The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.

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Note: S t = 100, τ = 1.0, r d = 0.03, r f = 0.0, σ = 0.2. Source: Clark [forthcoming].

The foreign exchange options market is one of the largest and most liquid OTC derivative markets in the world. Surprisingly, very little is known in the academic literature about the construction of the most important object in this market: The implied volatility smile. The smile construction procedure and the volatility quoting mechanisms are FX specific and differ significantly from other markets. We give a brief overview of these quoting mechanisms and provide a comprehensive introduction to the resulting smile construction problem. Furthermore, we provide a new formula which can be used for an efficient and robust FX smile construction. TECHNICAL PAPER^Ŵilmott magazine 59The introduced deltas can be stated as Black-Scholes type of formulas for puts and calls. For example, the premium-adjusted spot delta can be deduced fromwhere D S is the standard Black-Scholes delta. The resulting formulas are summarized in Table 2. At-the-MoneyThe at-the-money definition is not as obvious as one might think. If a volatility s ATM is quoted, and no corresponding strike, one has to identify which at-the-money quotation is used. Some common at-the-money definitions areIn addition to that, all notions of ATM involving delta will have sub-categories depending on which delta convention is used. The at-the-money spot quotation is well known. ATM-forward is very common for currency pairs with a large interest rate differential (emerging markets) or a large time to maturity. Choosing the strike in the ATM-delta-neutral sense ensures that a straddle with this strike has a zero delta (where delta has to be specified). This convention is considered as the default ATM notion for short-dated FX options. The formulas for different at-the-money strikes can be found in

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