On implied volatility recovery of a time-fractional Black-Scholes equation for double barrier options

ABSTRACT

This paper investigates a time-fractional Black-Scholes equation with an implied volatility which is assumed to be associated with underlying price. Two aspects are considered. One is for the forward problem, i.e. a robust $L_1$-CDIA ($L_1$-central difference implicit approximation) scheme is utilized to solve the initial boundary value problem effectively. Remarkably, convergence analysis for the forward problem solver is also given. The other is for the inverse problem, i.e. recovery of the implied volatility via additional data. By linearization which is similar as [Bouchouev et al. Quant. Finance, 2 (2002) 257–263.], a Fredholm integral equation of the first kind for the unknown volatility is obtained. Based upon the integral equation, a uniqueness of the recovery is shown under some assumptions and an efficient numerical reconstruction algorithm is proposed to reconstruct the coefficient. Numerical results for both direct problem and inverse problem are presented to illustrate the validity and effectiveness of the proposed schemes.

Keywords:

• Time-fractional partial differential equation
• Reconstruct volatility
• Fredholm integral equation
• Uniqueness

2010 Mathematics Subject Classifications:

• 45Qxx
• 45Bxx

Disclosure statement

No potential conflict of interest was reported by the authors.