Volatility is (mostly) path-dependent
Julien Guyon is a full professor of applied mathematics at Ecole des Ponts ParisTech, Paris. He is joining us at QuantMinds International to speak about 'The 4-Factor Path-Dependent Volatility Model'. Ahead of his session, he has shared an insight into his work with us. Explore it below and secure your spot in Barcelona, to learn more.
It is crucial for the pricing, hedging, and risk-management of portfolios of derivatives to understand and correctly capture the joint dynamics of the underlying assets and their implied volatilities. For example pricing and hedging portfolios of S&P 500 (SPX) derivatives requires precisely understanding the joint dynamics of the SPX and VIX indices. It is also very useful to build accurate predictors of future realized volatility. Such forecasts are used for instance in risk management, derivative hedging, market making, market timing, and portfolio selection. While stochastic volatility (SV) models view volatility as an extra stochastic process driven by its own sources of randomness (possibly correlated with those driving the dynamics of the asset price), it has been recently observed by several authors including Zumbach, Chicheportiche, Bouchaud, Blanc, Donier and myself, that financial markets exhibit a clear pattern of path-dependent volatility (PDV): the implied volatility and future realized volatility depend on the path followed by the asset price in the recent past. In some sense, while SV feeds the volatility level in the asset returns, PDV does quite the opposite: it feeds returns into volatility. The joint calibration of traditional flexible parametric SV models, such as the 2-factor Bergomi model, to at-the-money SPX skews and VIX implied volatilities also points to PDV models, by automatically selecting fully correlated Brownian motions. In the talk we provide extra philosophical, intuitive, theoretical, and empirical arguments supporting the use and relevance of PDV models.
In this joint work with Jordan Lekeufack (UC Berkeley), we aim to learn precisely how much of volatility is path-dependent, and how it depends on past asset returns. We thus aim to empirically explain volatility as an endogenous factor as best as we can. Our empirical study covers (a) learning implied volatility from past asset returns (e.g., learn the VIX from the SPX path), and (b) learning future realized volatility from past asset returns. We prove that volatility is mostly path-dependent. A surprisingly simple linear PDV model explains more than 85% or even more than 90% of the variance of the implied volatility (depending on the equity index), and around 60% of the variance of the (more noisy) future daily realized volatility, even on our very challenging test set (2019-2022). This proves that volatility is mostly endogenous. The explicit path-dependency that we uncover is remarkably simple: a linear combination of a weighted sum R1 of past daily returns and the square root of a weighted sum R2 of past daily squared returns with different time-shifted power-law (TSPL) weights.
We thus uncover a historical PDV or empirical PDV or P-PDV model. Its continuous-time version is a diffusion process that can also be fed risk-neutral parameters and turned into an implied PDV or risk- neural PDV or Q-PDV model if needed to calibrate to option prices. For practical engineering purposes, our historical PDV model can easily be made Markovian by approximating the TSPL kernels by superpositions of two or more exponential kernels; mixing two exponential kernels is usually enough. Alternatively, one can directly conduct the empirical study with combinations of exponential kernels rather than TSPL kernels. In particular, we propose a 4-factor Markovian PDV model which captures all the important stylized facts of volatility, produces remarkably realistic price and volatility paths, and produces remarkably realistic SPX and VIX smiles as well. Indeed the model captures volatility clustering as a result of the combined mean reversion of the observable factors R1 and R2; the leverage effect and strong negative SPX skews; the weak and strong Zumbach effects; some spurious roughness in daily realized volatility; and strong positive VIX skews despite a constant (lognormal) instantaneous volatility of instantaneous volatility. Remarkably, our model can even jointly calibrate to SPX and VIX smiles (as well as VIX futures) with a very good accuracy. To the best of our knowledge, it is the first time that a Markovian parametric SV (here, PDV) model is shown to jointly fit SPX and VIX smiles so accurately, i.e., to practically solve the joint SPX/VIX smile calibration problem.
An SV component may finally be added to the model if needed to account for the small remaining exogenous part of volatility, resulting in a Path-Dependent Stochastic Volatility (PDSV) model. A statistical study of the residuals of the empirical PDV model suggests a simple form for the extra SV component. Given the strong endogeneity of volatility, we believe that this is how volatility modelling should be approached: first explain volatility explicitly in an endogenous way as best as we can, then model the remaining (smaller) exogenous part.
Compared with local volatility (LV) and SV modelling, we believe that this is a better and more natural way of modelling volatility:
- First model the (large) purely endogenous part of volatility explicitly as best as we can, using PDV models that depend on observable factors, such as past asset returns.
- Then model the (small) exogenous part, if needed, based for instance on a statistical study of the residuals of the PDV model.
This comes in contrast with (a) LV modelling, which ignores the path-dependent nature of volatility, and (b) SV modelling, which starts from the exogenous part, postulates a dynamics for the instantaneous volatility that typically depends on unobservable factors (such as moving averages of past Brownian shocks), and can only generate some implicit, complicated path-dependency by correlating the Brownian motions that drive the dynamics of the asset price and the stochastic volatility. It is our hope that this PDV/PDSV approach opens an interesting line of future research in volatility modelling and stirs interesting debates and discussions among practitioners and academic researchers.
This article is an excerpt of an article first published here.
