Exercise 14.10 (Knock-Out Option) In the same setting as Exercise 14.8, consider a knock-out option that expires worthless if the underlying price ever hits the prespecified level H. That is, let τ be the first passage (hitting) time to the level H. Then, the premium of this option is given by

where S(0),K > H. Prove that

where S(0) = S, γ = 1 − 2r/σ² and

Note that the first term of the premium is the Black–Scholes equation. See Cox and Rubinstein (1985, page 411) for the interpretation of the second term.

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C = eTE [{S(T)- K}+{r>T}],