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Explicit asymptotics on first passage times of diffusion processes

Part of: Mathematical finance Stochastic processes Distribution theory

Published online by Cambridge University Press:  15 July 2020

Angelos Dassios  and
Luting Li
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Luting Li*
Affiliation:
London School of Economics and Political Science
*
*Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email addresses: a.dassios@lse.ac.uk, l.li27@lse.ac.uk
*Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email addresses: a.dassios@lse.ac.uk, l.li27@lse.ac.uk
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Abstract

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

Keywords

First passage timediffusion processperturbation theoryOrnstein–Uhlenbeck processBessel processexponential-Shiryaev processhypergeometric diffusionspecial functions

MSC classification

Primary: 91G60: Numerical methods (including Monte Carlo methods)
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62E17: Approximations to distributions (nonasymptotic) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 681 - 704
Copyright
© Applied Probability Trust 2020

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References

Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Computing 18, 408–421.CrossRefGoogle Scholar
Alili, L., Patie, P. and Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein–Uhlenbeck process. Stoch. Models 21, 967–980.CrossRefGoogle Scholar
Arbib, M. A. (1965). Hitting and martingale characterizations of one-dimensional diffusions. Z. Wahrscheinlichkeitsth 4, 232–247.10.1007/BF00533754CrossRefGoogle Scholar
Bachelier, L. (1900). Théorie de la spéculation. Gauthier-Villars, Paris.CrossRefGoogle Scholar
Baldi, P., Caramellino, L. and Iovino, M. G. (1999). Pricing general barrier options: a numerical approach using sharp large deviations. Math. Finance 9, 293–321.10.1111/1467-9965.t01-1-00071CrossRefGoogle Scholar
Bateman, H. (1954). Tables of Integral Transforms, Vol. I & II. McGraw–Hill, New York.Google Scholar
Blake, I. and Lindsey, W. (1973). Level-crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.CrossRefGoogle Scholar
Borodin, A. N. (2009). Hypergeometric diffusion. J. Math. Sci. 159, 295–304.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2012). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.Google Scholar
Carr, P. and Linetsky, V. (2006). A jump to default extended CEV model: an application of Bessel processes. Finance Stoch. 10, 303–330.10.1007/s00780-006-0012-6CrossRefGoogle Scholar
Chen, R.-R. and Scott, L. (1992). Pricing interest rate options in a two-factor Cox–Ingersoll–Ross model of the term structure. Rev. Financial Studies 5, 613–636.CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. (2005). A theory of the term structure of interest rates. In Theory of Valuation, World Scientific, Hackensack, NJ, pp. 129–164.CrossRefGoogle Scholar
Dassios, A. and Li, L. (2018). An economic bubble model and its first passage time. Preprint. Available at http://arxiv.org/abs/1803.08160.Google Scholar
Dassios, A. and Lim, J. W. (2018). Recursive formula for the double-barrier Parisian stopping time. J. Appl. Prob. 55, 282–301.CrossRefGoogle Scholar
Dassios, A., Qu, Y. and Lim, J. W. (2018). Azéma martingales and Parisian excursions of Bessel and CIR processes. Working paper.Google Scholar
Dassios, A. and Wu, S. (2010). Perturbed Brownian motion and its application to Parisian option pricing. Finance Stoch. 14, 473–494.CrossRefGoogle Scholar
Dassios, A. and Zhang, Y. Y. (2016). The joint distribution of Parisian and hitting times of Brownian motion with application to Parisian option pricing. Finance Stoch. 20, 773–804.CrossRefGoogle Scholar
Doob, J. L. (2012). Classical Potential Theory and Its Probabilistic Counterpart: Advanced Problems. Springer, New York.Google Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and Sølna, K. (2011). Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press.CrossRefGoogle Scholar
Göing-Jaeschke, A. and Yor, M. (2003). A clarification note about hitting times densities for Ornstein–Uhlenbeck processes. Finance Stoch. 7, 413–415.CrossRefGoogle Scholar
Hamana, Y. and Matsumoto, H. (2013). The probability distributions of the first hitting times of Bessel processes. 365, 5237–5257.CrossRefGoogle Scholar
Hartman, P. and Watson, G. S. (1974). ‘Normal’ distribution functions on spheres and the modified Bessel functions. Ann. Prob. 2, 593–607.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Holmes, M. H. (2012). Introduction to Perturbation Methods. Springer, New York.Google Scholar
Ichiba, T. and Kardaras, C. (2011). Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation. J. Appl. Prob. 48, 699–712.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (2014). Stochastic Differential Equations and Diffusion Processes. Elsevier, Amsterdam.Google Scholar
Itô, K. and McKean, H. P., Jr. (2012). Diffusion Processes and Their Sample Paths. Springer, Berlin, Heidelberg.Google Scholar
Jarrow, R. A. and Turnbull, S. M. (1995). Pricing derivatives on financial securities subject to credit risk. J. Finance 50, 53–85.CrossRefGoogle Scholar
Kent, J. T. (1978). Some probabilistic properties of Bessel functions. Ann. Prob. 6, 760–770.CrossRefGoogle Scholar
Kent, J. T. (1980). Eigenvalue expansions for diffusion hitting times. 52, 309–319.CrossRefGoogle Scholar
Kent, J. T. (1982). The spectral decomposition of a diffusion hitting time. Ann. Prob. 10, 207–219.CrossRefGoogle Scholar
Lánský, P., Sacerdote, L. and Tomassetti, F. (1995). On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biol. Cybernet. 73, 457–465.CrossRefGoogle ScholarPubMed
Li, L. (2019). First passage times of diffusion processes and their applications to finance. Doctoral Thesis, London School of Economics and Political Science.Google Scholar
Linetsky, V. (2004). Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models. J. Comput. Finance 7, 1–22.CrossRefGoogle Scholar
Lozier, D. W. (2003). NIST digital library of mathematical functions. Ann. Math. Artif. Intel. 38, 105–119.CrossRefGoogle Scholar
McKean, H. P., Jr. (1960). The Bessel motion and a singular integral equation. Mem. Coll. Sci. Univ. Kyoto A: Math. 33, 317–322.CrossRefGoogle Scholar
Novikov, A., Frishling, V. and Kordzakhia, N. (2003). Time-dependent barrier options and boundary crossing probabilities. Georgian Math. J. 10, 325–334.Google Scholar
Novikov, A. A. (1981). A martingale approach to first passage problems and a new condition for Wald’s identity. In Stochastic Differential Systems, Springer, Berlin, Heidelberg, pp. 146–156.CrossRefGoogle Scholar
Olver, F. W. J. (2010). NIST Handbook of Mathematical Functions (Hardback and CD-ROM). Cambridge University Press.Google Scholar
Patie, P. (2004). On some first passage time problems motivated by financial applications. Doctoral Thesis, Universität Zürich.Google Scholar
Peskir, G. (2006). On the fundamental solution of the Kolmogorov–Shiryaev equation. In From Stochastic Calculus to Mathematical Finance, Springer, Berlin, Heidelberg, pp. 535–546.CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Redner, S. (2001). A Guide to First-Passage Processes. Cambridge University Press.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2013). Continuous Martingales and Brownian Motion. Springer, Berlin, Heidelberg.Google Scholar
Ricciardi, L. M., Crescenzo, A. D., Giorno, V. and Nobile, A.G. (1999). An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math. Japon. 50, 247–322.Google Scholar
Schrödinger, E. (1915). Zur Theorie der Fall- und Steigversuche an Teilchen mit brownscher Bewegung. Phys. Z. 16, 289–295.Google Scholar
Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Ann. Phys. 385, 437–490.CrossRefGoogle Scholar
Schwartz, E. S. (1997). The stochastic behavior of commodity prices: implications for valuation and hedging. J. Finance 52, 923–973.CrossRefGoogle Scholar
Shiryaev, A. N. (2002). Quickest detection problems in the technical analysis of the financial data. In Mathematical Finance – Bachelier Congress 2000, Springer, Berlin, Heidelberg, pp. 487–521.CrossRefGoogle Scholar
Shiryaev, A. N. (1961). The problem of the most rapid detection of a disturbance in a stationary process. Soviet Math. Dokl. 2, 795–799.Google Scholar
Siegert, A. J. F. (1951). On the first passage time probability problem. Phys. Rev. 81, 617.CrossRefGoogle Scholar
Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of the Brownian motion. Phys. Rev. 36, 823.CrossRefGoogle Scholar
Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Econom. 5, 177–188.CrossRefGoogle Scholar
Wan, F. Y. M. and Tuckwell, H. C. (1982). Neuronal firing and input variability. J. Theoret. Neurobiol. 1, 197–218.Google Scholar
Wang, M. C. and Uhlenbeck, G. E. (1945). On the theory of the Brownian motion II. Rev. Mod. Phys. 17, 323.CrossRefGoogle Scholar