Return to the barrier: option pricing and calibration in foreign exchange markets

A new entrant into the over-the-counter (OTC) markets is the return barrier option. For these contracts, a large (daily) return event triggers a knockout, so they have a unique and direct exposure to jump risk. Here we investigate this new exotic derivative and present new analysis based on calibrated foreign exchange market examples. For breadth of analysis, we calibrate nine different models, covering Lévy processes, jump-diffusions, stochastic volatility, stochastic volatility with jumps and stochastic local volatility (stochastic alpha–beta–rho), and we investigate the behavior of these contracts in light of how well each model captures the market surface. The exposure of return barrier options to jumps induces a unique sensitivity to the model’s ability to properly calibrate the market-implied tails of the underlying. We show that failure to capture the market’s implied volatility surface can lead to significant mispricing of the return barrier option. Our analysis shows that jump-diffusions and Lévy processes alone are ill-suited to the problem, while a stochastic volatility model combined with jumps produces high-quality fits and sensible option prices in the foreign exchange markets.