Pricing Timer Options:Second Order Multiscale Stochastic Volatility Asymptotics

Pricing option is one of the most important problems in financial mathematics because the precision of pricing affects their effectiveness as derivative.In this paper,we study the pricing of timer option.A timer option is a derivative security with the feature that its expiration date is random,depending on the realized volatility of the underlying asset.The option can be exercised when cumulative realized variance reaches to the prescribed variance budget.This feature of the timer option can protect the investor from overpaying the price due to the high level implied volatility.In this paper,we study the pricing of timer options in a class of stochastic volatil-ity models.Numerous empirical studies have identified at least a fast time scale and a slowly varying factor affecting stock price volatility.In this class of stochastic volatility models,the volatility is not a constalt,but a variable driven by two diffusion process-es.In fact,this class of multiscale stochastic volatility models captures this feature of stocks in the real world.For the driving process of the volatility,we introduce t-wo parameters 0<(?),?<1 for the singular and regular perturbations.According to Feynman-Kac formula,the price of option can be transformed to the problem of cor-responding solving partial differential equations.We expand the price of timer option according to these two parameters,and obtain the full second order asymptotics using the singular and regular perturbation techniques.We obtain implied volatility approx-imation from the price asymptotic formula.The random maturity of the timer option can be transformed to a constant related to the ratio of variance budget to effective variance in the asymptotic analysis.We can establish the leading term of the approx-imated price via the Black-Scholes formula,and derive subsequent terms through the leading term and its derivatives.Numerical experiment shows that the approximation formulas have high level of numerical accuracy.In particular,the precision of the timer option pricing is higher when the variance budget is larger.