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

Kaustav Das, Nicolas Langrené

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)745-788
Number of pages44
JournalStochastics
Volume94
Issue number5
DOIs
Publication statusPublished - 2022

Keywords

  • 41A58
  • 65C20
  • 91G60
  • closed-form approximation
  • closed-form expansion
  • GARCH
  • Heston
  • Stochastic volatility

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