The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines
This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.
|Date of creation:||01 Mar 2008|
|Date of revision:|
|Publication status:||Published as: Chiarella, C., Kang, B., Meyer, G. H. and Ziogas, A., 2009, "The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines", International Journal of Theoretical and Applied Finance, 12(3), 393-425.|
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- Merton, Robert C., 1976.
"Option pricing when underlying stock returns are discontinuous,"
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- Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
- Thomas Adolfsson & Carl Chiarella & Andrew Ziogas & Jonathan Ziveyi, 2013. "Representation and Numerical Approximation of American Option Prices under Heston Stochastic Volatility Dynamics," Research Paper Series 327, Quantitative Finance Research Centre, University of Technology, Sydney.
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- S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
- Carl Chiarella & Andrew Ziogas, 2006. "American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach," Research Paper Series 174, Quantitative Finance Research Centre, University of Technology, Sydney.
- Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
- Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-63, December.
- Elias Tzavalis & Shijun Wang, 2003. "Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary," Working Papers 488, Queen Mary University of London, School of Economics and Finance.
- Gerald Cheang & Carl Chiarella & Andrew Ziogas, 2009. "An Analysis of American Options under Heston Stochastic Volatility and Jump-Diffusion Dynamics," Research Paper Series 256, Quantitative Finance Research Centre, University of Technology, Sydney.
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