Show that the knockout option as the same value as an

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Unformatted text preview: le that such a simple formula exists. 202 CHAPTER 6. MATHEMATICAL FINANCE Theorem 6.11. The price of the European call option (St S 0 (d 1 ) e rt K )+ is given by K (d 2 ) where the constants d1 = ln(S0 /K ) + (r + p t 2 /2)t p d2 = d1 t. Proof. Using the fact that log(St /S0 ) has a normal(µt, 2 t) distribution with 2 µ=r /2, we see that Z1 2 2 1 ⇤ rt + rt E (e (St K ) ) = e (S0 ey K ) p e (y µt) /2 t dy 2t 2⇡ log(K/S0 ) p Splitting the integral into two and then changing variables y = µt + w t, p dy = t dw the integral is equal to Z1 Z1 p 2 2 1 1 = e rt S0 eµt p ew t e w /2 dw e rt K p e w /2 dw (6.33) 2⇡ ↵ 2⇡ ↵ p where ↵ = (log(K/S0 ) µt)/ t. The handle the first term, we note that Z1 Z1 p p2 2 2 1 1 t) /2 p p e (w ew t e w /2 dw = et /2 dw 2⇡ ↵ 2⇡ ↵ p 2 = et /2 P (normal( t, 1) > ↵) The last probability can be written in terms of the distribution function normal(0,1) , i.e., (t) = P ( t), by noting p p P (normal( t, 1) > ↵) = P ( > ↵ t) p p = P...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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