CS43: Volatility by Diffusion: A Novel Approach to SABR

Organizer: Michał Barski (University of Warsaw)

Characterization of the Probability Distribution in the SABR Model

Maciej Wiśniewolski

In their famous paper, Hagan and co-authors introduced a stochastic volatility model called the SABR model, the name being an acronym derived from the model’s parameters: Stochastic Alpha, Beta, Rho. The model describes the price dynamics of a financial asset by allowing its volatility to be stochastic. This is implemented through a system of two stochastic differential equations (SDEs) with appropriate initial conditions.

Although the formulation of the model is relatively simple, the problem of providing a rigorous mathematical description of the solution to the associated SDEs has turned out to be surprisingly difficult and remains unsolved to this day.

In our talk, we present a new approach to characterizing the probability distribution in the SABR model. To achieve this, we perform an unconventional change of probability measure — unconventional in the sense of parameter dependence — which allows us to reduce the original problem of describing the solution to the initial SDEs to analyzing the probability distribution of a certain one-dimensional diffusion process that may explode.

Using the approach developed by Karatzas and Ruf \[2\] for explosive diffusions, we are able to compute the power transform of the price process in the SABR model. Remarkably, we are also able to deconstruct the parameter dependencies and invert the transform, obtaining the probability distribution of the SABR model as an explicitly given functional of a Brownian motion.

Bibliography

\([1]\) Hagan, P. S., Kumar, D., Lesniewski, A., Woodward, D. Managing Smile Risk. Wilmott Magazine, 2002.

\([2]\) Karazas, I., Ruf, J.: Distribution of the time to explosion for one-dimensional diffusions, (2016), Probability Theory and Related Fields, 164, 1027-1069.

Measuring volatility: deterministic and stochastic perspectives (regularization by noise)

Rafał Łochowski

In the talk I will review several results on convergence of functions of sums of increments of stochastic processes or deterministic paths to limiting objects, which are known under the names like: volatility, integrated volatility, quadratic variation, \(p\)-variation, \(\psi\)-variation etc. The emphasis will be on the sampling method used and the dependence of the existence and value of the limit on the sampling metod (partition scheme) chosen.

For example, while for semimartingales the sampling method may be almost arbitrary, (though the mode of convergence may depend on the method chosen), for deterministic functions it is not the case. I will also present some open questions related to the variations of fractional Brownian motions and the horizontal component of the Peano curve.

Bibliography

\([1]\) P. Das, R. Łochowski, T. Matsuda and N. Perkowski "Level crossings of fractional Brownian motion." Ann. Probab., to appear.

\([2]\) P. L. Zondi, D. Hove, R. Łochowski and F. J. Mhlanga "Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)." preprint, arXiv:2501.07966. .

Characterization of moments in the SABR model

Michał Barski

In this talk, I will present a new approach to computing moments in the SABR model by reducing the original system of stochastic differential equations to a one-dimensional diffusion process with possible explosion. The central idea is to introduce an auxiliary process θ, whose negative moments are directly related to the moments of the asset price in the SABR framework.

Using the specific polynomial structure of the coefficients in the SDE for θ, we derive a recursive formula for the integer moments of the asset price. This recurrence takes the form of a system of ordinary differential equations depending on the model parameters and can be solved explicitly in special parameter regimes, such as when \(\beta=0\) or \(\rho=0\). In the general case, the initial term of the recursion is characterized by its Laplace transform expressed as a power series.

This method provides new analytical tools for studying the moment structure of the SABR model and, in many cases, leads to a unique identification of the distribution of the underlying asset price.

The talk is based on a joint work with M. Wiśniewolski and B. Polaczyk.