Fast estimation of true bounds on Bermudan option prices under jump-diffusion processes

Abstract

Fast pricing of American-style options has been a difficult problem since it was first introduced to the financial markets in 1970s, especially when the underlying stocks' prices follow some jumpdiffusion processes. In this paper, we extend the 'true martingale algorithm' proposed by Belomestny et al. [Math. Finance, 2009, 19, 53-71] for the pure-diffusion models to the jump-diffusion models, to fast compute true tight upper bounds on the Bermudan option price in a non-nested simulation manner. By exploiting the martingale representation theorem on the optimal dual martingale driven by jump-diffusion processes, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improving the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our algorithm.

True martingales for upper bounds on Bermudan option prices under jump-diffusion processes

“…Remark 1 The proofs of all theorems, corollaries and propositions in this paper are provided in Zhu et al (2013).…”

Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions

“…The resultant bound is then the true upper bound. More recently, Zhu, Ye, & Zhou (2014) extend the method in Belomestny et al (2009) to a jump-diffusion model. Their theoretical analysis shows that the martingale property of the estimated optimal dual martingale is preserved and no nested simulation is used in their algorithm.…”

Solving the Dual Problems of Dynamic Programs via Regression

“…The naive approach is to replace the optimal value functions with approximate ones, and use nested simulation to estimate the conditional expectations; however, this approach often requires substantial computational effort and might cause the resulted approximation to lose the dual feasibility. Various methods have been proposed to improve the accuracy and efficiency of the approximation, including the non-nested simulation approach by Belomestny et al (2009) and Zhu et al (2015) in American-style option pricing, and the pathwise optimization techniques by Desai et al (2011) and Ye and Zhou (2012). The advantage of the pathwise optimization method is that it explores a subspace of feasible dual penalties by considering the best linear combination of the existing dual penalties.…”

“…We will show that this approach could be viewed as a special case of the proposed framework with a specific functional basis of the dual penalty space. Our framework is more universal and powerful, because it reveals the structure of the optimal dual penalty regardless of the underlying probability measure (i.e., not restricted to the Brownian measure in Belomestny et al (2009) or the Poisson random measure in Zhu et al (2015)). We will also show that the approximation scheme in Ye and Zhou (2015) could be viewed as a special case of the proposed framework as well.…”