Effective Methods for Volatility Modeling
Inspired by the work and several papers of Hagan, Lesniewski and Woodward as well as Andreasen, Huge and Antonov we develop the theory of effective methods for volatility in a three paper series:
M. Felpel, J. Kienitz & T. A. McWalter (2021) Effective stochastic volatility: applications to ZABR-type models, Quantitative Finance, 21:5, 837 852, DOI: 10.1080/14697688.2020.1814396
M. Felpel, J. Kienitz & T. A. McWalter (2022) Effective Markovian projection: application to CMS spread options and mid-curve swaptions, Quantitative Finance, DOI: 10.1080/14697688.2022.2043558
M. Felpel, J. Kienitz & T. A. McWalter (2022) Effective Local Stochastic Volatility models, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4016334
In our series of articles, we consider a general approach.
Called the effective method for volatility modeling, that includes all practically relevant stochastic volatility models (Heston, SABR, etc.) and introduces new variants and extensions (for e.g. ZABR models), multi dimensional versions and adding a local volatility component to the general models. With applications in mind, we provide numerical methods and consider their scope.
Often it is observed that a specific stochastic volatility model is chosen not for particular dynamical features, relevant for exotic payoff structures. But instead for convenience and ease of implementation. The SABR model, with its semi-closed form approximate solution for the prices of vanilla options, is a well-known example.
Further to the convenience for vanillas when pricing interest rate derivatives depending on multiple rates. Moreover, such as CMS spread or mid-curve options. The modeling is again triggered by simplicity rather than by the modeling needs.
Finally, practitioners wish to calibrate their models to all available vanilla quotes at once and they wish to incorporate the dynamics of the forward implied volatility smile. To this end local stochastic, we use volatility models. When calibrating such models, one of the key challenges lies in the proper calibration of the so-called leverage function. This often must occur during each time step of a Monte Carlo simulation.
In particular, we consider the mean-reverting ZABR and free ZABR models. We use the method of deriving an effective partial differential equation for the density. This approach leads to versions of the well known approximation formula for the SABR model, but for the general case and also provides expressions for arbitrage-free models.
For multi-dimensional modeling we introduce the Effective Markovian Projection, which allows us to apply the effective methods for single rates in a multi-dimensional setting. For incorporating a local volatility component into the general modeling an alternative approach using effective stochastic volatility techniques which allow a direct calibration of the leverage function is introduced and applied. Lastly, since we mainly work with rates this includes a local stochastic volatility approach for modeling interest rates.
We illustrate all our findings with numerical methods and Python code.
