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PRICING PERPETUAL TIMER OPTION UNDER THE STOCHASTIC VOLATILITY MODEL OF HULL–WHITE

Part of: Integral transforms, operational calculus Mathematical finance

Published online by Cambridge University Press:  16 May 2017

JICHAO ZHANG Open the ORCID record for JICHAO ZHANG [Opens in a new window] ,
XIAOPING LU Open the ORCID record for XIAOPING LU [Opens in a new window]  and
YUECAI HAN Open the ORCID record for YUECAI HAN [Opens in a new window]
JICHAO ZHANG
Affiliation:
School of Mathematics, Jilin University, Changchun, Jilin 130012, China email [email protected], [email protected]
XIAOPING LU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, WollongongNSW 2522, Australia email [email protected]
YUECAI HAN*
Affiliation:
School of Mathematics, Jilin University, Changchun, Jilin 130012, China email [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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The valuation of perpetual timer options under the Hull–White stochastic volatility model is discussed here. By exploring the connection between the Hull–White model and the Bessel process and using time-change techniques, the triple joint distribution for the instantaneous volatility, the cumulative reciprocal volatility and the cumulative realized variance is obtained. An explicit analytical solution for the price of perpetual timer call options is derived as a Black–Scholes–Merton-type formula.

Keywords

timer optionstochastic volatilityBessel process

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 44A10: Laplace transform
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 3-4 , April 2017 , pp. 406 - 416
Copyright
© 2017 Australian Mathematical Society 

References

Bernard, C. and Cui, Z., “Pricing timer options”, J. Comput. Finance 15 (2011) 69–104 ; doi:10.21314/JCF.2011.228.Google Scholar
Bick, A., “Quadratic-variation-based dynamic strategies”, Manag. Sci. 41 (1995) 722–732 ; doi:10.1287/mnsc.41.4.722.Google Scholar
Borodin, A. N. and Salminen, P., Handbook of Brownian motion – facts and formulae, 2nd edn, Probability and its Applications (Birkhäuser Basel, Berlin, 2002) ;doi:10.1007/978-3-0348-8163-0.CrossRefGoogle Scholar
Carr, P. and Lee, R., “Hedging variance options on continuous semimartingales”, Finance Stoch. 14 (2010) 179–207; doi:10.1007/s00780-009-0110-3.CrossRefGoogle Scholar
Fontenay, E. D., Hedge fund replication and structured products (Société Générale, 2007).Google Scholar
Fouque, J. P., Papanicolaou, G. and Sircar, K. R., Derivatives in financial markets with stochastic volatility (Cambridge University Press, Cambridge, 2000).Google Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financ. Stud. 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.Google Scholar
Hull, J. C. and White, A. D., “The pricing of options on assets with stochastic volatilities”, J. Finance 42 (1987) 281–300; doi:10.1111/j.1540-6261.1987.tb02568.x.Google Scholar
Itô, K., “Stochastic integral”, Proc. Imp. Acad. Tokyo 20 (1944) 519–524 ; doi:10.3792/pia/1195572786.Google Scholar
Karatzas, I. and Shreve, S. E., Brownian motion and stochastic calculus, 2nd edn. Volume 113 of Graduate Texts in Mathematics (Springer, New York, 1998) ;doi:10.1007/978-1-4612-0949-2.Google Scholar
Li, C., “Managing volatility risk”, Doctoral Dissertation, Columbia University, 2010.Google Scholar
Li, C., “Bessel processes, stochastic volatility, and timer options”, Math. Finance 26 (2016) 122–148; doi:10.1111/mafi.12041.Google Scholar
Li, M. and Mercurio, F., “Closed-form approximation of perpetual timer option prices”, Int. J. Theor. Appl. Finance 17 (2014) 1450026; doi:10.1142/S0219024914500265.Google Scholar
Liang, L. Z., Lemmens, D. and Tempere, J., “Path integral approach to the pricing of timer options with the Duru–Kleinert time transformation”, Phys. Rev. E 83 (2011) 056122 ; doi:10.1103/PhysRevE.83.056112.Google Scholar
Lintsky, V., “The spectral representation of Bessel processes with constant drift: applications in queueing and finance”, J. Appl. Probab. 41 (2004) 327–344; doi:10.1239/jap/1082999069.Google Scholar
Merton, R. C., “Theory of rational option pricing”, Bell J. Econ. Manag. Sci. 4 (1973) 141–183 ; doi:10.2307/3003143.Google Scholar
Merton, R. C., “Option pricing when underlying stock returns are discontinuous”, J. Financ. Econ. 3 (1976) 125–144; doi:10.1016/0304-405X(76)90022-2.Google Scholar
Pitman, J. and Yor, M., “Bessel processes and infinitely divisible laws”, in: Stochastic integrals, Volume 851 of Lecture Notes in Mathematics (ed. Williams, D.), (Springer, Berlin–Heidelberg, 1981); doi:10.1007/BFb0088732.Google Scholar
Revuz, D. and Yor, M., Continuous martingales and Brownian motion, 3rd edn. Volume 293 of Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 1999) ; doi:10.1007/978-3-662-06400-9.Google Scholar
Saunders, D., “Pricing timer options under fast mean-reverting stochastic volatility”, Can. Appl. Math. Q. 17 (2009) 737–753 ;http://www.math.ualberta.ca/ami/CAMQ/pdf_files/vol_17/17_4/17_4f.pdf.Google Scholar
Shiga, T. and Watanabe, S. Z., “Bessel diffusions as a one-parameter family of diffusion processes”, Wahrscheinlichkeitsth. verw. Geb. 27 (1973) 37–46; doi:10.1007/BF00736006.Google Scholar
Yor, M., Exponential functionals of Brownian motion and related processes, Springer Finance (Springer, Berlin–Heidelberg, 2001); doi:10.1007/978-3-642-56634-9.Google Scholar