Effective Methods for Volatility Modeling
Inspired by the works and several articles of Hagan, Lesniewski and Woodward as well as Andreasen, Huge and Antonov, we develop the theory of effective methods for volatility in a series of three articles:
M. Felpel, J. Kienitz & TA McWalter (2021) Effective stochastic volatility: applications to ZABR-like models, Quantitative Finance, 21:5, 837852, DOI: 10.1080/14697688.2020.1814396
M. Felpel, J. Kienitz and TA 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 and TA McWalter (2022) Effective models of local stochastic volatility, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4016334
In our series of articles, we consider a general approach.
Called on efficient method for modeling volatility, which includes all practically relevant stochastic volatility models (Heston, SABR, etc.) and introduces new variants and extensions (e.g. for ZABR models), multidimensional versions and the addition of a local volatility component to general models. For the sake of application, we propose numerical methods and consider their scope.
It is often observed that a specific stochastic volatility model is not chosen for particular dynamic characteristics, relevant for exotic payoff structures. But rather for convenience and ease of implementation. The SABR model, with its approximate solution in semi-closed form for vanilla option prices, is a well-known example.
In addition to convenience for vanillas when pricing interest rate derivatives based on multiple rates. Furthermoresuch as CMS spread or mid-curve options. JModeling is again triggered by simplicity rather than modeling needs.
Finally, practitioners want to calibrate their models on all available vanilla quotes at once and they want to incorporate the dynamics of the forward implied volatility smile. For this local stochastic purpose, we use volatility models. When calibrating such models, one of the main challenges is the correct calibration of the so-called leverage function. This must often happen at every time step of a Monte Carlo simulation.
In particular, we consider the mean-reversion ZABR and free ZABR models. We use the method of derivation of an efficient 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 multidimensional modeling, we introduce the effective Markovian projection, which allows us to apply the effective methods for single rates in a multidimensional setting. To incorporate a local volatility component into the general modeling, an alternative approach using efficient stochastic volatility techniques that allow direct calibration of the leverage function is introduced and applied. Finally, since we mainly work with rates, this includes a local stochastic volatility approach to model interest rates.
We illustrate all our results with numerical methods and Python code.