The Analysis of Time Series: An Introduction, 4th edn. By C. Chatfield. ISBN 0 412 31820 2. Chapman and Hall, London, 1989. 242 pp. £13.50.

The Analysis of Time Series: An Introduction

This paper aims to extend the analytical tractability of the Black--Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day.

This paper was accepted by Michael Fu, stochastic models and simulation.

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Option Pricing Under a Double Exponential Jump Diffusion Model

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

http://www.cmap.polytechnique.fr/~rama/pide.pdf

The analysis of time series: an introduction (4th edition), by C. Chatfield. Pp 241. £13·50. 1989. ISBN 0-412-31820-2 (Chapman and Hall)

An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed-point iteration scheme are presented. For typical model parameters, it is shown the error is reduced by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform method. Numerical tests of convergence for a variety of options are presented.

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Robust numerical methods for contingent claims under jump diffusion processes

The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence.

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A penalty method for American options with jump diffusion processes

A semi-Lagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one-dimensional partial integrodifferential equations (PIDEs) is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank--Nicolson and second-order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. With low volatility values, it is observed that the nonsmoothness at the strike in the payoff affects the convergence rate; a subquadratic convergence rate is observed.

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A Semi-Lagrangian Approach for American Asian Options under Jump Diffusion

The existing telecommunications infrastructure in most of the world is adequate to deliver voice and text applications, but demand for broadband services such as streaming video and large file transfer (e.g., movies) is accelerating. The explosion in Internet use has created a huge demand for telecommunications capacity. However, this demand is extremely volatile, making network planning difficult. In this paper, modern financial option pricing methods are applied to the problem of network investment decision timing. In particular, we study the optimal decision problem of building new network capacity in the presence of stochastic demand for services. Adding new capacity requires a capital investment, which must be balanced by uncertain future revenues. We study the underlying risk factor in the bandwidth market and then apply real options theory to the upgrade decision problem. We notice that sometimes it is optimal to wait until the maximum capacity of a line is nearly reached before upgrading directly to the line with the highest known transmission rate (skipping the intermediate lines). It appears that past upgrade practice underestimates the conflicting effects of growth and volatility. This explains the current overcapacity in available bandwidth. To the best of our knowledge, this real options approach has not been used previously in the area of network capacity planning. Consequently, we believe that this methodology can offer insights for network management.

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Managing capacity for telecommunications networks under uncertainty

Many debt issues contain an embedded call option that allows the issuer to redeem the bond at specified dates for a specified price. The issuer is typically required to provide advance notice of a decision to exercise this call option. The valuation of these contracts is an interesting numerical exercise because discontinuities may arise in the bond value or its derivative at call and/or notice dates. Recently, it has been suggested that finite difference methods cannot be used to price callable bonds requiring notice. Poor accuracy was attributed to discontinuities and difficulties in handling boundary conditions. As an alternative, a semi-analytical method using Green's functions for valuing callable bonds with notice was proposed. Unfortunately, the Green's function method is limited to special cases. Consequently, it is desirable to develop a more general approach. This is provided by using more advanced techniques such as flux limiters to obtain an accurate numerical partial differential equation met...

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Y. d'Halluin

Papers

Robust numerical methods for contingent claims under jump diffusion processes

A penalty method for American options with jump diffusion processes

A Semi-Lagrangian Approach for American Asian Options under Jump Diffusion

Managing capacity for telecommunications networks under uncertainty

A numerical PDE approach for pricing callable bonds