strangles, this because at-the-money straddles or out-of-the-money strangles have a delta exposure close to zero and the strategy is for this reason hedged against large market fluctuations (Deng, 2008, p. 2).

Because hedge funds prefer to conceal a profit-making strategy, it is unknown to what extent dispersion trading strategies are used and whether it is possible to make a realistic excess profit

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based on dispersion trading. Furthermore, it is the author’s personal belief that there are merely academic studies that take a pragmatic approach in the research to a realistic and frequently trading dispersion strategy. Probably the most realistic strategies are developed by Deng (2008) and Magnusson (2013). Although both find some significant trading opportunities and extended the general knowledge on this topic, the strategies are still elementary and open for evolution. For this reason this thesis aims to construct and evaluate a close to real life signal dispersion trading strategy on the CAC 40 index, where options are used to take advantage of the relative differences in implied volatilities of the index and the constituents.

There are many crucial elements in developing a successful and feasible quantitative trading strategy like dispersion trading, e.g. the choice of weighting schemes, positions and position limits, hedging the Greeks and the market impact of a trade. However, due to the scope of this thesis not all factors determining the profit and loss (P&L) of the strategy can be dealt with. The crucial question of this research is whether there are dispersion trading opportunities on the CAC 40 index with a naive dispersion trading strategy. On the way to close this question, it will be examined whether a weighting scheme based on a tracking portfolio constructed by evolutionary heuristics performs different than a tracking portfolio based on a linear dimension reduction method. The two naive trading strategies, one for each optimisation method, will then be adjusted to become more dynamic and realistic by allowing for entry signals, position signals, daily delta-hedging and an approximation for transaction costs.

The remainder of this thesis is organised as follows. Chapter 2 provides all the indispensable knowledge to conduct the research and lays the foundation for the subsequent chapters. Sub-sequently in Chapter 3 the methodology is presented. Chapter 4 describes the data used for the empirical analysis. Chapter 5 presents the preliminary results and analyses of the training period, whereupon the results of the trading period are portrayed and evaluated. Chapter 6 concludes.

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Chapter 2

Theory

A fortunate dispersion trade is established from a proper interplay between the volatilities of the assets underlying the option contracts. Therefore, in order to build a profitable dispersion strategy, a thorough knowledge of the possible financial derivatives used in a hedging portfolio must be created, together with their interactions. Consequently the goal of this chapter is to lay a strong theoretical foundation for the remainder of this thesis.

2.1

Dispersion trading

Before the more technical side of this chapter is touched, a short description of dispersion trading is given in this section.

2.1.1 The concept

Next to the return, probably the best-known and used concept in the financial world is the volatility of an asset, i.e. the standard deviation of the return series of an asset, hence a measure

for the variability of the price. This is partly due the fact that over the last decades the

demand for options has been booming and because of the emergence of more complex investment products, including structured products. On these financial derivatives volatility means the conditional standard deviation of the underlying asset’s return and some of these products’ payoffs are solely based on this volatility measure. Dispersion trading is a trading strategy that aims to profit from the discrepancy in implied volatilities between different products and hence notwithstanding its elegant name, it is a reasonably simple concept.

Dispersion trades can be set up using (combinations of) options and third generation volatil-ity products (e.g. volatilvolatil-ity/variance swaps) and in general there are two reasons to enter a dispersion trade. Naturally the first reason is because of statistical arbitrage opportunities, the second reason is to hedge correlation products. As will be explained in Subsection 2.1.2 a long position in a dispersion trade, i.e. a long position in the volatility of the components of an index and a short position on the volatility of the index itself, can practically be seen as a short position in correlation. However strictly speaking, as shown by Jacquier and Slaoui (2007),

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implied correlation from a dispersion trade with variance swaps tends to exceed the strike of a correlation swap. This result is important because financial institutions over the years have

sold structured products such as mountain range options1 and consequently by selling these

products a financial institution exposes itself with a short position in correlation. Therefore by taking a short position in a dispersion trade it neutralises the exposure in short correlation (FDAXhunter, 2004).

The concept is clarified with an example. Lets assume that an investor lives in a simple world with no transaction costs, which comprises of four stocks A, B, C and D, together with a weighted average index of these stocks X. He observes that the implied volatility of the index has a premium compared to the implied volatility of a same weighted portfolio of stocks A, B, C and D (derived from option prices). The goal of the investor is to create a hedged position which takes advantage of the relative value differences in the implied volatilities of the options, hence he decides to take a long volatility dispersion position, i.e.:

• A long position on the volatility of stocks A, B, C and D • A short position on the volatility of the index X

The investor thus initiates a short position in index options and a long position in options on the stocks A,B,C and D. Profits are realised in the following events (Marshall, 2008):

1. Implied volatilities return into equilibrium

2. The options expire and more is earned on the long position than the costs on the short position

As a matter of course, the first case is fairly straightforward because the investor observes that there is a disequilibrium in implied volatilities and buys the relatively cheap options on the constituents and sells the relatively expensive options on the index. When the implied volatilities converges back into equilibrium the investor makes a profit on (1) the long leg, (2) the short leg or in the most advantageous situation (3) a combination thereof.

If the disequilibrium between implied volatilities does not soften, the investor makes only a profit at the time of maturity when the long leg is worth more than the negative of the short leg (from the dispersion investor’s point of view). This situation is most likely to happen when during the period where the investor has a long dispersion position active in the market, there is minimal volatility on the index X and maximal volatility on the components of the index (stocks A,B, C and D). The next subsection 2.1.2 deals further with this issue.

1Options where the payoffs depend on the performance of a basket of underlying securities, e.g. Everest-,

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2.1.2 Optimal dispersion

In the previous example it was mentioned that in the case where the spread between the two implied volatilities does not mean-revert to its long-term mean zero, the most likely way that a long dispersion position would end up in the money is with minimal volatility on the index and maximal volatility on its constituents. This is only possible whenever the stocks comprising the index appear to be uncorrelated, meaning that the move of one stock is canceled out by the move of another stock with the result that the index stays close to put. The investor may wish to delta-hedge his volatility position on the individual stocks, however as the index hardly moves, no delta-hedging is required on the short position. Altogether, the investor makes a theta related profit on the index and a gamma profit on the individual stocks.

Although one can earn a lot of profit on a dispersion position with optimal dispersion on its constituents, it can be hazardous. When the stocks have perfect correlation more money is lost on the short position than earned in theta and this is exactly the reason why dispersion trading is closely related to the concept to correlation trading. A long position in a dispersion trade can be seen as a bet on low correlation, i.e. a short position in correlation, and vice versa for a short dispersion position.

Jan10 Apr10 Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12 Jul12 Oct12 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Date Implied Vol.

S&P 500 Model−Free (VIX) Simulated Constituents

Long Dispersion Trade Long Dispersion Trade

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2.1.3 Market neutrality

A market neutral strategy is a popular strategy taken by hedge funds and proprietary trading desks. Herein, a trader does not bet on broad market movements but rather the goal is to profit from a relative mispricing which exhibits in the market. This is done by taking a long position in the relatively cheap security and a short position in the relatively expensive security and therefore the strategy is hedged against specific market movements, hence the theoretical beta of such a strategy is equal to zero.

Dispersion trading in a world with no additional trading costs (e.g. transaction costs and market impact), where a trade is possible on all the constituents of the index, is a market neutral strategy. In the end, the market index can be seen as a weighted average of single stocks and thus both the long and the short leg are characterised with the same risk. However, because volatility pricing discrepancies in the market are small and consequently payoffs due to volatility dispersion are marginal, profits fade away after adjusting for trading costs. If one tackles this problem by taking a position on a portfolio which mimics the index instead of a weighted average of all constituents, a correlation risk between the tracking portfolio and the market index arises. Hence dispersion trading in its authentic form and in a perfect world is a market neutral strategy, in reality it is statistical arbitrage.

The weights of the volatility positions on the single stocks can be determined based on the preferences of the trader with respect to the portfolio’s market risk exposure, i.e. the Greeks, and the financial products used. One method is already described: using the weights of a tracking portfolio which mimics the characteristics of the index. Alternative weighting strategies can for example be based on vega-neutrality, gamma-neutrality or theta-neutrality. In the case of vega/gamma neutral weights, the vega/gamma of the index equals the sum of the single stock vegas/gammas. If one aims for theta-neutrality, a short position in vega and gamma is entered into.

2.1.4 Tracking P&L

When initiating a dispersion trade one needs to decide whether the aspire is to enter into a self-financing portfolio, which means that the market value of the short leg offsets the value of the long leg and accordingly the market value of the portfolio is equal to zero at inception. An imaginable way to open a self-financing portfolio is to first enter the short position and then directly go into the long position with the proceeds of the short leg. The advantage of a self-financing portfolio is that no initial investment needs to be made. Nevertheless more leverage is created because the value of the long leg is adjusted to the short leg at the starting point.

Another issue of the P&L of a strategy is the way of calculating the total simple return when a short position is present in the portfolio. A short sell can be translated as selling a financial product which is not owned by the seller, but borrowed from someone else in exchange for a borrow fee and an obligatory repayment of the financial asset at some future time. Thus a short seller has a financial liability in the future while receiving money at the start, meaning that the

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return on a short position can be calculated as the negative return of a long position. When a self-financing portfolio is initiated then the total simple return of the portfolio can be calculated as the ratio between the total value of the portfolio at maturity divided by the proceeds of the short position at the start. In all other cases the simple returns need to be reweighted to the size of the positions.

2.2

Options as hedging strategy

In this section the various aspects of options are explained and the way how they can be used in a dispersion hedging strategy.

2.2.1 Why options?

An option gives the holder the right but not the obligation to buy or sell an underlying asset at a specified strike price on or before a specified date, called the maturity date (Etheridge, 2008). At first sight, this definition suggests that options are some kind of extension of forwards/futures contracts, however, whereas it costs nothing to enter into a forward/futures contract, an option has a price because it gives the right and not the obligation to buy or sell the underlying asset. In the global financial world there are many different types of options and depending on its terms, some are sold on OTC markets and others on exchanges. Options in their simplest form are plain vanilla options and the more complex options are called exotic options. The most commonly traded options are European and American options; both are categorised as plain vanilla options and depending on its terms and conditions they are sold on OTC markets and on exchanges. The difference between American and European options is that American options may be exercised before the maturity date, whereas European options can only be exercised at maturity, i.e. when the contract expires. In general it applies that options on indices are of the European variant and single stock options are American, however some exchanges also provide European options for stocks.

The advantage of using plain vanilla options in a volatility dispersion strategy is that ex-changes (usually) offer a lot of different standardised options, i.e. standardised strike prices and maturities, on the same underlying asset. Because an exchange continuously publishes publicly option prices, it enables itself to attract many independent buyers to carry out a trade, which intensifies the volume and lowers the margins. Hence these options have fairly liquid markets (Etheridge, 2008). Another advantage is that a trader can combine options with different strike prices and maturities in a portfolio to hedge certain market factor exposure or to minimise undesirable future outcomes.

Although using options has its advantage in a dispersion trading strategy wherein a trader takes a portfolio in index options and in single stock options, this strategy has also its downsides. In particular this strategy is path-dependent and the volatility risk exposure of this portfolio can become unhedged as the market environment changes, moreover delta-hedging is required

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continuously. Especially variance/volatility swaps are a solution for the path-dependent issues of this strategy, see Subsection 2.3.1.

2.2.2 Price and value

In this subsection the price of a European option is given, the Black-Scholes pricing formula. The price of an American option is not considered here because an explicit formula only exist in a few special cases, which means that in general this option must be priced with numerical methods, such as with the Binomial Option Pricing model or with Monte Carlo estimation (Etheridge, 2008).

The price of a European option with constant volatility

Consider a market with a riskless cash bond, {Bt}t≥0 and a risky stock with stochastic process

{St}t≥0. It is assumed that the riskless borrowing rate is constant and that

dBt= rBtdt with B0= 1, (2.1)

dSt= µStdt + σStdWt, (2.2)

where {Wt}t≥0is (P, {Ft}t≥0) Brownian motion. Thus it is assumed that {St}t≥0is a geometric

Brownian motion with constant drift.

Now one may define the discounted price process { ˜St}t≥0, where ˜St= Bt−1St, from here it

can be derived that

d ˜St= (µ − r) ˜Stdt + σ ˜StdWt.

The process is defined as

Xt= Wt+ σ−1(µ − r)t,

and hence

dXt= dWt+ σ−1(µ − r)dt,

d ˜St= σ ˜StdXt.

By Girsanov’s theorem, under the risk-neutral measure Q, Xt follows a standard Brownian

motion and therefore ˜St a martingale. Now, by expressing the value of a European call or put

option as Vt = F (t, St) and ˜Vt = Bt−1Vt = e−rtVt and defining ˜F such that ˜V = ˜F (t, ˜St),

applying Itˆo’s formula to ˜V and using the zero drift condition for a martingale under Q,

∂ ˜F ∂t(t, x) = − 1 2 ∂2F˜ ∂2x(t, x)σ 2x2.

The following equation is obtained, which is the Black-Scholes PDE:

−rF (t, x) + ∂F ∂t(t, x) + rx ∂F ∂x(t, x) + 1 2 ∂2F ∂x2(t, x)x 2σ2 = 0. (2.3)

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The Black-Scholes PDE has an explicit solution for European options, the Black-Scholes pricing

formula, and at time t ∈ (0, T ), the value of this option, Vt, whose payoff at maturity is

VT = f (ST) with strike price K and θ = (T − t) is given by

Vt= e−rθ Z ∞ −∞ f  Stexp  r −1 2σ 2  θ + σz √ θ  ·√1 2πexp  −z2 2  dz. (2.4)

Now if we denote the price of a European call option as C(t, St; K) and a European put option

as P (t, St; K) at time t ∈ (0, T ) on a non-dividend paying stock with price St, using the same

notation it can be shown that

C(t, St) = StΦ(d1) − Ke−rθΦ(d2), (2.5) P (t, St) = Ke−rθΦ(−d2) − StΦ(−d1), (2.6) d1= log St K +  r +σ22θ σ√θ , (2.7) d2= d1− σ √ θ, (2.8) Φ(z) = √1 2π Z z −∞ e−z2/2dz. (2.9)

The price of a European option with time-varying volatility

The same process is assumed as in (2.1) and (2.2), only σ is replaced by σt, where the latter

satisfies thatR0T σt2dt is finite with P-probability one. Again Girsanov’s Theorem is used to find

a risk-neutral measure, Q, under which { ˜Wt}t≥0 is a standard Brownian motion, where

˜ Wt= Wt+ Z t 0 γsds, γt= (µ − r)/σt.

The discounted stock price process { ˜St}t≥0is characterised by the stochastic differential equation

d ˜St= (µ − r − σtγt) ˜Stdt + σtS˜td ˜Wt,

and { ˜St}t≥0 is a Q-martingale when the following boundedness assumptions are satisfied:

EP  exp 1 2 Z T 0 γt2dt  < ∞, EQ  exp 1 2 Z T 0 σt2dt  < ∞.

By defining the (Q, {Ft}t≥0)-martingale {Mt}t≥0, where Mt= EQBT−1CT|Ft, and by showing

that any claim CT can be replicated by φtunits of stocks and ψ = Mt− φtStunits of cash-bonds

at time t, the fair value of the claim is, Vt = EQe−r(T −t)CT|Ft. Because σt only depends on

(t, St), using the Feynman-Kac Stochastic Representation Theorem, the price can be expressed

as a solution to (2.3), with σ2= σ2(t, x). This means that in the Black-Scholes pricing formula

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The P&L of a delta-hedged portfolio

A trader may wish to combine certain put and call option in a portfolio to minimise the risk and the exposure to the Greeks, however before these combinations are treated it is pleasing to investigate the P&L for a single delta-hedged option with time-varying volatility. Assuming

the same process as in (2.1) and (2.2), and replacing σ by σt, a delta hedged portfolio Πt at

time t ∈ (0, T ), implies that one has two opposite positions in a derivative and the associating underlying asset. It is only enthralling to consider one of the two possible cases and therefore

assume that this portfolio Πt consists of a short position in the asset, and a long position in an

option with value Vt. The value of this portfolio changes over period τ ∈ R+ with

Πt+τ − Πt= Vt+τ − Vt− Z t+τ t ∂Cu ∂Su dSu− Z t+τ t r  Cu− ∂Cu ∂Su  Sudu, ∆Π = ∆Vt− δt∆St+ (δtSt− Vt)r∆t, (2.10)

and with δt= ∂C∂Stt. However to obtain a more insightful expression of the P&L over the period

τ , assumed infinitely small, the second-order Taylor expansion of dVt is taken. The next steps

are based on the derivations of Forde (2003) and Jacquier and Slaoui (2007);

dVt= ∂V ∂t(t, St)dt + ∂V ∂St (t, St)dSt+ ∂V ∂σt (t, St)dσt +1 2  ∂2V ∂S2 t (t, St)(dSt)2+ ∂2V ∂σ2 t (t, St)(dσt)2+ 2 ∂2V ∂St∂σt (t, St)dStdσt  ,

when there exists some risk-neutral measure, ˆP, such that the Black-Scholes implied volatility,

ˆ

σt, has a drift.

By rewriting the Black-Scholes PDE and replacing the unknown time-varying volatility for

the implied volatility, one can find an expression for rVtdt. After substituting in Eq. (2.10), we

obtain dΠt= ∂V ∂t(t, St)dt + ∂V ∂St (t, St)dSt+ ∂V ∂σt (t, St)dσt +1 2  ∂2V ∂St2(t, St)(dSt) 2+∂2V ∂σt2(t, St)(dσt) 2+ 2 ∂2V ∂St∂σt (t, St)dStdσt  − δtdSt+ rδtStdt −  ∂V ∂t(t, St) + 1 2 ∂2V ∂St2(t, St)S 2 tσˆt2+ rSt ∂V ∂St (t, St)  dt. This can be rewritten in terms of the Greeks Γ (Gamma), ν (Vega), Vanna and Vomma as

dΠt= 1 2ΓS 2 t "  dSt St 2 − ˆσt2dt # + νdσt+ 1 2V omma · (dσt) 2 + V anna · σtStρζdt, (2.11)

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the asset and the volatility. The Greeks are defined as Γ = ∂δt ∂St = ∂ 2V ∂2S t (t, St), (2.12) ν = ∂V ∂σt (t, St), (2.13) V omma = ∂ 2V ∂σ2t (t, St), (2.14) V anna = ∂δt ∂σt = ∂ 2V ∂St∂σt (t, St). (2.15)

Hence the P&L of a long delta-hedged dispersion strategy is found by summing the individual

stock P&Ls and subtracting the index P&L, i.e. dΠLDt =Pn

i=1dΠi,t− dΠI,t. Also, this

delta-hedged portfolio of single options can readily be extended to the P&L of combinations of options, such as straddles and strangles.

It can be shown that under the Black-Scholes framework, i.e. constant volatility, Eq. (2.11) can be simplified to dΠt= ∂V ∂t(t, St) "  dSt Stσ √ dt 2 − 1 # , (2.16)

where ∂V /∂t is theta and dSt/(Stσ

dt) can be interpreted as a standardised move of the underlying asset’s price over a specific time. If we now consider an index, I, together with its n

constituent stocks, with σi the volatility of the ith stock, wi its corresponding weight in index

I, pi the number of shares of stock i and ρij the correlation between the ith and the jth stock,

i, j ∈ (1, . . . , n), Eq. (2.16) in terms of the index is given by

dΠI,t= ∂V ∂t(t, SI,t)   dSI,t SI,tσI √ dt !2 − 1  . (2.17)

Writing zI,t = dSI,t/(SI,tσI

dt) and zi,t = dSi,t/(Si,tσi

dt) for the standardised move of the index and single stocks, respectively, then we can derive that

zI,t= dSI,t SI,tσI √ dt = Pn i=1pidSi,t σI √ dtPn j=1pjSj,t = 1 σIPnj=1pjSj,t · n X i=1 σipiSi,tdSi,t σiSi,t √ dt = 1 σIPnj=1pjSj,t · n X i=1 σipiSi,tzi,t = n X i=1 wiσi σI zi,t. (2.18)

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terms of its constituents as dΠI,t= ∂V ∂t(t, SI,t)z 2 I,t− 1 = ∂V ∂t(t, SI,t)   n X i=1 wizi,tσi σI !2 − 1   = 1 σ2I ∂V ∂t(t, SI,t)   n X i=1 w2iσi2z2i,t+ n X i=1,j6=i wiσiwjσjzi,tzj,t− σI2   = 1 σ2I ∂V ∂t(t, SI,t)   n X i=1 w2iσi2z2i,t+ n X i=1,j6=i wiσiwjσjzi,tzj,t−   n X i=1 w2iσ2i + n X i=1,j6=i wiσiwjσjρi,j     = 1 σ2I ∂V ∂t(t, SI,t)   n X i=1 w2iσi2 zi,t2 − 1 + n X i=1,j6=i wiσiwjσj(zi,tzj,t− ρi,j)  . (2.19) Therefore, the P&L of a long dispersion trade under the Black-Scholes framework is given by

dΠLDt = n X i=1 dΠi,t− dΠI,t = n X i=1 zi,t2 − 1 ∂V ∂t(t, Si,t) − w 2 iσi2 1 σ2 I ∂V ∂t(t, SI,t)  − 1 σI2 ∂V ∂t(t, SI,t) · n X i=1,j6=i wiσiwjσj(zi,tzj,t− ρi,j) . (2.20) 2.2.3 Combinations

A combination is an option strategy wherein a position is taken on both call and put options on the same underlying stock (Hull, 2012). The best-known combinations are strangles, straddles, strips and straps, but because the latter two are a bet on a specific market movement, they are not optimal for a dispersion trading strategy, which is market neutral in its purest form. For this reason only the straddle and the strangle are discussed below.

Straddle

This strategy involves a long position in both a European call and put option on the same underlying asset with the same strike price and time to maturity. The payoff is V-shaped, which means that a trader limits its downside risk by accepting a loss when the underlying asset does not move much in either direction. However a significant profit is made when at maturity the underlying ends up with a large distance from its initial value. Thus someone who enters into a straddle is uncertain in which way the underlying asset is going to move. It is a straightforward observation that a reverse position in a straddle (i.e. a short position) is very risky because the loss arising from a large move in the underlying asset is unlimited.

Like the fact that the Black-Scholes model gives under certain parameter conditions the price of a European put or call option, the Greeks of European options do also have an exact expression. Therefore by using simple calculus rules for taking derivatives, the Greeks of a

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straddle can be found from the Black-Scholes formula (Eq. (2.4)). If we denote the value of a

straddle with Πt at time t, then

∆ = ∂Πt ∂St = 2Φ(d1) − 1, (2.21) Γ = ∂ 2Π t ∂St2 = 2φ(d1) Stσ √ T − t, (2.22) Θ = ∂Πt ∂t = − Stφ(d1)σ T − t − rKe −r(T −t) (2Φ(d2) − 1), (2.23) ν = ∂Πt ∂σ = 2Stφ(d1) √ T − t, (2.24)

where φ(z) is the first derivative of Φ(z). Strangle

In a strangle an investor goes long in a European call and put option with the same time to maturity, however the difference with a straddle is the fact that the strike prices of the two options differ. A strangle yields less downside risk than a straddle and as a consequence the underlying asset must move more intense to make a profit. The Greeks for a strangle can be found in an analogous way as for the straddle and by comparing an at-the-money (ATM) straddle with an out-of-the-money (OTM) strangle (the most common combinations), although both have a very small initial delta exposure, the OTM strangle has less delta exposure than the ATM straddle for small movements of the underlying and is more preferred in a delta optimal point of view. The Greeks of a straddle are defined as

∆ = ∂Πt ∂St = Φ(d1c) + Φ(d1p) − 1, (2.25) Γ =∂ 2Π t ∂S2 t = φ(d1 c) + φ(d 1p) Stσ √ T − t , (2.26) Θ = ∂Πt ∂t = − Stσ(φ(d1c) + φ(d1p)) 2√T − t − rKe −r(T −t)(Φ(d 2c) − Φ(−d2p)), (2.27) ν = ∂Πt ∂σ = St(φ(d1 c) + φ(d 1p)) √ T − t, (2.28)

where the subscript denotes whether the variable is evaluated with respect to the call or the put option.

P&L combinations

In Subsection 2.2.2 the P&L of a delta-hedged long dispersion strategy was presented, assuming that the underlying stock followed a geometric Brownian motion with constant drift and (time-varying) volatility and with a constant riskless borrowing rate. Naturally, this solution can be extended to the combinations considered in this subsection by summing the (reweighted) individual P&Ls of the options within the portfolio.

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2.2.4 The volatility surface

The Black-Scholes pricing model assumes that the price process of the option’s underlying asset is ruled by a geometrical Brownian motion, which theoretically implies that options on the same asset should trade at the same implied volatility regardless of the time to maturity and the strike price. This assumption is not observed in real financial markets however; in reality there is empirical evidence that the assets return distribution exhibits excess kurtosis and is skewed compared to the lognormal distribution (Hull, 2012). Hence, implied volatilities differ between options on the same underlying but with a different strike price (volatility smile) and with distinct time to maturities (term structure of volatility).

Different kinds of assets display different kinds of behaviour in their prices. For example, the asset class equity (stocks) shows in general the so-called leverage effect, where a negative price shock (e.g. stock market crash) has a larger effect on the future volatility than a large positive price shock. As a large negative stock return leads to a decrease in equity value for the company, its leverage increases, i.e. the debt-to-equity ratio increases, and hence a larger

return on equity is expected.2 But if this effect is indeed a common stylised fact for stocks,

then the implied volatility can be seen as a decreasing convex function of the strike price and therefore this type of volatility smile is also known as the volatility skew. Another example where there is no constant implied volatility from options as a function of strike prices are exchange rates. Typically the time path of an exchange rate is rough and exhibits jumps, furthermore the volatility shows time varying properties and consequently extreme outcomes are more likely to occur. Hence, generally the implied volatility is an increasing convex function of the absolute distance between the current exchange rate and the strike prices.

When short-dated volatilities are low it is expected that the volatility will increase in the future and vice versa. Combining this effect with the volatility smile is called the volatility surface, i.e. the implied volatility as a function of both the time to maturity as the strike prices of an option. A ramification of the existence of this volatility surface is that the formulas of the Greeks derived from the Black-Scholes pricing model and given in the previous subsection are no longer correct. For example by taking the volatility surface into account, the delta of a call option is given by ∆ = ∂C ∂S + ∂C ∂σimp ∂σimp ∂S .

Because normally an option does not yield a constant implied volatility as a function of the standardised strike prices (K/S) (Etheridge, 2008),

∂C

∂σimp

∂σimp

∂S 6= 0,

and is in most of the cases positive for equity options. Nonetheless, the changes in the volatility surface observed in the market are usually small and the Greeks of the Black-Scholes model can be used as a reasonable approximation (Hull, 2012).

2

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80 85 90 95 100 105 110 115 120 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Price Underlying Delta 4 dtm 12 dtm 30 dtm

(a) Delta straddleK=100, σ=0.3, r=0.75%

80 85 90 95 100 105 110 115 120 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Price Underlying Delta 4 dtm 12 dtm 30 dtm (b) Delta strangleK=100, σ=0.3, r=0.75% 80 85 90 95 100 105 110 115 120 0 0.05 0.1 0.15 0.2 0.25 Price Underlying Gamma 4 dtm 12 dtm 30 dtm (c) Gamma straddleK=100, σ=0.3, r=0.75% 80 85 90 95 100 105 110 115 120 0 0.02 0.04 0.06 0.08 0.1 0.12 Price Underlying Gamma 4 dtm 12 dtm 30 dtm (d) Gamma strangleK=100, σ=0.3, r=0.75% 80 85 90 95 100 105 110 115 120 0 5 10 15 20 25 30 Price Underlying Vega 4 dtm 12 dtm 30 dtm

(e) Vega straddleK=100, σ=0.3, r=0.75%

80 85 90 95 100 105 110 115 120 0 5 10 15 20 25 Price Underlying Vega 4 dtm 12 dtm 30 dtm (f) Vega strangleK=100, σ=0.3, r=0.75% 80 85 90 95 100 105 110 115 120 −100 −80 −60 −40 −20 0 20 Price Underlying Theta 4 dtm 12 dtm 30 dtm (g) Theta straddleK=100, σ=0.3, r=0.75% 80 85 90 95 100 105 110 115 120 −60 −50 −40 −30 −20 −10 0 10 Price Underlying Theta 4 dtm 12 dtm 30 dtm (h) Theta strangleK=100, σ=0.3, r=0.75%

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2.3

Swaps as hedging strategy

In this section the different components of specific swaps are explained, and how they can be used in a dispersion hedging strategy.

2.3.1 Why swaps?

Next to options, other financial derivatives which have good properties for volatility dispersion trading are variance/volatility swaps. A volatility swap is an OTC product similar to a forward contract where one can speculate on the amount of realised volatility of an asset over a specific prespecified period. The payoff of this swap is the difference between the realised volatility of the asset and a fixed amount of volatility determined at the beginning, multiplied by a notional principal. The variance swap is analogous to the volatility swap, only the variance is used instead of the volatility. Both products are designed to give a direct exposure to the volatility/variance of an asset for hedging and risk-management purposes, however because the payoff of a variance swap can typically be replicated by a portfolio of vanilla options (see Subsection 2.3.2) they are easier to value and more popular (liquid) than volatility swaps (Carr and Lee, 2009). On the other hand, the advantage of a volatility swap is that the payoff is a linear function of the realised volatility of an asset and hence they give direct exposure to vega.

Taking a long position in an option always has strictly positive costs, except in the special case that an option is worthless. Initiating a position in a volatility/variance swap however, has zero costs because the fixed amount in these swaps represents the expected value of the realised volatility/variance under the risk-neutral distribution of the underlying. Another important difference between options and these specific swaps is that the latter instruments are a pure play on the realised volatility, meaning that delta-hedging is not labour intensive. On the contrary, the delta in a strategy with options is path-dependent and must be hedged in theory continuously. However Martin (2013) explains that the market for variance swaps collapsed during the Global Financial crisis because the prices of most of the underlying assets showed discontinuous jumps, and variance swaps are not able to be replicated with options in this situation. Moreover, the market for variance swaps has not recovered since then.

2.3.2 Price and value

In this subsection the theoretical strike price of a variance swap is given together with the Greeks, the volatility swap is not considered here because the payoff can not be replicated by a portfolio of options and hence is hard to value.

A variance swap is an agreement to exchange realised variance 1 T n X i=1  log Sti Sti−1 2 , (2.29)

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variance) at some future time T . In the limit, δt → 0, this implies that the payoff, V, of a variance swap with notional principal N is given by

V = N 1 T Z T 0 σt2dt − K  . (2.30)

It is conventional to set N = Nσ/(2K), where Nσ is the vega notional of a volatility swap.

Demeterfi et al. (1999) show that under nice3 behaviour of the underlying asset, the price

of a variance swap can be replicated by an infinite number of put options with strike prices

Kput ∈ [0, S∗] and an infinite number of call options with strike prices Kcall ∈ [S∗, ∞). The fair

fixed variance K is equal to

K = 2 T  rT − S0 S∗ erT − 1  − logS∗ S0 + erT Z S∗ 0 1 K2P (S0, K, T )dK + e rT Z ∞ S∗ 1 K2C(S0, K, T )dK  , (2.31)

where S∗ is a parameter which defines the boundary between call and put options. It can be

shown that in the case that this boundary parameter is equal to the fair forward value of the

underlying asset price, i.e. S∗= S0erT, Eq. (2.31) can be simplified to

K = 2e rT T Z S∗ 0 1 K2P (S0, K, T )dK + Z ∞ S∗ 1 K2C(S0, K, T )dK  . (2.32)

The greeks of a variance swap are given by H¨ardle and Silyakova (2010) as

∆ = 2T−1 S∗−1− St−1 , (2.33) Γ = 2St−2T−1, (2.34) Θ = −σ2T−1, (2.35) ν = 2σ(T − t)T−1. (2.36) 80 85 90 95 100 105 110 115 120 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 Price Underlying Delta 4 dtm 12 dtm 30 dtm

(a) Delta Variance SwapK=100, σ=0.3, r=0.75%

80 85 90 95 100 105 110 115 120 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Price Underlying Gamma 4 dtm 12 dtm 30 dtm

(b) Gamma Variance SwapK=100, σ=0.3, r=0.75%

Figure 2.3: Evolution of the Greeks for the Variance Swap.

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2.3.3 Volatility dispersion trading and correlation trading

Volatility and variance swaps provide pure exposure to volatility with low sensitivity to the direction of the underlying asset, i.e. low delta and gamma risk. Therefore, a dispersion trade initialised with variance swaps can disclose some important properties of the relationship be-tween dispersion trading and correlation trading. The payoff at time T of a long dispersion trade using variance swaps is given by

ΠLD = n X i=1 Ni 2Ki (σ2i − Ki2) ! − NI 2KI (σI2− KI2) = 1 2 n X i=1 Niσi2 Ki −NIσ 2 I KI ! +1 2 NIKI− n X i=1 NiKi ! , (2.37)

where NI is the vega notional of a volatility swap on the index and Ni is the vega notional of

a volatility swap on stock component i ∈ (1, . . . , n). As will be explained in the next section, ¯

ρ = σI/ (Pni=1wiσi), can be seen as a proxy for the average correlation of a market index.

When this statistic is substituted in the last line of Eq. (2.37), it is obtained that

ΠLD = 1 2 n X i=1 Niσi2 Ki − NIρ¯ 2(Pn i=1wiσi) 2 KI ! +1 2 NIKI− n X i=1 NiKi ! . (2.38)

Differentiating Eq. (2.38) with respect to ¯ρ gives

∂ΠLD ∂ ¯ρ = − NIρ (¯ Pn i=1wiσi)2 KI ≤ 0, (2.39)

and hence a long volatility dispersion trade corresponds to short selling correlation. Eq. (2.38) is also differentiated with respect to the single stock volatility,

∂ΠLD ∂σj = Njσj Kj −wjNIρ¯ 2Pn i=1wiσi KI . (2.40)

If it is assumed that that the sum of the single stock vega notional is equal to the negative of

the index vega notional, i.e. the dispersion trade is vega neutral, and wi = Ni/NI is the weight

for vega-neutrality for stock component i ∈ (1, . . . , n), Eq. (2.40) can be simplified as

∂ΠLD ∂σj = Nj  σj Kj −ρ¯ 2Pn i=1wiσi KI  . (2.41)

When the proxy for average correlation is evaluated with the implied volatilities, denoted by ˆρ,

and assuming t = 0 such that the value of the variance swap equals zero, the latter equation can be written as ∂ΠLD ∂σj = Nj  σj Kj −ρ¯ 2Pn i=1wiσi ˆ ρPn i=1wiKi  = Nj  εj− ¯ ρ2ζ ˆ ρ  , (2.42)

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where εj = σj/Kj is the ratio of realised and implied volatility for stock component j ∈

(1, . . . , n), and ζ = (Pn

i=1wiσi) (Pi=1n wiKi)−1. Hence the payoff is non-decreasing in the

volatility of stock j if εjρ ≥ ¯ˆ ρ2ζ. In the special case that the implied volatility risk premium

is roughly a constant proportion of the implied volatility at t = 0, i.e. v = (Ki− σi)/Ki, ∀i ∈

(1, . . . , n), hence the bias in the implied volatility is the same for each stock and index, we can rewrite Eq. (2.42) as ∂ΠLD ∂σj = Nj  (1 − v)Kj Kj −ρ¯ 2Pn i=1wi(1 − v)Ki ˆ ρPn i=1wiKi  = Nj(1 − v)  1 −ρ¯ 2 ˆ ρ  . (2.43)

The right-hand side of the latter equation is positive when ¯ρ2 < ˆρ, which is most likely the case

because it is reasonable to assume that the implied correlation is close to the realised correlation.

Furthermore, the only non-trivial way in which Eq. (2.43) is equal to zero is when ¯ρ2 = ˆρ = 1.

It may be clear that in general the single stock volatility exposure is non-zero and thus a long volatility dispersion trade is not equal to a perfect short correlation trade.

2.4

Volatility and correlation

Determining the price of a basket of options is not an effortless exercise. In general there does not exists an explicit expression for the value of a weighted sum of options due to the correlation between the price movements of the underlying assets. Unless these underlying assets are perfectly correlated, an index option typically costs less than the basket of options on each of the individual assets within the index. The concepts of volatility and correlation of assets will be explored in this section.

2.4.1 Portfolio variance

The variance of a portfolio consisting out of n securities, with σi the volatility of the ith security,

wi its corresponding weight in the portfolio and ρij the correlation between the ith and the jth

security, using the modern portfolio theory of Markowitz (1952) can be written as

σp2= n X i=1 w2iσi2+ n X i=1,j6=i wiwjσiσjρij. (2.44)

Because the correlation between two securities, ρij, is in absolute value between zero and one,

the maximum variance of the portfolio is attained when all underlying securities are perfectly

positively correlated and thus ρij = 1 ∀i, j. Hence the variance of a portfolio is reduced by

including mutually uncorrelated securities and this embodies the concept of diversification and the reason why index options generally do not have the same price as the corresponding weighted sum of single security options.

The standard deviation of a portfolio can be computed by taking the square root of the portfolio variance (2.44). However, in the case of a market portfolio, which can be seen as a

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portfolio where the unsystematic risk is diversified away and hence only exhibits systematic risk, a simpler approach can be used as shown by Marshall (2009). By using the properties of

beta, which is a measure of systematic risk in equity markets,4 she shows that the volatility of

a portfolio containing only systematic risk can be written as5

σm =

n

X

i=1

wiσiρi,m, (2.45)

and this is called the modified Markowitz equation.

2.4.2 Implied correlation

The implied correlation is an average correlation measure and adequately indicates the difference between the index implied volatility and the weighted average implied volatility of the basket of underlying assets. For an index which is not necessarily a market index the average correlation

can be found by solving Eq. (2.44) for a constant correlation parameter ¯ρ, hence it is given by

¯ ρ = σ 2 p − Pn i=1wi2σi2 Pn i=1,j6=iwiwjσiσj . (2.46)

One obtains the implied correlation by evaluating Eq. (2.46) with the implied volatilities. More-over, using the the modified Markowitz equation (2.45), a good proxy for the average correlation of a market index is given by

¯

ρ = Pnσm

i=1wiσi

, (2.47)

and the implied correlation is approximated by the evaluation of the latter equality with implied volatilities.

It was found that a vega-neutral dispersion trade is not equal to a pure correlation trade. Having defined the implied correlation it is it is insightful to approximate this spread using Eq. (2.44) and an index variance swap with one variance unit (i.e. N = 1),

σp2− ˆσp2= n X i=1 w2iσ2i − n X i=1 wi2σˆi2+ n X i=1,j6=i wiwjσiσjρ − n X i=1,j6=i wiwjˆσiσˆjρˆ = n X i=1 w2i σ2i − ˆσ2i + n X i=1,j6=i wiwj(σiσjρ − ˆσiσˆjρ)ˆ = n X i=1 w2i σ2i − ˆσ2i + n X i=1,j6=i wiwj(σiσj(ρ − ˆρ) − (ˆσiσˆj− σiσj) ˆρ) = n X i=1 w2i σ2i − ˆσ2i + n X i=1,j6=i wiwjσiσj(ρ − ˆρ) − n X i=1,j6=i wiwj(ˆσiσˆj − σiσj) ˆρ, (2.48) 4

This simplification is not generally applicable to other markets than equity markets by the definition of beta.

5

The subscript of the portfolio has changed from p to m compared to Eq. (2.44) to pinpoint that the portfolio only contains the systematic market risk.

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where σi and ˆσi are the realised and implied volatility for i ∈ (1, . . . , n), respectively, and ρ

and ˆρ are the average correlations evaluated with the realised and implied volatilities. Now the

latter equation can be rewritten as

n X i=1,j6=i wiwj(ˆσiσˆj − σiσj) ˆρ = n X i=1 w2i σi2− ˆσi2 + n X i=1,j6=i wiwjσiσj(ρ − ˆρ) − (σ2p− ˆσ2p). (2.49)

The right-hand side of Eq. (2.49) can be interpreted as the payoff of a short index variance swap, long single stock variance swaps and a correlation swap. In other words, the payoff of a specific long dispersion trade, i.e. a short position in correlation, and a long correlation swap. Hence the left-hand side defines the considered spread.

The implied correlation is a measure for the market’s expectations of future correlation and it reflects the changes in the relative premium between index and stock options. Also, it indirectly expresses the implied volatility spread, but with the advantage that it is independent of the current level of volatility. It can therefore be used to identify opportunities in which a mispricing of implied volatility has created a disparity between the implied volatility of the index and its components. Another measure of identifying remunerative volatility dispersion trades is found

by setting ρij in Eq. (2.44) equal to one, in this way one derives an upper bound for the

variance and volatility of a portfolio. The difference between the square root of both sides of this expression can be seen as the volatility dispersion statistic,

D = σp−

n

X

i=1

wiσi. (2.50)

The implied correlation and the volatility dispersion statistic can both be used to describe the dispersion trading opportunities. A long dispersion position corresponds roughly to a short position in correlation and vice versa. The volatility dispersion statistic directly specifies an upper bound for the volatility spread and can trivially be used for identifying a dispersion trade, however it is a distance measure between two implied volatilities and therefore contains less information than the implied correlation.

2.5

Tracking portfolio

Until now it was assumed that a dispersion trade was done by taking positions in derivatives on both the index as well as the corresponding basket of constituents. However, if discrepancies in implied volatility exist, they are marginal and a dispersion trade on all the constituents of an index is not profitable due to the transactions costs from initiating and hedging the positions. Referring to Subsection 2.1.3, one possible solution to this problem is creating a tracking portfolio which mimics the characteristics of the index with a minimum amount of securities and hence this comes down to a trade-off between transaction costs and a correlation risk of the tracking portfolio with the market index. Because a tracking portfolio is based on the history of returns of the constituents of an index, it is backward looking and hence choosing

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the amount of securities in the tracking portfolio too small can lead to uncorrelated behaviour of the short and long leg in a dispersion trade and therefore uncontrolled payoffs.

In the process of constructing a tracking portfolio one needs to decide which securities to include and which weight they should have. Because index tracking is a low-cost alternative to active portfolio management and financial institutions even publicly offer index tracking funds, such as ETFs, the problem of creating a tracking portfolio is well investigated by academics and practitioners alike. The complexity of this problem can widely vary, depending on the given restrictions and objectives. However in the general case, without a prespecified number of securities in the tracking portfolio, this corresponds to a conjunction of a combinatorial and a continuous numerical problem, where both problems need to be approached simultaneously (Krink et al., 2009).

A relatively simple approach to creating a tracking portfolio is based on principal component analysis (PCA), herein one decomposes the sample covariance matrix of the returns into pairs of eigenvalues and eigenvectors ordered by importance. Hence the ith principal component of a

return vector r is the linear combination yi = wi0r that maximises V ar(yi) with the constraints

that wi0wi= 1 and Cov(yi, yj) = 0 ∀i 6= j (Tsay, 2010). Then the first n principal components

are chosen such that the cumulative proportion of variance of these principal components is large

enough.6 More advanced methods such as DECS7 (Krink et al., 2009) use search heuristics to

encounter simultaneously the combinatorial problem of choosing the number of securities.

6

e.g. more than 90%.

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Chapter 3

Methodology

A dispersion trading strategy can be implemented and tested in various ways, yielding different results and conclusions on the P&L of a dispersion trade. In this chapter it is explained what methods are used for this research and how they can be blended into a complete dispersion strategy; furthermore, testing methods for the results are described.

3.1

Overview

Practically initiating a dispersion trade comes down to two steps. The first step is selecting a weighting scheme. In this thesis a tracking portfolio is used which is expected to display the same characteristics as the market index over the period where a dispersion trade is active. Thereafter, the second step is to formalise, execute and maintain a trading strategy based on a specific information set available, which could be no information whatsoever and consequently a naive trading strategy is entered into, or the information set could contain several signals of the market available on that date. In most of the existant literature on dispersion trading, the focus is on whether the market shows volatility dispersion, and if so, naive positions in financial derivatives are used to take advantage of this discrepancy in the market. However, there hardly exists any academic research on how market signals can be used in determining the position in a financial product and the purpose of this thesis is to contribute to the knowledge of dispersion trading in this direction. Furthermore it is of interest to know whether different optimisation methods for a tracking portfolio yield significant different results. In this chapter each component of the research method followed is discussed separately.

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3.2

Tracking portfolio

In this study two tracking portfolio optimisation methods are used: the PCA method and the DECS method. Both were already touched upon in Section 2.5, but in this section they are explained in further detail.

3.2.1 PCA analysis

Principal component analysis (PCA) is a statistical method to explain the structure of the covariance matrix or equivalently the correlation matrix of a multidimensional random variable with a few linear combinations of the components of this random variable. Likewise, it is a popular tool for dimension reduction of a multidimensional random variable without significant loss of information (Tsay, 2010). The main procedure of the PCA method is decomposing the covariance-/correlation matrix into its eigenvalues and eigenvectors, where the eigenvalues and corresponding eigenvectors are ordered by their importance. The principal components are then defined as the product of the eigenvectors and the multidimensional random variable minus its mean vector (Su, 2005).

To be more specific, consider a k-dimensional return vector r and wi = (wi,1, ..., wi,k)0 a

k-dimensional real-valued vector, i = 1, ..., k, satisfying (Tsay, 2010):

E(r) = µ and V(r) = Σr,

yi = wi0r i = 1, ..., k,

Cov(yi, yj) = wi0Σrwj, i, j = 1, ..., k.

The idea of PCA is to find linear combinations wi such that yi has maximal variance and yi and

yjare uncorrelated for i 6= j. But since the covariance matrix is positive definite, it has a spectral

decomposition1 and therefore wi = ei for i = 1, ..., k, where ei is the ith normalised eigenvector

corresponding to the ith eigenvalue λi of Σr, ordered with respect to their importance.

Practically, for this study this means that at the moment of time when a tracking portfolio is created, one needs to decompose the covariance matrix into its eigenvalues ordered in significance and the associating eigenvectors. The first m principal components are then chosen such that the

cumulative variance proportion2 is greater than or equal to a pre-specified percentage. Within

these m principal components, the N∗ most prevailing stocks of the index are chosen to form

the tracking portfolio by evaluating their cumulative squared correlation (Su, 2005)

N∗ X j=1 qi,j2 = PN∗ j=1λjγi,j2 σi2 ,

where γi,j is the (i, j)th element of the ordered eigenvector matrix of Σr.

1

Σr = P ΛP0, where Λ and P are the diagonal eigenvalue matrix and the corresponding eigenvector matrix

respectively, with the eigenvalues in descending order.

2i.e. the sum of the first m eigenvalues divided by the total sum of all the eigenvalues, because in this case

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3.2.2 Differential evolution and combinatorial search

Although the simplicity and the low computational complexity are the main advantages of principal component analysis, it is very restrictive in the sense that it does not tackle the index tracking problem in a simultaneous search for a selection of optimal assets in combination with associating optimal weights. This is important because in determining whether a selection of assets is optimal in mimicking the index characteristics, the result depends on the positions in the assets taken and vice versa. Moreover, non-linear constraints such as minimum and maximum holding positions in a single asset can not be solved with PCA. Because in a dispersion trading strategy the tracking portfolio is a critical element, it is of great interest to investigate whether a more sophisticated method for constructing a tracking portfolio would yield significantly different results from the PCA method.

Because of the duality in the optimisation problem, quadratic programming can not be used. An alternative is therefore to use search heuristics, which iteratively searches for a superior solu-tion within a problem. The main advantage of using search heuristics is that various constraints can easily be implemented, and that optimisation is based on a single evaluation criterion, such

as a distance measure.3 The disadvantage of search heuristics is that a problem may require

many iterations and that its rate of convergence and consequently the accuracy of the solution is poor. There are many notorious examples of search heuristic optimisation methods, e.g. particle swarm optimisation, genetic algorithms, simulated annealing and differential evolution, and although most of them are inspired by biological and sociological motivations (Abraham et al., 2008), they can resolve many different problems. However as shown by Kennedy and Eber-hart (1995), differential evolution has very good performance in continuous numerical problems compared to the others, and Krink et al. (2009) complements on this field by proposing a hybrid model for index tracking, namely the differential evolution and combinatorial search (DECS) algorithm. This method combines differential evolution with combinatorial search to determine the optimal subset of assets in a tracking portfolio. Because Krink et al. show that it can deal with non-continuous numerical problems and moreover the focus of this method is to construct a tracking portfolio of an index, DECS will be used in this study as the competitor of PCA. DECS is a variant on differential evolution, where the latter is a population based search heuris-tic. This means that it generates an initial population, P, of possible solutions which it itera-tively refines by the following procedure:

1. For each element of the population P (j), three other elements: P (x), P (y) and P (z), are selected randomly. Subsequently a new candidate solution, c, is created by a combination of the three random selected candidate solutions of the population, together with scaling factor, f :

c = f · (P (x) - P (y)) + P (z).

3

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2. c is substituted by a recombination between P (j) and c, where each component, o, of the recombination is equal to P (j, o) with probability 1-cf and c(o) with probability cf , cf is the crossover factor.

3. The new candidate solution c∗ is substituted in the population for P (j), if c∗ has better

fitness in the criterion function.

This general procedure is in the DECS extended with a position swapping procedure in which with a certain probability an asset with a zero weight allocation is swapped with an asset with a non-zero weight allocation. Furthermore it is supplemented with constraint violation

handlings.4

The index tracking problem which is considered in this thesis as a test against PCA and solved with DECS is given by (Krink et al., 2009)

minimise w f0(w) = v u u t1 T T X t=1 (RPt − RB t )2, subject to n X i=1 wi = 1, |wi| ≤ 1, i = 1, . . . , n, εiδ(wi) ≤ wi ≤ ξiδ(wi), δ(wi) =    1 if wi> 0 0 else i = 1, . . . , n, L ≤ n X i=1 δ(wi) ≤ K, i = 1, . . . , n, X i:wi>Lb wi ≤ U b, n X i=1 |∆wi| ≤ M, where:

t = 1, . . . , T Time period considered.

Rxt Return of either the benchmark (B) or the tracking portfolio (P).

w Real valued vector of asset weights in tracking portfolio, n-dimensional.

εi, ξi Lower and upper bounds for single asset weights.

L, K Lower and upper bounds for the total assets in tracking portfolio.

Lb Lower threshold for classifying asset weights as large.

U b Maximum proportion of large asset weights in tracking portfolio.

M Maximum deviation from previous weight allocation.

Thus the tracking error is the criterion function in which the population is evaluated and

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iteratively improved based on realistic constraints on the asset allocations, to limit market impact and transaction costs. The PCA method can not tackle these constraints. The specific values of the parameters are given in Appendix B.

3.3

Implied volatility

The volatility of a stock is not directly observable by the market data. If high-frequency data is used in estimating the volatility of a stock, e.g. intraday data, the prices are generally measured

with market microstructure noise,5 leading to an MA(1) effect in the returns and hence the

estimated realised volatility over one trading-day is inaccurate.6 However in option markets, if

one assumes that the prices are ruled by an econometric model such as the Black-Scholes model, then using the price of an option one is able to solve the volatility parameter from the model, i.e. the implied volatility. The advantage of implied volatility compared to the realised volatility is that it is forward-looking, assuming that information is processed in the market immediately. A disadvantage of using implied volatility is that a mathematical option pricing model is used and therefore a specific process of the underlying asset is assumed, the implied volatility may therefore differ from the actual volatility. In this section it is explained which pricing model is used for index and single stock options and how the implied volatility is derived from such a model.

3.3.1 Index options

Most index options are of the European type. Referring to Section 2.2, European options are plain-vanilla options and relatively easy to value because their pay-off at maturity is not path dependent. Furthermore assuming that the underlying asset is governed by a geometric Brownian motion this option can be priced with the Black-Scholes pricing formula. Because the price of an option is increasing in the volatility of the underlying, having observed the price of an option one is able to back out the unique implied volatility from the Black-Scholes

model by an iterative search procedure. If put-call parity7 is satisfied then the implied volatility

derived from European put and call options on the same underlying index, with the same time to maturity and strike price, is the same (Hull, 2012). However by the volatility smile and term structure generally observed in equity options, the implied volatility differs for options on the same index but with different strike prices and maturities.

Because an index can be seen as a portfolio of a certain number of stocks, the main difficulty in pricing index options is the dividend estimate of the index. The underlying stocks pay different amount of dividends on different dates and only the dividend payments within the option’s life must be considered and weighted accordingly. A relatively simple method to deal

5

e.g. bid-ask spread, discreteness of price changes etc.

6

Financial Econometrics, University of Amsterdam, catalogue number 6414M0007Y.

7

p + S0e−qT = c + Ke−rT, where c and p are European call and put prices, q is the dividend yield, K is the

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