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Numerical Solution of Stochastic Differential Equations with Jumps in Finance

verfasst von: Eckhard Platen, Nicola Bruti-Liberati

Verlag: Springer Berlin Heidelberg

Buchreihe : Stochastic Modelling and Applied Probability

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

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Über dieses Buch

In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992). The present monograph builds on the above-mentioned work and provides an introduction to stochastic differential equations with jumps, in both theory and application, emphasizing the numerical methods needed to solve such equations. It presents many new results on higher-order methods for scenario and Monte Carlo simulation, including implicit, predictor corrector, extrapolation, Markov chain and variance reduction methods, stressing the importance of their numerical stability. Furthermore, it includes chapters on exact simulation, estimation and filtering. Besides serving as a basic text on quantitative methods, it offers ready access to a large number of potential research problems in an area that is widely applicable and rapidly expanding. Finance is chosen as the area of application because much of the recent research on stochastic numerical methods has been driven by challenges in quantitative finance. Moreover, the volume introduces readers to the modern benchmark approach that provides a general framework for modeling in finance and insurance beyond the standard risk-neutral approach. It requires undergraduate background in mathematical or quantitative methods, is accessible to a broad readership, including those who are only seeking numerical recipes, and includes exercises that help the reader develop a deeper understanding of the underlying mathematics.

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Inhaltsverzeichnis

Frontmatter

1. Stochastic Differential Equations with Jumps

Abstract

Stochastic differential equations (SDEs) with jumps provide the most flexible, numerically accessible, mathematical framework that allows us to model the evolution of financial and other random quantities over time. In particular, feedback effects can be easily modeled and jumps enable us to model events. This chapter introduces SDEs driven by Wiener processes, Poisson processes and Poisson random measures. We also discuss the Itô formula, the Feyman-Kac formula and the existence and uniqueness of solutions of SDEs. These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps.

Eckhard Platen, Nicola Bruti-Liberati

2. Exact Simulation of Solutions of SDEs

Abstract

Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find applications in the statistics of stochastic processes and many applied areas, in particular in finance. They play a crucial role when used in standard models in various areas. These models often dominate the communication and thinking in a particular field of application, even though they may be too simple for advanced tasks. Various simulation techniques have been developed over the years. However, the simulation of solutions of some stochastic differential equations can still be problematic. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems of those simulation methods that use discrete-time approximations. This chapter follows closely Platen & Rendek (2009a) and provides methods for the exact simulation of paths of multi-dimensional solutions of stochastic differential equations, including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Lévy type processes. Other papers that could be considered to be related with exact simulation include Lewis & Shedler (1979), Beskos & Roberts (2005), Broadie & Kaya (2006), Kahl & Jäckel (2006), Smith (2007), Andersen (2008), Burq & Jones (2008) and Chen (2008).

Eckhard Platen, Nicola Bruti-Liberati

3. Benchmark Approach to Finance and Insurance

Abstract

This chapter introduces a unified continuous time framework for financial and insurance modeling. It is applicable to portfolio optimization, derivative pricing, actuarial pricing and risk measurement when security price processes are modeled via SDEs with jumps. It follows the benchmark approach developed in Platen & Heath (2006). The jumps allow for the modeling of event risk in finance, insurance and other areas. The natural benchmark for asset allocation and the natural numéraire for pricing is represented by the best performing, strictly positive portfolio. This is shown to be the growth optimal portfolio (GOP) which maximizes expected growth. Any nonnegative portfolio, when expressed in units of the GOP, turns out to be a supermartingale. This fundamental property leads to real world pricing which identifies the minimal replicating price. An equivalent risk neutral probability measure need not exist under the benchmark approach, which provides significant freedom for modeling when compared to the classical approach.

Eckhard Platen, Nicola Bruti-Liberati

4. Stochastic Expansions

Abstract

In this rather demanding chapter we present the Wagner-Platen expansion for solutions of SDEs with jumps. This stochastic expansion generalizes the deterministic Taylor formula and the Wagner-Platen expansion for diffusions to the case of SDEs with jumps. It allows expanding the increments of smooth functions of Itô processes in terms of multiple stochastic integrals. As we will see, it is the key tool for the construction of stochastic numerical methods and very convenient for other approximation tasks.

Eckhard Platen, Nicola Bruti-Liberati

5. Introduction to Scenario Simulation

Abstract

In this chapter we introduce scenario simulation methods for SDEs. We consider pathwise converging simulation methods that use discrete-time approximations. This chapter follows classical ideas, as described in Kloeden & Platen (1999), and examines different numerical schemes. It also considers some examples of typical problems that can be handled by the simulation of approximating discrete-time trajectories.

Eckhard Platen, Nicola Bruti-Liberati

6. Regular Strong Taylor Approximations with Jumps

Abstract

In this chapter we start to go beyond the work described in Kloeden & Platen (1999) on the numerical solution of SDEs. We now allow the driving noise of the SDEs to have jumps. We present regular strong approximations obtained directly from a truncated Wagner-Platen expansion with jumps. The term regular refers to the time discretizations used to construct these approximations. These do not include the jump times of the Poisson random measure, as opposed to the jump-adapted strong approximations that will be presented later in Chap. 8. A convergence theorem for approximations of a given strong order of convergence will be presented at the end of this chapter. The reader who aims to simulate a solution of an SDE with low jump intensity is referred directly to Chap. 8 which describes jump-adapted schemes that are convenient to use.

Eckhard Platen, Nicola Bruti-Liberati

7. Regular Strong Itô Approximations

Abstract

In this chapter we describe strong approximations on a regular time discretization that are more general than the regular strong Taylor approximations presented in the previous chapter. These approximations belong to the class of regular strong Itô schemes, which includes derivative-free, implicit and predictor-corrector schemes. More details on some of the results to be presented in this chapter can be found in Bruti-Liberati, Nikitopoulos-Sklibosios & Platen (2006) and Bruti-Liberati & Platen (2008).

Eckhard Platen, Nicola Bruti-Liberati

8. Jump-Adapted Strong Approximations

Abstract

This chapter describes jump-adapted strong schemes. The term jump-adapted refers to the time discretizations used to construct these schemes. These discretizations include all jump times generated by the Poisson jump measure. The form of the resulting schemes is much simpler than that of the regular schemes presented in Chaps. 6 and 7 which are based on regular time discretizations. The idea of jump-adapted time discretization goes back to Platen (1982a). It appeared later in various literature, for instance, Maghsoodi (1996). Jump-adapted schemes are not very efficient for SDEs driven by a Poisson measure with a high total intensity. In this case, regular schemes would usually be preferred. Some of the results of this chapter can be found in Bruti-Liberati et al. (2006) and in Bruti-Liberati & Platen (2007b). Results presented already in Kloeden & Platen (1999) and Chap. 5 are employed in the following when approximating the diffusion part of the solution of an SDE.

Eckhard Platen, Nicola Bruti-Liberati

9. Estimating Discretely Observed Diffusions

Abstract

In many areas of application the estimation of parameters in SDEs is an important practical task. This estimation is almost like inverting the problem of scenario simulation and can benefit from the application of Wagner-Platen expansions. We have already mentioned that it is important to apply scenario simulation when checking empirically the usefulness of a proposed estimation methods. This chapter introduces estimation techniques for discretely observed diffusion processes. Transform functions are applied to the data in order to obtain estimators of both the drift and diffusion coefficients. Consistency and asymptotic normality of the resulting estimators is investigated. Power transforms are used to estimate the parameters of affine diffusions, for which explicit estimators are obtained.

Eckhard Platen, Nicola Bruti-Liberati

10. Filtering

Abstract

A very powerful approach that allows us to extract, in an adaptive manner, information from observed date is that of filtering. The aim of this chapter is to introduce filtering of information about hidden variables that evolve over time. These variables may follow continuous time hidden Markov chains or may satisfy certain hidden SDEs. Their observation is considered to be perturbed by the noise of Wiener or other processes. Approximate discrete-time filters driven by observation processes will be constructed for different purposes.

Eckhard Platen, Nicola Bruti-Liberati

11. Monte Carlo Simulation of SDEs

Abstract

This chapter introduces what is commonly known as Monte Carlo simulation for stochastic differential equations. We explain that Monte Carlo simulation is a much simpler task than scenario simulation, discussed in the previous chapters. A weak convergence criterion will be introduced that allows us to classify various discrete-time approximations and numerical schemes for the purpose of Monte Carlo simulation. For simplicity, we focus on the case without jumps in this introductory chapter. The case with jumps is more complicated and will be described in Chaps. 12 and 13.

Eckhard Platen, Nicola Bruti-Liberati

12. Regular Weak Taylor Approximations

Abstract

In this chapter we present for the case with jumps regular weak approximations obtained directly from a truncated Wagner-Platen expansion. The desired weak order of convergence determines which terms of the stochastic expansion have to be included in the approximation. These weak Taylor schemes are different from the regular strong Taylor schemes presented in Chap. 6. We will see that the construction of weak schemes requires a separate analysis. A convergence theorem, which allows us to construct weak Taylor approximations of any given weak order, will be presented at the end of this chapter.

Eckhard Platen, Nicola Bruti-Liberati

13. Jump-Adapted Weak Approximations

Abstract

In this chapter we consider weak approximations constructed on jump-adapted time discretizations similar to those presented in Chap. 8. Since a jump-adapted discretization includes the jump times of the Poisson measure, we can use various approximations for the pure diffusion part between discretization points. Higher order jump-adapted weak schemes avoid multiple stochastic integrals that involve the Poisson random measure. Only multiple stochastic integrals with respect to time and Wiener processes, or their equivalents, are required. This leads to easily implementable schemes. It needs to be emphasized that jump-adapted weak approximations become computationally demanding when the intensity of the Poisson measure is high.

Eckhard Platen, Nicola Bruti-Liberati

14. Numerical Stability

Abstract

When simulating discrete-time approximations of solutions of SDEs, in particular martingales, numerical stability is clearly more important than higher order of convergence. The stability criterion presented is designed to handle both scenario and Monte Carlo simulation, that is, both strong and weak approximation methods. Stability regions for various schemes are visualized. The result being that schemes, which have implicitness in both the drift and the diffusion terms, exhibit the largest stability regions. Refining the time step size in a simulation can lead to numerical instabilities, which is not what one experiences in deterministic numerical analysis. This chapter follows closely Platen & Shi (2008).

Eckhard Platen, Nicola Bruti-Liberati

15. Martingale Representations and Hedge Ratios

Abstract

The calculation of hedge ratios is fundamental to both the valuation of derivative securities and also the risk management procedures needed to replicate these instruments. In Monte Carlo simulation the following results on martingale representations and hedge ratios will be highly relevant. In this chapter we follow closely Heath (1995) and consider the problem of finding explicit Itô integral representations of the payoff structure of derivative securities. If such a representation can be found, then the corresponding hedge ratio can be identified and numerically calculated. For simplicity, we focus here on the case without jumps. The case with jumps is very similar.

Eckhard Platen, Nicola Bruti-Liberati

16. Variance Reduction Techniques

Abstract

The evaluation of the expectation of a given function of a solution of an SDE with jumps provides via the Feynman-Kac formula, see Sect. 2.7, the solution of a partial integro differential equation. In many applications it is of major interest to obtain numerically these expectations, in particular in multi-dimensional settings. Monte Carlo simulation appears to be a method that may be able to provide answers to this question under rather general circumstances. However, raw Monte Carlo estimates of the expectation of a payoff structure, for instance for derivative security prices, can be very expensive in terms of computer resource usage. In this chapter we investigate the problem of constructing variance reduced estimators for the expectation of functionals of solutions of SDEs that can speed up the simulation enormously. We follow again closely Heath (1995). As we will see, variance reduction is more of an art and can be applied in many ways. This chapter shall enable the reader to design her or his own variance reduction method for a given problem at hand.

Eckhard Platen, Nicola Bruti-Liberati

17. Trees and Markov Chain Approximations

Abstract

This final chapter discusses numerical effects on tree methods. Furthermore, binomial, trinomial and multinomial trees will be interpreted as Markov chain approximations of solutions of SDEs. General higher order Markov chain approximations with given weak order of convergence will be described. In addition, the relationship with finite difference methods will be highlighted towards the end of the chapter. It is worth pointing out that jump diffusions can be rather easily approximated in a similar manner as will be described.

Eckhard Platen, Nicola Bruti-Liberati

18. Solutions for Exercises

Abstract

Solutions for Exercises

Eckhard Platen, Nicola Bruti-Liberati

Backmatter

Titel: Numerical Solution of Stochastic Differential Equations with Jumps in Finance
verfasst von: Eckhard Platen
Nicola Bruti-Liberati
Copyright-Jahr: 2010
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-13694-8
Print ISBN: 978-3-642-12057-2
DOI: https://doi.org/10.1007/978-3-642-13694-8

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