Recovery of the time-dependent implied volatility of time fractional Black–Scholes equation using linearization technique
Sajad Iqbal, Yujie Wei
Abstract
Abstract This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {L_{1}} -forward difference implicit approximation for the forward problem. Numerical results using <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {L_{1}} -forward difference implicit approximation ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {L_{1}} -FDIA) for the inverse problem are also discussed briefly.