Discrete-time hedging in incomplete markets : a comparison of two quadratic hedging strategies

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Discrete-time Hedging in Incomplete

Markets

A Comparison of two Quadratic Hedging Strategies

Mick Peetoom

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Mick Peetoom Student nr: 10618023

Email: Mick-Peetoom@hotmail.com Date: August 15, 2018

Supervisor: Prof. dr. Roger Laeven Second reader: dr. Servaas van Bilsen

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This document is written by Student Mick Peetoom who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Discrete-time Hedging in Incomplete Markets — Mick Peetoom iii

Abstract

This study examines hedging in incomplete markets. We focus on two popular so-called quadratic hedging strategies: local risk minimiza-tion and mean-variance hedging. The name originates from the fact that both methods find an optimal trading strategy after solving a quadratic criterion function, which deals with the residual risk arising from incompleteness in a different way. In this thesis, we start with hedging and pricing in a complete market using binomial trees and the Black Scholes equation. Then, we make the translation to pricing and hedging in a martingale framework. This framework is useful for the introduction of both methods and gives us the possibility to present the estimation methods. We also give an example of incomplete market pricing and hedging in the final chapter.

Keywords

Incomplete Markets, Hedging, Local Risk Minimization, Mean-Variance Hedging, Black Sc-holes equation, Martingale Measures

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Preface v

1 Introduction 1

2 Basic Pricing and Hedging 3

2.1 Discrete Time . . . 3

2.1.1 A Single Period Binomial Model . . . 3

2.1.2 Multiple Periods . . . 4

2.2 Continuous Time . . . 5

2.2.1 The Stock Price for ∆t → 0 . . . 5

2.2.2 The Contingent Claim for ∆t → 0 . . . 6

2.2.3 Shortcomings . . . 8

3 Hedging in Incomplete Markets 9 3.1 General Theory . . . 9

3.2 Local Risk Minimization . . . 12

3.3 Mean-Variance Hedging Strategy . . . 13

4 Estimation 16 4.1 Mean-Variance Hedging Estimation. . . 16

4.1.1 Pricing the Contingent Claim . . . 16

4.1.2 Dynamic Programming . . . 18

4.2 Local Risk Minimization Estimation . . . 20

4.3 Comparing Both Strategies . . . 24

5 Hedging in Incomplete Markets: An Example 25 5.1 Example Set-Up . . . 25

5.2 Results and Analysis . . . 26

6 Conclusion 30

References 32

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Preface

This master thesis is the last step in obtaining a master’s degree in Actuarial Science and Mathematical Finance. To be honest, it was a struggle. The theory for this thesis was much harder than expected and it would be impossible without my supervisor Roger Laeven. So I would like to thank him for his input and suggestions.

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Chapter 1

Introduction

It was called the holy grail for investors and compared with betting on a horse halfway through the race (Stewart, 2012): the Black Scholes equation. The equation was named after economists Fischer Black and Myron Scholes after their paper in 1973 and gave investors the ability to price and hedge their financial product at any moment of time. But it was too good to be true and the equation was not fitting the financial market. It was well suited in a complete market, where the risks of all financial products in the investor’s portfolio can be replicated. Perfect replication turned out to be not realistic and a market is in almost all cases incomplete. This means that there exists a remaining risk, which has to be dealt with.

Unfortunately, many were keen to use the Black Scholes equation. Eventually, this was one of the reasons that the financial crises in 2008 could happen as it caused the banks to crash (Stewart, 2012).

And now, in an ever increasing financial market, where the newest trading products are even more complex than before, an investor needs to invests his or hers wealth better than ever. A lot of research has been done in the last couple of decades in the field of pricing and hedging in incomplete markets. Next to this, a lot of research has also been done in the method for modelling the underlying stock price, allowing a stochastic volatility (Heston, 1993) or even self-exciting jumps (Boswijk et. al., 2016). In this thesis we want to introduce two popular quadratic hedging methods suitable for incomplete markets. The first one is local risk minimization introduced by F¨ollmer and Sondermann in 1986 and secondly we examine mean-variance hedging. Most of the research in these two hedging strategies are done in a continuous-time frame, but this thesis is limited to the discrete-time. The main interest is the figure out an answer to the question: How can we find an hedging strategy for an incomplete market and how do we apply this in real-life?

To start this thesis, we recall pricing and hedging in a complete market case in chapter 2. This forms the basis of pricing and hedging in incomplete market case. We think that including this makes the translation better to understand as we also point out its weak assumptions in a real-world financial market.

Then, in chapter3, we make a translation towards the martingale framework of 1

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pricing and hedging as both quadratic hedging strategies rely on this framework. This chapter also include the introduction of both strategies and some decompositions and results.

In chapter4we derive, following ˇCern´y (2004) and Van Gastel (2013), the estimation methods for both quadratic hedging strategies. We give the theoretical background and a step-by-step overview for the estimation towards the optimal trading strategy.

Estimation is demonstrated for mean-variance hedging in chapter 5 for a simple stochastic process and option. Since the provided example is also well suitable in a complete market, we compare the results of mean-variance hedging with the Black Scholes equation. Finally, we give in chapter 6 our conclusion over quadratic hedging strategies and hedging in incomplete markets.

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Chapter 2

Basic Pricing and Hedging

In this chapter, pricing and hedging of a contingent claim is discussed in case of a complete market. A contingent claim can be best represented as a contract where its future value depends on future values of other contracts. The price of such a contract is related of other assets and risk can be mitigated by portfolio of other contract. In a complete market, the risk can even be neutralized completely by replication. Therefore, the market must be arbitrage-free and frictionless. We consider both discrete time (sec-tion 2.1) as continuous time (section 2.2) using the notation of Hull (2011). We only consider European call and put options in this chapter. This chapter is considered as basic knowledge.

2.1 Discrete Time

2.1.1 A Single Period Binomial Model

European call and put options are types of contingent claims. It is riskier to invest in contingent claims than in ordinary stocks or bonds and we need to invest in such a manner that we minimize our risks. These contingent claims give the buyer the option to buy or sell stocks at time T for a predetermined value (strike price K). In order to price these contracts we first examine a single period binomial model with stepsize n = 1 (T_n = ∆t), where our portfolio consists of a single stock S and a risk-free bond B. For a call option, the situation can be sketched in a binomial tree as follows:

uS S dS Ber∆t B Ber∆t (uS − K)+= Hu H (dS − K)+= Hd

In this sketch, the stock price is expected to move upwards with factor u or downwards with factor d (usually denoted as 1/u). The risk-free bond is in both the upward or downward scenario multiplied by the exponential rate with risk-free rate r over the given time interval ∆t. The payout of the contingent claim is in the upward scenario equal to (uS − K)+ = max(uS − K, 0) and in the downward scenario to (dS − K)+=

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max(dS − K, 0). For a European put option, the first two trees are the same but now the contingent claim tree changes in upward scenario equal to (K − uS)+ = max(K − uS, 0)

and in downward scenario to (K − dS)+= max(K − dS, 0).

An aspect of a complete market is that all outcomes of contingent claims are replicable and that we can find a portfolio without being exposed to any risk at all. For the single period binomial model above, we have to solve:

φuS + ϕBer∆t= Hu

φdS + ϕBer∆t= Hd

This results in φ = Hu−Hd

S(u−d) and ϕ = e

−r∆t uHd−dHu

u−d . Here, φ is the amount invested

in stocks and therefore called the Delta (∆) or Delta-Hedge. To price H, we write down: H = φS + ϕ = Hu− Hd u − d + e −r∆tuHd− dHu u − d = e−r∆t er∆t− d u − d Hu+ u − er∆t u − d Hd = e−r∆t qHu+ (1 − q)Hd (2.1) where q = er∆t_u−d−d. The definition of q is important, since so far we did not use any-thing of the underlying probability of the stock in our pricing and hedging methods. The probability q is a new probability measure Q called risk-neutral probabilities and is different than the real-world probability measure P. More over these risk measures are covered in chapter 3.

2.1.2 Multiple Periods

In section 2.1.1we discussed a single period model. This period could be for instance one year. It is unrealistic to assume that a stock only has two values at the end of the year, so to make the model more realistic, we can change n = 1 to n = ₁₂1 (monthly) or even n = ₃₆₅1 (daily). Consequence, the binomial tree also grows:

. . . u2S uS . . . S udS dS . . . d2S . . .

The u’s and d’s in this tree are not the same as in the tree of section 2.1.1 and we have to find a new function for these upward and downward factors. Therefore we need the real-world probabilities p ∈ P (which are not used anymore after finding u and d).

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Discrete-time Hedging in Incomplete Markets — Mick Peetoom 5

Consider the logarithmic return of one step ln(St+∆t

St ) and call the mean of growth per

year ν and the variance of growth σ2. The ∆t period mean and variance can be written as follows: E[ln(St+∆t St )] = pln(u) + (1 − p)ln(d) = ν∆t Var[ln(St+∆t St )] = p(ln(u))2+ (1 − p)(ln(d))2− (pln(u) + (1 − p)ln(d))2 = σ2∆t Solving this gives us p = 1₂+ 1₂ν_σ√∆t, u ∼ e

√

σ∆t _{and d ∼ e}−√σ∆t_.

A similar tree can be made for the contingent claim and we price this tree using backward induction, since we assume to know all possible values at T . We then use equation (2.1) to price all other nodes to find H.

2.2 Continuous Time

In this section we want to extend the theory of section 2.1by looking to what happens when ∆t → 0, or n → ∞. This results into an even more realistic model where we can rehedge risks more often, such that replication will be more accurate. In order to do this we need the definitions of the risk-neutral probabilities q ∈ Q from section 2.1.1

and the definitions of u and d of section 2.1.2.

2.2.1 The Stock Price for ∆t → 0

We assume that an upward move or a downward move of the stock is independent of the previous move of the stock, so we can evaluate each move at each step as a coin flip. And since we have a series of coin flips we can say that, following the Central Limit Theorem, the possible outcomes of the logarithmic returns of the stock converges to a normal distribution. This means that the stock price itself converges to a lognormal distribution.

Let’s consider again one step of a stock evolution and also define the logarithm of the stock: uS S dS lnS+lnu lnS lnS+lnd

We can now write lnS(t + ∆t) = lnS(t)+Z(t), with

Z(t) =    σ√∆t with probability q −σ√∆t with probability 1 − q

We assume that Z(t) is random (a coin flip), or we say that Z(t) is an i.i.d. variable under the probability measure Q. To derive the mean and variance of Z(t), we first want to derive what q will be when ∆t → ∞, since q is time-dependent. This can be

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approximated using a first degree Taylor polynomial (more degrees are due to the small ∆t not necessary): q = e r∆t_{− e}−σ√∆t eσ√∆t_{− e}−σ√∆t ' 1 + r∆t − (1 − σ∆t + 1 2σ2∆t) (1 + σ√∆t +1₂σ2_{∆t)) − (1 − σ}√_{∆t) +}1 2σ2∆t) ' 1 2 + 1 2ζ √ ∆t, with ζ = r − 1 2σ 2 σ

Given this expression for q, the mean and variance under Q of Z(t) are: EQZ(t) = qσ √ ∆t − (1 − q)σ √ ∆t = (2q − 1)σ √ ∆t ' ζ√∆tσ√∆t = (r −1 2σ 2_)∆t VarQZ(t) = EQZ(t)2− (EQZ(t))2 ' σ2∆t − ((r − 1 2σ 2_)∆t)2 _{' σ}2_∆t

Now we have more insight in the behaviour of Z(t) for really small ∆t and we can work towards an expression for S(T ). Since the i.i.d. assumption we can use recursion to find:

lnS(T ) = lnS(0) + n−1 X i=0 Z(i) ⇒ lnS(T ) lnS(0) = n−1 X i=0 Z(i)

For large n, the Central Limit Theorem concludes that: lnS(T ) lnS(0) distr. −−−→ (r −1 2σ 2_{)T + σ}√_{T z}Q

where zQ _{has a standard normal distribution under Q. Therefore we find for S(T ) the}

following form:

S(T )−−−→ S(0)edistr. (r−12σ

2_{)T +σ}√_{T z}Q

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2.2.2 The Contingent Claim for ∆t → 0

In this section we look at the behaviour of the contingent claim H when ∆t → 0. Let’s write the contingent claim as a function of time and the underlying stock: H(t, S). Recall equation (2.1) for one step:

H(t, S) = e−r∆t[qH(t + ∆t, uS) + (1 − q)H(t + ∆t, dS)] (2.3) To find an expression for H(t + ∆t, uS) and H(t + ∆t, dS), we take the first degree partial derivatives with respect of t (higher degrees are again not necessary), the first and second degree partial derivative of the stock and we use again Taylor approximations for q, u and d: H(t + ∆t, uS) ' H(t, S) +∂H ∂t ∆t + ∂H ∂S(uS − S) + 1 2 ∂2H ∂S2(uS − S) 2

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Discrete-time Hedging in Incomplete Markets — Mick Peetoom 7 ' H(t, S) + ∂H ∂t ∆t + ∂H ∂SS(σ √ ∆t + 1 2σ 2_{∆t) +}1 2 ∂2H ∂S2S 2_σ2_∆t H(t + ∆t, dS) ' H(t, S) +∂H ∂t ∆t + ∂H ∂S(dS − S) + 1 2 ∂2_H ∂S2(dS − S) 2 ' H(t, S) + ∂H ∂t ∆t + ∂H ∂SS(−σ √ ∆t + 1 2σ 2_{∆t) +}1 2 ∂2_H ∂S2S 2_σ2_∆t

Rearrange equation (2.3) and substitute the expressions above in it gives: H(t, S)er∆t = [qH(t + ∆t, uS) + (1 − q)H(t + ∆t, dS)] = H(t + ∆t, dS) + q(H(t + ∆t, uS) − H(t + ∆t, dS)) = H(t, S) +∂H ∂t ∆t + ∂H ∂SS(−σ √ ∆t +1 2σ 2_{∆t) +} 1 2 ∂2H ∂S2S 2_σ2_∆t +q(2∂H ∂SS(σ √ ∆t + 1 2σ 2_∆t)) = H(t, S) +∂H ∂t ∆t + ∂H ∂SS(−σ √ ∆t +1 2σ 2_{∆t) +} 1 2 ∂2H ∂S2S 2_σ2_∆t +1 2(1 + ( r σ − σ 2) √ ∆t)(2∂H ∂SSσ √ ∆t)

If we now rearrange the terms, neglect all higher orders of ∆t and divide by ∆t, we arrive by the following equation:

rH = ∂H ∂t + rS ∂H ∂S + 1 2σ 2_S2∂2H ∂S2 (2.4)

which is called the Black-Scholes-Merton partial differential equation (PDE).

This PDE gives us the opportunity to derive the value of the contingent claim H(t, S) for all values t if we have the final values H(T, S). For a European call option, the final payoff is C(T, S) = (S − K)+ and we can use this (without proof) to extract the

Black-Scholes formula: C(t, S) = SN (d1) − Ke−r(T −t)N (d2) (2.5) d1= ln(_KS) + (r +1₂σ2)(T − t) σ√T − t , d2 = d1− σ √ T − t

For the European put option, P (T, S) = (K − S)+, we use the put-call parity:

max(S, K) = S + max(0, K − S) = K + max(0, S − K) which is the same as

S + P (T, S) = K + C(T, S), or C(t, S) − P (t, S) = S − er(T −t)K Therefore, we price the European put option as:

P (t, S) = Ke−r(T −t)N (−d2) − SN (−d1) (2.6)

We now know how contingent claims are priced in continuous time, so we now want to know how we find a hedge. The hedge ∆ is derived by taking the derivative to S of the contingent claim. So the amount of stocks for a call (∆C(t,S)) and a put (∆P (t,S)) are:

∆_C(t,S) = ∂C(t, S)

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∆P (t,S)=

∂P (t, S)

∂S = −N (−d1)

Due to continuity, we can also look at the second derivative of a contingent claim Γ(t, S) = ∂2H(t,S)_∂S2 and this represents the sensitivity of the hedge. Γ(t, S) is useful for

rehedging as it captures movements of the stock and determines whether an option is out-of-the-money, at-the-money or in-the-money.

This ends the discussion of basic pricing and hedging and in section2.2.3some problems are stated.

2.2.3 Shortcomings

In the real world, the Black-Scholes equation has a lot of shortcomings due it’s to many assumptions. So far, we tackled the problem of pricing and hedging of a contingent claim in a complete market in both discrete as continuous time. The problem is that the market is almost never complete and all the above is typically destroyed as soon as the model is modified and when the market becomes incomplete, there is no such thing as a unique probability measure (risk-neutral), which prices and hedges in a preference-independent fashion (Schweizer, 1999). Perfect replication is then impossible and risk can not be minimized to 0, resulting in a residual risk to be dealt with.

To name a couple: The Black and Scholes model does not allow a stochastic volatility and the presence of jumps. Allowing these phenomena makes the market incomplete, but more realistic. A constant volatility is not realistic, since movements of the stock are much larger in a turbulent time-period, such as a recession, compared with a ”normal” period. Also, one would expect that market endures shocks in turbulent time-periods, where the stock price makes a (persistent) jump.

Another modification of the Black-Scholes equation is choosing a non-Gaussian distri-bution for the underlying logarithmic return of the stock.

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Chapter 3

Hedging in Incomplete Markets

In this section, we make the translation to the incomplete market case. When the market does not have enough instruments to cover the riskiness, incompleteness arises. In chapter 2 we discussed completeness and gave some causes of incompleteness. Due to the incompleteness, there is no unique method to price and hedge anymore such that the risk is minimized to 0. In fact, there will always be a remaining risk and there are multiple methods to price, which can be chosen by preference. We focus on two methods: local risk minimization strategy and mean-variance hedging. Due to complexity, we stick to a discrete-time set-up.

The theory in this chapter is mainly inspired by the work of Schweizer, the PhD thesis of Vandaele (2010) and the master thesis of Van Gastel (2013).

3.1 General Theory

Consider a probability space (Ω, F , P), where F = (Ft)0≤t is the filtration, which

basi-cally is the information available at time t. P is the probability measure, which transforms all possible outcomes Ω to a value on the unit interval. In chapter 2 we already spoke about two different probability measure: real-world probability measure P and risk-neutral probability measure Q. Also consider again that the market is arbitrage-free and frictionless.

Definition 3.1. A stochastic process X = (Xt)0≤t≤T or X : Ω x T −→ R, t ∈ T =

{1, . . . , T }, is a sample path or trajectory of a random variable.

Definition 3.2. If stochastic process X is Ft-measurable, then this process is called

adapted.

Definition 3.3. Stochastic process X is called predictable if Xtis Ft−1-measurable for

all t.

Definition 3.4. A martingale Xt is an adapted process for all t, such that Xs =

E(Xt|Fs) for s ≤ t.

Definition 3.5. A martingale Xtis called integrable if E(Xt) < ∞, likewise it is called

square-integrable if E(|Xt|2) < ∞.

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Definition 3.6. Two martingales X and Y are strongly orthogonal if their product X · Y is also a martingale, which is an equal definition to Cov(∆Xt+1, ∆Yt+1|Ft) = 0,

where ∆Xi+1= Xi+1− Xi.

Similar as in the binomial tree of chapter 2, we want that the stochastic process X at time t − 1 to t, for all t, has multiple outcomes. We therefore have to define a nondegeneracy condition.

Definition 3.7. X satisfies the nondegeneracy condition (ND) if there exists a constant δ ∈ (0, 1) such that

E[∆Xt|Ft−1]

2

≤ δE[∆Xt2|Ft−1], for t = 1 . . . T

Definition 3.8. An integrable adapted stochastic process Xt given the probability

space (Ω, F , P) can be decomposed by the Doob decomposition as X = A + M

where M = (Mt)0≤t≤T is a square-integrable martingale under P with M0 = X0 and

A = (At)0≤t≤T a square-integrable predictable process with A0= 0

A prove of uniqueness of this decomposition is given in the thesis of Van Gastel (2013) given

∆At:= E[∆Xt|Ft−1], ∆Mt:= ∆Xt− ∆At

Definition 3.9. Given the ND condition, we define the mean-variance tradeoff process ˜ K = ( ˜Kt)t=0,1,...,T as ˜ Kt:= t X i=1 E[∆Xi|Fi−1] 2 Var[∆Xi|Fi−1] , for t = 0, 1, . . . , T which is assumed to be uniformly bounded as

E[∆Xi|Fi−1]

2

Var[∆Xi|Fi−1]

is P-a.s. bounded and uniformly in t Let us have d+1 assets, with price processes Si _{= (S}i

t)0≤t. S0 is taken as a numeraire

such that Xi = _SS0i are the discounted prices. A European call option with strike price K

and expiration date T can be expressed a random amount Hi=max(Xi− K, 0)=(Xi₋

K)+. Similar as in chapter 2, there are two things to do with the contingent claim H (Heath et. al., 2001): valuation/pricing (assign a value to H at times t < T ) and hedging (cover oneself against potential losses arising from sale of H).

A way to approach these problems is to consider dynamic portfolio strategies of the form ϕ=(ϑ, η)=(ϑt, ηt)0≤t≤T, where ϑ ∈ L2 is a d-dimensional predictable process and

η is adapted. L2 = L2(Ω, F , P) is defined as the Hilbert space of Rd square-integrable predictable processes with scalar product (U, Z) = E[U Z] and norm ||U || =pE[U2]. Definition 3.10. The value process Vt(ϕ) of the portfolio at time t can be written as

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The change in value ∆Vt+1 (= Vt+1− Vt) in time-period [t, t + 1] can be written as

∆Vt+1 = ϑt+1∆Xt+1+ (∆ϑt+1Xt+ ∆ηt+1)

where the first term are the gains made in [t, t + 1] and the second term the re-hedging costs. The total costs in this time-period is then defined as the difference of the rehedging costs and the incoming payment stream H = (Ht)0≤t≤T, which is an adapted square

integrable process

∆C_t+1H (ϕ) := ∆ϑt+1Xt+ ∆ηt+1− ∆Ht+1

= ∆Vt+1− ϑt+1∆Xt+1− ∆Ht+1

Definition 3.11. The cumulative gains process of trading strategy ϕ is defined by Gt(ϕ) =

t

X

i=1

ϑi∆Xi

Definition 3.12. The cumulative cost process at time t is given by Ct= Vt−

t

X

i=1

ϑi∆Xi

with C0 = V0 = η0 as the initial price.

Definition 3.13. The cost process up to time t given the payment stream H is defined by C_tH(ϕ) = Vt(ϕ) − t X i=1 ϑi∆Xi− Ht= Vt(ϕ) − Gt(ϕ) − Ht such that CH

0 = V0 = η0 is the initial price of the trading strategy and after this

∆C_tH− ∆Ht denotes the profits or losses.

Definition 3.14. A trading strategy ϕ is called zero-achieving if P-a.s. VT(ϕ) = 0 at

time T .

Definition 3.15. A trading strategy ϕ is called self-financing if the cost process P-a.s. constant. This is in complete market case.

Definition 3.16. A trading strategy ϕ is called mean-self-financing if the cost process is a martingale under P. This is in incomplete market case.

So far, we formulated option pricing and hedging in a different way than in chapter

2, but the results are similar. This means that when the market is complete we still can structure our riskless portfolio: V (H) = EQ_{[H]. As said, completeness is unlikely to}

exists and we now face V (H) = EQ_{[H]+ risky part. In the next two sections, we discuss}

the two quadratic hedging strategies by introducing their quadratic criteria functions and some derivations. These strategies are more suited for incomplete markets but they are dealing with the risky part differently. The main difference between the two strategies is that, as the name says, local risk minimization provides a local solution which is less complex than the global one provided by mean-variance. The latter provides in its own a solution with more control over the total costs and risks but computations are much more complex.

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3.2 Local Risk Minimization

The first quadratic hedging strategy is local risk minimization. F¨ollmer and Sondermann (1986) were the ones to introduce this method using a sequential regression scheme. After this, the topic has been studied intensively by the likes of Martin Schweizer, especially in continuous time. The idea is simple: if a contingent claim is no longer attainable by a self-financing portfolio, find a strategy which is at least attainable in a mean-self-financing portfolio (Van Gastel, 2013).

Consider the cost increment at time [t, t + 1], we find ∆C_t+1H (ϕ) = ∆ϑt+1Xt+ ∆ηt+1− ∆Ht+1

= Vt+1(ϕ) − Vt(ϕ) − ϑt+1(Xt+1− Xt) − ∆Ht+1

= ∆Vt+1(ϕ) − ϑt+1∆Xt+1− ∆Ht+1

For local risk-minimization, we want to find a solution for

min ϑ E h (C_t+1H (ϕ) − C_tH(ϕ))2|Ft i for each t

Definition 3.17. We define the local risk process R for t ∈ {1, . . . , T } by R0 = 0 and

∆Rt(ϕ) := E(∆CtH(ϕ))2|Ft−1

Then, if ϕ∗is a risk-minimizing strategy with VT(ϕ∗) = H P-a.s. and X a martingale,

we know that the cost process is a P-martingale such that

Rt(ϕ∗) = Var[Ct(ϕ∗)|Ft] = Var " H − T X i=t+1 ϑ∗_i∆Xi|Ft # which satisfies Var h H − T X j=t+1 ϑj∆Xj|Ft i = Rk(ϕ) ≥ Rk(ϕ∗) = Var h H − T X j=t+1 ϑ∗_j∆Xj|Ft i

Using backward recursion, a solution to this problem (proof can be found in Schweizer (1995) or the thesis of Van Gastel (2013)) is given by

ϑt+1= Cov(Vt+1 (ϕ) − Ht+1, ∆Xt+1|Ft) Var(∆Xt+1|Ft) (3.1) Vt(ϕ) = Ht+ EVt+1(ϕ) − Ht+1− ϑt+1∆Xt+1|Ft (3.2) ηt= Vt(ϕ) − ϑtXt (3.3)

This is a similar principle as we did in the binomial tree in chapter 2 as the values at time T are known.

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To compute ϑt, we use the fact that H can be decomposed as

H = V0+ T

X

j=1

ϑH_j ∆Xj+ LHT (3.4)

where LH = (LH_t )t=0,1,...,T is a martingale and orthogonal to X

E[∆LHt ∆Xt|Ft−1] = 0

such that ϑt= ϑHt is the optimal trading strategy and the value at time t is denoted as

Vt= V0+ t X j=1 ϑH_j ∆Xj+ LHt with Rt−1= E[(∆LHt )2|Ft−1] + (ϑt− ϑHt )2E[(∆XtH)2|Ft−1]

Decomposition (3.4) is called the F¨ollmer-Schweizer decomposition (FS-decomposition) and it turns out to be an important feature since finding the decomposition is equivalent to finding the locally risk-minimizing strategy for the given contingent claim (F¨ollmer & Schweizer, 1991). For this to be true, the mean-variance process ˜K has to be uniformly bounded.

The FS-decomposition basically decomposes the contingent claim in a unique way into a hedgable part and a non-hedgable part, where a self-financing hedging strategy is used for the hedgable part and a unique price for the costs of the hedge (unique due to no-arbitrage).

Definition 3.18. The unique market price of risk given X = A + M is predictable process λ λt:= E[∆Xt |Ft−1] Var[∆Xt|Ft−1] = ∆At E[(∆Mt)2|Ft−1] for t = 0, 1, . . . , T

In the next chapter, we present a method to estimate the FS-decomposition, but we first introduce the mean-variance hedging strategy in the section below.

3.3 Mean-Variance Hedging Strategy

In this section, we focus on mean-variance hedging in discrete time. We closely follow the paper of Schweizer (1995) and the general theory already explained in this chapter. In mean-variance hedging we, similar as in local risk minimization, want to find an optimal trading strategy by minimizing a quadratic criteria function given that the strategy is in L2(P) and given some initial capital c.

The mean-variance basic problem can be defined as min c,ϑ0,...,ϑT −1 E h (H − c − GT(ϕ))2 i

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The contingent claim H ∈ L2(P) considered fixed and can be, similar as done with local risk minimization, decomposed using the FS-decomposition (Schweizer, 1995):

H = H0+ T

X

j=1

ϑH_j ∆Xj+ LHT

with H0 is a constant, ϑH predictable and LH = (LHt )t=0,1,...,T a square-integrable

P-martingale with E[LH0 ] = 0 strongly orthogonal to M . Then, processes ϑH and LH are

defined by ϑt:= Cov(H −PTj=t+1ϑHj ∆Xj, ∆Xt|Ft−1) Var(∆Xt|Ft−1) for t = 1, . . . , T LH_t := E h H − T X j=t+1 ϑH_j ∆Xj|Ft i − EhH − T X j=t+1 ϑH_j ∆Xj i for t = 1, . . . , T In the complete market case, we already discussed that valuation of the contingent claim can be done by using the risk-neutral probability measure Q. This method is unique but only valid for completeness. In case of incompleteness, we can choose our probability measure by preference, which might lead to different strategies (F¨ollmer and Sondermann, 1986). Therefore, Schweizer and F¨ollmer introduced the minimal martin-gale measure (MMM) in 1991, which computed the optimal strategy for P in terms of ˆ

P and is optimal for local risk minimization. Also, Schweizer (1995) introduced another probability measure ˜P as the variance optimal martingale measure and this turns out to be the appropriate measure for mean-variance hedging (Caldentey and Haugh, 2006). We first introduce a signed measure.

Definition 3.19. A signed measure Q on (Ω, F) is called a signed martingale measure for X if all possible outcomes Ω are valued in probability measure Q, such that Q is equivalent to original probability measure P on F (and therefore by The Fundamental Theorem of Asset Pricing arbitrage-free) with dQ/dP ∈ L2(P) and X a Q-martingale meaning E hdQ dP∆Xt|Ft−1 i = 0 P-a.s. for t = 1, . . . , T If so, Q is called an equivalent martingale measure.

Next are two types of equivalent martingale measures. Define process ˆZ = ( ˆZt)t=0,1,...,T by ˆ Zt:= t Y j=1 1 − αj∆Xj 1 − αj∆Aj = t Y j=1 1 − αj 1 − αj∆Aj ∆Mj

where process α = (αt)t=1,...,T is defined as

αt:=

∆At

E[∆Xt2|Ft−1]

for t = 1 . . . T

and X satisfies the nondegeneracy condition, such that ˆZ is a square-integrable P-martingale.

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Also define process ˜Z = ( ˜Zt)t=0,1,...,T as

˜

Z := Z˜

0

E[Z˜0] where random variable ˜Z0 is

˜ Z0 := T Y j=1 (1 − βj∆Xj) ∈ L2(P)

and β = (βt)t=1,...,T a predictable process.

Definition 3.20. Define the signed measures ˆP and ˜P on (Ω, F) by setting dˆP dP := ˆZT and d˜P dP := ˜Z As we have E hdˆ_P dP∆Xt|Ft−1 i = Eh ˆZt∆Xt|Ft−1 i = Zˆk−1 1 − αt∆AtE h (1 − αt∆Xt)∆Xt|Fk−1 i = 0 P-a.s. for t = 1, . . . , T .

Schweizer (1995) called ˜P the variance-optimal signed martingale measure for X and ˆ

P the discrete-time version of the minimal martingale measure (F¨ollmer and Schweizer, 1991). These two measures are coinciding with each other if the mean-variance tradeoff process ˜K is deterministic as the process β and α then coincide.

We can now look at a different value process ˆV = ( ˆVt)t=1,...,T using the FS-decomposition

ˆ

Vt:= H0+ Gt(ϑH) + LHt for t = 0, 1, . . . , T

such that ˆV is a ˆP-martingale with P-a.s. V = ˆˆ E[H|Ft] and

E hdˆ_P

dP∆ ˆVt|Ft−1 i

= 0

for k = 1, . . . , T . Process ˆV can be interpreted as the intrinsic value process associated to H (Hofmann et. al., 1992).

Schweizer (1995) provided a decomposition (with prove) of ϑ(c)_t for all values c ∈ R

ϑ(c)_t = ϑH_t + αt ˆVt−1− c − Gt−1(ϑ(c) + (βt− αt) ˆVt−1− c − Gt−1(ϑ(c) + γt ! with γt= E h (LH_T − LH t−1)∆XtQT_j=t+1(1 − βj∆Xj)|Ft−1 i E h ∆X_t2QT j=t+1(1 − βj∆Xj)2|Ft−1 i

The first term in this decomposition ϑH_t is similar as done in local risk-minimization and determined uniquely and can therefore be seen as the pure hedging demand. The second term and third term are respectively the demands for the mean-variance purposes and the stochastic fluctuations in the mean variance ratio. The latter vanishes if mean-variance tradeoff process ˜K is assumed deterministic.

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Estimation

In this section we focus on the estimation and a numerical implementation of the two methods discussed in chapter3. For mean-variance hedging, the main inspiration is the paper of ˇCern´y (2004) in which he uses dynamic programming to compute the optimal strategy for mean-variance hedging in discrete time. For local risk minimization, Van Gastel (2013) uses a backward Monte-Carlo scheme based on work by Potters et al. (2001) to compute the F¨olmer-Schweizer decomposition and this is the main inspiration for this strategy

We start this section with the estimation method for mean-variance hedging in section4.1followed by the estimation method for local risk minimization in section4.2.

4.1 Mean-Variance Hedging Estimation

4.1.1 Pricing the Contingent Claim

Before we discuss dynamic programming we need to describe our set-up for contingent claim H. We stick to the method of ˇCern´y (2004) as it is feasible for many structures of stochastic processes and payoffs. His set-up is a multinomial tree in a six weeks time-period and only one trade each week. The logarithmic returns are regularly spaced with 2% gaps. He limits his example to only two assets Stand S0t, where S0 is the num´eraire,

and assumes i.i.d. returns. The num´eraire is set to be the risk-free return, which can be stochastic or fixed. Also, ˇCern´y used a stochastic interest rate and let the stocks have a filtered probability space (Ω, F , P, (Ft)t=0,1,...,T) with the expectation conditional on

the information at time t denoted as

EPt[X] := EP[X|Ft]

Then, the discount process up to time t can be written as S_t0= Rf,0× Rf,1× . . . Rf,t−1

S₀0 = 1 16

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The discounted gain process of our asset is then denoted similar as we already said in

3.1

Xt=

St

S_t0 and ∆Xt= Xt− Xt−1 Recall the value process of section3.1

Vt(ϕ) = ϑtXt+ ηt

and rewrite this process in a recursive way

Vt(ϕ) = Rf,t−1Vt−1(ϕ) + St0ϑtrt−1∆Xt ∆ V (ϕ) S0 ! t−1 = ϑtr_t−1∆Xt V (ϕ) S0 ! t = c + t−1 X i=0 ϑtr_i−1∆Xi+1

Given these notations, our optimal mean-variance hedging strategy is the solution of min c,ϑ0,...,ϑT −1 EP0 " S_T0 c + t−1 X i=0 ϑtr_i−1∆Xi+1 − HT 2 #

In the valuation process, parameters like volatility σ, initial stock value S0, strike

price K can be estimated from historical data and we only need to define the different martingale measures as we briefly did in section 3.3. We slightly change the notation for this part to ˇCern´y’s, but it is equivalent to the theory of Schweizer (1995).

In his paper, ˇCern´y values H by the mean value process H using the variance-optimal probabilities ˜P. He calculates the corresponding change of measure by the following formula

d˜P

dP = ˜m1|0m˜2|1m˜3|2. . . ˜mT |T −1 with conditional densities

˜ mt+1|t = qt+1|t pt+1|t = 1 − a(Rt+1|t− Rf) b a = E P t[Rt+1− Rf] EPt(Rt+1− Rf)2 and b = 1 − EPt[Rt+1− Rf] 2 EPt(Rt+1− Rf)2

Valuation of the mean-value process is similar as done in the complete market case: recursively computing Ht using risk-neutral probabilities qt+1|t= mt+1|tpt+1|tand

known option payoff HT

Ht= E ˜ P t hH_t+1 Rf i for t = 0, . . . , T − 1

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4.1.2 Dynamic Programming

This section provides the dynamic structure and estimation of mean-variance hedging given the mean value process of section4.1.1. We first give some theories from the paper of ˇCern´y (2004).

Definition 4.1. A random variable X : Ω −→ R is called exogenous if for every fixed ω ∈ Ω the value X(ω) does not depend on the choice of V₀c,ϑ(ω), ϑ0(ω), ..., ϑT −1(ω).

Recall the mean-variance minimization problem and assume that kt and Ht are

Ft-measurable, exogenous with kt > 0 P-a.s., no arbitrage and linear independency of

excess returns of basis assets. Then, if conditional matrix EP

t−1[kt∆Xt∆Xttr] is invertible

at each node of Ft−1, the problem

min c,ϑ0,...,ϑt−1 EP0 " kt V_tc,ϑ− H_t2 #

has the same optimal controls c, ϑ0, . . . , ϑt−2 as the problem

min c,ϑ0,...,ϑt−2 EP0 " kt−1 V_t−1c,ϑ− H_t−12 #

with kt−1 and Ht−1 Ft−1-measurable and exogenous, we have (proof can be found in

Appendix A of ˇCern´y (2004)) kt−1 R2 f,t−1 = EP t−1[kt] − EPt−1[kt∆Xttr] EPt−1[kt∆Xt∆Xttr] −1 EPt−1[kt∆Xt] Ht−1= EPt−1 h kt− EPt−1[kt∆Xttr](EPt−1[kt∆Xt∆Xttr])−1kt∆Xt Ht Rf,t−1 kt−1 R2 f,t−1

such that we dynamically find optimal value of ϑt−1 as

ϑD_t−1= −EPt−1[kt∆Xt∆Xttr] −1 EPt−1 " kt∆Xt V_t−1c,ϑ S_t−10 − Ht S0_t !# (4.1)

Also consider the following problem with kT > 0 and HT be FT-measurable random

variables min ϑ0,...,ϑT −1 EPt " kT V_Tc,ϑ− H_T2 # = kt V_tc,ϑ− Ht + 2_t where {} a F -adapted exogenous process with

2_t = EP

t[2t+1] + EPtkt+1(Rf,tHt+ St+10 ( ¯ϑDt )tr∆Xt+1− Ht+1)2

2_T = 0

where ¯ϑD_t is the solution of equation (4.1) after substituting contingent claim Ht−1 for

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Given the formula above, the complete mean-variance hedging problem is character-ized and if we start from t = T with kT = 1, all values of kt and Ht for 0 ≤ t ≤ T can

be calculated. Also, the initial wealth turns out to be c = H0 and the expected squared

hedging error is equal to 2

0 ( ˇCern´y, 2004). For this, rewrite the martingale measure

property as

d˜P dP = ˜mT and let’s say that

˜ mT ˜ mt = ˜m_{T |T −1}... ˜m_t+1|t such that E˜Pt[X] = EPt[ ˜mTX] ˜ mt := E P t[ ˜mT |T −1... ˜mt+1|tX] and therefore Ht S_t0 = E P t h ˜ mT |T −1... ˜mt+1|t HT S_T0 i = EP˜ t hH_T S_T0 i = H0 = ˆc

From theory above, the locally optimal hedging strategy ϑD_t is obtained from the minimization of the one-step-ahead hedging error EP

t(Vt+1− Ht+1)2. Now let Vt+1 =

RfVt+ ϑDt St(Rt+1− Rf) be a self-financing condition and the squared error is written

as

EPt

h

RfVt+ ϑDt St(Rt+1− Rf) − Ht+1

2i

So, the optimal value depends on both Ht+1 and Vt and it turns out that we find

Vt= Ht and ϑt= ¯ϑDt =

EPt[(Ht+1− RfHt)(Rt+1− Rf)]

StEPt[(Rt+1− Rf)2]

It’s optimum is in this case always Ht, which is not true for an incomplete market.

Therefore an adjustment between the difference is made between Vt and Ht

ϑD_t = ¯ϑD_t + aRf

Ht− Vt

St

with a equal to the definition in section4.1.1.

With this information, we can prove that this martingale measure notation is equal of the one in chapter 3. Define

at−1:= (EPt−1[kt∆Xt∆Xttr])−1EPt−1[kt∆Xt] (4.2)

such that the conditional change of measure is expressed as ˜

m_t|t−1= R

2

f,t−1kt(1 − atrt−1∆Xt)

kt−1

Substitute this expression in the martingale measure property to find d˜P dP = ˜mT = T Y t=1 R2_f,t−1kt(1 − atrt−1∆Xt) kt−1 = (S 0 T)2 k0 T Y t=1 (1 − atr_t−1∆Xt)

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Now, we can make two statements about kt. First, by construction we have EPt[ ˜mT = 1] such that k0= EP0 h (S0_T)2 T Y t=1 (1 − atr_t−1∆Xt) i

And second, we have

Substitute the above back into equation (4.2) brings us at at−1= EPt−1 h ∆Xt∆XttrEPt hS0 T S_t0 2 _YT j=t+1 (1−atr_j−1∆Xj ii−1 EPt−1 h ∆XtEPt hS0 T S_t0 2 _YT j=t+1 (1−atr_j−1∆Xj ii =EPt−1 h ∆Xt∆Xttr S0 T S_t0 2 YT j=t+1 (1−atr_j−1∆Xj i−1 EPt−1 h ∆Xt S0 T S_t0 2 YT j=t+1 (1−atr_j−1∆Xj i

And this coincides with the variance-optimal measure ˜Z of Schweizer (1995).

Besides the variance optimal martingale measure, ˇCern´y provided the special case of kt= EPt−1[R2f,t. . . R2f,T −1] resulting into the minimal martingale measure

dˆP dP = ˆm1|0mˆ2|1mˆ3|2. . . ˆmT |T −1 with ˆ mt|t−1= 1 − EP t[∆Xttr] EPt[∆Xt∆Xttr] −1 ∆Xt 1 − EP t[∆Xttr] EPt[∆Xt∆Xttr] −1 EPt[∆Xt]

Unfortunately, we are not sure that we can use this measure (and method) for the local risk minimization strategy, even though the minimal martingale measure is well suited for this measure. The problem contains in the fact that dynamic programming gives a global solution and not the desired local solution. For this reason we decide to stick with the estimation techniques provided by Van Gastel (2013) in the next section.

4.2 Local Risk Minimization Estimation

For this strategy we rely on the estimation of the decomposition of F¨ollmer and Schweizer described by Van Gastel (2013). This procedure is divided into two parts: the estimation of the Doob decomposition and the estimation of the FS-decomposition.

To approximate the conditional expectations of section 3.2, one needs to simulate a set of scenarios using Monte Carlo. After simulation, one can use the least squares

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method of Longstaff and Schwarz (2001). In their approach, they replaced in the L2 -space of functions in an infinite expansion of a function by a truncation of weighted Laguerre polynomials. Besides this, they also approximated the conditional expectations and variances by sample means and variances based on the Monte Carlo simulations.

The infinite expansion of the Hilbert L2-space can be written as f (x) =

∞

X

j=0

βjPj(x)

with coefficients estimated by regression as βj =

< f, Pj >

< Pj, Pj >

And then, we replace this by a truncated expansion:

f (x) ≈

M

X

j=0

βjPj(x)

given polynomials P0. . . PM, which can be chosen by preference. Van Gastel (2013) used

the same polynomials as Longstaff and Schwarz (2001), which are the weighted Laguerre polynomials: P0(x) = exp (−x/2) P1(x) = exp (−x/2)(1 − x) P2(x) = exp (−x/2)(1 − 2x + x2 2 ) Pn(x) = exp (−x/2) ex n! dn dXn(x n_{exp (−x)}

Now, assume we have done N scenario simulations and let this be the input for the integrable process X. By definition 3.8, X can be approximated by the Doob decompo-sition

X = ˜A + ˜M

with ˜A a predictable process and ˜M an adapted process that is approximately a mar-tingale. We can approximate the predictive term of this decomposition by

∆At+1= EPt[∆Xt+1]

where the conditional expectation is Ft-measurable. This expression can be

approxi-mated by the truncated function of weighted Laguerre polynomials such that

∆ ˜At+1 := f (t, Xt) = M

X

j=0

βj(t)Pj(Xt)

where βj(t) is yet to be estimated by minimizing the variance over βj(t):

min

βj

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To do this, we use the second adjustment of Longstaff and Schwarz: using the sample expectations and sample variances

¯ x := 1 N N X i=0 xi and s2:= 1 N − 1 N X i=0 (xi− ¯x)2

Validity of this is due to the Central Limit Theorem (Van Gastel, 2013) as the sample distribution (X_t+1i )1≤i≤N converges to a normal distribution. If done properly, both

expected values will coincide:

EPt[∆Xt+1− ˜At+1] = 0

The second term of the Doob decomposition can be written as

N

X

i=0

∆ ˜M_ki = 0

due to its martingale property. This means that we can write the conditional mean of this term as

EPt[∆ ˜Mt+1] ≈ 0

By approximation and due to the martingale property of the second term of the Doob decomposition we can determine βj(t) by

min βj N X i=1 ∆X_t+1i − M X j=0 βj(t)Pj(Xti) 2

From econometric literature, we know that this is similar to an Ordinary Least Squares regression of ∆Xt+1 on the Laguerre polynomials. And this expression can be solved

using a Monte Carlo version of the Doob decomposition (Van Gastel, 2013):

X = ˜A + ˜M (4.3)

In this decomposition, X is again an integrable process generate from a set N = (ωi_t)i=1,...,N_t=0,...,T starting all from X0. The two components of this decomposition are

de-fined as ˜ At:= t X j=0 ∆ ˜Aj and ˜M := X − ˜A

where ˜A a predictable process with ˜A0 = 0 and ˜M an adapted process with ˜M0 = X0

and N -approximately a martingale such that sample expectation _N1 PN

j=1∆ ˜M = 0. A

proof of validation of this decomposition can be found in Van Gastel (2013).

Now recall the value process V and let it be a martingale process. This process relies on the integrable process X and future values can at a given time of V can be calculated after having information on X till that given time. An advantaged is that we do not have to know anything of X as we use the Monte Carlo scenarios. We approximate V by ˜ Vt:= M X j=0 βj(t)Pj(Xt)

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for which βj(t) can be estimated again with

min βj N X i=1 ˜ V_t+1i − M X j=0 βj(t)Pj(Xki) 2 for ˜ V_ti := M X j=0 βj(t)Pj(Xti)

And this process is adapted and it is N -approximately a martingale given the integrable process X generated by the set N = (ωi_t)i=1,...,N_t=0,...,T and X0= 0.

Now, we can approximate the F¨ollmer-Schweizer decomposition starting from ˜ Vt:= M X j=0 βj(t)Pj(t)

and define parameter estimate ˜ ϑt+1:= M X j=0 γj(t)Pj(Xt)

where parameter γj is just like βj still to be determined. For this, we want to solve

recursively the minimization min

βj,γj

VarPt(∆Vt+1− ϑt+1∆Xt+1− ∆Ht+1)

where H is the payment stream. This expression can be approximated from the Monte Carlo scenarios as follows (see section3.2 for formula)

min β,γ N X i=0 ˜ V_t+1i − ∆ ˜H_t+1i − M X j=0 βj(t)Pj(Xt) − M X j=0 γj(t)Pj(Xti)∆Xt+1i 2 (4.4)

and after solving this expression for βj and γj, we find the optimal strategy ˜ϕ = ( ˜ϑ, ˜η).

In conclusion, having the integrable process X, given the set of scenarios N = (ωi

t) i=1,...,N

t=0,...,T starting all from X0, gives us a predictable hedge coefficient ˜ϑ and an

adapted process ˜V . From these two processes, trading strategy ˜ϕ is derived given ˜

ηt:= ˜Vt− ˜ϑtXt.

After sequential regression, the error term of the regression is defined as the cost process of trading strategy ˜ϕ

∆CH( ˜ϕ)i_t= ˜V_t+1i − ∆H_t+1i − M X j=0 βj(t)Pj(Xt) − M X j=0 γj(t)Pj(Xti)∆Xt+1i (4.5)

which has a sample mean equal to zero. This means that this approximate the op-timal local risk minimizing strategy such that the cost process, given H, CH( ˜ϕ) is N -approximately a martingale. Also,

CovN(∆CH( ˜ϕ)t, ∆Xt) = N

X

i=1

∆CH( ˜ϕ)i_t∆ ˜M_ti= 0

since ∆ ˜Ai_t is known at time t and PN

i=1∆ ˜Mti = 0. This means that the cost process is

N -approximately strongly orthogonal to ˜M from the Doob decomposition of X. And this basically is the estimation method for local risk minimization.

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4.3 Comparing Both Strategies

So far, we introduced local risk minimization in section3.2and mean-variance hedging in section3.3. Both method deal with the residual risk differently as can be seen in their quadratic criteria function. They can be represented in the FS-decomposition as well as through their recursive structure. The main difference is that mean-variance hedging optimizes its criteria in a global way and local risk minimization in a local way. This becomes more clear in sections4.1 and 4.2.

In the estimation process, we tackled the problem of finding the solution of the criteria functions and finding the conditionals in two different ways. While we rely for mean-variance hedging more on the dynamic structure, we solely focused on the determination of the FS-decomposition for local risk minimization. Both methods have their advantages and disadvantages.

A big advantage of the method we used for local risk minimization is that we do not need any information of the contingent claim and we do have to know this for mean-variance hedging. Therefore we need to determine the variance optimal martingale measure also. For local risk minimization we only rely on the Monte Carlo simulation. A disadvantage is that we have to determine the polynomials P0, . . . , PM, which can be

done in multiple ways, like the weighted Laguerre polynomials, Fourier series etcetera (Van Gastel, 2013). Main influence of this choice is the rate of convergence to a robust estimate.

Both methods allow quite some room in the stochastic structure of the stock price and the payoff structure of the contingent claim. While one can apply the Monte Carlo simulation to any stochastic structure for local risk minimization, one can also apply real data in the method described by ˇCern´y (2004) as well as on a Monte Carlo simulation. In the next chapter, we provide an example of the use of mean-variance hedging based on ˇCern´y (2004). The example limits to a basic stochastic structure and a call option and it can therefore compared with the continuous Black-Scholes estimates.

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Chapter 5

Hedging in Incomplete Markets:

An Example

In this chapter we provide an example of how pricing and hedging can be applied in practice. For this, we stick again to ˇCern´y (2004) with an example of the mean-variance hedging strategy using dynamic programming (see section4.1).

5.1 Example Set-Up

In this example, we want to compare mean-variance with the continuous time ∆-hedge of Black and Scholes (1973) for a European call option. Recall their results from chapter

2: C(t, S) = SN (d1) − Ke−r(T −t)N (d2) (5.1) d1= ln(_KS) + (r +1₂σ2)(T − t) σ√T − t , d2 = d1− σ √ T − t

And their hedging result

∆_C(t,S) = ∂C(t, S)

∂S = N (d1)

We compare this with the priced value of the contingent claim Ht, the

parame-ter ¯ϑD_t and the one-step-ahead squared hedging error ESREt(Ht+1) = EPt

h

(RfHt+

¯

ϑD_t St(Rt+1− Rf) − Ht+1)2

i

valued in t = 0. We also assume no-arbitrage, no transac-tion costs and i.i.d. returns just like ˇCern´y. Because of this we can construct our weekly stock returns based FTSE 100 data (see ˇCern´y, 2004):

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Rt=                                   

exp (0.06) with probability 0.013 exp (0.04) with probability 0.067 exp (0.02) with probability 0.273 exp (0.00) with probability 0.384 exp (−0.02) with probability 0.199 exp (−0.04) with probability 0.050 exp (−0.06) with probability 0.014

, for all t

The stock price is set at 100 and we evaluate multiple tree sizes with multiple strikes next to the example of the 6 week’s time-period tree used by ˇCern´y. Also, the risk-free rate is set at 1.00075. Calculations are done in the statistical programming language R.

5.2 Results and Analysis

In this section, we present the tables and statistics of the example. We want to compare the prices and parameters for 7 different trees, increasing the size of the tree by one step each time. For each tree we also compare multiple strike prices. The evolvement of the stock price and mean value process for a time-period of 6 weeks can be found in the paper of ˇCern´y (2004) and we only present the values at t = 0.

Below, all tables are given: Single step tree

Number of steps in tree H0 ϑ¯D0 C0 ∆ ESRE0

K=95 5.085 0.9851 5.077 0.9919 0.0083 K=98 2.247 0.8460 2.265 0.8353 0.2554 K=100 0.8197 0.5356 0.9045 0.5181 0.5601 K=102 0.1698 0.1852 0.2298 0.1932 0.3170 K=105 0.0123 0.0195 0.0105 0.0139 0.0162 K=106 0.0019 0.0030 0.0028 0.0042 0.0004

Two steps tree

K=95 5.211 0.9505 5.195 0.9585 0.0168 K=98 2.555 0.7750 2.577 0.7647 0.1407 K=100 1.244 0.5376 1.301 0.5256 0.2325 K=102 0.4693 0.2820 0.5268 0.2809 0.1766 K=105 0.0814 0.0691 0.0843 0.0638 0.0307 K=110 0.00092 0.0013 0.00102 0.0012 3.79e-05

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Three steps tree

K=95 5.361 0.9198 5.341 0.9253 0.0213 K=98 2.831 0.7398 2.851 0.7308 0.1018 K=100 1.569 0.5411 1.615 0.5313 0.1495 K=102 0.7399 0.3308 0.7910 0.3272 0.1273 K=105 0.1935 0.1163 0.2004 0.1116 0.0383 K=110 0.0077 0.0072 0.0086 0.0071 0.0005 K=115 5.47e-05 7.51e-05 0.00012 0.00014 1.42e-07

Four steps tree

K=95 5.519 0.8948 5.499 0.8981 0.0230 K=98 3.079 0.7187 3.099 0.7107 0.0801 K=100 1.845 0.5446 1.885 0.5361 0.1100 K=102 0.9839 0.3619 1.030 0.3578 0.0995 K=105 0.3248 0.1554 0.3340 0.1510 0.0406 K=110 0.0247 0.0177 0.0276 0.0178 0.0016 K=115 0.0006 0.0007 0.0010 0.0009 5.22e-06

Five steps tree

K=95 5.678 0.8745 5.659 0.8764 0.0233 K=98 3.308 0.7047 3.327 0.6975 0.0662 K=100 2.092 0.5479 2.127 0.5404 0.0869 K=102 1.208 0.3843 1.250 0.3798 0.8159 K=105 0.4642 0.1876 0.4748 0.1833 0.0403 K=110 0.0532 0.0311 0.0585 0.0314 0.0031 K=115 0.0028 0.0022 0.0040 0.0028 3.58e-05

Six steps tree

K=95 5.838 0.8579 5.819 0.8588 0.0228 K=98 3.521 0.6948 3.5389 0.6882 0.0563 K=100 2.317 0.5512 2.350 0.5442 0.0718 K=102 1.417 0.4017 1.456 0.3971 0.0691 K=105 0.6064 0.2144 0.6177 0.2102 0.0390 K=110 0.0925 0.0461 0.1003 0.0465 0.0046 K=115 0.0753 0.0051 0.0100 0.0060 0.0001

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Seven steps tree

K=95 5.995 0.8441 5.978 0.8443 0.0220 K=98 3.721 0.6875 3.739 0.6813 0.0490 K=100 2.528 0.5541 2.558 0.5478 0.0611 K=102 1.613 0.4158 1.651 0.4112 0.0598 K=105 0.7488 0.2371 0.7607 0.2331 0.0372 K=110 0.1411 0.0617 0.1511 0.0620 0.0061 K=115 0.0158 0.0092 0.0200 0.0104 0.0003

We see a couple of movements in the results. First we want to recall that the set-up is probably estimated well under a complete market situation as the contingent claim is just a simple call option not containing anything like jumps of a stochastic volatility component, so you could expect that both the mean-variance hedge and the continuous Black and Scholes hedge are performing similar. This is something the results confirm as we do not see differences of a big magnitude. What is surprising is that in general the mean-variance hedging strategy results into a cheaper contingent claim price and higher share in risky stocks than the continuous Black and Scholes, except for strike price K = 95. Apparently, the lower strike price seems more risky for the mean-variance hedge (as a higher price means more risks so less shares in the risky stock).

Besides the value and parameters in general, we see a price and value drop as the strike price increases, which is also as expected, since it is less likely that the contingent claim has a positive outcome as the strike price is higher. Also, as the tree expands, the price increases and the share in the risky stock decreases.

For the one-step-ahead squared hedging error, the results are less consistent. It is expected that it is higher for situations when the strike price is close to the initial stock price, as the mean value process contains a lot more values and one could also expect it to be increasing as the number of nodes increases. The latter is not true in this case as the ESRE is the largest at the 5 step tree and decreases afterwards.

Overall, we see that the mean-variance hedging strategy can be implemented and in this example, it results in similar outcomes as the continuous Black and Scholes equation for this simple stochastic structure and payoff. A big set-back is that the dynamic structure is computational hard to implement as it has no algorithm in polynomial time. This makes it harder to expand the tree, for example: generating the stock price for the seven steps tree will take about 0.25 seconds, while the 10 steps tree will take about 75 seconds till it is generated. This makes it a lot harder to expand. Also, we now have a tree with seven possible outcomes at each node and each time, but increasing this will also cause computational problems, similar to the one of increasing the amount of nodes. Logically, increasing both the amount of nodes as the amount of outcomes will give even more computational problems. Nevertheless, the mean-variance hedging strategy has a lot of advantages as one can implement many features of the incomplete

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market: stochastic volatility, jump components, complex payoff structures, etc. This gives the method a more realistic and risk minimizing solution than the Black and Scholes equation. A proof is not included in this thesis.

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Conclusion

In conclusion, the main interests of this thesis were to figure out how an investor can price and hedge its portfolio in an incomplete market properly in a theoretical way as well in the practical way.

We started this thesis with recalling pricing and hedging in the complete market case in chapter 2. First we described the single period binomial model and derived the trees for a stock, a risk-free bond and a contingent claim. We used replication for this situation to derive a current price for the contingent claim and a hedging demand. Then, we expended the tree towards a multiple period tree and also explained how one can price and hedge in that situation. We then took it a step further towards the continuous case. We derived how both the stock price and the contingent claim behaves for infinitely small time steps and this led to the results of Black and Scholes (1973). The assumptions of this chapter turned out to be not realistic as it was all meant for a complete market.

In chapter 3, we introduced two strategies for dealing with pricing and hedging in an incomplete market in discrete-time: local risk minimization and mean-variance hedging. But doing so, we first needed to introduce the martingale framework and some important definitions. The two strategies are called quadratic hedging strategies as both are a result of solving a quadratic criterion function. Local risk minimization results into an optimal trading strategy by minimizing the quadratic cost increments and mean-variance hedging minimized the portfolio’s total quadratic differences. In both cases, research of F¨ollmer and Schweizer (1991) could be applied as the contingent claim can be decomposed into the so-called F¨ollmer-Schweizer decomposition. This decomposition turned out to be very useful as determining this decomposition also led to the optimal strategy.

After introducing the theory in chapter4, we gave a step-by-step estimation proce-dure for both methods. We decided to choose a different approach for the methods: for mean-variance hedging we used dynamic programming inspired by ˇCern´y (2004) and for local risk minimization we used a sequential regression scheme in order to estimate the FS-decomposition, inspired by Van Gastel (2013). Both methods have their advantages and disadvantages, some explained in section4.3, but the thing what makes them truly

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efficient is the freedom and flexibility of choice for any underlying stochastic process for the stock price and payoff structure of the contingent claim.

Even though chapter 3and 4 gave enough information for our main interest in this thesis. We provided a small example of how you can price and hedge in an incomplete market in chapter 5. We have chosen for a simple stochastic structure assuming i.i.d. returns and a European call option as payoff structure. Since this example is similar to the example used in chapter2, we compared the results of mean-variance hedging with the continuous Black Scholes equation. As expected, results were similar and in-line with the theory. The only surprising result is that mean-variance hedging values the risks slightly lower than the Black Scholes equation for all strike prices and tree sizes, except for a strike equal to 95.

There is a lot of space for future research. We decided to keep the thesis simple and gave an overview of already existing literature. An obvious thing to do is to change the setting into the continuous time as most research is done in this time-setting. Chapter5

can also be extended by adding the local risk minimization example and apply it on more advanced stochastic structures (like adding stochastic volatility, a jump component or even a self-exciting jump component) or more advanced products like Van Gastel (2013) did.

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