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Many mathematical assumptions on which classical derivative pricing methods are based have come under scrutiny in recent years. The present volume offers an introduction to deterministic algorithms for the fast and accurate pricing of derivative contracts in modern finance. This unified, non-Monte-Carlo computational pricing methodology is capable of handling rather general classes of stochastic market models with jumps, including, in particular, all currently used Lévy and stochastic volatility models. It allows us e.g. to quantify model risk in computed prices on plain vanilla, as well as on various types of exotic contracts. The algorithms are developed in classical Black-Scholes markets, and then extended to market models based on multiscale stochastic volatility, to Lévy, additive and certain classes of Feller processes.

This book is intended for graduate students and researchers, as well as for practitioners in the fields of quantitative finance and applied and computational mathematics with a solid background in mathematics, statistics or economics.​

Inhaltsverzeichnis

Frontmatter

Basic Techniques and Models

Frontmatter

Chapter 1. Notions of Mathematical Finance

Abstract
The present notes deal with topics of computational finance, with focus on the analysis and implementation of numerical schemes for pricing derivative contracts. There are two broad groups of numerical schemes for pricing: stochastic (Monte Carlo) type methods and deterministic methods based on the numerical solution of the Fokker–Planck (or Kolmogorov) partial integro-differential equations for the price process. Here, we focus on the latter class of methods and address finite difference and finite element methods for the most basic types of contracts for a number of stochastic models for the log returns of risky assets. We cover both, models with (almost surely) continuous sample paths as well as models which are based on price processes with jumps. Even though emphasis will be placed on the (partial integro)differential equation approach, some background information on the market models and on the derivation of these models will be useful particularly for readers with a background in numerical analysis.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 2. Elements of Numerical Methods for PDEs

Abstract
In this chapter, we present some elements of numerical methods for partial differential equations (PDEs). The PDEs are classified into elliptic, parabolic and hyperbolic equations, and we indicate the corresponding type of problems that they model. PDEs arising in option pricing problems in finance are mostly parabolic. Occasionally, however, elliptic PDEs arise in connection with so-called “infinite horizon problems”, and hyperbolic PDEs may appear in certain pure jump models with dominating drift.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 3. Finite Element Methods for Parabolic Problems

Abstract
The finite element methods are an alternative to the finite difference discretization of partial differential equations. The advantage of finite elements is that they give convergent deterministic approximations of option prices under realistic, low smoothness assumptions on the payoff function as, e.g. for binary contracts. The basis for finite element discretization of the pricing PDE is a variational formulation of the equation. Therefore, we introduce the Sobolev spaces needed in the variational formulation and give an abstract setting for the parabolic PDEs.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 4. European Options in BS Markets

Abstract
In the last chapters, we explained various methods to solve partial differential equations. These methods are now applied to obtain the price of a European option. We assume that the stock price follows a geometric Brownian motion and show that the option price satisfies a parabolic PDE. The unbounded log-price domain is localized to a bounded domain and the error incurred by the truncation is estimated. It is shown that the variational formulation has a unique solution and the discretization schemes for finite element and finite differences are derived. Furthermore, we describe extensions of the Black–Scholes model, like the constant elasticity of variance (CEV) and the local volatility model.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 5. American Options

Abstract
Pricing American contracts requires, due to the early exercise feature of such contracts, the solution of optimal stopping problems for the price process. Similar to the pricing of European contracts, the solutions of these problems have a deterministic characterization. Unlike in the European case, the pricing function of an American option does not satisfy a partial differential equation, but a partial differential inequality, or to be more precise, a system of inequalities. We consider the discretization of this inequality both by the finite difference and the finite element method where the latter is approximating the solutions of variational inequalities. The discretization in both cases leads to a sequence of linear complementarity problems (LCPs). These LCPs are then solved iteratively by the PSOR algorithm. Thus, from an algorithmic point of view, the pricing of an American option differs from the pricing of a European option only as in the latter we have to solve linear systems, whereas in the former we need to solve linear complementarity problems.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 6. Exotic Options

Abstract
Options with more sophisticated rules than those for plain vanillas are called exotic options. There are different types. Path dependent options depend on the whole history of the underlying and not just on the realization at maturity. In particular, we consider barrier options which depend on price levels being attained over a period and Asian options which depend on the average price of the option’s underlying over a period. Furthermore, we look at options which have different exercise styles like compound options which are options on options and swing options which have multiple exercise rights. We assume that the dynamics of the stock price is modeled by a geometric Brownian motion.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 7. Interest Rate Models

Abstract
We consider options on interest rates and present commonly used short rate models to model the time-evolution of the interest rate. Many interest rate derivatives in fixed income markets can then be priced numerically using the computational techniques described in the previous chapter, i.e. they can be interpreted as compound options on bonds.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 8. Multi-asset Options

Abstract
In Chap. 6, we considered exotic options written on a single underlying. Further examples of exotic options are given by the so-called multi-asset options. These are options derived from d≥2 underlying risky assets, whose price movement can be described by a system of SDEs. The pricing functions of multi-asset options are multivariate functions satisfying a parabolic partial differential equation in d dimensions, together with an appropriate terminal value depending on the type of the option. We distinguish between different types of European multi-asset options. We distinguish between different types of multi-asset options like basket, rainbow or quanto options.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 9. Stochastic Volatility Models

Abstract
In Sect. 4.​5, we considered local volatility models as an extension of the Black–Scholes model. These models replace the constant volatility by a deterministic volatility function, i.e. the volatility is a deterministic function of s and t. In stochastic volatility (SV) models, the volatility is modeled as a function of at least one additional stochastic process. Such models can explain some of the empirical properties of asset returns, such as volatility clustering and the leverage effect. These models can also account for long term smiles and skews.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 10. Lévy Models

Abstract
One problem with the Black–Scholes model is that empirically observed log returns of risky assets are not normally distributed, but exhibit significant skewness and kurtosis. If large movements in the asset price occur more frequently than in the BS-model of the same variance, the tails of the distribution, should be “fatter” than in the Black–Scholes case. Another problem is that observed log-returns occasionally appear to change discontinuously. Empirically, certain price processes with no continuous component have been found to allow for a considerably better fit of observed log returns than the classical BS model. Pricing derivative contracts on such underlyings becomes more involved mathematically and also numerically since partial integro-differential equations must be solved. We consider a class of price processes which can be purely discontinuous and which contains the Wiener process as special case, the class of Lévy processes. Lévy processes contain most processes proposed as realistic models for log-returns.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 11. Sensitivities and Greeks

Abstract
A key task in financial engineering is the fast and accurate calculation of sensitivities of market models with respect to model parameters. This becomes necessary, for example, in model calibration, risk analysis and in the pricing and hedging of certain derivative contracts. Classical examples are variations of option prices with respect to the spot price or with respect to time-to-maturity, the so-called “Greeks” of the model. For classical, diffusion type models and plain vanilla type contracts, the Greeks can be obtained analytically. With the trends to more general market models of jump–diffusion type and to more complicated contracts, closed form solutions are generally not available for pricing and calibration. Thus, prices and model sensitivities have to be approximated numerically.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Advanced Techniques and Models

Frontmatter

Chapter 12. Wavelet Methods

Abstract
In the previous sections, we developed various algorithms for the efficient pricing of derivative contracts when the price of the underlying is a one-dimensional diffusion, a multidimensional diffusion, a general stochastic volatility or a one-dimensional Lévy process. In this part, we introduce variational numerical methods for pricing under yet more general processes with the aim of achieving linear complexity.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 13. Multidimensional Diffusion Models

Abstract
In the present chapter, we develop efficient pricing algorithms for multivariate problems, such as the pricing of multi-asset options and the pricing of options in stochastic volatility models, which exploit a third feature of the wavelet basis, namely that wavelets constitute a hierarchic basis of the univariate finite element space. This allows constructing the so-called sparse tensor product subspaces for the numerical solution of d-dimensional pricing problems with complexity essentially equal to that of one-dimensional problems.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 14. Multidimensional Lévy Models

Abstract
In this chapter, we extend the one-dimensional Lévy models described in Chap. 10 to multidimensional Lévy models. Since the law of a Lévy process is time-homogeneous, it is completely characterized by its characteristic triplet. The drift has no effect on the dependence structure between the components. The dependence structure of the Brownian motion part of the Lévy process is given by its covariance matrix. For purposes of financial modeling, it remains to specify a parametric dependence structure of the purely discontinuous part which can be done by using Lévy copulas.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 15. Stochastic Volatility Models with Jumps

Abstract
In Chap. 9, we considered pure diffusion stochastic volatility models. We extend these models by adding jumps and derive numerical solutions for different models such as Bates or Barndorff-Nielsen and Shephard.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Chapter 16. Multidimensional Feller Processes

Abstract
In this chapter, we extend the setting of Chap. 14 to a more general class of processes. We consider a large class of Markov processes in the following. Under certain assumptions we can apply the theory of pseudodifferential operators in order to analyse the arising pricing equations. The dependence structure of the purely discontinuous part of the market model X is described using Lévy copulas. Wavelets are used for the discretization and preconditioning of the arising PIDEs, which are of variable order with the order depending on the jump state.
Norbert Hilber, Oleg Reichmann, Christoph Schwab, Christoph Winter

Backmatter

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