Explicit Constructions of Martingales Calibrated to Given Implied Volatility Smiles

Abstract

The construction of martingales with given marginal distributions at given times is a recurrent problem in financial mathematics. From a theoretical point of view, this problem is well known, as necessary and sufficient conditions for the existence of such martingales have been described. Moreover, several explicit constructions can even be derived from solutions to the Skorokhod embedding problem. However, these solutions have not been adopted by practitioners, who still prefer to construct the whole implied volatility surface and use the explicit constructions of calibrated (jump-) diffusions, available in the literature, when a continuum of marginal distributions is known. In this paper, we describe several new constructions of calibrated martingales, which do not rely on a potentially risky interpolation of the marginal distributions but only on the input marginal distributions. These calibrated martingales are intuitive since the continuous-time versions of our constructions can be interpreted as time-changed (jump-) diffusions. Moreover, we show that the valuation of claims, depending only on the values of the underlying process at maturities where the marginal distributions are known, can be extremely efficient in this setting. For example, path-independent claims of this type can be valued by solving a finite number of ordinary (integro-) differential equations. Finally, an example of calibration to the S&P 500 market is provided.