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Approximating the Mean-Reverting SABR Pricing PDE with Lie Group Methods

Article arXiv papers · Author: Siyan Zhang et al.
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Summary

The document develops analytical tools for partial differential equations used to price contingent claims under the mean-reverting SABR stochastic volatility model. The fundamental solution is not available in closed form, so the work seeks an approximation suitable for numerical methods.

It expands the pricing solution in volatility of volatility and studies the zero-volatility-of-volatility operator, which is degenerate and falls outside classical results. Commutator calculations and solvable Lie group methods yield an exact solution operator for this limiting equation, while a perturbation comparison connects it to the full model’s semigroup. The analysis restricts volatility to bounded domains and also examines semigroups associated with parabolic and hyperbolic problems. The excerpt establishes mathematical methods rather than reporting pricing benchmarks or empirical validation.

Key ideas

  • The mean-reverting SABR pricing equation lacks a known closed-form fundamental solution.
  • An expansion in volatility of volatility provides a route to approximating its solutions.
  • Lie algebra techniques give an exact solution operator for the zero-volatility-of-volatility case.
  • A perturbation result relates the limiting operator to the full pricing model.
  • The existence analysis is conducted on bounded volatility domains.

Tags

OptionsVolatilityDerivatives pricingStatistics
Full text
# Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model


# Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model









We use commutator techniques and calculations in solvable Lie groups to investigate certain evolution Partial Differential Equations (PDEs for short) that arise in the study of stochastic volatility models for pricing contingent claims on risky assets. In particular, by restricting to domains of bounded volatility, we establish the existence of the semi-groups generated by the spatial part of the operators in these models, concentrating on those arising in the so-called "SABR stochastic volatility model with mean reversion." The main goal of this work is to approximate the solutions of the Cauchy problem for the SABR PDE with mean reversion, a parabolic problem the generator of which is denoted by $L$. The fundamental solution for this problem is not known in closed form. We obtain an approximate solution by performing an expansion in the so-called volvol or volatility of the volatility, which leads us to study a degenerate elliptic operator $L_0$, corresponding the the zero-volvol case of the SABR model with mean reversion, to which the classical results do not apply. However, using Lie algebra techniques we are able to derive an exact formula for the solution operator of the PDE $\partial_t u - L_0 u = 0$. We then compare the semi-group generated by $L$--the existence of which does follows from standard arguments--to that generated by $L_0$, thus establishing a perturbation result that is useful for numerical methods for the SABR PDE with mean reversion. In the process, we are led to study semigroups arising from both a strongly parabolic and a hyperbolic problem.

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