Analytic Approximation of Finite-Maturity Timer Option Prices

Li, Minqiang (2014): Analytic Approximation of Finite-Maturity Timer Option Prices.

Abstract

We develop an approximation technique for pricing finite-maturity timer options under Heston-like stochastic volatility models. By approximating the distributions of the accumulated variance and the random variance budget exceeding time, we obtain analytic expressions for timer option prices under zero correlation. For nonzero correlation, we use a simple linear combination approximation which matches the asymptotic correlation behavior. Numerical analysis using the Heston model shows that the method is fairly accurate, especially when the volatility of variance is small or the maximum maturity is large.

• Item Type: MPRA Paper
• Language: English
• Keywords: Finite-Maturity Timer Options; Analytic Approximation; Perturbation; Hitting time; Integrated Diffusion
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Ahn, D.H., and B. Gao. 1999. A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12, 721–762.

Bernard, C., Z. Cui. 2011. Pricing timer options. Journal of Computational Finance 15(1), 69–104.

Bick, A. 1995. Quadratic-variation-based dynamic strategies. Management Science 41(4), 722–732.

Carr, P., and R. Lee. 2010. Hedging variance option on continuous semimartingales. Finance and Stochastic 14, 179–207.

Folland, G.B. 1999. Real Analysis: Modern Techniques and Their Applications. Wiley.

Hawkins, D., and S. Krol. 2008. Product overview: Timer options. Lehman Brothers Equity Derivatives note.

Heston, S.L. 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327-343.

Hull, J., and A. White. 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 42(2), 281–300.

Lee, R. 2012. Timer options for risk-controlled variance exposure. Global Derivatives USA 2012 conference presentation.

Lewis, A.L. 2000. Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach, California.

Li, C.X. 2010. Managing volatility risk. Doctoral Dissertation, Columbia University.

Li, C.X. 2013. Bessel processes, stochastic volatility, and timer options. Mathematical Finance, forthcoming. DOI:10.1111/mafi.12041

Li, M. 2013. Derivatives pricing on integrated diffusion processes: A general perturbation approach. Working paper. Bloomberg LP.

Li, M., F. Mercurio. 2013a. Closed-form approximation of timer option prices under general stochastic volatility models. Working paper. Bloomberg LP.

Liang, L.Z.J., D. Lemmens, and J. Tempere. 2011. Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation. Physical Review E (83).

Lipton, A. 2001. Mathematical Methods for Foreign Exchange: a Financial Engineers Approach. World Scientific.

Nayfeh, A.H. 2000. Perturbation Methods. Wiley-Interscience.

Neuberger, A. 1990. Option pricing: A non-stochastic approach. Working paper. London Business School.

Romano, M., and N. Touzi. 1997. Contingent claims and market completeness in a stochastic volatility model. Mathematical Finance 7(4), 399–412.

Royden, H. 1988. Real Analysis. Pearson. Saunders, D. 2010. Pricing timer options under fast mean-reverting stochastic volatility. Working paper.

Sawyer, N. 2007. SG CIB launches timer options. Risk 20(7), 6.

Sawyer, N. 2008. Equity derivative house of the year: Soci´et´e G´en´erale. Risk 21(1).

Toft, K.B. 1996. On the mean-variance tradeoff in option replication with transactions costs. Journal of Financial and Quantitative Analysis 31(2), 233–263.