Elisa Alòs is an associate professor in the Department of Economics and Business at the Universitat Pompeu Fabra (Barcelona). She completed her Ph.D. in Mathematics in 1998 at the University of Barcelona, with a dissertation based on Malliavin Calculus techniques applied to the study of stochastic integral equations.
Her research relies on the applications of stochastic analysis in mathematical finance. In particular, it is focused on the application of Malliavin calculus techniques and the use of fractional noises in market modeling. Recently, she has started to study stochastic epidemiological models, and she is also interested in the fractal properties of biological systems.
She currently serves as an Associate Editor at SIAM Journal on Financial Mathematics, Stochastic Analysis and Applications, Stochastic Processes and their Applications, and Mathematics.
Our editor Dr Paul Bilokon interviewed Dr Alòs on 21 May, 2025, about the second edition of her book with David Garcia Lorite, Malliavin Calculus in Finance: Theory and Practice (CRC Press).
PB: Paul Malliavin (1925-2010) was a French mathematician who came to stochastics from harmonic analysis. How can you succinctly describe the Malliavin calculus and why is it treated, within stochastic analysis, separately from the now near-classical Ito and Stratonovich calculus?
EA: Malliavin calculus is an extension of the classical calculus of variations, to the field of stochastic processes. Its first motivation was the study of densities (that can be expressed in terms of Malliavin derivatives). In particular, the development of a probabilistic proof of the Hörmander theorem of smoothness of the density of the solution of a second order stochastic differential equation.
But Malliavin calculus quickly exceeded this initial application. Some tools such as the Malliavin derivative operator, the Skorohod integral, the integration by parts formula, the Clark-Ocone-Haussman representation, and the anticipating Itô formula proved to be extremely useful in a huge set of applications.
One of the main differences from classical Itô calculus is that it is posible to deal with non-adapted processes (such as for example the future integrated volatility). In this way, the anticipating Itô formula is an extension of the classical Itô formula where the integral is replaced by the Skorohod integral (that still has zero expectation) and we have an ‘extra term’ given by the Malliavin derivative of the non-adapted process.
PB: I see that your book has a chapter on fractional Brownian motion, which is nowadays an ingredient of several popular models (rough Heston, rough Bergomi). What is the connection between the Malliavin calculus and fractional Brownian motion? Does one really need to understand the Malliavin calculus to deal with rough volatility models?
EA: What we commonly know as ‘rough volatilities’ are models where the volatility is driven by a fractional Brownian motion with a Hurst parameter H<0.5. Fractional volatilities with H>0.5 were first introduced by Comte and Renault in 1998 to better explain the behaviour of the long-end of the implied volatility surface. It was in 2007 when, working with my colleagues Jorge A. León (CINVESTAV, Ciudad de Mexico), and Josep Vives (Universitat de Barcelona), we found that the short-end skew slope of this implied volatility surface is determined by the short-end behaviour of the Malliavin derivative of the volatility process. That is, if we want this slope to blow up (as observed in financial practice) we should consider a model with this blow-up in its Malliavin derivatives, as it is the case of fractional volatilities with H<0.5.
That is, historically, the first time (to my knowledge) that these fBm volatility models appear in the literature is because of Malliavin calculus. That is, in many cases we can express ‘desired properties’ of models in terms of their Malliavin derivatives and then this leads to the design of a model.
Is it posible to understand rough volatilities without Malliavin calculus? Yes, you can get similar resuls, for example, if you work with the conditional expected integrated volatility (that is adapted) and then you apply classical Itô calculus. But then you have to identify the corresponding martingale representation. This martingale representation can be obtained via the Clark-Ocone-Haussman theorem (and then we are applying again Malliavin calculus) or directly from the expression of the concrete model (and in this case, this leads to particular results). Based on my experience, the main advantage of Malliavin calculus is that it allows us to get general results in terms of the Malliavin derivative of the underlying volatility process.
PB: In Chapter 4 of your book you introduce the key tools of the Malliavin calculus, including the Clark-Ocone-Haussman theorem, integration by parts, and the anticipating Ito’s formula. How rich is this conceptual framework? What are the main applications?
EA: As pointed out before, the Clark-Ocone-Haussman theorem gives the explicit martingale representation of a random variable. This leads to some classical results such as in the determination of hedging strategies. Moreover, it is a valuable tool when dealing with spot and integrated volatilities (for example, we can lead to interesting expressions for convexity adjustments between variance and volatility swaps, or when working with the VIX). The integration by parts formula is perhaps the best-known result of Malliavin calculus, and it can be applied to the computation of the Greeks. The anticipating Itô formula is mainly applied to expressions where the future integrated volatility appears, leading to useful extensions of the classical Hull and White formula that allow us to identify the effect of correlation on option prices and implied volatilities.
PB: You have mentioned that one of the main applications of the Malliavin calculus is the computation of Greeks. AAD is nowadays a popular method for computing these sensitivities “automatically”. Are there any interesting connections between the Malliavin calculus and AAD?
EA: To my knowledge, there is no literature comparing these both methods and it will be something interesting to explore. Obviously AAD is a more general method, but we do not know to what extent, in some particular cases, Malliavin calculus can be faster. This is something to explore with my coauthor David García-Lorite.
PB: In the new, second edition of your book you have added Chapter 11, which is about the Bachelier implied volatility. Isn’t the work of Bachelier supposed to be superseded by the Black-Scholes framework? Where is the Bachelier implied volatility used and why is it interesting?
EA: The Bachelier model allows prices to be negative, and recently, negative prices have been registered for interest rates (from the 2008 crisis) and commodities (like oil futures prices after the COVID-19 pandemia). Some markets, such as the CME, moved temporarily to the Bachelier framework. Even when the probability of negative prices was neglected some decades ago, now it seems necessary to have an adequate framework for this scenario.
PB: Do you have any advice for the students of stochastics and quantitative and computational finance as to the study of Malliavin calculus?
EA: I think it is important, first, to learn the basics: mathematics and computational tools. Then, being able to learn everyday, and to learn from others and their experience. I really believe that collaboration is something extremely valuable.