University of Amsterdam
MSc Stochastics and Financial Mathematics
Master Thesis
Optimal Hedging in Stochastic
Black-Box Models
Author:
Wouter van Krieken Supervisors:
Dr. K.E. Bouwman (Cardano) Dr. ir. R. Lord (Cardano)
Dr. P.J.C. Spreij (UvA) Second reader:
Dr. A.J. van Es (UvA)
Abstract
Derivatives are, in general, hard to include in scenario sets that are generated by a com-plex or unknown stochastic model. We investigate whether the Hedged Monte Carlo (HMC) method can be used to price and hedge derivatives in scenario sets without us-ing specific knowledge about the scenario generatus-ing model. We provide a mathematical framework and justification for the use of the method, to determine risk-neutral prices and min-variance hedges. Thereby there is made use of a new notion, ‘self-financing in expectation’. HMC is analyzed in the Black-Scholes and the Heston model using differ-ent derivatives. Moreover, methodology to analyse and improve the method’s accuracy in determining prices and hedges is developed. These analyses focus on the basis func-tions the method uses for regressions. Furthermore, variafunc-tions on HMC are proposed. A direct minimization which restricts to self-financing portfolios is developed. This enables the use of time-dependent basis functions, which can result in more accurate optimal hedges. In addition, adjustments for the inclusion of stochastic interest rates in HMC are proposed. Features and limitations of the developed methodology are demonstrated in the KNW model, which is used for pension regulations by the Dutch central bank (DNB). Keywords: Hedged Monte Carlo, scenario sets, min-variance hedging, KNW model
Preface
For the final research of my Master ‘Stochastics and Financial Mathematics’, I wanted to do an internship in the financial industry. I was curious to experience how the things I learned during my study would relate to practice. Cardano Rotterdam gave me the great opportunity of working in an organisation for which financial mathematics is daily practice. For six months, I did research for this thesis as an intern in the Quantitative Analytics (QA) team.
I found the people of the QA team a very enthusiastic and competent group of ex-perts. I appreciate that all the ‘quants’ took the effort to teach me various lessons about quantitative analytics, pension funds and programming. The interesting discussions and laughs I had with them contributed to the amazing time I had at Cardano. I would espe-cially like thank my supervisors at Cardano, dr. ir. Roger Lord and dr. Kees Bouwman. They where always full of ideas and able to push me in the right directions when needed. I would also like to thank my supervisor from the UvA, dr. Peter Spreij, who was a great help for my thesis as well. Finally, I would like to thank my fellow student Kay Bogerd, who read along during the process and gave a lot of useful feedback.
Wouter van Krieken Haarlem, August 1, 2015
Contents
1 Introduction 6
2 Risk-neutral pricing and risk 8
2.1 Set-up . . . 8
2.2 Risk-neutral measure and completeness . . . 10
2.3 Black-Scholes model . . . 11 2.4 Heston model . . . 12 2.5 Risk . . . 15 3 Hedging 18 3.1 Optimal hedge . . . 18 3.2 Continuous hedging . . . 20 3.3 Static hedging . . . 20 3.4 Discrete hedging . . . 23
4 The Hedged Monte Carlo (HMC) Method 25 4.1 Introduction to the method . . . 26
4.2 Discussion of the HMC method . . . 28
4.3 Results in the Black-Scholes model . . . 33
4.4 Results in the Heston model . . . 41
4.5 Goodness of fit . . . 44
4.6 Basis functions . . . 51
4.7 Computational time . . . 54
5 Variations on the HMC method 55 5.1 Direct minimization . . . 55
5.2 Stochastic interest rates . . . 58
6 Hedging in DNB scenario sets 63 6.1 Introduction of the KNW model . . . 63
6.2 Simulating the KNW model . . . 64
6.3 Applying HMC . . . 67
A Descriptions of financial instruments 78 A.1 Options . . . 78 A.2 Interest rate instruments . . . 79 B Three well-known risk-neutral derivative pricing methods 80 B.1 Monte Carlo (MC) Method . . . 80 B.2 Least Squares Method (LSM) . . . 83 B.3 Finite Difference (FD) Method . . . 84
C Basis functions 86
Chapter 1
Introduction
Scenario-based studies are widely used in the financial industry. These studies consist of a (great) amount of, often stochastically generated, trajectories of relevant economic variables. Although these scenario sets can have several purposes, for example pricing of derivatives, they serve mainly as a financial risk management tool. The future profit and loss (P&L) of a certain portfolio, liability, balance sheet, or derivative can be tracked. In particular, asset and liability management (ALM) studies heavily rely on scenario sets. ALM is used by pension funds, banks and insurance companies, in order to study the balance sheet’s risk. These studies are used to determine consequences of strategic decisions or the impact of certain scenarios. In addition, if a probabilistic model is used, probabilities can be assigned to certain events.
Obviously, whatever a scenario set’s purpose is, the way scenarios are chosen is crucial. Often a stochastic model is used, which will naturally be the starting point of this thesis. In order to have realistic scenarios, these models tend to become increasingly complex. A direct consequence of this is that it is not straightforward how to price financial deriva-tives in a scenario set. Even if pricing formulas are available, including derivaderiva-tives in a scenario set can become problematic, as den Iseger and Potters pointed out [1]. This is because the scenario model and the model used to price the derivative are often incon-sistent, for two reasons. First, pricing must be done under a risk neutral measure, while scenario sets are often generated under the ‘real’ measure. Second, the scenario model’s economic assumptions can be different than the assumptions underlying the pricing for-mulas. Typically, scenario sets are based on certain views on the future development of the market, while these views are not necessary consistent with how instruments are priced in the market.
Creating realistic and useful scenario sets is a whole study itself, but this will not be the point of interest in this thesis. Different parties are specialized in developing scenario studies, mainly for ALM purposes. They supply scenario sets for external risk managers. This introduces two challenges for users who want to perform extensive analysis on a scenario set. First, the scenario generating model is often complex as various economic views are included. Second, the model might not be fully known to the user, for which the model is then virtually a ‘black box’. Therefore, pricing derivatives in a scenario context is often non-trivial.
goes hand in hand with hedging. A hedge of a derivative, is a portfolio that matches the future’s derivative’s payoff as accurate as possible. Besides that hedge portfolios can be used to determine prices, they have practical uses itself as well. Hedges are used to reduce a portfolio’s risk. However, like pricing in a black-box or complex setting, constructing hedge portfolios is far from trivial. This motivates to developing a widely applicable numerical method to determine prices and hedges of derivatives in a scenario set, which will be this thesis’ main objective. Potters, Bouchaud and Sestovic’s Hedged Monte Carlo (HMC ) method will be used as a starting point to tackle the above problem of including derivatives in scenario sets [2].
Various numerical methods are developed for pricing (e.g. Monte Carlo and Finite Difference) and for hedging (e.g. Dynamic Programming and Optimal control ). We see great potential in the HMC method, since both prices and hedges are computed. Moreover, the method relies on scenario sets, which allows us to apply the method in the setting described above, where the model is complex or unknown.
This thesis’ contribution to research on HMC is threefold. First, since HMC is a relatively new method, this thesis contributes by presenting results in different set-ups and proposing methods how to track and improve the method’s performance. Second, a mathematical justification of the method is presented, which can not be found in litera-ture. Third, some variations on the method are introduced, which enables to apply the method to a larger set of problems.
A possible application of the research done for this thesis lies in the research of pen-sion funds. Due the recent financial crisis, penpen-sion funds’ financial positions worsened. In addition, changes in society, such as public health, raise a demand for pension schemes which are sustainable in the uncertain future. Therefore, there are discussions within the pension fund sector, regulators and politics about how pension designs should be restruc-tured. One of the main topics of debate is how and whether risk should be shared between different pension participants. We think the methodology developed in this thesis can be used to compare different pension solutions, by replicating the pension outcomes with some relevant assets. In particular, replicating portfolios could be used to determine to what extend an individual bank account can achieve the same results as a more complex pension scheme in which risk is shared within different groups of participants.
Chapter 2 and 3 introduce the reader to the basic principles of pricing and hedging and develop the mathematical framework used in the thesis. The Hedged Monte Carlo (HMC) method is introduced and analyzed in detail in Chapter 4. Finally, some variations on the HMC method are proposed in Chapter 5 and the methodology is applied on an ALM model in Chapter 6.
We expect the reader to be familiar with financial derivatives. In Appendix A the payoffs of the instruments used in this thesis are described. In Appendix B three well-known derivative pricing methods are discussed, namely the Monte Carlo (MC) method, the Least Squares Method (LSM) and the Finite Difference (FD) method. The HMC method has similarities with these methods so it is convenient for the reader to understand these methods. Appendix C discusses the basis functions used in this thesis and Appendix D describes how options are priced in the Heston model in this thesis.
Chapter 2
Risk-neutral pricing and risk
Innumerable methods have been developed in order to determine prices of financial instru-ments. Some well-known methods are discussed in Appendix B. Although the techniques and underlying models tend to have great differences, almost all pricing methods are based on the fundamental principles of asset pricing. Probably the most important as-sumption is the postulate of an arbitrage-free market. As we will see in Section 2.2 this is equivalent to the existence of so-called risk-neutral measure.
In this chapter some definitions, axioms and notations are introduced in order to build a mathematical framework, which enables us to speak of prices and hedges of financial instruments. These principles form the basis of modern portfolio theory and will be used throughout this thesis. We illustrate arbitrage-free pricing in the Black-Scholes model , which is certainly the most well-known pricing model based on the discussed principle [3]. That the arbitrage principles are not sufficient to determine prices in every model, we will show by introducing another well-known model, namely the Heston model . For determining prices and hedges in such models, the notion of risk becomes important, which will be the final topic in this chapter. Risk plays a fundamental role in this thesis, since it will enable us to compare hedge portfolios.
2.1
Set-up
We use a set-up based on F¨ollmer and Schied [4], which is standard in the context of stochastic portfolio theory. Let (Ω, F , P) be the probability space, where we consider P as the ‘real world’ measure. We consider a market consisting of a + 1 assets which can be traded at times t ≥ 0. The random vector ¯St= (St0, . . . , Sta) denotes the prices of the
assets at a certain time t, from which we will assume that the time instants 0 = t0, t1, t2, . . .
are discrete. We will often abbreviate ¯Sti with ¯Si, as it is in general clear what is meant
by a time index. Although in practice it is not possible to trade at infinitely small time instants, a few times in this thesis we will consider a continuous time line R≥0, which
will enable us to elaborate on certain concepts of asset pricing and hedging. At time t, all prices up to t are known. In mathematical terms this means that ¯St is adapted to a
certain filtration (Ft) ⊂ F . Opposed to the random assets S1, . . . , Sn, the zeroth asset
will be considered as risk free (a discount factor), so S0 will be assumed to be completely deterministic or, more general, predictable with respect to the filtration. We write ¯S and
S when, respectively, including and excluding the zeroth risk-free asset, so ¯S = (S0, S).
Additionally, if the opposite is not stated, we will often assume that S has the Markov property under P. This means that the distribution of the future state only depends on the current state.
A portfolio process ¯ξ = (ξ0, . . . , ξn) is an adapted process that denotes weights for
each asset. The value (price) of a portfolio is given by the adapted process Wt = ¯ξt· ¯St,
which we will refer to as the wealth process. In this thesis, ‘·’ denotes the inner product. A portfolio process is self-financing (SF ) if there are no external cash flows. In the dis-crete case this means that ¯ξi· ¯Si = ¯ξi−1· ¯Si. In the continuous case this is expressed as
d( ¯ξt· ¯St) = ¯ξt· d ¯St.
Note that this set-up includes no bid-ask spread, so it implies a perfectly liquid and efficient market. In addition, it is assumed that any portfolio can be achieved by an investor and this does not influence the prices. We will make use of the notion of arbitrage. This means that it is not possible to have a portfolio that earns more than the risk-free asset, without taking risk:
Definition 2.1. A self-financing portfolio process ¯ξ is an arbitrage opportunity if for a fixed time T we have
P WT S0 T > W0 S0 0 > 0 and P WT S0 T ≥ W0 S0 0 = 1.
I.e., the discounted profit is non-negative and positive with positive probability. Here, W denotes the portfolio’s wealth process, Wt = ¯ξt· ¯St
A market with no arbitrage opportunities is called an arbitrage free market .
For a fixed time T , a contingent claim CT is an FT-measurable random variable,
representing an uncertain cash flow at time T . A contingent claim is called a derivative if it only depends on the stock prices up to time T , so if it is σ((St)t≤T)-measurable.
All derivatives and other financial instruments used in this thesis are described in Ap-pendix A. A contingent claim is called attainable if there exists a self-financing portfolio process such that CT = WT a.s. In this thesis, we call such a portfolio a perfect hedge.
We define subhedge and a superhedge as a self-financing portfolio process such that, re-spectively, WT ≤ CT a.s. and WT ≥ CT a.s.
It is in general not clear what a ‘fair price’ of a contingent claim is. Clearly, a minimum requirement of this notion would be that it is an arbitrage free price, i.e. adding the claim to the market would not result in an arbitrage opportunity. The following proposition shows that this definition is in essence a restricting of prices in terms of hedges:
Proposition 2.2. Suppose that the market is arbitrage free. Let C be a contingent claim. 1. C is attainable if and only if C has a unique arbitrage free price.
2. If C is not attainable, then its arbitrage free prices are given by the following well-defined open interval:
Proof. (Sketch) Note that the first assertion is a (almost) direct consequence of the sec-ond. Suppose that C is not attainable. It can be shown that the set of arbitrage free prices is convex. Denote by I the open interval above, which can be proved to be well-defined. Let ξ0 be a subhedge. Since C is not attainable and by definition of a subhedge, WT0 > CT with positive probability. Therefore, adding C with price ξ00 · S0 would
in-troduce an arbitrage opportunity. Similar reasoning holds for superhedges, so I at least contains all arbitrage free prices. Now let p ∈ I. Then any portfolio ξ0 with price p is neither a sub- or super hedge. Construct ξ00 by adding a short position of C to ξ0. Note that ξ00 has price zero, and since ξ0 is neither a sub- or super hedge, ξ00 is not an arbi-trage opportunity. So there can not be constructed an arbiarbi-trage opportunity by having a portfolio holding C with price p. Therefore, the interval I contains only arbitrage-free prices, which completes the proof.
Chapter 3 elaborates on hedging. There it becomes clear that the above concept of arbitrage free pricing is not sufficient, since in general for unattainable contingent claims the interval of arbitrage free prices is large. In Section 2.5, some basics of risks are introduced, which will enable us to further specify the concept of a fair price.
2.2
Risk-neutral measure and completeness
It is convenient to define prices as the discounted expectation of the value of a contingent claim. The real world measure P is, however, not suitable for this purpose. This is because risks can be priced differently in the market. Prices can, hence, not simply be computed by considering only the assets’ expected return. Therefore, adjusted the measures are used such that prices are discounted conditional expectations:
Definition 2.3. A probability measure Q on (P, F) is a risk-neutral measure (also called martingale measure) if EQs[St] S0 t = Ss S0 s , Q-a.s., ∀s ≤ t. (2.1) Here EQ
s denotes the expectation with respect to Q conditional on Fs. In most literature,
the set of risk-neutral measures is restricted to the measure which are equivalent to P. This means that Q and P are absolutely continuous with respect to each other, denoted by Q ∼ P.
Intuitively, a risk-neutral measure is thus a ‘probability measure conform the market prices’. The use of a risk-neutral measure can be motivated by the fact that it indeed exists in an arbitrage-free discrete market, which is stated in the following important theorem:
Theorem 2.4. The First Fundamental Theorem of Asset Pricing: A market is arbitrage free if, and only if, there exists a risk-neutral measure that is equivalent to the measure P.
We omit the proof of this theorem as well as those of the following two, as they can be found in any introduction to modern portfolio theory (e.g. [4]). The second fundamental theorem characterizes a complete market . That is, a market in which every contingent claim is attainable.
Theorem 2.5. The Second Fundamental Theorem of Asset Pricing: An arbitrage-free market is complete if, and only if, there exists a unique risk-neutral measure that is equivalent to the measure P.
In the literature, often the above two fundamental theorems are extended by char-acterizing markets as the amount of atoms in the probability space. An atom is a set A ∈ F such that any measurable subset of A has probability zero or equal probability as A. One can show that the amount of atoms is limited in a discrete complete market: Theorem 2.6. Let d denote the amount of atoms in which (Ω, F , P) can be divided. Consider a market where (a + 1) stocks ¯S take values on T + 1 discrete time instants. Then the market is complete if and only if d ≤ (a + 1)T.
This indicates that the setting of a complete discrete market is relatively simple and unrealistic. In this case, all possible asset paths can be represented in a binomial tree. Therefore, we will typically not consider complete discrete markets in this thesis. For continuous markets, however, the notion of completeness is more relevant. In the con-tinuous case, a similar result can be obtained. Loosely, we can state that a concon-tinuous market is complete if, and only if, the amount of tradeable assets is greater or equal than the amount of risk drivers [5]. We will often consider a model which describes a complete continuous market, but restrict hedge portfolios to re-balance on discrete time instants.
2.3
Black-Scholes model
Black and Scholes introduced a method to price options, using the concepts of arbitrage free pricing [3]. Since then, this method has become commonly known as the Black-Scholes model or the Black-Black-Scholes-Merton model, due to R.C. Merton’s contributions to this pricing method. Although the model, due to its assumptions that are in practice too simplistic, is often not directly applicable, the model is still important nowadays as the principles of the pricing method are widely used in various financial models. Further-more, as the model is relatively simple, the pricing formulas can be used as a benchmark for more complex models, which is exactly the way this thesis uses the Black-Scholes model as well.
In the Black-Scholes model, a stock is modelled which returns are normally dis-tributed. Hence, the asset price process S follows a geometric Brownian Motion:
St= S0exp (µ − σ 2 2 )t + σWt .
Here Wt denotes a standard Brownian Motion and the constant parameters µ and σ
denote, respectively, the drift and volatility of the stock price. The interest rate r is assumed to be constant as well. In addition, it is possible to have a hedge portfolio that can be re-balanced continuously.
Let Vt be the time t value of an option that matures at time T . Note that VT equals
t ≤ T . There are various ways to derive this formula. We follow the method using a Stochastic Differential Equation (SDE). A portfolio is constructed by having ∆t stocks
at time t, so the value Πt of the portfolio is given by
Πt = Vt− ∆tSt.
We assume a self-financing portfolio, so d(∆tSt) = ∆tdSt. Now by Itˆo’s lemma we have
dΠt= ∂Vt ∂t dt + ∂Vt ∂St dSt+ σ2 2 S 2∂2V ∂S2dt − ∆tdSt.
The hedge weight ∆t is chosen such that the above is completely deterministic. Thus
∆t= ∂S∂Vtt, such that the dStpart of the above equation vanishes. Since the above portfolio
is now less and we assume no-arbitrage, the expected return of Π must equal the risk-free rate r. So we have, dΠt = rΠt, which leads to the famous Black-Scholes equation:
∂V ∂t + rS ∂V ∂S + σ2 2 S 2∂2V ∂S2 − rV = 0. (2.2)
This is a partial differential equation with a boundary condition at t = T , at which the value of the option is known. Solving this partial differential equation for a call-option with strike K, yields the Black-Scholes formula:
Vt= SΦ(d1) − Ke−r(T −t)Φ(d2). (2.3)
Here Φ denotes the standard cumulative normal distribution and d1,2 =
ln(S/K) ± (r + σ2/2)(T − t)
σ√T − t . (2.4) Remark 2.7. In the above approach, the hedge weight on the risk-free asset is not directly used, as only the hedge weight on the risky asset is computed. The process Π is not as a the wealth process W introduced in Section 2.1. However, a perfect hedge can be computed by investing in the bank account process S0
t = B(t) = ert as well.
A typical result of this arbitrage-free method of pricing, is that the option price is independent of the drift parameter of the stock. However, we will see in Chapter 3 that this does not hold when the assumption of continuous hedging is dropped. Furthermore, the assumption of constant drift, volatility and interest rate in practice does not hold in the financial markets. More complex models are, therefore, developed in order to price derivatives. For more general stock process models and derivatives there do not exist pricing formulas in closed form. Therefore, various numerical methods are developed to price derivatives, from which some of the well-known are discussed in Appendix B.
2.4
Heston model
As we would like to consider incomplete models in this thesis as well, we now introduce another well-known model for a stock price. In the Black-Scholes model, the stock’s volatility is constant, leading to normal distributed stock returns. Heston generalized
this model, introducing the following PDE for a stock price S and stochastic (‘local’) volatility √v [6]:
dS(t) = µSdt +pv(t)SdW1(t),
dv(t) = k(θ − v(t))dt + σpv(t)dW2(t).
This model, referred to as the Heston Model , is an example of a Stochastic Volatility Model . In this model, the stock’s volatility is square root mean reverting, with mean θ and rate of reversion k. The parameter σ denotes the ‘volatility of the volatility’ and W1
and W2 are Brownian Motions with constant correlation ρ. The latter means that the
Brownian motions have quadratic covariation hW1, W2i t = ρ.
Remark 2.8. In contrast to the Black-Scholes model, the Heston Model can not be exactly simulated by a simple Euler discretization. The latter would even result in a positive probability of a negative volatility, which is not possible in the continuous version. This problem can be overcome by taking taking v0(t) = max(v(t), 0) at each time instant, but then the dynamics still do not follow the Heston Model perfectly. Various research is done on efficient and accurate discretization of the Heston model [7–9]. In this paper we will use Lord et al’s full truncation scheme in order to exclude the possibility of a negative volatility without losing to much accuracy [7]. Furthermore, simulation is done under smaller time instants than the hedging is performed, such that the error due Euler discretization is reduced.
2.4.1
Incompleteness
Typically, stochastic volatility models, like the Heston model, are not complete as the volatility is not a trade-able asset. The stochastic volatility can be seen as an extra risk driver, which can not be hedged. If a volatility dependent asset is in the market, however, the market can become complete. Romano and Touzi showed, for example, for a large class of stochastic volatility models including Heston, that the market can be ‘completed’ by adding any (but just one) European option [10] on the stock.
Thus, if the market contains no volatility dependent assets, the Heston model is incomplete and prices of contingent claims in the Heston model are not always unique. In order to price options, therefore, extra assumptions need to be made how assets with volatility risk are valued. This can be done, for example, by considering option prices in the market. In the following, we will assume that the Heston parameters can then be chosen in such a way, that the Heston dynamics are risk-neutral. This implies that the stock’s drift equals the risk-free rate and derivatives can be priced by taking the discounted expectation.
2.4.2
Option pricing in the Heston model
If we assume the Heston model’s parameters to be chosen such that the dynamics are in a certain risk-neutral measure Q, pricing can be done, having a constant interest rate r, by
Figure 2.9: The price and implied volatility as a function of call option’s strike on a stock with Heston dynamics, with initial stock price 100, long-term and initial volatility θ, v(0) = 30%, interest rate r = 5%, revert rate k = 2, correlation ρ = −0.3. Different values of the parameter σ are used, 0.0 (blue), 0.5 (red), 1.0 (yellow), 1.5 (purple) and 2.0 (green).
It is, however, not trivial how to compute ECT. For the simulations in this thesis we use a
common approach which makes use of Fourier Inversion, a pricing method introduced in Stein and Stein [11] and Heston [6]. This pricing method is briefly discussed in Appendix D.
The distribution of stock returns in the Heston model is well-known to have, generally, ‘fatter tails’ than the normally distributed stock returns in the Black-Scholes model. The difference of both models can be illustrated by plotting an option’s implied volatility for different strikes, like in figure 2.9. The implied volatility is the unique volatility such that the Black-Scholes formula is consistent with a certain option price. Clearly this is a constant for any strike in the Black-Scholes model. In the Heston model, however, implied volatilities for different strikes typically show a convex shape, known as the volatility smile. This behavior turns out to be in line with observations in the market, which is probably the main motivation for use of the Heston model. Viewing figure 2.9, the reader should be warned though, that a different parameter set-up can lead to qualitative different results.
Remark 2.10. Although implied volatility is a widely-used notion, the computation of the above smile may not be straightforward to the reader. For those readers, we clarify
what is done for computing Figure 2.9 in a chronological order:
1. Certain Heston model parameters are chosen and several option strikes are chosen. 2. The Heston option prices are calculated using Equation (D.1).
3. Using these prices, the implied volatility are computed such that they match the prices in the Black-Scholes formula.
Thus, from parameters we convert to option prices, using Heston, and than back to volatilities, using Black-Scholes.
Although the Black-Scholes model is not accepted as a correct pricing model, it is used a lot to express market option prices in volatilities.
The Heston model is a good example of a model in which perfect hedges not always exist. We are, however, interested in computing portfolios that replicate contingent claims as well as possible. To be able to compare portfolios in their ability to replicate payoffs, we will need the notion of risk , which will be introduced in the following section.
2.5
Risk
Risk is an abstract concept as there can be various drivers and the word can be referring to different concepts. As far as this thesis is concerned, our focus will be to have a quantitative and suitable measure of the risks of contingent claims. Our main motivation for being able to quantify risk is that we would like to rate hedge portfolios in the ability to reduce the risk of a certain contingent claim. Artzner, Delbaen, Eber and Heath introduced a widely accepted definition of a risk measure [12]. Artzner et al. proposed axioms for measures on random variables in L2. The latter space consist of all the random
variables for which the expectation and variance exist and are finite, which seems to be a natural requirement.
Definition 2.11. For a positive integer n, the space Ln consists of all random variables
X in (Ω, F , P) with E|X|n< ∞.
Since hedging is about minimizing the difference in the outcome of a contingent claim and a hedge portfolio, it is convenient to use a deviation risk measure, introduced by Rockafellar, Uryasev and Zabarankin [13]. This measure is similar to Artzner et al.’s risk measures, but reflects more a random variable’s uncertainty. The axioms of a deviation risk measure are given by:
Definition 2.12. A deviation risk measure is a function R : L2 → [0, ∞) satisfying the following properties, for any X, Y ∈ L2:
1. Shift-invariant: R(X + a) = R(X), for all a ∈ R. 2. Positive homogeneous: R(λX) = λR(X) for all λ > 0. 3. Sublinearity: R(X + Y ) ≤ R(X) + R(Y ).
The first and the fourth axioms expose that deviation risk measure focuses on un-certainty. The second axiom can be interpreted, loosely speaking, as ‘risk is probability times impact’. The third axiom makes sure that no extra risk is introduced when combin-ing multiple portfolios. A fifth axiom can be imposed, in which case we have a symmetric deviation risk measure:
5. Symmetry: R(X) = R(−X).
A symmetric deviation risk measure can also be obtained by replacing the second axiom by R(λX) = |λ|R(X) for any λ ∈ R.
Remark 2.13. Artzner et al’s risk measure takes values in (−∞, ∞] and is obtained slightly different. In addition, they introduce the notion of a coherent risk measure, which will not be discussed in this thesis. Rockafellar et al. showed that there is a one to one relationship for risk measures ρ and deviation risk measures R:
ρ(X) = R(X) − E(X), R(X) = ρ (X − E(X)) .
The choice of use of a risk measure or a deviation risk measure is thus simply a matter of convenience, as they can easily be exchanged with each other. Since this thesis only makes uses of the above type of risk measure (R), the word ‘deviation’ will sometimes be omitted.
Example 2.14. For any a ∈ (0, 1), the well-known Value at Risk (VaR) is given by: VaRa(X) = inf{λ ∈ R : P(X > λ) ≤ 1 − a}.
One might want to define a deviation risk measure by: RVaRa (X) = VaRa(X − EX).
However, the third (sublinearity) and fourth (positivity) axiom of a deviation risk measure are not necessarily satisfied [13]. Since these axioms seem to be a natural requirement of a risk measure, the VaR risk measure is not used in this thesis.
The standard deviation is the obvious example of a symmetric risk measure, and it is widely used as well. Besides that it is easy to compute, one can justify the use with the Central Limit Theorem, by composing the outcome of a contingent claim CT into a large
sum of independent random variables [14]. It is questionable, however, whether such an approach is justifiable. Certainly not all stock returns are exactly normally distributed. Furthermore, it is not clear whether a risk measure should indeed be symmetric, as a risk averse trader might be more interested in the amount which can be lost than the amount that can be won. If however, there is no information available whether a contingent claim should be considered as a long of short position, it makes sense to use a symmetric risk measure (see [15, §2.5] for a discussion in more detail).
Example 2.15. Suppose that one constructs a portfolio holding w and 1 − w of claims X and Y with correlation ρ, with w ∈ [0, 1]. Then the square of the standard deviation risk measure of the payoff is:
Suppose that ρ > 0 and Var(X) = Var(Y ). Then the risk is minimized if w = 1/2, so the minimization of the standard deviation risk measure imposes diversification.
The standard deviation strongly relates to a normal distribution. In practice, stock return distributions are well-known to have ‘fatter tales’, since there are sometimes events where there is a huge jump in the stock price. The main critique on the standard deviation as a risk measure is, therefore, that huge outliers are not properly taken into account, such that risk is underestimated.
Despite some disadvantages of the standard deviation as a risk measure it is used a lot in the literature. This thesis will follow this practice as this risk measure is convenient to use, because of its tractability. It should be stressed though, that in some contexts a different risk measure should be used. When considering risk in the perspective of an individual entity, for example, risk might be quantified differently.
In an incomplete setting like, for example, the Heston model, it is not directly clear what the fair price of an unattainable contingent claim is. Clearly, if a hedge portfolio exist that replicates the claim’s payoff closely, with respect to a certain risk measure, the price of the contingent claim should be close to the price of the hedge. Various research is done for generalizing no-arbitrage pricing using the notion of risk. For example, Cerny and Hodges provided a framework, referred to as good-deal-pricing, for ‘tigher-than-no-arbitrage price bounds’ [16].
Chapter 3
Hedging
In Chapter 2 we saw, considering arbitrage opportunities, that hedging plays a fundamen-tal role in pricing of financial instruments. A hedge portfolio itself can have a practical use as well. A hedge can be performed, for example, by a trader who sold a derivative, in order to reduce the risk of the trade. A second example is a company with uncertain future liabilities, which reduces the risk by trading certain correlated assets. Regulators even prescribes banks, pension funds and other institutions to be secured for the risk they are exposed to, which is often done by buying derivatives. Generally, this thesis refers to a hedge as a portfolio that is constructed with the intention to reduce the risk of a certain contingent claim.
First, we will discuss the notion of an optimal hedge, as it will not always be possible to hedge a contingent claim perfectly. A hedge strategy can be characterized by the amount of re-balancing moments on the time line, which can be done continuously, only once, or finitely many times. Second, each of these cases will be discussed in more detail in this chapter.
3.1
Optimal hedge
Suppose we have a contingent claim CT, which we would like to hedge with a self-financing
portfolio process ¯ξ. The objective is to choose ¯ξ such that the accessory wealth WT is
close to CT. Therefore, in order to compare the performance of different hedge portfolios,
we need to choose a certain metric on L. Since hedging is about reducing uncertainty, it is natural to normalize the space L to the space L0 with the map X 7→ X − EX, so considering only random variables with zero expectation. Note that the problem of finding an optimal hedge can now be expressed in terms of a deviation risk measure R:
Find a self-financing portfolio process ¯ξ that minimizes R(WT − CT). (3.1)
In the case R equals the standard deviation risk measure, the resulting portfolio process is referred to as the variance-optimal hedge. Computing this hedge will be our main concern in this thesis.
Using the self-financing condition, in the case of discrete time instants, we can extract ξ0
0, the amount invested in the risk-free asset at time zero, from WT:
WT = W1+ T −1 X i=1 ¯ ξi· ( ¯Si+1− ¯Si) = ξ00S 0 1 + ξ0· S1+ T −1 X i=1 ¯ ξi· ( ¯Si+1− ¯Si).
Since ξ00S10 is deterministic, we see by definition 2.12 of a deviation risk measure, that the problem of finding an optimal hedge (Equation (3.1)) is independent of ξ0
0. In other
words, the initial value of the hedge portfolio W0 can be freely chosen. The most natural
assumption is to have the portfolio’s initial value such that the portfolio’s expected value at time T equals the derivative’s expected payoff:
EWT = ECT.
In the case of a attainable contingent claim, the above approach leads to a perfect hedge: Proposition 3.1. Let CT be an attainable contingent claim. Let ¯ξ? be a variance-optimal
hedge with respect to a deviation risk measure R, such that EWT? = ECT. Then ¯ξ? is a
perfect hedge.
Proof. Since CT is attainable, there exists a portfolio process ξ0 such that WT0 = CT a.s.
By the positivity property of a deviant risk measure, we have then R(WT0 − CT) = 0.
Since ¯ξ? minimizes R(W?
T − CT), we obtain R(WT?− CT) = 0 as well. Therefore, we have
W?
T − CT = c a.s. for certain c ∈ R. Since EWT = ECT, we obtain c = 0. We see that ¯ξ?
is a perfect hedge, which completes the proof.
Moreover, imposing the condition EWT = ECT, it can be proved that the problem is
now equivalent to the following minimization problem:
Find a self-financing portfolio process ξ that minimizes E(WT − CT)2. (3.2)
So now we are minimizing with respect to the so called mean squared error . That the minimum variance approach is indeed equivalent to the mean squares approach, will be the content of proposition 3.8. There, the result is proven for the static hedge case, but the same can easily be done for the case of discrete hedging or continuous hedging. Remark 3.2. Commonly known concepts are delta hedging and gamma hedging. In the case of a complete market, like we saw in the Black-Scholes case, the perfect hedge weight of the stock was given by ∆ = ∂ V∂S. Then, any movement in the stock price is compensated by the hedge portfolio. In the case of an incomplete market, this can obviously not be achieved. In that case, a delta hedge can be defined as the hedge such that is has the same derivative with respect to S in expectation.
Gamma hedging means that the hedge portfolio is constructed, such that the second order derivative ∂∂S2V2 is neutralized.
3.2
Continuous hedging
In Section 2.3 we saw that an option can be hedged perfectly in the Black-Scholes model. Here, the hedge portfolio was constructed by holding ∆ := ∂C∂S of the stock (delta hedg-ing). It can be shown that in the Black-Scholes Model, any simple contingent claim can be hedged perfectly [5, chap. 7]. In this context, a simple contingent claim is one that only depends on a stock’s value at a certain time. The Black-Scholes model is therefore a complete model. However, not necessarily every model that allows for continuously hedg-ing is complete. For example, the Heston model introduced in Section 2.4 is incomplete if the volatility is a non-tradable risk driver. In addition, in models for which stock prices do not have continuous paths, perfect hedging is often impossible. Various research is done on variance-optimal hedging in continuous time. Refer, for example, to [17, 18].
For the remainder of this thesis, continuous hedging is not considered as it is in practice not applicable. Some Black-Scholes results will, however, be used as a benchmark for the performance of introduced pricing and hedging methods.
3.3
Static hedging
In the case of a static hedge, a hedge portfolio is chosen at time t = 0 and remains fixed until maturity T of the contingent claim. In the case of a model with one risky asset, the variance-optimal hedge is relatively simple to compute, which is well-known in the commodity future market as the optimal hedge ratio [19]:
Proposition 3.3. Consider a one time-step model with one risky asset S. Let CT be a
contingent claim with Std(CT) > 0. The variance-optimal hedge ratio is given by:
∆∗ = Cov(ST, CT) Var(ST) = Corr(ST, CT) Stdv(CT) Stdv(ST) . (3.3) Proof. The time T payoff variance of a portfolio holding ∆ risky assets and the contingent claim −CT is given by
Var(∆ST − CT) = (∆)2Var(ST) + Var(CT) − 2∆Cov(ST, CT).
The above is quadratic in ∆ with minimum at ∆∗ = Cov(ST, CT)
Var(ST)
.
This can be generalized to multiple dimensions. Note that, in the case the assets are independent, the hedge ratios equal to the hedge ratios in the one-dimensional case: Proposition 3.4. Consider a one time-step model with one risky assets S = (S1, . . . , Sn)T.
If S is linearly independent, then variance-optimal hedge ∆∗ is given by ∆∗ = Cov(S, CT)TCov(S)−1.
Proof. By differentiating Var (∆ · ST − CT) = n X i=1 (∆i)2Var(STi) + Var(CT) + 2 n−1 X i=1 X i<j≤n ∆i∆jCov(Si, Sj) − 2 n X i=1 ∆iCov(Si, CT)
with respect to ∆, and setting it equal to zero, we obtain ∆T ∗ Cov(S) = Cov(S, CT).
Since S is linearly independent, Cov(S) is invertable. The result follows.
The above results show that having the standard deviations and correlations, finding a variance-optimal static hedge is relatively straight forward.
Remark 3.5. In the case the market has only one risky asset S, the variance-optimal hedge ratio ∆ of a claim CT ∈ FT results in the hedge error variance
(1 − Corr(ST, CT)2)Var(CT).
The ‘one-period’ market is thus complete if and only if ST and X are perfectly correlated
for any X ∈ FT with Std(X) > 0. In this case, any function f (ST) is perfectly correlated
with ST, which is only possible if ST can attain only one or two values. This illustrates
theorem 2.6, as F can then consist of a maximum of two atoms.
Example 3.6. Suppose we would like to perform a variance-optimal static hedge on a call option CT = (ST − K)
+
in the Black-Scholes model. In this case, ST is log-normal
distributed with parameters µ0 = log S0+ (µ − σ2/2)T and σ0 = σ
√
T . For determining ∆∗ we need to compute:
Cov(ST, CT) = E[STCT] − E[ST]E[CT].
Note that E[ST] = S0exp(µT ) and E[CT] can computed by using the Black-Scholes
formula with µ as the interest rate constant:
E[ST]E[CT] = S0eµTeµT S0Φ(dµ(0)) − Ke−µTΦ(dµ(−1)) = S02e2µTΦ(dµ(0)) − KS0eµTΦ(dµ(−1)), where dµ(n) is given by dµ(n) = log(S0/K) + (µ + σ2/2)T σ√T + nσ √ T . Now we compute, using the fact that S2
T is log-normal distributed with parameters 2µ 0
and 2σ0: E[STCT] = E[ST (ST − K) 1ST>K] = E[ST21S2 T>K2] − KE[ST1ST>K] = e2µ0+(2σ0)2/2Φ − log K + µ 0+ 2σ02 σ0 − Keµ0+σ02/2Φ − log K + µ 0+ σ02 σ0 = S02e(2µ+σ2)TΦ log( S0 K) + (µ + 3σ 2/2)T σ√T ! − KS0eµTΦ log(S0 K) + (µ + σ 2/2)T σ√T ! = S02e(2µ+σ2)TΦ(dµ(1)) − KS0eµTΦ(dµ(0)). Using Var(ST) =
eσ2T − 1S02e2µT, we can now conclude that the variance-optimal hedge ratio is given by
∆∗ = 1 eσ2T − 1 eσ2TΦ(dµ(1)) − K S0 e−µT (Φ(dµ(0)) − Φ(dµ(−1))) − Φ(dµ(0)) . (3.4) The option can now be priced by putting C0 = W0 and ECT = EWT:
C0 = W0
= e−rTE[CT] + W0− e−rTE[CT]
= e−rTE[CT] + ¯ξ0 · S0− e−rTE[ξ¯T · ST]
= e−rTE[CT] + ∆∗S0 1 − e(µ−r)T .
Note that if r = µ we obtain the Black-Scholes price e−rTE[CT]. The Black-Scholes hedge
delta equals dr(0), and is different from the hedge obtained here, however. For example,
if µ = r = 0.5, σ = 30%, K = S0 = 100, the Black-Scholes delta equals 0.56 while the
static delta equals 0.59. Figure 3.7 shows the resulting price and hedge ratio as a function of the drift.
If r 6= µ, the stock’s dynamics are not risk-neutral. One can argue that the price of the hedge in this case does not necessarily equal the option price. In this case, the quantity C0 can be seen as ‘production cost of replicating the payoff’, as argued by Bertsimas et
al. [15, §2.5].
In this thesis, risk is often minimized using a mean-squared risk measure. The follow-ing proposition shows that this approach is equivalent to a minimization of the variance or standard deviation. The minimization with squares, in a Monte Carlo setting, turns out to be more straightforward than a variance minimization. Furthermore, a price is computed.
Proposition 3.8. The following minimization results in a variance-optimal hedge ratio: min
C0,∆E
(C0+ ∆ST − CT)2 . (3.5)
Moreover, C0 = ∆E[ST] − E[CT], so the expected result of the hedge equals the derivative’s
Figure 3.7: The optimal hedge portfolio price and hedge ratio as function of the drift of a call option on a Black-Scholes stock, with initial stock price 100, volatility 30%, interest rate 5%.
Proof. By definition of the variance, we have
E(C0+ ∆ST − CT)2 = Var [∆ST − CT] + E [C0+ ∆ST − CT]2.
Using Bellman’s principle of optimization, we can first minimize with respect to C0:
min C0,∆ Var [∆ST − CT] + E [C0+ ∆ST − CT]2 = min ∆ Var [∆ST − CT] + min C0 E [C 0+ ∆ST − CT]2 .
Now clearly C0 = ∆E[ST] − E[CT] is optimal, and the second term vanishes. So we have
showed that the minimization is now equivalent to the problem of finding the minimum-variance optimal hedge:
min
∆ Var [∆ST − CT] = minC0,∆E
(C0+ ∆ST − CT)2 .
3.4
Discrete hedging
In the case of discrete hedging (also referred to as dynamic hedging), the portfolio is re-balanced at time instants t0, t1, . . . , tT −1. Obviously a discrete hedge should be able to
continuous case. In view of theorem 2.6, only relatively simple models can be complete under discrete hedging.
For small time increment sizes, the results from continuously hedging can be used, in order to compute a nearly perfect hedge, and thus close to optimal hedge. However, if time increments are not small, formulas obtained from continuous models can result in wrong prices and sub-optimal hedges, as we already saw in the case of static hedging. Furthermore, in many cases no perfect hedge exist and there is some risk left-over even after performing an optimal hedge, which is sometimes referred to as basis risk .
Wilmott used a Taylor expansion based approach in order to determine minimum-variance hedges of options in the Black-Scholes model [20]. Wilmott derived an option price as well, but the formula is wrong due to a mistake, as he remarked on his forum [21]. As far as this thesis’s author knows, no analytic formula exists. An interesting result from Wilmott is, however, that the price and hedge are dependent on the stock’s drift, in contrast to the continuous case. Moreover, this correction to the drift is proportional to the size of the time instants. If re-balancing is done relatively frequently, ordinary results from the continuous case can be used. If however, the stock’s drift is not equal to the interest rate and hedging takes place on discrete time instants, the option’s price and hedge depend on the stock’s drift parameter.
Schweizer published several papers on variance-optimal hedging in discrete time in-stants, for example [17, 22, 23]. Here, existence of the optimal hedge and expressions for the hedge portfolio are derived under different conditions. These expressions are, unfortunately, not directly applicable to specific models, for example the discrete version of the Black-Scholes model. Bertsimas, Kogan and Lo derived expressions for optimal hedges in more specific cases [15], in the case of a Markov model. These results, still require numerical methods in order to determine the prices and optimal hedges. In this thesis, therefore, different methods are used to determine performance to certain hedge methods. Often we will use the Monte Carlo (MC) method as a benchmark, introduced in Appendix B.
Chapter 4
The Hedged Monte Carlo (HMC)
Method
The (Optimal) Hedged Monte Carlo (HMC) Method, introduced by Potters, Bouchaud and Sestovic [2], is inspired by the Least Squares Monte Carlo (LSM ) method (Sec-tion B.2). Both methods use regressions on Monte Carlo (Sec(Sec-tion B.1) simula(Sec-tions. A difference is that the LSM method uses regressions to determine when an American-style derivative needs to be exercised, while the HMC method uses regressions to determine the value and hedge weights at each time instant.
An interesting feature of HMC is that it uses paths which are not necessarily gener-ated under the risk-neutral measure. Therefore, in contrary to many widely used pricing methods (see Appendix B), a model can be used that generates scenarios from the ‘real’ measure, which is typically the case with ALM. Moreover, since the method makes use of scenarios, no knowledge is needed about the scenario generating model. Therefore, the method can be applied to our main problem described in the introduction Chapter 1, namely pricing and hedging in models which are unknown or to complex to derive analytic equations for.
Probably since the HMC method is relatively new, it is not as well known as other pricing or hedging methods. Moreover, there is little research published on HMC. Petrelli et al. applied and analyzed the method for multi-asset derivatives [24]. Moreover, they listed various advantages of the HMC approach and discussed the inclusions of transaction costs and conditions on the hedge portfolios’ volatility [25]. Transaction costs and hedge portfolio volatilities are not taken into account in this thesis.
This chapter contributes to the little research available on HMC by applying the method to different models and analyzing various aspects of the method. In addition, we provide a mathematical justification of the method by introducing the new notion self-financing in expectation (SFE ).
This chapter is structured as follows. First, the HMC is introduced and discussed in detail. After that, the method is applied to both the Black-Scholes model and the Heston model. Furthermore, the method’s use of basis functions is tested and analyses are done on the regressions. By doing different analyses, we develop insights in how well the method works, how it can be used, and how we can track the performance of the method in pricing and determining hedge portfolios. The chapter is closed with some
reports on computational time.
4.1
Introduction to the method
We return to Chapter 2’s set-up, so we have an adapted stock price process St =
(S1
t, . . . , Stn) and an adapted portfolio processes ξt = ξ1t, . . . , ξnt to be constructed. We
write ¯St and ¯ξt if we include the zeroth risk-free asset in the vectors as well. Suppose we
want to price and hedge a derivative CT with payoff at time T . Recall from Chapter 3
that our objective is to compute minimum variance hedges, and that this is equivalent to finding the least squares hedge. So our main objective is the following minimization problem:
min
¯
ξ E
(CT − HT)2 . (4.1)
Here the ¯ξ are portfolio processes that are re-balanced at discrete time instants ti. The
random variable CT denotes a certain contingent claim’s payoff and the process Htdenotes
the result of the hedge portfolio ¯ξ: Ht = C0+ t−1 X i=0 ¯ ξi· ¯Si+1− ¯Si .
Here we have defined a new hedge process H, instead of using the wealth process W , since we do not consider self-financing portfolios at this stage. The local risk Ri at time
ti is defined by Ri = E h Ci+1− ¯ξi· ¯Si+1 2i1/2 . (4.2)
Here Ci denotes the estimated value of the claim at time ti, which is given by
Ci =
( ¯ξ
i· ¯Si i < T
CT i = T.
(4.3) Clearly, Ci is Ft-adapted. Moreover, having chosen any ¯ξi, equation (4.2)’s conditional
expectation can not be calculated easily. In a MC context, however, the expectation can be estimated as an average over all paths.
Since CT is given, backward induction can be used to construct a hedge portfolio
weights ¯ξi as a function of Si, such that at each time instant the local risk is minimized:
¯
ξi = arg min ¯ ξi
Ri, i = 0, . . . , T − 1. (4.4)
Recall that the portfolio process ¯ξ has to be adapted, so the ¯ξi are restricted to be Fi
-measurable. In many cases, S is the only stochastic process that is observed, and it is Markov under P as well. In that case, we can assume the ¯ξi to be Si-measurable, which we
will assume in the discussion of the HMC method. In order to include this dependency of the stock process, a set of l basis functions Lkis used (see appendix C on basis functions).
At time ti, the jth asset portfolio weight, is then given by: ξij = l X k=1 aji,kLk(Si). (4.5)
For our first analyses, we use polynomial basis functions and consider only one random asset. So the basis functions are given by
Lk(s) = sk−1.
Summarizing, the HMC method uses backward induction and performs a linear least squares regression at each time instant, minimizing local risks.
Remark 4.1. Potters et al.’s paper uses a slightly different approach for the definition and optimization of the local risk. In their paper the deterministic asset is not in the price vector S, and a constant interest rate r over each time instant is chosen. Instead of letting the derivative price be given by ¯ξ · ¯S, the price Ci at time ti is approximated by
basis functions. Equation (4.2) is then given by
R0i = Eh e−rCi+1(Si+1) − Ci(Si) + ξi(Si) · (Si− e−rSi+1)
2i1/2 .
Note that the above includes ξ and not ¯ξ, so the risk-free asset weight is not estimated. We now show that both approaches are equivalent, if the interest rate r over each time increment is constant. We put, for any i < T :
Ci = ξi0S 0
i + ξi· Si. (4.6)
In Potters et al.’s paper, ξi and Ci are determined by regression at each time instant,
while in this thesis ξi0 and ξi are computed. Therefore, this thesis’ ξi0 can be derived from
Potters et al.’s Ci and vice versa. Furthermore, from equation (4.6) we see ξ0i and Ci can
be expressed in each other in any system of basis functions.
Using Si0 = e−rSi+10 , we can now show that that Ri is a multiple of R0i:
Ci+1− ¯ξi· ¯Si+1 = Ci+1− erξ¯i· ¯Si− ¯ξi· ¯Si+1+ erξ¯i· ¯Si
= Ci+1− erCi− ¯ξi· ¯Si+1+ erξ¯i· ¯Si
= Ci+1− erCi+ ¯ξi· erS¯i− ¯Si+1
= ere−rCi+1− Ci+ ξi· Si− e−rSi+1 ,
so Ri = erR0i. Hence, we have shown that both minimization problems are equivalent
if interest rates are constant, by showing that R0 is basically the discounted version of R. In the case of stochastic interest rates, both approaches can differ. The inclusion of stochastic interest rates in HMC is discusses in more detail in Section 5.2.
The notation used in this thesis is more convenient for the adjustments made in this thesis. Using Potters et al.’s definition of R0, however, it is more clear that in fact prices are determined along each path.
4.2
Discussion of the HMC method
Potters et al. present HMC as a method for both pricing and hedging, but omitted the mathematical details in their paper. In this section we justify the method mathemati-cally, by introducing the notion of ‘self-financing in expectation’. As far as this thesis’ author’s knowledge reaches, this approach can not be found in the literature. In this section’s discussions, the errors introduced by the regressions are not taken into account, as this will be the topic of Section 4.5. After having discussed the HMC minimization in detail, several ways to generalize the method to more complex problems are discussed.
In our analyses we will consider three minimization problems, which were introduced in the previous section:
1. Equation (4.1), the direct computation of the optimal hedge.
2. Equation (4.4), the backward induction version of the previous problem.
3. Equation (4.5), the HMC minimization, which is the previous problem estimated with basis functions.
For all these three minimization problems, it is easy to see that they have a solution. All domains are closed spaces, and the objective has lower limit zero.
4.2.1
Self-financing in expectation
The HMC method does not necessarily compute self-financing portfolios. In Chapter 5, a variation of the method is presented that restricts to self-financing portfolio’s. We will show that HMC produces portfolios having the following notion of self-financing:
Definition 4.2. Let ¯ξ be a portfolio process and CT be a contingent claim. Let (Ci)0≤i≤T
be defined as Equation (4.3). Then we call ¯ξ self-financing in expectation (SFE or P-SFE) if
Ei
¯
ξi· ¯Si+1 = Ei[Ci+1] . (4.7)
An interpretation of a SFE hedge portfolio can be, that hedge errors are immediately compensated with an inflow or outflow of cash. The following results show that the backward induction approach indeed computes a SFE portfolio process:
Lemma 4.3. Let a ∈ R \ {0} be fixed, f : X → Rm a function with domain X . Suppose
that the following minimization problem is well-defined (the minimum exists): min
c∈R,X∈Xkac1 − f (X)k 2 .
Here 1 denotes the unit vector in Rm and k · k the standard norm on Rm. Then for any
solution (c, X) ac = 1 m X f (X) holds.
Proof. The minimization problem can be solved sequentially, so we can first consider min c∈R kac − f (X)k 2 = min c∈R n m(ac)2+ kf (X)k2− 2acXf (X))o. Differentiating with respect to c, we find that for an optimal solution
2ma2c − 2aXf (X) = 0 must hold. Since we assumed a 6= 0, we obtain the result.
Lemma 4.4. Let (P, F, Ω) be a probability space, and let G ⊂ F be a σ-algebra. Let Y ∈ F be a random variable with E[Y2] < ∞. Suppose that the following minimization problem is well-defined:
X = arg min
X∈GE(Y − X) 2 .
Then X = EG[Y ], P-a.s.
Proof. We prove by contradiction. Suppose that X 6= EG[Y ]. Assume, without lost of
generality, that there exists an > 0 and A ∈ G such that E1AX < E1AY +
and P(A) > 0. We define X0 = X + 1A. Note that X0 ∈ G. What is left to show is that
E(Y − X)2 − E (Y − X0)2 = E h (Y − X)2− (Y − X0)2i = Eh1A(Y − X)2− 1A(Y − X0) 2i = E1A(Y − X)2− 1A(Y − X − )2 = E1A 2(Y − X) − 2 > P(A)2 > 0.
Proposition 4.5. Let CT be a contingent claim and let ¯ξ be a portfolio process defined
by Equation 4.4 (the min-variance backward induction approach). Then ¯ξ is SFE. Proof. Let i be fixed. Then ¯ξi is defined such that Equation (4.2) is minimized, thus
ξi0 = arg min ξ0 i∈Fi E h Ci+1− ξi · Si+1− ξi0S 0 i+1 2i . Since S0
i+1> 0, the above problem can be written in the form of lemma 4.4:
ξi0 = arg min ξ0 i∈Fi E Ci+1− ξi· Si+1 S0 i+1 − ξ0 i . By the lemma we obtain
EiCi+1− ¯ξi · ¯Si+1 = 0.
Now, a similar result can be obtained for HMC. A difference is, however, that there is not minimized over a certain expectation, but over the average in a set of scenarios. We will, therefore, write Emc for the MC average over the scenarios:
Proposition 4.6. Suppose that there is a linear combination of basis functions that is constant. The HMC methods computes a portfolio process that is self-financing in Emc-expectation.
Proof. Assume, without lost generality, that the basis functions contain a constant func-tion Lk. Similar to Proposition 4.5, we can now apply Lemma 4.4 to obtain the result
(minimizing over the weight a0i,k).
4.2.2
HMC as a pricing method
Clearly, if the algorithm computes an (almost) perfect hedge, the estimated price is the cost of the hedge, which equals the unique arbitrage-free price. Therefore, the method computes correct prices and hedges in a (almost) complete model. The following propo-sition shows that, in a risk-neutral setting where perfect hedging is not possible, prices can be estimated as well. Here we assume, for ease of notation, a constant interest rate, but the result can easily be generalized.
Proposition 4.7. Let Q be a risk-neutral measure, hence ¯
Si = e−rEQi
¯ Si+1 .
Then ¯ξ is self-financing in Q-expectation if, and only if, C equals the risk-neutral price process of CT.
Proof. The process ¯ξ is self-financing if and only if EQ
i [ ¯ξi· ¯Si+1] = EQi [Ci+1]. Since Q is
risk-neutral, this is equivalent to erC
i = EQi [Ci+1], which completes the proof.
Hence, if the HMC method is applied in a risk-neutral setting, the computed accessory price process C is an unbiased estimator of the risk-neutral prices of the contingent claim. So far, we have not discussed accuracy. Potters et al. showed that, in a Black-Scholes set-up, HMC computes an option price with higher accuracy than standard MC. We replicate this result in Section 4.3. Now, we are going to argue, heuristically, why the HMC method can compute more accurate prices than a standard MC. Thereby we ig-nore the error introduced by the use of regressions, which will be topic of discussion in Section 4.5.
Since HMC computes Emc-SFE portfolios, for each time instant i, HMC computes
portfolio weights such that
Emc[ ¯ξ · ¯Si+1] = Emc[Ci+1].
Here Emc means that we are averaging over scenario set paths. The error
i in the above
estimation should be close to i = Emc
h
Ci+1− ¯ξi· ¯Si+1
2i1/2 ≈ Ri.
The accuracy can thus be estimated as √i
N, where N denotes the number of MC paths
used. The standard deviation of the hedge error thus provides an estimation of the pricing error E at t = 0:
E = r P
i2i
N .
So, while the MC method’s error depends on the contingent claim’s payoff variance, the HMC reduces this variance by computing a hedge which reduces the payoffs uncertainty. An extra error is, however, introduced by the use of basis functions.
Remark 4.8. The above heuristics illustrate that HMC has similarities with the MC’s control variate method (see Section B.1). An advantage of HMC over the control variate method is that portfolio weights and prices along paths are computed. An advantage of the control variate method in comparison with HMC is, however, that it does not rely on regressions.
In a non-risk-neutral setting it is questionable whether C0 can be viewed as a price
if some basis risk is left. In that case, the quantity C0 can better be seen as ‘production
cost of replicating the payoff’, as Bertsimas et al. pointed out [15, §2.5].
4.2.3
HMC as a hedge method
HMC computes the hedge portfolio by backward induction. The following theorem jus-tifies this approach, if the stock process ¯S is risk-neutral.
Theorem 4.9. Let Q be a risk-neutral measure. Let all F-adapted portfolio processes which are Q-SFE be denoted by Ξ. Then an optimal portfolio process ξ? which is defined by ξ? = arg min ξ∈Ξ E Q(C T − HT?) 2 can be computed by backward induction:
ξi? = arg min ξi∈FiE Q i h Ci+1− ξiS¯i+1 2i , i = T − 1, T − 2 . . . , 0. Proof. For any Q-SFE portfolio process:
CT − HT = CT − ¯ξ0· ¯S0− T −1 X i=0 ¯ ξi· ¯Si+1− ¯Si = CT − ¯ξ0· ¯S0− T −1 X i=0 ¯ ξi· ¯Si+1+ T −1 X i=0 ¯ ξi· ¯Si = CT − ¯ξ0· ¯S0− T −1 X i=0 ¯ ξi· ¯Si+1+ T −1 X i=0 Ci = T −1 X i=1 Ci+1− ¯ξi· ¯Si+1 .
If we take the square and the expectation, all cross terms equal zero, since for any i < j we have EQ Ci+1− ¯ξi· ¯Si+1 Cj+1− ¯ξj · ¯Sj+1 =EQ EQj Ci+1− ¯ξi· ¯Si+1 Cj+1− ¯ξj · ¯Sj+1 Tower rule =EQ C i+1− ¯ξi· ¯Si+1 EQj Cj+1− ¯ξj · ¯Sj+1
ξi, Si+1, Ci+1 ∈ Fi+1⊂ Fj
=0 ξ is SFE.¯
Combining both results we obtain EQ[(CT − HT)2] = EQ[ T −1 X i=0 Ci+1− ¯ξi· ¯Si+1 2 ] = T −1 X i=0 EQ[ Ci+1− ¯ξi· ¯Si+1 2 ] = R2i. In addition, by Proposition 4.7 we see that the process C is independent of the choice of SFE portfolio process ¯ξ. Therefore, the above can be minimized by backward induction. The above proof uses Proposition 4.7, which only holds in the risk-neutral measure. This raises the question whether backward induction is indeed equivalent to direct min-imization in a non-risk-neutral setting, for example the ‘real measure’. In general, the answer is negative.
Remark 4.10. The variance risk measure is well-known to be time-inconsistent . That means that optimal hedge portfolio processes computed at t = 0 is not always optimal at t > 0. This is counterintu¨ıtive, since an investor has incentive to deviate from the optimal strategy initially computed, although the optimal strategy at t = 0 outperforms the strategies which are allowed to be changed. This phenomenon is introduced in literature by Strotz [26]. For more background on time-consistent mean-variance hedging, please refer to Basak and Chabakauri [27, 28]. In the latter three papers referred to, it is argued that in many cases only time-consistent strategies should be considered, since these are the only ones that are actually performed.
If the model is close to complete or close to risk-neutral, the direct solution and the backward solution are close to each-other. Moreover, it can be argued (Remark 4.10, that backward induction is the right way to compute portfolios. Therefore, we do not expect that any problems are introduced by the use of backward induction. Section 5.1 elaborates on direct minimization.
4.2.4
Extensions
As Potters et al. pointed out, the HMC method allows for some natural generalizations. First, the vectors ξt and St can be set to hold multiple assets instead of just one
risk-free asset and one uncertain asset [24]. Furthermore, the basis functions can depend on more state variables than just the asset prices, letting the portfolio depend on more (nontradable) factors. This way, path-dependent derivatives can be included. For exam-ple, barrier options can be priced by adding a state variable which indicates whether the barrier is hit. Second, the HMC method can be adjusted for American-style derivatives.
The latter property will not be discussed in this thesis, as this is already done by Potters et al. Fourth, Petrelli et al. discussed how transactions costs and risk premiums can be included [25]. In Chapter 5 some variations on the HMC method are discussed.
4.3
Results in the Black-Scholes model
In this section, HMC results are presented in a relatively simple setting, namely the Black-Scholes model. Having shown that the method indeed estimates prices and optimal hedges, the next goal is to investigate the accuracy of the method. This section’s aim is to develop quantitative as well as qualitative tests on the method’s performance. The insights gained can later be used for hedging in more complex models and using different methods.
4.3.1
Black-Scholes option prices
First we verify the method by pricing an option, replicating Potters et al.’s results. Like in their paper, we use 8 polynomial basis functions (so order 7). We simulate a stock price with Black-Scholes dynamics, having the drift equal to the interest rate, such that the price under the discrete hedging equals the usual Black-Scholes price (Section 3.4). Moreover, having the drift term equal to the risk-free rate makes the model risk-neutral, so results can be compared with Section B.1’s MC method as well. Figure 4.12 shows the comparison of the prices of an option using both the HMC and the MC method. Over a number of 500 simulations, the HMC method results in an average price of 6.56 with a standard deviation of 0.07. We obtain similar results as in Potters et al.’s paper. The MC computes an average price of 6.63, with standard deviation 0.43. Thus, both methods converge to the same Black-Scholes price 6.58, but the HMC method’s error is about six times smaller than for the MC method, which is a similar result as Potters et al. obtained.
Remark 4.11. The HMC regressions take some extra computation time in comparison with the MC method. One could, therefore, argue that computation time should be taken into account when comparing the HMC and MC accuracy. However, since linear regressions are used, computation time of HMC is relatively small. This will be discussed in more detail in Section 4.7.
Moreover, if there is only limited data available and one is not able to simulate extra paths, a variance reduction can be very valuable. In Chapter 6 we will consider such setting.
Interestingly, the HMC method does not necessarily need to be applied in a risk neutral set-up, like is needed for the MC method. In other words, the stock’s drift is not necessarily equal to the risk-free rate. In figure 4.13, the HMC is plotted for different drift constants. We see that if the drift term is close to the risk-free rate, the method produces a price close to Black-Scholes’ price. If the drift term deviates from the risk-free rate, however, a lower price is calculated. This is in line with Section 3.4’s discussion on discrete hedging. As we have seen there, this ‘smile effect’ is dependent of the amount of re-balancing time instants that are chosen. In the case of a static hedge, we saw in
Figure 4.12: Option prices after 500 applications of the HMC method and the MC method. Both methods are applied on 500 simulations of 500 stock price paths. The derivative used is an at-the-money option on a Black-Scholes modeled stock, with initial stock price 100, volatility 30%, and the drift equal to the interest rate of 5%. The HMC method is applied using 20 hedging time intervals and eight polynomial basis functions. The green and red histogram correspond, respectively, to the HMC and the MC method. The blue dotted line indicates the Black-Scholes price of the option, which equals 6.58.
