Perturbation and Symmetry Techniques Applied to Finance

Perturbation and Symmetry Techniques Applied to Finance A Dissertation submitted to obtain the title of Doctor of Philosophy (Dr. rer. pol.) from the Frankfurt School of Finance & Management. Specialty: Quantitative Finance Submitted by: Stephen Taylor Dissertation Advisor: Jan Vecer March 2010 1 2 Declaration of Authorship: I hereby certify that unless otherwise indicated in the text of references, or acknowledged, this dissertation is entirely the product of my own work. Stephen Taylor 3 Contents 1. Introduction 3 2. Background Material 7 2.1. Probability Theory and Stochastic Processes 7 2.2. Stochastic Differential Equations 9 2.3. Financial Applications of Stochastic Differential Equations 10 2.4. Riemannian Geometry 14 2.5. Finance/Geometry Relation 17 2.6. Heat Kernel Expansion Formula 20 3. Local Volatility Model Applications 23 3.1. Black-Scholes-Merton Example 23 3.2. Constant Elasticity of Variance Model 27 3.3. Quadratic Local Volatility Model 35 3.4. Cubic Local Volatility Model 36 3.5. Affine-Affine Short Rate Model 37 3.6. Generalized CEV Model 40 4. Two Dimensional Stochastic Volatility Models 46 4.1. Geometry of A Class of Two Dimensional Stochastic Volatility Models 46 4.2. Geometry of Extended Stochastic Volatility Models 51 4.3. The History of SABR Approximations with Examples 54 4.4. Heat Kernel Methods for the SABR Model 63 4.5. Paulot's Conversion Technique 68 4.6. Lee's Moment Formula 73 4.7. The Heston Model 74 4.8. General Class of Stochastic Volatility Models 75 4.9. Higher Dimensions and Limitations 76 4.10. An Alternative Perturbation Theory for Stochastic Volatility 76 5. Symmetries of Finance Equations 81 5.1. Introduction to Symmetry Analysis 81 5.2. Correspondence with the Standard Theory 86 5.3. Symmetry Analysis of the Black-Scholes-Merton Equation 88 5.4. Symmetries of the Classical Asian Equation 94 5.5. Symmetries of Non Standard Asian Equations 103 5.6. Symmetries of New Asian Equations 109 References 113 4 1. Introduction A significant portion of the mathematical finance literature is devoted to constructing exact pricing formulas for a variety of contingent claims whose underlying assets are assumed to evolve according to a specified stochastic process. In particular, the Black-Scholes-Merton [18], [119], [120], pricing formula for a European call option on an asset which undergoes lognormal dynamics has served as a cornerstone for much of the development of modern pricing theory. Although the lognormal model has long been a guide for gaining insight into how derivatives' prices depend on market parameters, it is often too simplistic for practical use and has many drawbacks. Specifically, asset prices generally do not follow lognormal dynamics, and one is typically unable to calibrate this model in a manner that is consistent with current market option prices. These issues, amongst others, led Dupire [48] to consider the more general class of local volatility models. A local volatility model allows an asset process to evolve by a one- dimensional It¯oprocess, where the diffusion coefficient function is chosen so that the model's present day European call prices agree with current market quotes. Unfortunately, one can- not determine the transition density for a general It¯oprocess (which is equivalent to constructing an explicit formula (up to quadrature) for a call option). There are only a handful of local volatility models where it is possible to derive an expression for the transition density given by a composition of elementary and special functions. Constructing such a solution is equivalent to solving a single linear parabolic equation of one spatial and one temporal variable. Generally, as the functional form of the diffusion coefficient becomes more compli- cated, the prospect of finding an exact solution to the associated transition density equation diminishes. In lieu of exact formulas, there are several widely used approximate pricing techniques. For example, tree pricing, Monte Carlo, and finite difference methods are currently the most popular approximation methods used in practice. Alternatively, one can attempt to construct explicit approximation formulas for the transition density of a local volatility process that generically will only be valid in certain model parameter regimes. There are a variety of approximation techniques that have been applied to one dimensional local volatility models, (c.f. Corielli et. al. [39], Cheng et. al. [31], Kristensen and Mele [97], and Ekstrom and Tysk [52] for example) The advantage of such formulas is that they are considerably more computationally efficient, relative to their previously mentioned counterparts, and, at times, retain qualitative model information in their functional form. The main drawback of these approximations is that they can be inaccurate for certain model parameters, and a thorough error analysis against an established approximation method is usually required prior to real world use. The literature regarding explicit approximation formulas, which are generically constructed using different types of perturbation theory, has been growing over the past several years. In particular, heat kernel perturbation theory has proven to be effective in creating some of the most accurate approximation formulas to date. Roughly speaking, heat kernel perturbation theory utilizes a Taylor series expansion ansatz in time together with a geometrically moti- vated prefactor which is chosen to simplify subsequent calculations. The heat kernel ansatz is most naturally stated using the language of differential geometry. Hagan and Lesnewski [78], [79], [80], [106] were the first to introduce differential geometric methods in finance. Henry-Labordere [98], [99], [101], used heat kernel methods to construct implied volatility formulas for local volatility models, the SABR model, and the SABR Libor Market Model. More recently, Gatheral, et. al. [64] have also considered heat kernel expansions in the context of local volatility models. Improving upon the work of Henry-Labordere, Paulot 5 [134] constructed (to our knowledge) the most accurate explicit implied volatility formula for the SABR model. Forde [58, 59] has rigorized and extended this work to a larger class of stochastic volatility models. We refer the reader to Medvedev's work [114] for a list of several references to the finance perturbation theory literature. There are two main advantages of heat kernel methods over other forms of perturbation theory. First, the only error in a formula constructed solely using a heat kernel ansatz is due to a Taylor series assumption in time. In particular, an expression for a transition density constructed using heat kernel perturbation theory will increase in accuracy as time decreases. Consequently, implied volatility smile approximation formulas constructed from a heat kernel expansion tend to be more accurate at out of the money strikes than others constructed with alternative formalisms that are perturbative in both time and strike. Secondly, the heat kernel ansatz solves its associated parabolic equation exactly to zeroth order in time. Thus in some sense, one does not carry along zeroth order complexities when trying to establish a first order correction for the transition density of a process. Heat kernel perturbation theory is not without its limitations. Even in the case of two dimensional models, the heat kernel ansatz can not be expressed explicitly. This is due to the fact that one can not obtain an explicit distance function on a generic two dimensional geometry. There are other techniques in differential equations which one can use to attempt to reduce the complexity of a given pricing equation. One such technique is Lie symmetry analysis. Lie symmetry analysis provides a means to reduce the dimension of a partial differential equation by using the symmetry of the equation to construct a natural coordinate system in which it takes a simpler form. This can be particularly useful in finance as one can reduce the dimension of a given pricing equation and then numerically simulate the reduced equation at a significantly faster speed than the original. We study the symmetry groups of several equations relevant to finance including the heat equation, Black-Scholes-Merton equation, the classical Asian equation in the lognormal model, and several related Asian type equations. We examine both the heat and Black- Scholes-Merton equations in order to demonstrate our method for computing symmetry groups of partial difference equations. We then apply this method to the standard Asian equation and find many interesting symmetries which allow one to reduce the equation to a lower dimensional problem. In particular, we first review the construction of the symmetry group of the Asian equation (c.f. Glasgow and Taylor [154]) and show that there are several ways one can reduce this equation by one spatial dimension. We next consider several additional equations which can be constructed the from Asian equation by a change of numeraire. Finally, we examine several relations between these different equations. This work is organized in the following manner: In section 2, we provide basic conventions and background and review the construction of the heat kernel ansatz. Specifically, we first review topics in probability theory, stochastic differential equations, and Riemannian geometry which we will need throughout the remainder of the text. In addition, we review the heat kernel ansatz for a partial differential equation and then demonstrate how it can be used to construct explicit approximation formulas for the transition density of stochastic processes. Such formulas can be used in conjunction with integral approximation techniques to enable one to construct explicit formulas for path independent derivatives. In section 3, we state the explicit form of the heat kernel ansatz in one dimension. We then apply this ansatz to several local volatility models. We first consider the Black-Scholes- Merton model, where we show that one can use the heat kernel method to construct the exact solution of the European call equation whose underlying evolves according to lognormal 6 dynamics.

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