Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

Abstract

We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, eg., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems. , but for the homogeneous boundary conditions. Also, here we present full derivation of the explicit value of the solution spatial gradient u x at the lower x = y(τ ) and upper x = z(τ ) boundaries. This derivation differs from that in (Lipton and Kaushansky, 2018) (and is closer in sense to (Tikhonov and Samarskii, 1963)), but provides a similar result. Also, all the results obtained in this paper are new.The rest of the paper is organized as follows. Section 1 describes the double barrier pricing problem for the time-dependent barriers and rebates at hit and shows that it can be reduced to solving inhomogeneous PDE with homogeneous boundary conditions. Section 2 describes in detail the solution of this problem by using the GIT method. We provide two alternative integral representations of the solution -one via the Jacobi theta functions, and the other one -using the Poisson summation formula. Despite these solutions are equal in a sense of infinite series, their convergence properties are different. A system of the Volterra equations for the gradient of the solution at both boundaries is obtained for both representations. Section 2 provides the same development but using the HP method. The final section concludes.