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.

I would like to share my research and thoughts about stochastic volatility models and, in particular, about the log-normal stochastic volatility model that I have been developing in a series of papers (see introductory paper with Piotr Karasinski in 2012, the extension to include quadratic drift with Parviz Rakhmonov in 2022, and application of the model to Cheyette interest rate model and  to factor HJM framework.

In my new working paper with Parviz Rakhmonov, which is available on SSRN here, we address specification of the functional form for the dynamics of stochastic volatility (SV) driver including affine, log-normal, and rough specifications. We propose the four principles which, in our opinion, determine the applicability of an SV model for valuation of derivative securities for different asset classes including equities, rates, commodities, FX and cryptocurrencies. We emphasise that the invariance of an SV under different numeraires is crucial for the model applications for modeling volatility of different asset classes. We argue that currently only the two SV dynamics satisfy these universality conditions: affine Heston SV model and log-normal SV model with quadratic drift. We discuss that both models are analytically tractable for valuation of vanilla options and model calibration when applying these models in different asset classes. We also present some empirical evidence for the considered models and discuss their link with contemporary research topics such as volatility skew-stickiness. We conclude that log-normal SV model with quadratic drift is robust because it does not require special conditions (such as Feller condition for Heston model) for numerical implementation of the model using MC and PDE methods.

For illustrations, we use the implied volatilities of the core assets for equity indices, rates and commodities:

1.  S&P 500 index and its implied volatilities proxied with VIX index;

2. 10y US treasury rate and its implied volatilities proxied with MOVE index;

3. Oil futures (using USO ETF) and its implied volatilities proxied with OVX index;

4. Bitcoin (denoted as BTC) and its implied at-the-money (ATM) volatility for options with time-to-maturity of 7 days (We use historical options data of Deribit exchange with the data set starting on April 2019).

We formulate the following principles for universality and feasibility of a stochastic volatility (SV) model. Our primary focus is based on specifying the parametric form of the dynamics of the volatility driver so that we leave aside important but, in our opinion, secondary features of a universal volatility model including jumps, local volatility, etc.

1. The dynamics of volatility must have the same marginal distribution under statistical measure P and risk-neutral valuation measure Q. This point ensures that the model can be used under the both statistical and pricing measures. More generally, this requirement implies that the model can be used with different numeraires specific to different asset classes, including equities, rates, commodities, FX and cryptocurrencies. For universality of a SV driver, the SV model dynamics must be functionally invariant     under different numeraires. For an example, for interest rates derivatives it is necessary that the volatility dynamics are invariant under the annuity measure, while for options on FX and cryptocurrencies the model must be invariant under the price numeraire.

2. The price process augmented with stochastic volatility must remain a strict martingale under different model specification. This particular point is important for the model application for assets with positive implied volatility skews and, as a result, with positive return-volatility correlation.

Lions and Musiela in their 2007 paper  show that most of one-factor SV models fail to produce strict martingale dynamics when return-volatility correlation between SV and return drivers is positive. This point can be overlooked in equity derivatives where return-volatility correlation is strongly negative but it cannot neglected for other asset classes where return-volatility is positive on many occasions. In Figure 1, we show that volatility beta may become positive (so that return-volatility correlation is positive too) for interest rates and commodities. We show here that return-volatility correlation may become positive for cryptocurrencies too.

Figure 1: Volatility beta estimated using EWMA regression model with span of 65 days: (A) time series from inceptions, (B) Empirical PDF

3. The dynamics of volatility must be well behaved: the volatility process must be strictly positive without explosions, the stationary distribution of the volatility must exist. This point ensures that the model can implemented efficiently with analytical and numerical methods.

4. The model is relatively easy to implement both analytically (for model calibration to market data) and numerically (through Monte-Carlo and PDEs) for valuation of exotic options and structured products.

We make the following conclusions about our four principals applied to well-known SV models (the references are given in our paper).

Stein-Stein SV model does not admit a valid change of measure. While it is still possible to use this model by directly specifying it under either Q or P measures, the scope of the model is limited. For example price numeraire (for FX and cryptocurrency derivatives) or annuity numeraire (for interest rate derivatives) cannot be applied for this model.

Exp-OU SV model, Bergomi one-factor model, and log-normal volatility model with linear drift allow for change of measures, but the functional form of model dynamics changes because of an additional term which arises in the drift of the volatility due to measure change. These models do not admit strong martingale dynamics when  return-volatility correlation is positive. In our opinion, these models are originally designed for applications in equity derivatives and their application to other asset classes is rather limited.

Rough SV model is an extension of Exp-OU using the power kernel for Brownian driver in the volatility dynamics. While rough volatility may provide good fit to empirical auto-correlation function (ACF) as we show in the Figure 2 below, the marginal improvement over a one-factor SV model is rather low when using ACF fit metric (the absolute difference is less than 0.1). Rough OU-based SV models inherit drawbacks of Exp-OU models: first, the difficulty in changing measures consistently and, second, the lack of martingale property when return-volatility correlation is positive. In our opinion, rough SV models are designed exclusively for equity markets and it may not be feasible to apply them for other asset classes. On the implementation side, rough Exp-OU models can only be implemented with MC methods.

Figure 2: Auto-correlation of implied volatilities as function of lag periods (in days) for the three implies volatility indices: A) VIX for the S&P 500 index, B) MOVE for the 10y UST rate, C) OVX for oil ETF, D) Bitcoin (BTC) ATM implied volatilities for options with maturities of 7 day. Empirical is the empirical estimate, Log SV is the fitted auto-correlation of the lognormal SV model, Rough is the rough auto-correlation with fitted decay power alpha.

Heston SV model allows for consistent measure changes under different numeraires. The model also produces true martingale dynamics when return-volatility correlation is positive and the variance cannot hit zero as long as the Feller condition is satisfied. On the implementation side, Heston model admits a closed-form solution for valuation of vanilla options, which makes it easy for model calibration. These facts undoubtedly have made Heston model applicable to multiple asset classes. However, numerical implementation of Heston model using MC or numerical PDE methods is rather complicated, especially when Feller condition is not satisfied. There is a great deal of literature on how to make Heston model work in practice.

Heston model also implied the stationary distribution of the volatility which has a thin right tail which is inconsistent with empirical data shown in Figure 3 below.

Figure 3: Steady-state PDF of the logarithm of the volatility (y-axis is shown in log-scale).

Log-normal SV model with quadratic drift allows for consistent measure changes using different numeraires. For positive return-volatility correlation, the model produces true martingale dynamics as long as the quadratic mean-reversion coefficient exceeds volatility beta. For model calibration, we develop a closed-form and accurate solution for valuation of vanilla options under this model in this paper. For numerical implementation using MC methods, we also develop a first-order MC scheme using the log-transform of the volatility to unbounded domain. Since in log-coordinates the valuation problem in log-volatility is defined on unrestricted domain, the problem can be solved efficiently using PDE methods for such domain (see my old workshop slides here). As a result, log-normal SV model with quadratic drift can be considered as a robust choice for modeling price dynamics for different asset classes.

Enjoy the reading of our paper in full and feel free to provide comments.

Python code for producing figures is available on Github in stochvolmodels package https://github.com/ArturSepp/StochVolModels and in module for the paper https://github.com/ArturSepp/StochVolModels/tree/main/my_papers/volatility_models