Artur Sepp Blog on Quantitative Investment Strategies

Our article “Log-normal Stochastic Volatility Model with Quadratic Drift” co-authored with Parviz Rakhmonov is published in International Journal of Theoretical and Applied Finance with open access https://www.worldscientific.com/doi/10.1142/S0219024924500031

The log-normality of realised and implied volatilities of asset returns is a well-documented empirical feature. For example, see Christoffersen-Jacobs-Mimouni (2010) for equity indices and Andersen-Lund (1997) for short-term interest rates. Yet, the difficulty in implementing log-normal stochastic volatility (SV) models in practice is that these models are not analytically tractable due to being non-affine, so that standard techniques for affine SV model cannot be applied here. Our key contribution is the closed-form accurate and fast approach for valuation of vanilla options under the log-normal SV model.

I have started working on the log-normal SV model back in 2012 together with Piotr Karasinski and we published a joint paper in Risk. Over years, I have developed the affine expansion for log-normal SV model which is analytic (up to solving a system of ODEs) and which provides a very accurate solution to the moment generating function (MGF) arising in log-normal SV models. Be means of this solution to the MGF, we can value vanilla options using methods developed for valuation under affine models, including the Lipton-Lewis formula.

With Parviz, we have extended Karasinki-Sepp stochastic volatility model by adding a quadratic mean-reversion to the drift, which turns to be important for the model to be functionally invariant under different numeraire measures (see our paper on this topic). We have provided detailed proofs on important aspect of our model including the positivity and finiteness of the volatility process, the martingality of the price dynamics, the existence of the solution for valuation equation in this model, and the stability of the affine expansion. The most parts of our paper are technical to address these necessary topics.

We have also included the illustration of model calibration to options data on Bitcoin from April 2019 to October 2023. For this extended period, we show that the model can fit accurately to the market data across different market regimes with low/high volatilities, positive/negative skews, and  steep/flat convexities of market implied volatilities.

A big advantage of using the log-normal SV model in a traditional quant valuation setup, it that the model is easy to implement for Monte-Carlo (MC) simulations and for numerical PDE solvers using the the logarithm of the volatility as a modelling variable, which is defined on unrestricted domain. In contrast, affine models require to handle the positivity of the volatility in MC simulations and PDEs solvers, which is not trivial. The availability of closed-form solution for vanilla options enables fast model calibration to market data.

Finally, our log-normal SV model is conceptually robust because if can be applied for valuation of derivatives on different asset classes. In particular, we apply this model to interest rates (see application to Cheyette model here and to Factor HJM model here), whereas traditional SV models have many limitations when it comes to modeling dynamics of fixed income  derivatives.

For transparency and as a courtesy to the readers, Python implementation of the analytics from the paper for valuation under our log-normal SV model is available in Github.

Enjoy reading and testing our model.