Abstract
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.
Similar content being viewed by others
References
Applebaum D (2004) Lévy process and stochastic calculus. Cambridge University Press, New York
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, Levendorskii S (2001) Feller processes of normal inverse gaussian type. Quant Finance 1:318–331
Barndorff-Nielsen OE, Shephard N (2001) Normal modified stable processes. Economics Series Working Papers from University of Oxford, Department of Economics, 72
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654
Boguslavskaya E (2014) Solving optimal stopping problems for Lévy processes in infinite horizon via \(A\)-transform. ArXiv e-prints, March
Boyarchenko SI, Levendorskiĭ SZ (2000) Option pricing for truncated Lévy processes. Int J Theor Appl Finance 3:549–552
Boyarchenko SI, Levendorskiĭ SZ (2002a) Non-Gaussian Merton–Black–Scholes theory. World Scientific, Singapore
Boyarchenko SI, Levendorskiĭ SZ (2002b) Perpetual american options under Lévy processes. SIAM J Control Optim 40(6):1663–1696
Boyarchenko SI, Levendorskiĭ SZ (2002c) Barrier options and touch-and-out options under regular lvy processes of exponential type. Ann Appl Probab 12(4):1261–1298
Carr P, Madan D (1999) Option valuation using the fast fourier transform. J Comput Finance 2(4):61–73
Carr P, Geman H, Madan D, Yor M (2002) The fine structure of asset returns: an empirical investigation. J Bus 75(2):305–332
Chandra SR, Mukherjee D (2016) Barrier option under Levy model: a pide and Mellin transform approach. Mathematics 4(1):2. https://doi.org/10.3390/math4010002
Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Hall, London
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)
Gerber HU, Shiu ESW (1994) Martingale approach to pricing perpetual american options. Astin Bull 24:195–220
Hull JC (2015) Options, futures and other derivatives, 9th edn. Prentice-Hall, Englewood Cliffs
Hurd TR, Kuznetsov A (2009) On the first passage time for brownian motion subordinated by a Lévy process. J Appl Probab 46(1):181–198. https://doi.org/10.1239/jap/1238592124
Kim SI, Kim YS (2018) Normal tempered stable structural model. Rev Deriv Res 21(1):119–148
Kim YS (2005) The modified tempered stable processes with application to finance. Doctoral Dissertation, Sogang University
Kim YS, Lee JH (2006) The relative entropy in CGMY processes and its applications to finance. Math Methods Oper Res 66(2):327–338
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)
Koponen I (1995) Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys Rev E 52:1197–1199
Lewis AL (2001) A simple option formula for general jump-diffusion and other exponential Lévy processes. https://doi.org/10.2139/ssrn.282110
Rachev ST, Kim YS, Bianch ML, Fabozzi FJ (2011) Financial models with Lévy processes and volatility clustering. Wiley, London
Rogers LCG (2000) Evaluating first-passage probabilities for spectrally one-sided Levy processes. J Appl Probab 37(4):1173–1180. https://doi.org/10.1239/jap/1014843099
Schoutens W (2003) Lévy processes in finance. Wiley, London
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
As “Appendix”, we discuss perpetual American option pricing and barrier option pricing under the Lévy market model.
1.1 Perpetual American option
The perpetual call and put option price on Lévy model can be obtained by the martingale method introduced in Gerber and Shiu (1994). In this section, we just follow the martingale method for the Lévy market price model. We consider a perpetual American call option with strike price K. If the option holder exercise the call at a time T, then the holder obtain \((S(T)-K)^+\) where \(x^+ = \max \{0,x\}\). Let L be a real number with \(L\ge K\). The holder will exercise the call when the asset price first become greater than or equal to the level L. We define the first passage time
where \(l=\log (L/S(0))>0\). Then the current value of the perpetual American call is
Let
which is the Laplace transform of \(\tau (l)\). Applying Lemma 1, we can obtain the Laplace transform as
where \(\eta ^+(ir)\) is the value satisfying (2) and (3) for \(l>0\) and \(u=ir\). Hence we have
By solving
we find the optimal value \(L^+\)
Hence, we obtain the maximum value
If \(L^+< S(0)\) then the call is immediately exercised so we have price \(S(0)-K\). Therefore the perpetual call price is equal to
We consider a perpetual American put option with strike price K. If the option holder exercise the put at a time T, then the holder obtain \((K-S(T))^+\). Let L be a real number with \(0<L\le K\). The holder will exercise the put when the asset price first become less than or equal to the level L. We define the first passage time
where \(l=\log (L/S(0))<0\). Then the current value of the put is
which is the Laplace transform of \(\tau (l)\). For the same arguments as the call option case, we find the optimal value \(L^-\)
where \(\eta ^-(ir)\) is the value satisfying (2) and (3) for \(l<0\) and \(u=ir\). Hence the perpetual put price is equal to
1.2 Barrier option
Let \(\varPi \) be the payoff function of European options. For example, the European call and put options with strike price K are given by \(\varPi (S(T))=(S(T)-K)^+\) and \(\varPi (S(T))=(K-S(T))^+\), respectively. The knock-in barrier option with the barrier level B, time to maturity T is priced by the following equation
where \(l = \log (B/S(0))\). Note that \(l<0\) for the down-and-in barrier option and \(l>0\) for the up-and-in barrier option. Since we have
the knock-out barrier option price can be obtained by the following equation
where \(V = e^{-rT}E\left[ \varPi (S(T))\right] \). Note that \(l<0\) for the down-and-out barrier option and \(l>0\) for the up-and-out barrier option.
Case 1 \(\varPi (S(T)) = \varPi (S(T))1_{\tau (l)<T}\)
If \(\varPi (S(T)) = \varPi (S(T))1_{\tau (l)<T}\) then the barrier option price is the same as option prices without the barrier:
For example (1) up-and-in call option with \(K>B\), we have \((S(T)-K)^+=(S(T)-K)^+1_{\tau (l)<T}\), and (2) down-and-in put option with \(K<B\), we have \((K-S(T))^+=(K-S(T))^+1_{\tau (l)<T}\).
Case 2 \(\varPi (S(T)) \ne \varPi (S(T))1_{\tau (l)<T}\)
If \(\varPi (S(T))\ne \varPi (S(T))1_{\tau (l)<T}\), we have
where \(f_{\tau (l)}\) is the pdf of \(\tau (l)\). Since we have \(S(0)e^l = B\) and
the \(c_{i}\) becomes
By European option pricing formula using Fourier transform [see Carr and Madan (1999), Lewis (2001) and Rachev et al. (2011)], we have
where \(\hat{\varPi }(z)=\int _{-\infty }^\infty e^{-izx}\varPi (e^x)dx\) for complex number z and \(\rho \) is a real constant such that \(\psi _X(u+i\rho )\) and \(\hat{\varPi }(u+i\rho )\) are well defined for all \(u\in \mathbb {R}\). Hence we have
Let
then
European call and put options
For the call option payoff \(\varPi (S(T))=(S(T)-K)^+\), we have
and for the put option payoff \(\varPi (S(T))=(K-S(T))^+\), we have
The down-and-in call option price (\(c_{di}\)) and up-and-in put option price (\(p_{ui}\)) are always in Case 2. Therefore, we have their prices as
and
For the up-and-in call option price (\(c_{ui}\)) and the down-and-in put option price (\(p_{di}\)), we consider Case 1, and finally obtain
and
The up-and-out and down-and-out calls corresponding to the up-and-in and down-and-in calls above are priced by \(c_{uo} = c - c_{ui}\), and \(c_{do} = c - c_{di}\), respectively, where c is the vanilla call option price with strike price K, and time to maturity T. The up-and-out and down-and-out puts corresponding to the up-and-in and down-and-in puts above are priced by \(p_{uo} = p - p_{ui}\), and \(p_{do} = p - p_{di}\), respectively, where p is the vanilla put option price with strike price K, and time to maturity T.
Rights and permissions
About this article
Cite this article
Kim, Y.S. Tempered stable process, first passage time, and path-dependent option pricing. Comput Manag Sci 16, 187–215 (2019). https://doi.org/10.1007/s10287-018-0326-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10287-018-0326-9
Keywords
- Lévy process
- Tempered stable process
- First passage time
- Barrier option pricing
- Perpetual American option pricing