Bridging Diffusing-Diffusivity and Stochastic Volatility: Insights from Physics to Finance
Rohit Jain, Dr Rui Qin, Ashwani Roy
The Black-Scholes model assumes constant volatility, while real markets show volatility clustering, fat tails and leverage effects. Stochastic volatility models address this, and the mathematics behind them closely resembles a problem statistical physics has already solved: a particle whose diffusion coefficient changes as it moves through a heterogeneous environment. This paper maps one onto the other and uses the physics to derive results in finance.
Main contributions
- The structural parallel established between geometric Brownian motion in asset pricing and the Langevin equation describing a particle under stochastic forces, where drift acts as friction and the volatility term as a thermal force.
- Diffusing diffusivity mapped to stochastic volatility, with asset volatility playing the role of the diffusion coefficient, both constrained to positive values and both requiring averaging over stochastic realisations.
- Two models solved analytically: volatility driven by a mean-reverting Ornstein-Uhlenbeck process, which reduces to a special case of the Stein and Stein model, and volatility driven by a Lévy process.
- Exact stock price distributions derived under stochastic volatility, with the implications set out for derivative pricing, risk management and the shape of observed market behaviour.
- Fokker-Planck methods carried across from physics into financial mathematics, demonstrating what a cross-disciplinary approach makes tractable.
Click below to download the full paper.
Download this paper
Enter your details to get the PDF.