Tempered stable process, first passage time, and path-dependent option pricing
In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Lévy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. For the numerical illustration, we calibrate risk neutral process parameters using S&P 500 index option prices and apply those parameters to find prices of perpetual American option and barrier option.
KeywordsLévy process Tempered stable process First passage time Barrier option pricing Perpetual American option pricing
JEL ClassificationG13 C21 C42
I am grateful to Professor Kyuong Jin Choi, in Haskayne School of Business, University of Calgary, who gave the motivation to complete of this research. Also, all remaining errors are entirely my own.
- Barndorff-Nielsen O (1977) Exponentially decreasing distributions for the logarithm of particle size. Proc R Soc Lond A 353(1674):401–419. ISSN 0080-4630. https://doi.org/10.1098/rspa.1977.0041
- Barndorff-Nielsen OE, Shephard N (2001) Normal modified stable processes. Economics Series Working Papers from University of Oxford, Department of Economics, 72Google Scholar
- Boguslavskaya E (2014) Solving optimal stopping problems for Lévy processes in infinite horizon via \(A\)-transform. ArXiv e-prints, MarchGoogle Scholar
- Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Hall, LondonGoogle Scholar
- Ferreiro-Castilla A, Schoutens W (2012) The \(\beta \)-meixner model. J Comput Appl Math 236(9):2466–2476. https://doi.org/10.1016/j.cam.2011.12.004. (ISSN 0377-0427)
- Hull JC (2015) Options, futures and other derivatives, 9th edn. Prentice-Hall, Englewood CliffsGoogle Scholar
- Kim YS (2005) The modified tempered stable processes with application to finance. Doctoral Dissertation, Sogang UniversityGoogle Scholar
- Kim YS, Lee J, Mittnik S, Park J (2015) Quanto option pricing in the presence of fat tails and asymmetric dependence. J Econom 187(2):512–520. ISSN 0304-4076. https://doi.org/10.1016/j.jeconom.2015.02.035. (Econometric Analysis of Financial Derivatives)
- Lewis AL (2001) A simple option formula for general jump-diffusion and other exponential Lévy processes. https://doi.org/10.2139/ssrn.282110