Pricing Bermudan options under Merton jump-diffusion asset dynamics

Abstract

In this paper, a recently developed regression-based option pricing method, the Stochastic Grid Bundling Method (SGBM), is considered for pricing multidimensional Bermudan options. We compare SGBM with a traditional regression-based pricing approach and present detailed insight in the application of SGBM, including how to configure it and how to reduce the uncertainty of its estimates by control variates. We consider the Merton jump-diffusion model, which performs better than the geometric Brownian motion in modelling the heavy-tailed features of asset price distributions. Our numerical tests show that SGBM with appropriate set-up works highly satisfactorily for pricing multidimensional options under jump-diffusion asset dynamics.

Keywords:

• Monte Carlo simulation
• least-squares regression
• jump-diffusion process
• Bermudan option
• high-dimensional problem

2010 AMS Subject Classifications:

• 65C05
• 80M31
• 93E24
• 60E75
• 41A63

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. It denotes the realization at time T with the values of the option's underlying assets equal to ${\mathit{S}}_T$.

2. We write ${\mathit{S}}_{t_m}$, which means that the stock price at time $t_m$ is equal to ${\mathit{S}}_{t_m}$. In the following discussions, the condition of the expectation may also be formulated as ${\mathit{S}}_{t_m}=\hat{\mathit{S}}$ to emphasize that the stock price at time $t_m$ is known as a realization $\hat{\mathit{S}}$.

3. The authors of [3] show that Regress-Later is fundamentally different from Regress-Now, noticing that the former does not introduce a projection error between two time steps in the regression stage. As a result, Regress-Later achieves a faster convergence rate than Regress-Now in terms of the sample size.

4. If we want to bundle the paths into two parts, the partition point for ‘equal-size bundling’ is the median of the asset prices while that for ‘equal-range bundling’ is the mean of the asset prices.

5. For simplicity, we neglect the discounting term $D_{t_{m-1}}$.

6. Since $\pi \left({\mathit{S}}_{t_m}\right)$ is defined by the density function of ${\mathit{S}}_{t_m}$ conditioned on ${\mathit{S}}_{t_{m-1}}=\hat{\mathit{S}}$, the simulation of ${\mathit{S}}_{t_m}$ with respect to this density function can be treated as a sub-simulation from the unique state ${\mathit{S}}_{t_{m-1}}=\hat{\mathit{S}}$.

7. In the following sections, the other empirical density functions can be defined in a similar fashion.

8. SGBM is also feasible for another well-known jump-diffusion model, the Kou model [16]. Since the only distinction between the MJD model and the Kou model is the distribution of jump sizes, all discussions in this paper about SGBM can be extended to the Kou model as well.

9. The jumps in the dynamics of each asset should, however, follow the same Poisson process.

10. In fact, the number of basis functions is also a tuning parameter, but large number of basis functions may result in an over-fitting problem in the regression step. We therefore choose the number of basis functions just three or four in the one-dimensional case.