Consider a European down-and-out partial barrier call option where the barrier provision is activated only between the option’s starting date (time 0) and t₁. Here, t₁ is some time earlier than the expiration date T , where 0 1

under the risk neutral measure Q. Assuming S₀ > B, show that the down-andout call price is given by (Heynen and Kat, 1994a)

where

Show that the above price formula reduces to the price function defined in (4.1.12a,b) when t₁ is set equal to T.

Modify the price formula in Problem 4.6 by setting ρ = √t₁/T σ₁ = σ and σ₂ = √t₁/T σ so that γ₁₂ = 1.

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d St St = rdt + o Z