# Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility

## TL;DR

This paper develops closed-form approximations for European put options under stochastic volatility models, enabling fast calibration and accurate pricing with error bounds, by expanding around the mean of a Black-Scholes expectation.

## Contribution

It introduces a second-order Taylor expansion approach for option pricing under stochastic volatility with piecewise-constant parameters, providing explicit formulas and error bounds.

## Key findings

- Errors are within acceptable ranges for practical use
- Derived bounds on Taylor expansion remainder
- Fast calibration scheme demonstrated

## Abstract

We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. The difficulties then faced are simplifying a number of expectations induced by the Taylor expansion. Under the assumption of piecewise-constant parameters, we derive closed-form pricing formulas and devise a fast calibration scheme. Furthermore, we perform a numerical error and sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term generated by the Taylor expansion.