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Calculation of the Greeks by Malliavin Calculus Stephan Sturm Calculation of the Greeks by Malliavin Calculus Diplomarbeit zur Erlangung des Akademischen Grades ”Magister der Naturwissenschaften” Betreuer: Walter Schachermayer Institut f¨ur Matematik Universit¨atWien Februar 2004 M¨udigkeit sp¨urte er keine, nur war es ihm manchmal unangenehm, dass er nicht auf dem Kopf gehn konnte. Georg Buchner¨ , Lenz Preface It would not be correct to describe the work on this thesis, now having it fin- ished successfully, as a short and easy task. On the contrary, it took me a long time and in the couse of the work I had a lot of doubts about my own mathematical abilities. It was my old friend Josef Teichmann who proposed me to write a diploma thesis on Malliavin Calculus under his guidance. It was for me quite a hard, but also very joyful path starting out as a student whith hardly any knowledge in probability theory to the heights of stochastic analysis. The goal of my diploma thesis was a profound understanding of the calcu- lation of the Greeks by Malliavin calculus in the n-dimensional, elliptic case as first presented by Fourni´ee.a. (1998) and a generalization to hypoellipticity; this generalization is the focus of current research, see e.g. the works of Malli- avin and Thalmaier (2003, preprint), Gobet and Munos (2002, preprint) and Teichmann and Touzi (working paper). Malliavin calculus, i.e. the stochastic calculus of variations which is build up on the notion of a weak derivative on the Wiener space, the Malliavin derivative, lies in the core of the intersection of stochastic analysis, functional analysis and differential geometry. It is the perfect tool for a calculation of the sensitivity of the price of an option with respect to small changes in the parameters, i.e. the Greeks. The abstract notions of functional analysis allow us to write the Greeks as the expectation of the product of the original payoff function with a specific factor, the Malliavin weight which is in fact a Skorohod integral, the adjoint operator of the Malliavin derivative and a generalization of the notion of the Itˆointegral. After a short introduction to martingale theory I will give the foundations of stochastic analysis, introduce the Itˆoand the Stratonovich notion of the stochas- tic integral (with respect to a Brownian motion, but also in the more general continuous semimartingale case) and present the classical Girsanov theory of transformations of the probability measure. The proof of the unique existence of the solution of a stochastic differential equation follows an introduction to its first derivative with respect to the initial value, the first variation process. The introduction of the Wiener chaos decomposition allows me to under- stand multiple Wiener-Itˆointegrals as iterated (classical) Itˆointegrals and hence to look at stochastic integration as a process of climbing up the Wiener chaos. The Malliavin derivative is introduced as the inverse climbing down and I will prove its (functional) analytic properties up to the Clark-Haussmann-Ocone for- 2 Stephan Sturm: Calculation of the Greeks by Malliavin Calculus 3 mula, in the core the chain rule. The divergence operator (or Skorohod integral) is introduced as its adjoint operator and it is shown that it coincides for pro- gressively measurable processes with the Itˆointegral. As last theoretical point I will show the connection between the first variation process and the Malliavin derivative which leads us to some closing remarks on the existence and smooth- ness of densities of random variables. The hitherto developed mathematical theory is used to answer a specific question of mathematical finance: What is the behavior of an option if we vary the parameters a little bit? In the jargon of finance this means to calculate the Greeks. The idea behind the notion of ”calculating” is here to develop a formula which is better fitted for a numerical evaluation than the simple difference quo- tient. The method is to express the derivative of the expectation as expectation of a product of the original payoff function and some weight. For the n-dimensional elliptic case we follow the already classical paper by Fourni´ee.a. to show that we can calculate the derivatives with respect to the interest rate by classical Girsanov theory while for those with respect to the initial value and the volatility the Malliavin calculus is of great use. So we can write the weights as Skorohod integral whose integrands depend only on the underlying processes. Dropping the ellipticity condition we will then show in an hypoelliptic setting with d Brownian motions for an n-dimensional process that we can calculate the Greeks also here. As one concrete application of this method we calculate the Greeks in the Black-Scholes model and show the strength of the hypoelliptic formula by using it for an approximation of the Hobson-Rogers delta. In particular the obtained formulas allow simple numerical algorithms to approximate the solutions of hypoelliptic partial differential equations. This feature is applied in Hubalek, Teichmann, Tompkins (2004) to fit parameters of a model to real market data without using sophisticated PDE techniques. The main sources for me were the book of Revuz and Yor [RY 91] for martin- gale theory and stochastic integration with respect to continuous semimartin- gales, the manuscripts of Teichmann [Tei 02] and [Tei 03] for stochastic integra- tion, the theory of SDEs, Wiener Chaos and Malliavin Calculus, Bass [Ba 98] for the first variation process and Nualart’s book on Malliavin Calculus [Nua 95]. General reference was Kallenberg [Kal 02], the calculation of the Greeks in the elliptic setting is due to Fourni´ee.a. [FLLLT 99], the hypoelliptic treatment was inspired by Teichmann and Touzi [TT]. First of all I have to thank Josef Teichmann who initiated me to this subject and was always there if I had to discuss some problems of my work. Sebas- tian Markt made some linguistical suggestion and thus helped me to master my problems with the English language. All possible faults obviously remain in my responsibility. The Department of Financial and Actuarial Mathematics at the Technical Stephan Sturm: Calculation of the Greeks by Malliavin Calculus 4 University of Vienna under the direction of Walter Schachermayer gave me an ideal environment for my work in a very pleasant atmosphere. I am also very grateful that the department gave my the possibility to participate in the “Berlin Workshop on Mathematical Finance for Young Researchers”. Thanks also to CIMPA and Prof. S.G. Dani who gave me the possibility to spend two intensive weeks at the Summer School “Probability Measures on Groups” at the Tata Institute for Fundamental Research (TIFR), Bombay. Last but not least I have to thank all those people who spent their days and nights with me, laughing and discussing, cooking, eating (and sometimes too much) drinking, on the mountains, in caf´esor in our flats, shortly: my friends. In particular I want to mention Herwig Czech, Ulrike Girardi, Florian Huber, Sebastian Markt, Martina Punz, Christian Selinger, Martin, Susanne and Wil- helm Sturm, Josef Teichmann and Florian Wenninger. I want to dedicate this work to three people who always looked on my for- mation and on my studies, but sadly could not see the fruits of all their care, love and help: My aunt Anna Kopeˇcny (1914-2001) and my grandparents Maria Sturm (1920-2003) and Eduard Sturm (1920-2000). Chapter 1 Preliminaries Modern probability theory was founded by Andrei Nikolaevich Kolmogorov who, claiming that “the theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra”, was the first to treat this subject from an axiomatic, measure theory based point of view. It is here not the place to go into the measure theoretic details of the foundations of modern probability theory; we will refer to the lit- erature where needed. We give here only an introduction to stochastic process, in particular martingales, and a very short recapitulation of the basics of the theory of tensors on Hilbert spaces. 1.1 Stochastic Processes and Martingales In this section we will give an introduction to stochastic processes, i.e. families of random variables, and martingales, stochastic processes which can be thought as “fair games”. 1.1.1 Stochastic Processes Definition 1.1.1 (Stochastic Process) Given an index set T , a stochastic process is a family of measurable mappings Xt, t ∈ T ⊂ R≥0 ∪ {∞}, from a probability space (Ω, F,P ) to a measurable space (E, G), the state space. Under the path (or trajectory) of a stochastic process we understand the mapping t → Xt(ω). A process is called continuous iff for almost all ω’s the paths are continuous. We introduce the following notions of “proximity” of stochastic processes: Definition 1.1.2 (Modifications and Indistinguishability) Two stochastic processes Xt, Yt, t ∈ T defined on the same probability space (Ω, F,P ) are called 5 Stephan Sturm: Calculation of the Greeks by Malliavin Calculus 6 (i) modifications of each other iff for each t ∈ T we have Xt = Yt a.s. (ii) indistinguishable iff for almost all ω ∈ Ω we have Xt = Yt for all t ∈ T . So the difference is that for a modification the null set where both processes differ is dependent on t, while for indistinguishability it is required to be inde- pendent of the concrete choice of t. Theorem 1.1.3 (Conditions for Indistinguishability) Given two stochastic processes Xt, Yt, modifications of each other. If both pro- cesses have right continuous paths a.s., then they are indistinguishable. Proof Let A, B the null sets where Xt resp. Yt are not right continuous.
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