2-2 - Implementation Problems and Solutions in Stochastic...

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Unformatted text preview: Implementation Problems and Solutions in Stochastic Volatility Models of the Heston Type Jia-Hau Guo* and Mao-Wei Hung** August 2007 ABSTRACT In Heston s stochastic volatility framework, the main problem for implementing Heston s semi-analytic formulae for European-style financial claims is the inverse Fourier integration. The numerical integration scheme of a logarithm function with complex arguments has puzzled practitioners for many years. Without good implementation procedures, the numerical results obtained from Heston s formulae may not be robust, even for customarily-used Heston parameters, as the time to maturity is increased. In this paper, we compare three major approaches to solve the numerical instability problem inherent in the fundamental solution of the Heston model. ___________________________ *Guo is from the School of Business, Soochow University, Taipei, Taiwan. **Hung is from the College of Management, National Taiwan University, Taipei, Taiwan. Address correspondence to: Professor Mao-Wei Hung, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan. Tel: +886-2-3366-4988 Fax: +886-2-2369-0833 E-mail:hung@management.ntu.edu.tw 1 I. Introduction The randomness of the variance process varies as the square root of variance in the Heston s stochastic volatility framework. The literature on asset pricing using the Heston model has expanded dramatically over the last decade to successfully describe the empirical leptokurtic distributions of asset price returns. However, the implementations of Heston s formulae are not as straightforward as they may appear and most numerical procedures are not reported in detail (see Lee [2005]). The complex logarithm contained in the formula of the Heston model is the primary problem. This paper compares three main approaches to this problem: rotation-corrected angle, direct integration, and simple adjusted formula. Recently, the robustness of Heston s formula has become one of the main issues on option pricing. It is a well-known fact that the logarithm of a complex variable q i re z = is multi-valued, i.e., ) 2 ) (arg( | | ln ln n z i z z p + + = where [ ) p p , ) arg(- z and Z n . If one restricts the logarithm to its principal branch by setting = n (similar to most software packages, such as C++, Gauss, Mathematica, and others), it is necessarily discontinuous at the cut (see Figure 1). The Heston model is represented in Lewis illustration, in which the type of financial claim is entirely decoupled from the calculation of the Green function. Different payoffs are then managed through elementary contour integration over functions and contours that depend on the payoff. In this way, one can see that the 2 issue is fundamentally related to the Green function component of the solution. Once the implementation problems in the Green function component of the solution have been solved, the robustness of the formulae for all European-style financial claims in...
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This note was uploaded on 07/26/2011 for the course ECON 101 taught by Professor Markspenser during the Spring '11 term at Webster FL.

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2-2 - Implementation Problems and Solutions in Stochastic...

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