No Access

PRICING OF PERPETUAL AMERICAN OPTIONS IN A MODEL WITH PARTIAL INFORMATION

PAVEL V. GAPEEV

Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom

Supported by ESF AMaMeF Short Visit Grant No. 1505.

https://doi.org/10.1142/S0219024911006450Cited by:14 (Source: Crossref)

Abstract

We study the perpetual American call option pricing problem in a model of a financial market in which the firm issuing a traded asset can regulate the dividend rate by switching it between two constant values. The firm dividend policy is unknown for small investors, who can only observe the prices available from the market. The asset price dynamics are described by a geometric Brownian motion with a random drift rate modeled by a continuous time Markov chain with two states. The optimal exercise time of the option for small investors is found as the first time at which the asset price hits a boundary depending on the current state of the filtering dividend rate estimate. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also provide closed form estimates for the rational option price and the optimal exercise boundary.

Keywords:

References

M. Abramovitz and I. A. Stegun , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( Wiley , New York , 1972 ) . Google Scholar
H. Bateman and A. Erdélyi , Higher Transcendental Functions ( Mc Graw-Hill , New York , 1953 ) . Google Scholar
M. Beibel and H. R. Lerche, Statistica Sinica 7, 93 (1997). Web of Science, Google Scholar
A. Bensoussan and J. L. Lions , Applications of Variational Inequalities in Stochastic Control ( North Holland , Amsterdam , 1982 ) . Google Scholar
D. C. Brody, L. P. Hughston and A. Macrina, International Journal of Theoretical and Applied Finance 11, 107 (2008). Link, Web of Science, Google Scholar
R. C. Dalang and M.-O. Hongler, Annals of Applied Probability 14, 2167 (2004), DOI: 10.1214/105051604000000747. Google Scholar
R. J. Elliott , L. Aggoun and J. B. Moore , Hidden Markov Models: Estimation and Control ( Springer , New York , 1995 ) . Google Scholar
R. J. Elliott and C. A. Wilson, Hidden Markov Models in Finance, eds. R. S. Mamon and R. J. Elliott (Springer, New York, 2007) pp. 15–30. Crossref, Google Scholar
A. Friedman , Stochastic Differential Equations and Applications, Volumes I, II ( Academic Press , New York , 1976 ) . Google Scholar
P. V. Gapeev and M. Jeanblanc, International Journal of Theoretical and Applied Finance 13(7), 1001 (2010). Link, Google Scholar
P. V. Gapeev and A. N. Shiryaev, Stochastics: An International Journal of Probability and Stochastic Processes 83(6), 519 (2011), DOI: 10.1080/17442508.2010.530349. Crossref, Web of Science, Google Scholar
P. V. Gapeev and A. N. Shiryaev, Bayesian quickest detection problems for some diffusion processes, [math.ST] (25 pp) , arXiv:1010.3430v1 . Google Scholar
R. Geske, Journal of Finance 33(2), 617 (1978), DOI: 10.2307/2326573. Crossref, Web of Science, Google Scholar
B. I. Grigelionis and A. N. Shiryaev, Theory of Probability and its Applications 11, 541 (1966), DOI: 10.1137/1111060. Crossref, Web of Science, Google Scholar
X. Guo, Journal of Applied Probability 38, 464 (2001). Crossref, Web of Science, Google Scholar
X. Guo and Q. Zhang, IEEE Transactions on Automatic Control 50, 1450 (2005), DOI: 10.1109/TAC.2005.854657. Crossref, Web of Science, Google Scholar
Z. Jiang and M. R. Pistorius, Finance and Stochastics 12, 331 (2008), DOI: 10.1007/s00780-008-0065-9. Crossref, Web of Science, Google Scholar
A. Jobert and L. C. G. Rogers, SIAM Journal on Control and Optimization 44, 2063 (2006), DOI: 10.1137/050623279. Crossref, Web of Science, Google Scholar
A. N. Kolmogorov , Selected Works of A. N. Kolmogorov, Volume II, Probability Theory and Mathematical Statistics , ed. A. N. Shiryaev ( Kluwer , Dordrecht , 1992 ) . Google Scholar
N. V. Krylov , Controlled Diffusion Processes ( Springer , New York , 1980 ) . Crossref, Google Scholar
R. S. Liptser and A. N. Shiryaev , Statistics of Random Processes I ( Springer , Berlin , 1977 ) . Crossref, Google Scholar
H. P. McKean, Industrial Management Review 6, 32 (1965). Web of Science, Google Scholar
B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, 5th edn. (Springer, Berlin, 1998). Crossref, Google Scholar
G. Peskir, Séminaire de Probabilité XL, Lecture Notes in Mathematics 1899 (Springer, 2007) pp. 69–96. Google Scholar
G. Peskir and A. N. Shiryaev , Optimal Stopping and Free-Boundary Problems ( Birkhäuser , Basel , 2006 ) . Google Scholar
R. Rishel, Communications in Information and Systems 6, 193 (2001). Crossref, Web of Science, Google Scholar
A. N. Shiryaev , Optimal Stopping Rules ( Springer , Berlin , 1978 ) . Google Scholar
A. N. Shiryaev , Essentials of Stochastic Finance ( World Scientific , Singapore , 1999 ) . Link, Google Scholar
A. N. Shiryaev and A. A. Novikov, Statistics and Decisions 26, 289 (2008), DOI: 10.1524/stnd.2008.1025. Crossref, Google Scholar
M. H. Vellekoop and J. W. Nieuwenhuis, Applied Mathematical Finance 13, 265 (2006), DOI: 10.1080/13504860600563077. Crossref, Google Scholar