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

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.

Keywords:

• Stochastic volatility
• closed-form expansion
• closed-form approximation
• Heston
• GARCH

MSC:

• 91G60
• 41A58
• 65C20

Acknowledgement

This research is supported by an Australian Government Research Training Program (RTP) Scholarship.

9. Data availability statement

USD/JPY FX option price data with expiries 1, 3, 6, 12 months and notional of 1000,000 was retrieved from the Bloomberg Terminal on 9 July 2018.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by an Australian Government Research Training Program (RTP) Scholarship.