Unformatted text preview: ) + m=1 ! Bt + (r t/h
pX
h
h
EXm
m=1 µ+ 2 /2)t so under the risk neutral measure, P ⇤ ,
St = S0 · exp((r 2 /2)t + Bt ) The value of the option g (ST ) in the discrete approximation is given by the
expected value under its risk neutral measure. Ignoring the detail of proving
that the limit of expected values is the expected value of the limit, we have
proved the desired result. The BlackScholes partial di↵erential equation
We continue to suppose that the option payo↵ at time T is g (ST ). Let V (t, s)
be the value of the option at time t < T when the stock price is s. Reasoning
with the discrete time approximation and ignoring the fact that the value in
this case depends on h,
V (t h, s) = 1
[p⇤ V (t, su) + (1
1 + rh p⇤ )V (t, sd)] 201 6.7. CALLS AND PUTS
Doing some algebra we have
V (t, s) (1 + rh)V (t s, h) = p⇤ [V (t, s)
+ (1 V (t, su)] p )[V (t, s)
⇤ V (t, sd)] Dividing by h we have
V (t, s) V (t h, s)
rV (t h, s)
h V (t, su)
V (t, s) V (t, sd)
⇤ V (t, s)
⇤
=p
+ (1 p )
h
h (6.30) Letting h ! 0 the lefthand side...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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