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George,Y

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Registered: 2006-03-12

Posts: 1,379

Black-Scholes Formula Derivation

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However something needs to be clarified about B[sub]t[/sub] and dc

B[sub]t[/sub] follows a Brownian motion in time t

You can cut equal time intervals and each interval B changes by a normal random variable with 0 mean and time length variance; each B change in a time interval has nothing to do with another B change in a different interval.

And

hence

As Δt gets small, mathematicians argue the square of ΔB has a quite steady mean Δt with a negligible (higher order) variance 2*Δt[sup]2[/sup]. Hence here comes

Ito's lemma

dB*dB=dt;
dt*dt=o(dt);
dB*dt=o(dt)

Thus, to approach a difference of a function which involves a brownian motion B in it directly or indirectly, you have to use Taylor expansion with order 2 to capture the innegligible dB[sup]2[/sup].

thus here come the dc

where

o(dt) is higher order term of dt, negligible the same way we do our normal calculus. And the final formual for dc is:

Last edited by George,Y (2008-04-19 05:09:29)

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X'(y-Xβ)=0