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  • I am grateful to Professors Dilip Madan (Editor) and Jerome Detemple (Co-Editor), an Associate Editor, and two anonymous referees for their extensive and constructive comments. This research was supported by the Guanghua School of Management, the Center for Statistical Sciences, and the Key Laboratory of Mathematical Economics and Quantitative Finance (Ministry of Education) at Peking University, as well as the National Natural Science Foundation of China (Project 11201009). It was also initially supported in part by the faculty fellowship at Columbia University during the author's PhD study.


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.