The Fine Structure of Currency Returns
Implied in One-Touch Options

Abstract

Under Black-Scholes assumptions, the returns process for an option’s underlying asset follows a logarithmic diffusion. The volatility parameter determines all aspects of risk. But Black-Scholes falls short of explaining real world returns, as well as real world option prices. Extensions to the model add stochastic volatility, possibly through a stochastic time-change, or jumps; or might even replace the diffusion entirely with an infinity of jumps of different sizes as in the variance gamma (VG) model. These models are flexible enough that they all can fit a sample of returns and price plain vanilla options quite well. This article brings into consideration another type of option contract: the one-touch option. The payoffs on these path-dependent contracts are determined not by the level of the underlying at expiration but by the extreme values along the price path from the initial date to expiration. The option pays when, and only if, the underlying hits a prespecified level at some point during its lifetime. Buesser notes that the various returns processes have quite different path behavior, so calibrating against one-touch options provides a significant source of additional information about the market dynamics investors expect. Comparing relative performance across models for the FX market, plain vanilla options on USDEUR and JPYUSD appear to embed a substantially different returns process than is implied by one-touch options.