Abstract
In this paper, in view of application to pricing of Barrier options under a stochastic volatility model, we study a reflection principle for the hyperbolic Brownian motion, and introduce a hyperbolic version of Imamura-Ishigaki-Okumura’s symmetrization. Some results of numerical experiments, which imply the efficiency of the numerical scheme based on the symmetrization, are given.
Highlights
Reflection principle and the static hedge of barrier options The reflection principle of standard Brownian motion relates the probability distribution of a first hitting time to a boundary to the 1-dimensional marginal distribution of the process
The formula has a direct application in continuous-time finance, that is, the static hedging of barrier options1
The static hedge of the knock-out option consists of two plain-vanilla (=without knock-out condition) options, long position of call option with pay-off (ST − K)+, and short position of “put option” whose value
Summary
Reflection principle and the static hedge of barrier options The reflection principle of standard Brownian motion relates the probability distribution of a first hitting time to a boundary to the 1-dimensional marginal distribution of the process. The formula has a direct application in continuous-time finance, that is, the static hedging of barrier options. Τ := inf{s > 0 : Ss < K }, the first hitting time of S to K , the knock-out boundary, with K < K. The static hedge of the knock-out option consists of two plain-vanilla (=without knock-out condition) options, long position of call option with pay-off (ST − K)+, and short position of “put option” whose value. At τ equals the call, and is zero at T on τ > T. Zero at T on τ ≤ T since at τ it is liquidated, and (ST − K) at T on τ > T
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