Explicit approximate analytic formulas for timer option pricing with stochastic interest rates

• Timer option price is formulated as a four-dimensional PDE using Δ-hedging approach.
• A dimension-reduction technique is then proposed to reduce the four-dimensional PDE into a two-dimensional PDE.
• A perturbation approach is developed to solve the reduced two-dimensional nonlinear PDE.
• An explicit approximate analytic formula for power style timer option is derived.
• Numerical examples of pricing power style timer options are provided.

The interest rate risk is an important factor in the valuation of timer options. Since the valuation of timer options with interest rate risk is a four-dimensional problem, the dimensionality curse causes tremendous difficulty in finding analytic solutions to the pricing of timer options. In this paper, a fast approximate analytic method is developed to price power style timer options with Vasicek interest rate model. The valuation of timer options with interest rate risk is formulated as a four-dimensional partial differential equation (PDE) using Δ-hedging approach. A dimension-reduction technique is then proposed to reduce the four-dimensional PDE into a two-dimensional nonlinear PDE. A perturbation approach is developed to solve the reduced two-dimensional nonlinear PDEs and then an explicit approximate analytic formula for the timer option is obtained. In particular, explicit approximate analytic formulas for timer options under both Heston and Hull–White models are further derived. Numerical examples of pricing timer options under the above two models are provided. Both the approximate analytic method and the crude Monte Carlo simulation method are used for the examples. The numerical results show that prices of timer options by both methods are close and the approximate analytic method is much faster than the crude Monte Carlo method.