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2011 | Buch

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PDE and Martingale Methods in Option Pricing

verfasst von: Andrea Pascucci

Verlag: Springer Milan

Buchreihe : B&SS — Bocconi & Springer Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers an introduction to the mathematical, probabilistic and numerical methods used in the modern theory of option pricing. The text is designed for readers with a basic mathematical background. The first part contains a presentation of the arbitrage theory in discrete time. In the second part, the theories of stochastic calculus and parabolic PDEs are developed in detail and the classical arbitrage theory is analyzed in a Markovian setting by means of of PDEs techniques. After the martingale representation theorems and the Girsanov theory have been presented, arbitrage pricing is revisited in the martingale theory optics. General tools from PDE and martingale theories are also used in the analysis of volatility modeling. The book also contains an Introduction to Lévy processes and Malliavin calculus. The last part is devoted to the description of the numerical methods used in option pricing: Monte Carlo, binomial trees, finite differences and Fourier transform.

Inhaltsverzeichnis

Frontmatter

1. Derivatives and arbitrage pricing

Abstract

A financial derivative is a contract whose value depends on one or more securities or assets, called underlying assets. Typically the underlying asset is a stock, a bond, a currency exchange rate or the quotation of commodities such as gold, oil or wheat.

Andrea Pascucci

2. Discrete market models

Abstract

In this chapter we describe market models in discrete time to price and hedge European and American-style derivatives. We present the classical model introduced by Cox, Ross and Rubinstein in [78] and we mention briefly the pricing problem in incomplete markets. General references on topics covered in this chapter are Dana and Jeanblanc [84], Föllmer and Schied [134], Lamberton and Lapeyre [226], Pliska [282], Shreve [310], van der Hoek and Elliott [329]: we also mention Pascucci and Runggaldier [277] where several examples and exercises can be found.

Andrea Pascucci

3. Continuous-time stochastic processes

Abstract

In this chapter we introduce the elements of the theory of stochastic processes that we will use in continuous-time financial models. After a general presentation, we define the one-dimensional Brownian motion and we discuss some equivalence notions among stochastic processes. The most substantial part of the chapter is devoted to the study of the first and the second variation of a process: such a concept is introduced at first in the framework of the classical function theory and for Riemann-Stieltjes integration. Afterwards, we extend our analysis to the Brownian motion by determining its quadratic-variation process.

Andrea Pascucci

4. Brownian integration

Abstract

In this chapter we introduce the elements of stochastic integration theory that are necessary to treat some financial models in continuous time. In Paragraph 3.4 we gave grounds for the interest in the study of the limit of a Riemann-Stieltjes sum of the form

$$ \sum\limits_{k = 1}^N {u_{t_{k - 1} } \left( {W_{t_k } - W_{t_{k - 1} } } \right)} $$

(4.1)

as the refinement parameter of the partition {t₀, . . . , t_N} tends to zero. In (4.1) W is a real Brownian motion that represents a risky asset and u is an adapted process that represents an investment strategy: if the strategy is self-financing, the limit of the sum in (4.1) is equal to the value of the investment.

Andrea Pascucci

5. Itô calculus

Abstract

As for the Riemann and Lebesgue integral, the definition of stochastic integral is theoretical and it is not possible to use it directly for practical purposes, apart from some particular cases. Classical results reduce the problem of the computation of a Riemann integral to the determination of a primitive of the integrand function; in stochastic integration theory, the concept of primitive is translated into “integral terms” by the Itô-Doeblin formula¹. This formula extends Theorem 3.70 in a probabilistic framework and lays the grounds for differential calculus for Brownian motion: as we have already seen the Brownian motion paths are generally irregular and so an integral interpretation of differential calculus for stochastic processes is natural.

Andrea Pascucci

6. Parabolic PDEs with variable coefficients: uniqueness

Abstract

In this chapter we consider elliptic-parabolic equations with variable coefficients of the form

$$ L_a u: = Lu - au = 0, $$

(6.1)

where L is the second order operator

$$ L = \frac{1} {2}\sum\limits_{j,k = 1}^N {c_{jk} \partial _{x_j x_k } + } \sum\limits_{j = 1}^N {b_j \partial _{x_j } - } \partial _t , (t,x) \in \mathbb{R}^{N + 1} . $$

(6.2)

Andrea Pascucci

7. Black-Scholes model

Abstract

In this chapter we present some of the fundamental ideas of arbitrage pricing in continuous time, illustrating Black-Scholes theory from a point of view that is, as far as possible, elementary and close to the original ideas in the papers by Merton [250], Black and Scholes [49]. In Chapter 10 the topic will be treated in a more general fashion, fully exploiting martingale and PDEs theories.

Andrea Pascucci

8. Parabolic PDEs with variable coefficients: existence

Abstract

The Black-Scholes model is based upon the results of existence and uniqueness for parabolic equations with constant coefficients, in particular for the heat equation. The study of more sophisticated diffusion models requires analogous results for differential operators with variable coefficients.

Andrea Pascucci

9. Stochastic differential equations

Abstract

In this chapter we present some basic results on stochastic differential equations, hereafter shortened to SDEs, and we examine the connection to the theory of parabolic partial differential equations.

Andrea Pascucci

10. Continuous market models

Abstract

In this chapter we present the theory of derivative pricing and hedging for continuous-time diffusion models. As in the discrete-time case, the concept of martingale measure plays a central role: we prove that any equivalent martingale measure (EMM) is associated to a market price of risk and determines a risk-neutral price for derivatives, that avoids the introduction of arbitrage opportunities. In this setting we generalize the theory in discrete time of Chapter 2 and extend the Markovian formulation of Chapter 7, based upon parabolic equations.

Andrea Pascucci

11. American options

Abstract

We present the main results on the pricing and hedging of American derivatives by extending to continuous time the ideas introduced in the discrete-market setting in Section 2.5. Even in the simplest case of the Black-Scholes market model, the hedging and pricing problems for American options need very refined mathematical tools. In the complete-market setting, Bensoussan [41] and Karatzas [198], [199] developed a probabilistic approach based upon the notion of Snell envelope in continuous time and upon the Doob-Meyer decomposition. The problem was also studied by Jaillet, Lamberton and Lapeyre [185] who employed variational techniques, and by Oksendal and Reikvam [273], Gatarek and Świech [149] in the framework of the theory of viscosity solutions. American options for models with jumps were studied among others by Zhang [345], Mulinacci [260], Pham [279], Levendorskii [235], Ekström [119], Ivanov [181], Lamberton and Mikou [227], Bayraktar and Xing [36].

Andrea Pascucci

12. Numerical methods

Abstract

In this chapter we present some methods for the numerical solution of deterministic and stochastic differential equations. The numerical approximation is necessary when it is not possible to determine explicitly the solution of an equation (i.e. nearly always).

Andrea Pascucci

13. Introduction to Lévy processes

with Rossella Agliardi

Abstract

The classical Black-Scholes model employs the Brownian motion as the driving stochastic process of asset prices. Empirical evidence has pointed out that such an assumption does not provide an accurate description of financial data and has promoted the development of more flexible models. This chapter presents the fundamentals of Lévy processes and option pricing under such stochastic processes. Since this chapter is intended as an informal introduction to Lévy processes, many of the proofs are omitted: for a complete treatment of the theory we refer to the classical monographs by Bertoin [44], Sato [297], Jacod and Shiryaev [184].

Andrea Pascucci

14. Stochastic calculus for jump processes

Abstract

In this chapter we introduce the basics of stochastic calculus for jump processes. We follow the approaches proposed by Protter [287] for the general theory of stochastic integration and by Applebaum [11] for the presentation of Lévy-type stochastic integrals. We extend to this framework, the analysis performed in the previous chapters for continuous processes: in particular, we prove Itô formula and a Feynman-Kač type representation theorem for solutions to SDEs with jumps. For simplicity, most statements are given in the one-dimensional case. Then we show how to derive the integro-differential equation for a quite general exponential model driven by the solution of a SDE with jumps: these results open the way for the use of deterministic and probabilistic numerical methods, such as finite difference schemes (see, for instance, Cyganowski, Grüne and Kloeden [82]), Galerkin schemes (see, for instance, Platen and Bruti-Liberati [281]) and Monte Carlo methods (see, for instance, Glasserman [158]). In the last part of the chapter, we examine some stochastic volatility models with jumps: in particular, we present the Bates and the Barndorff-Nielsen and Shephard models.

Andrea Pascucci

15. Fourier methods

Abstract

As already explained in the previous chapters, in order to reproduce the real market dynamics it is necessary to introduce more sophisticated models than the Black-Scholes one. These models have to be calibrated to the market in order to approximate the quoted implied volatility surfaces: once this is done, they can give prices to exotic derivatives that are consistent with plain vanilla options.

Andrea Pascucci

16. Elements of Malliavin calculus

Abstract

This chapter offers a brief introduction to Malliavin calculus and its applications to mathematical finance, in particular the computation of the Greeks by the Monte Carlo method. As we have seen in Section 12.4.2, the simplest way to compute sensitivities by the Monte Carlo method consists in approximating the derivatives by incremental ratios obtained by simulating the payoffs corresponding to close values of the underlying asset. If the payoff function is not regular (for example, in the case of a digital option with strike K and payoff function 1_[K,+∞[) this technique is not efficient since the incremental ratio has typically a very large variance. In Section 12.4.2 we have seen that the problem can be solved by integrating by parts and differentiating the density function of the underlying asset, provided it is sufficiently regular: if the underlying asset follows a geometric Brownian motion, this is possible since the explicit expression of the density is known.

Andrea Pascucci

Backmatter

Titel: PDE and Martingale Methods in Option Pricing
verfasst von: Andrea Pascucci
Verlag: Springer Milan
Electronic ISBN: 978-88-470-1781-8
Print ISBN: 978-88-470-1780-1
DOI: https://doi.org/10.1007/978-88-470-1781-8