TY - JOUR
T1 - Towards a General Local Volatility Model for All Asset Classes
JF - The Journal of Derivatives
DO - 10.3905/jod.2019.1.079
SP - jod.2019.1.079
AU - Gatarek, Dariusz
AU - Jabłecki, Juliusz
Y1 - 2019/07/05
UR - http://jod.pm-research.com/content/early/2019/08/27/jod.2019.1.079.abstract
N2 - We propose a unified approach to local volatility modeling, encompassing all asset classes, with straightforward application to equity and interest rate underlyings. Specifically, we 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. We 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 us to analyze equity and interest rate instruments, for example, as special cases of the same general local volatility model, rather than two separate models. We then derive a Markovian projection for the general model, with an approximate local volatility diffusion for the Margrabe option underlying. Finally, we discuss a specific application of the model to swaptions qua asset-for-asset options, where we 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, we 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.TOPICS: Options, statistical methods
ER -