Consider a European down-and-out call option where the terminal payoff depends on the payoff state variable S₁ and knock-out occurs when the barrier state variable S₂ breaches the downstream barrier B₂. Assume that under the risk neutral measure Q, the dynamics of S₁,t and S₂,t are given by

Let X₁ denote the option’s strike price. Show that the price of this down-andout call with an external barrier is given by (Kwok, Wu and Yu, 1998)

where 

Transcribed Image Text:

d Si,t Sist rdt + o¡ dZi,t, i=1,2 and dZ₁dZ₂ = pdt.