Bessel Processes, Stochastic Volatility, and Timer Options

27 Pages Posted: 13 Jan 2016

See all articles by Chenxu Li

Chenxu Li

Peking University - Guanghua School of Management

Date Written: January 2016


Motivated by analytical valuation of timer options (an important innovation in realized variance‐based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first‐passage time problem on realized variance, and generalize the standard risk‐neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first‐passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton‐type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed.

Keywords: timer options, volatility derivatives, realized variance, stochastic volatility models, Bessel processes

Suggested Citation

Li, Chenxu, Bessel Processes, Stochastic Volatility, and Timer Options (January 2016). Mathematical Finance, Vol. 26, Issue 1, pp. 122-148, 2016, Available at SSRN: or

Chenxu Li (Contact Author)

Peking University - Guanghua School of Management ( email )

Guanghua School of Management
Beijing, 100871

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