# A compute the value function vn a and the replicating

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Unformatted text preview: of (6.30) converges to @V (t, s) @t rV (t, s) (6.31) Expanding V (t, s) in a power series in s V (t, s0 ) V (t, s) ⇡ @V (t, s)(s0 @x @2V (s0 s)2 (t, s) @ x2 2 s) + Using the last equations with s0 = su and s0 = sd, the right-hand side of (6.30) is ⇡ @V (t, s)s[(1 u)p⇤ + (1 d)(1 @x 1 @2V (t, s)s2 [p⇤ (1 u)2 + (1 2 @ x2 p⇤ )]/h p⇤ )(1 d)2 ]/h From (6.28) (1 (1 u)p⇤ + (1 d)(1 p⇤ ) ⇡ h u)2 p⇤ + (1 d)2 (1 p⇤ ) ⇡ h ✓ 2 2 ◆ +µ = r 2 so taking the limit, the right-hand side of (6.30) is @V 1 @2V (t, s)s[ rh] + (t, s)s2 @x 2 @ x2 2 h Combining the last equation with (6.31) and (6.30) we have that the value function satisﬁes @V @t rV (t, s) + rs @V 1 (t, s) + @x 2 2 2@ s 2 V (t, s) = 0 @ x2 (6.32) for 0 t &lt; T with boundary condition V (T, s) = g (s). 6.7 Calls and Puts We will now apply the theory developed in the previous section to the concrete examples of calls and puts. At ﬁrst glance the formula in the ﬁrst result may look complicated, but given that the value is deﬁned by soolving a PDE, it is remarkab...
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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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