American and exotic option pricing with jump diffusions and other Lévy processes

In general, no analytical formulas exist for pricing discretely monitored exotic options, even when a geometric Brownian motion governs the risk-neutral underlying. While specialized numerical algorithms exist for pricing particular contracts, few can be applied universally with consistent success and with general Lévy dynamics. This paper develops a general methodology for pricing early exercise and exotic financial options by extending the recently developed PROJ method. We are able to efficiently obtain accurate values for complex products including Bermudan/ American options, Bermudan barrier options, survival probabilities and credit default swaps by value recursion; European barrier and lookback/hindsight options by density recursion; and arithmetic Asian options by characteristic function recursion. This paper presents a unified approach to tackling these and related problems. Algorithms are provided for each option type, along with a demonstration of convergence. We also provide a large set of reference prices for exotic, American and European options under Black–Scholes–Merton, normal inverse Gaussian, Kou’s double exponential jump diffusion, Carr–Madan–Geman–Yor (also known as KoBoL) and Merton’s jump-diffusion models.