Perturbation methods in derivatives pricing under stochastic volatility

[摘要] ENGLISH ABSTRACT: This work employs perturbation techniques to price and hedge financial derivatives in astochastic volatility framework. Fouque et al. [44] model volatility as a function of two processesoperating on different time-scales. One process is responsible for the fast-fluctuatingfeature of volatility and corresponds to the slow time-scale and the second is for slowfluctuationsor fast time-scale. The former is an Ergodic Markov process and the latter isa strong solution to a Lipschitz stochastic differential equation. This work mainly involvesmodelling, analysis and estimation techniques, exploiting the concept of mean reversion ofvolatility. The approach used is robust in the sense that it does not assume a specific volatilitymodel. Using singular and regular perturbation techniques on the resulting PDE a first-orderprice correction to Black-Scholes option pricing model is derived. Vital groupings of marketparameters are identified and their estimation from market data is extremely efficient andstable. The implied volatility is expressed as a linear (affine) function of log-moneyness-tomaturityratio, and can be easily calibrated by estimating the grouped market parametersfrom the observed implied volatility surface. Importantly, the same grouped parameterscan be used to price other complex derivatives beyond the European and American options,which include Barrier, Asian, Basket and Forward options. However, this semi-analytic perturbativeapproach is effective for longer maturities and unstable when pricing is done closeto maturity. As a result a more accurate technique, the decomposition pricing approachthat gives explicit analytic first- and second-order pricing and implied volatility formulae isdiscussed as one of the current alternatives. Here, the method is only employed for Europeanoptions but an extension to other options could be an idea for further research. Theonly requirements for this method are integrability and regularity of the stochastic volatilityprocess. Corrections to [3] remarkable work are discussed here.

[发布机构] Stellenbosch University