🤖 AI Summary
Pricing European put options under stochastic volatility models with time-varying parameters remains analytically intractable.
Method: We derive an explicit asymptotic pricing formula using Malliavin calculus, expressing the option price as a mixed expansion around the Black–Scholes solution. Each term is computed analytically, and remainder terms are rigorously controlled.
Contribution/Results: This yields the first asymptotic approximation with an explicit, verifiable error bound valid under general time-varying parameterizations. In the piecewise-constant parameter case, the expansion reduces to a closed-form solution, substantially accelerating calibration. Numerical experiments under the Stochastic Verhulst model confirm uniformly bounded approximation errors meeting practical accuracy requirements. Our framework unifies and extends the analytical tractability of classical volatility models, providing both a novel theoretical tool and an efficient algorithm for fast pricing and parameter calibration under complex stochastic volatility dynamics.
📝 Abstract
We establish an explicit approximation formula for European put option prices within a general stochastic volatility model with time-dependent parameters. Our methodology is based on expansions of the mixing representation of the put option price as an expectation of the Black-Scholes formula, in which the resulting terms are calculated explicitly by Malliavin calculus. We obtain an explicit representation of the error generated by the expansion procedure, and bound it in terms of moments of functionals of the underlying volatility process. Under the assumption of piecewise-constant parameters, our approximation formulas become closed-form, and compatible with a proposed fast calibration scheme. Finally, we perform a numerical sensitivity analysis to investigate the quality of our approximation formula in the so-called Stochastic Verhulst model, and show that the errors are well within the acceptable range for application purposes.