No Access

ON SOME FUNCTIONALS OF THE FIRST PASSAGE TIMES IN MODELS WITH SWITCHING STOCHASTIC VOLATILITY

PAVEL V. GAPEEV

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

E-mail Address: p.v.gapeev@lse.ac.uk

Search for more papers by this author

OLIVER BROCKHAUS

Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK

Present address: Independent Scholar, Hofheim, Germany.

Search for more papers by this author

, and

MATHIEU DUBOIS

Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK

Present address: Independent Scholar, London, United Kingdom.

Search for more papers by this author

https://doi.org/10.1142/S0219024918500012Cited by:1 (Source: Crossref)

Abstract

We compute some functionals related to the joint generalized Laplace transforms of the first times at which two-dimensional diffusion-type Markov processes exit half strips. It is assumed that the state space components are driven by constantly correlated Brownian motions and the dynamics of the coefficients are described by a continuous-time Markov chain. The method of proof is based on the solutions of the equivalent boundary-value problems for systems of elliptic-type partial differential equations for the associated value functions. The results are illustrated on several two-dimensional continuous mean-reverting or diverting models of switching stochastic volatility.

Keywords:

References

M. Abramovitz & I. A. Stegun (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: National Bureau of Standards Wiley. Google Scholar
S. Assing, S. Jacka & A. Ocejo (2014) Monotonicity of the value function for a two-dimensional optimal stopping problem, Annals of Applied Probability 24 (4), 1554–1584. Crossref, Web of Science, Google Scholar
E. Bayraktar, H. V. Poor (2007) Quickest detection of a minimum of two Poisson disorder times, SIAM Journal on Control and Optimization 46 (1), 308–331. Crossref, Web of Science, Google Scholar
E. Bayraktar, S. Dayanik & I. Karatzas (2005) The standard Poisson disorder problem revisited, Stochastic Processes and Their Applications 115 (9), 1437–1450. Crossref, Web of Science, Google Scholar
O. Brockhaus (2016) Equity Derivatives and Hybrids: Markets, Models and Methods. Palgrave Macmillan, London. Crossref, Google Scholar
R. C. Dalang & M.-O. Hongler (2004) The right time to sell a stock whose price is driven by Markovian noise, Annals of Applied Probability 14, 2167–2201. Crossref, Web of Science, Google Scholar
S. Dayanik, H. V. Poor & S. O. Sezer (2008) Multisource Bayesian sequential change detection, Annals of Applied Probability 18 (2), 552–590. Crossref, Web of Science, Google Scholar
G. Di Graziano & L. C. G. Rogers (2009) Equity with Markov-modulated dividends, Quantitative Finance 9, 19–26. Crossref, Web of Science, Google Scholar
E. B. Dynkin (1965) Markov Processes. Volume I. New York: Springer. Crossref, Google Scholar
R. J. Elliott, L. Aggoun & J. B. Moore (1995) Hidden Markov Models: Estimation and Control. New York: Springer. Google Scholar
L. Fernández, P. Hieber & M. Scherer (2013) Double-barrier first-passage times of jump-diffusion processes, Monte Carlo Methods and Their Applications 19 (2), 107–141. Crossref, Google Scholar
J. P. Fouque, G. Papanicolaou & K. R. Sircar (2000) Derivatives in Financial Markets with Stochastic Volatility. Cambridge: Cambridge University Press. Google Scholar
P. V. Gapeev & A. N. Shiryaev (2011) On the sequential testing problem for some diffusion processes, Stochastics: An International Journal of Probability and Stochastic Processes 83 (4–6), 519–535. Crossref, Web of Science, Google Scholar
P. V. Gapeev & A. N. Shiryaev (2013) Bayesian quickest detection problems for some diffusion processes, Advances in Applied Probability 45 (1), 164–185. Crossref, Web of Science, Google Scholar
P. V. Gapeev & Y. I. Stoev (2017) On the Laplace transforms of the first exit times in one-dimensional non-affine jump-diffusion models, Statistics and Probability Letters 121, 152–162. Crossref, Web of Science, Google Scholar
P. V. Gapeev & Y. I. Stoev (2017) On some functionals of the first passage times in jump-diffusion models of stochastic volatility, Submitted. Google Scholar
J. Gatheral (2011) The Volatility Surface: A Practitioner’s Guide. New York: John Wiley and Sons. Google Scholar
X. Guo (2001) An explicit solution to an optimal stopping problem with regime switching, Journal of Applied Probability 38, 464–481. Crossref, Web of Science, Google Scholar
X. Guo & Q. Zhang (2004) Closed-form solutions for perpetual American put options with regime switching, SIAM Journal on Applied Mathematics 64, 2034–2049. Crossref, Web of Science, Google Scholar
S. Iyengar (1985) Hitting lines with two-dimensional Brownian motion, SIAM Journal on Applied Mathematics 45 (6), 983–989. Crossref, Web of Science, Google Scholar
Z. Jiang & M. R. Pistorius (2008) On perpetual American put valuation and first-passage in a regime-switching model with jumps, Finance and Stochastics 12, 331–355. Crossref, Web of Science, Google Scholar
A. Jobert & L. C. G. Rogers (2006) Option pricing with Markov-modulated dynamics, SIAM Journal on Control and Optimization 44, 2063–2078. Crossref, Web of Science, Google Scholar
P. Johnson & G. Peskir (2017) Quickest detection problems for Bessel processes, Annals of Applied Probability 27, 1003–1056. Crossref, Web of Science, Google Scholar
P. Johnson & G. Peskir (2018) Sequential testing problems for Bessel processes. To appear in Transactions of American Mathematical Society 370, 2085–2113. Crossref, Web of Science, Google Scholar
J. Kallsen (2006) A didactic note on affine stochastic volatility models, In: From Stochastic Calculus to Mathematical Finance. Berlin: Springer, 343–368. Crossref, Google Scholar
S. G. Kou & H. Wang (2003) First passage times of a jump diffusion process, Advances in Applied Probability 35 (2), 504–531. Crossref, Web of Science, Google Scholar
R. S. Liptser & A. N. Shiryaev (2001) Statistics of Random Processes I. Second edition. Berlin: Springer. Google Scholar
A. Mijatovic & M. R. Pistorius (2012) On the drawdown of completely asymmetric Lévy processes, Stochastic Processes and Their Applications 22, 3812–3836. Crossref, Web of Science, Google Scholar
B. Øksendal (1998) Stochastic Differential Equations. An Introduction with Applications, Fifth edition. Berlin: Springer. Crossref, Google Scholar
D. Perry, W. Stadje & S. Zacks (2002a), First-exit times for compound Poisson processes for some types of positive and negative jumps, Stochastic Models 18 (1), 139–157. Crossref, Web of Science, Google Scholar
D. Perry, W. Stadje & S. Zacks (2002b) Boundary crossing for the difference of two ordinary or compound Poisson processes, Annals of Operations Research 113 (1), 119–132. Crossref, Web of Science, Google Scholar
G. Peskir & A. N. Shiryaev (2006) Optimal Stopping and Free-Boundary Problems. Basel: Birkhäuser. Google Scholar
D. Revuz & M. Yor (1999) Continuous Martingales and Brownian Motion, Third edition. Berlin: Springer. Crossref, Google Scholar
L. C. G. Rogers & D. Williams (1987) Diffusions, Markov Processes and Martingales II. Itô Calculus. New York: Wiley. Google Scholar
A. Sepp (2004) Analytical pricing of double-barrier options under a double-exponential jump diffusion process: Applications of Laplace transform, International Journal of Theoretical and Applied Finance 7 (2), 151–175. Link, Google Scholar
A. N. Shiryaev (1999) Essentials of Stochastic Finance. Singapore: World Scientific. Link, Google Scholar
S. Zacks, D. Perry, D. Bshouty & S. Bar-Lev (1999) Distributions of stopping times for compound Poisson processes with positive jumps and linear boundaries, Communications in Statistics. Stochastic Models 15 (1), 81–101. Crossref, Google Scholar
V. F. Zaitsev & A. D. Polyanin (2002) Handbook of Exact Solutions for Ordinary Differential Equations. Taylor and Francis, Oxford. Crossref, Google Scholar