This paper develops a method for the exact simulation of a skeleton, a hitting time, and other functionals of a one-dimensional jump diffusion with state-dependent drift, volatility, jump intensity, and jump size. The method requires the drift function to be [C.sup.1], the volatility function to be [C.sup.2] and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features.
Subject classifications: jump-diffusion process; stochastic differential equation; exact simulation; exact sampling; unbiased simulation estimator.
Area of review: Financial Engineering.
History: Received October 2010; revisions received August 2011, July 2012; accepted February 2013. Published online in Articles in Advance August 7, 2013.
Jump diffusions are widely used as models for the time evolution of prices and rates in equity, fixed income, commodity, foreign exchange, energy, and other markets. They also serve as models of default timing in portfolios of credit-sensitive assets such as loans and corporate bonds. Monte Carlo simulation is an important tool to address the pricing, risk management, and inference problems arising in this context. Relative to alternative numerical approaches, simulation has the widest scope. It requires few restrictions on the coefficients of the jump diffusion and the functional to be evaluated.
This paper develops a method for the exact simulation of a one-dimensional jump diffusion with state-dependent drift, volatility, jump intensity, and jump size. The method requires the drift function to be [C.sup.1], the volatility function to be [C.sup.2], and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method generalizes an acceptance/rejection scheme developed for certain one-dimensional diffusions X by Beskos and Roberts (2005) and its extension to a wider class of diffusions developed by Chen and Huang (2013). The basic idea is to sample a skeleton of a standard Brownian motion W, and to accept that skeleton as one of X with a probability proportional to the Radon-Nikodym derivative between the law of X and the law of W. The generation of the acceptance indicator is based on a thinning mechanism, exploiting the observation that the Radon-Nikodym derivative is analogous to the conditional probability that a doubly stochastic Poisson process does not jump during some interval. We extend this approach to address the presence of jumps in X that arrive at a state-dependent intensity and have a state-dependent magnitude. At the same time, we relax some of the conditions on the drift and volatility functions assumed in the aforementioned articles, thereby widening the scope of the acceptance/rejection scheme even for diffusions. Moreover, we enlarge the class of expectations that can be treated exactly. These include expectations of functionals depending on a skeleton of X, the complete path of the jump component, a hitting time, and the value of X at the hitting time, as well as the expectation of the exponential of the time integral of X.
The exact method is...