On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr’s approximation for American puts. (English) Zbl 1060.91061

Florin, Avram; Chan, Terence; Usabel, Miguel

Stochastic Processes Appl. 100, No. 1-2, 75-107 (2002).

The paper studies pricing formulae for barrier options (including American puts) when the underlying asset value follows a spectrally one-sided Lévy process (i.e. the support of the Lévy measure of its jumps is contained in either the positive or the negative half-axis). The approach to pricing is an extension of P. Carr’s method of the so-called Canadian options (the latter have random expiration times with a Poisson distribution). For a Canadian-American put option, the optimal exercise boundary is a constant. Therefore, it becomes possible to restrict the analysis to constant barrier contingent claims.
The P. Carr’s method of American put price approximation, based on approximating the original contingent claim fixed expiration time by a sequence of Poissonian expiration moments converging to it in expectation, is extended from diffusion to one-side jump Lévy underlying asset processes. For the latter, the authors study the cumulant generating function of its “surplus over the barrier” process. The pricing formula uses the Laplace transform of the approximating Canadian option value (in the initial underlying value) in terms of the said cumulant generating function.
A special section applies the general results of the paper to the special case of a Canadian put on a purely Poissonian underlying process with exponential jump sizes.

Reviewer: Alexis Derviz (Praha)

MSC:

91B28	Finance etc. (MSC2000)
60G40	Stopping times; optimal stopping problems; gambling theory
60G48	Generalizations of martingales
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

Laplace transform

References:

[1]	Asmussen, S., Avram, F., Usabel, M., 2000. Finite time multivariate ruin distribution with one Laplace inversion. Preprint.
[2]	Barndorff-Nielsen, O.E.; Sheppard, N., Modelling by Lévy processes for financial econometrics, () · Zbl 0991.62089
[3]	Bertoin, J., Lévy processes, (1996), Cambridge University Press Cambridge
[4]	Bingham, N.H., Fluctuation theory in continuous time, Adv. appl. probab., 7, 705-766, (1975) · Zbl 0322.60068
[5]	Boyarchenko, S.I., Levendorskii, S.Z., 2000. Perpetual American options under Lévy processes. Preprint. · Zbl 1025.60021
[6]	Broadie, M.; Detemple, J., American option valuation: new bounds and approximations, Rev. financial stud., 9, 1211-1250, (1996)
[7]	Carr, P., Randomization and the American put, Rev. financial stud., 11, 597-626, (1998) · Zbl 1386.91134
[8]	Carr, P.; Jarrow, R.; Myneni, R., Alternative characterizations of American put options, Math. finance, 2, 87-105, (1992) · Zbl 0900.90004
[9]	Chan, T., Pricing contingent claims on stocks driven by Lévy processes, Ann. appl. probab., 9, 504-528, (1999) · Zbl 1054.91033
[10]	Chan, T., 1999b. Pricing perpetual American options driven by spectrally one-sided Lévy processes. Preprint.
[11]	Eberlein, E.; Keller, U., Hyperbolic distributions in finance, Bernoulli, 1, 281-299, (1995) · Zbl 0836.62107
[12]	Geman, H., Madan, D., Yor, M., 1999. Asset prices are Brownian motion: only in business time. Preprint, University of Maryland. · Zbl 1134.91019
[13]	Gerber, H.; Shiu, E., Option pricing by esscher transforms, Trans. soc. actuaries, XLVI, 99-139, (1994)
[14]	Gerber, H.; Landry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance math. econom., 22, 263-276, (1998) · Zbl 0924.60075
[15]	Gerber, H.; Shiu, E., Pricing perpetual options for jump processes, North American act. J., 2, 101-112, (1998) · Zbl 1081.91528
[16]	Jacka, S.D., Optimal stopping and the American put, Math. finance, 1, 1-14, (1991) · Zbl 0900.90109
[17]	Ju, N., Pricing an American option by approximating its exercise boundary as a multipiece exponential, Rev. financial stud., 11, 627-646, (1998)
[18]	Ju, N.; Zhong, R., An approximate formula for pricing American options, J. derivatives, 7, 31-40, (1999)
[19]	Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer New York · Zbl 0734.60060
[20]	Kim, J., The analytic valuation of American puts, Rev. financial stud., 3, 547-572, (1990)
[21]	Kwok, Y.K., Mathematical models of financial derivatives, (1998), Springer Berlin · Zbl 0931.91018
[22]	Madan, D., 1999. Purely discontinuous asset price processes. Preprint, University of Maryland. · Zbl 1005.91047
[23]	McKean, H.P.; Samuelson, P.A., A free boundary problem arising in mathematical economics, Ind. manage. rev., 6, 32-39, (1965)
[24]	McMillan, L., Analytic approximation for the American put option, Adv. futures options res., 1, 119-140, (1986)
[25]	Omberg, E., The valuation of American put options, Adv. futures options res., 1, 117-142, (1987)
[26]	Pham, H., Optimal stopping, free boundary and American option in a jump-diffusion model, Appl. math. optim., 35, 145-164, (1997) · Zbl 0866.60038
[27]	Revuz, D.; Yor, M., Continuous martingales and Brownian motion, (1991), Springer Berlin · Zbl 0731.60002
[28]	Usabel, M., 1999. Multivariate finite time ruin probabilities for hyper-exponential waiting times. Documento de Trabajo, Universidad Carlos III de Madrid. · Zbl 1028.91561
[29]	Zhang, X.L., Formules quasi-explicites pour LES options américaines dans un modèle de diffusion avec sauts, Math. comput. simulation, 38, 151-161, (1995) · Zbl 0828.60040
[30]	Zhang, X.L., Numerical analysis of American option pricing in a jump-diffusion model, Math. oper. res., 22, 3, 668-690, (1997) · Zbl 0883.90021
[31]	Zhang, X.L., Discussion to H. gerber and E. shiu (1998), “pricing perpetual options for jump processes”, North American act. J., 2, 101-112, (1998)