Consider the European zero-rebate up-and-out put option with an exponential barrier: B() = Be , where

Consider the European zero-rebate up-and-out put option with an exponential barrier: B(τ) = Be^−γτ , where B(τ) > X for all τ . Show that the price of this barrier put option is given by

where p_E (S, τ) is the price of the corresponding European vanilla put option. Deduce the price of the corresponding European up-and-in put option with the same barrier.

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8-1 B(t)² y) - [B(r)] - ' PE (B(5) ² x). 8 = ² (r =1) 9 S S 02 p(S, T) = PE(S, T) —