Abstract
In modelling financial derivatives, the pricing of barrier options are complicated as a result of their path-dependency and discontinuous payoffs. In the case of rebate knock-out barrier options, d...
Highlights
Introduction and literature reviewDerivative securities over the years have offered investors an increased expected return, as well as a reduction in risk exposures
If a speculator perceives that the underlying asset price will stay at a specific price range, the knock-out barrier option will offer more profit potential compared to the vanilla option
In the finite difference methods (FDM), the choice of Smax, which is an artificial limit is yet to be known and a proper choice will lead to a faster convergence and more accurate result
Summary
Let Vðt; SÞ be the value of a non-dividend paying down-and-out (DO) barrier option which pays a rebate when the barrier is breached. Under the Black-Scholes framework (Black & Scholes, 1973), the option price Vðt; SÞ satisfies the Black–Scholes PDE below:. Applying the risk-neutrality concept of option pricing, we obtain the solution of Equation (2.2) as ð1. Where f is the density function for the underlying and g is the first passage time density of the underlying S, at which a downstream barrier B is first hit by the Brownian motion WðtÞ. The function gðp; BÞdp 1⁄4 PðτB 2 dpÞ; where τB 1⁄4 infft : SðtÞ 1⁄4 Bg. The first part of the integral occurs when the barrier is not breached and the second follows from the rebate features. The functions f and g are known and the integrals above can be valued by applying the concept of reflection principle and method of images (Yue-Kuen, 1998)
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