Towards a general local volatility model for all asset classes

The authors propose a unified approach to local volatility modeling, encompassing all asset classes, with straightforward application to equity and interest rate underlyings. Specifically, they consider a local volatility model for asset-for-asset or Margrabe (1978) options under general conditions that underlying dynamics follow Itô processes and derive a closed-form non-parametric local volatility formula. They then show that many standard contracts—European equity, FX, and interest rate options—can be seen as particular examples of the Margrabe-type payoff, which allows them to analyze equity and interest rate instruments, for example, as special cases of the same general local volatility model, rather than two separate models. They then derive a Markovian projection for the general model, with an approximate local volatility diffusion for the Margrabe option underlying. Finally, they discuss a specific application of the model to swaptions qua asset-for-asset options, where they consider the Markovian projection with some frozen parameters as a minimal “poor man’s” model, featuring equity-like dynamics for the swap rate with its own “short rate” and the “dividend” implied from the term structure of interest rates. Using a number of numerical examples, they compare the minimal model to a fully fledged Cheyette local volatility model and the market benchmark Hull–White one-factor model (Hull and White 1990), demonstrating the adequacy of the “poor man’s” model for pricing European and Bermudan payoffs.