Quanto adjustments in the presence of stochastic volatility

It is well known that the quanto adjustment in the drift of the underlying has a significant impact on the prices of quanto options. Alexander Giese points out that an additional quanto adjustment in the underlying’s volatility needs to be considered in the presence of stochastic volatility. By deriving closed-form solutions for standard quanto options, he demonstrates that this additional quanto adjustment also has a material impact on quanto options

Quanto options are options where the payout is paid in a currency different from the currency in which the underlying asset is traded and where the applied foreign exchange rate between the two currencies is set to one. The fixed forex rate allows the holder of a quanto option to participate in the performance of the underlying without carrying the risk of a changing forex rate. However, pricing and risk-managing quanto options on foreign equities has become increasingly challenging in recent years due to high levels of equity/forex correlation and high volatility. Both market parameters determine the well-known quanto adjustment in the drift of the underlying, as derived by Reiner (1992) in the classical Black-Scholes model. While most of the research on quanto options has focused on the Black-Scholes framework, researchers recently started to study quanto options in the context of stochastic volatility models, which allow skews and smiles in the implied volatility surface of the underlying asset. Dimitroff, Szimayer & Wagner (2009) assume the Heston (1993) model and Jäckel (2010) uses a stochastic local volatility model in their studies on quanto options. While both studies conclude that the quanto option prices in a stochastic volatility model differ from the Black-Scholes price, they provide little explanation or intuition for the observed differences. Furthermore, in both papers the model prices for quanto options need to be calculated using either Monte Carlo methods or numerical solutions of the pricing partial differential equation (PDE) due to the absence of closed-form solutions.

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