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## #1 2008-04-18 09:14:09

George,Y
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### Black-Scholes Formula Derivation

However something needs to be clarified about Bt and dc

Bt 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*Δt2. 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 dB2.

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 (2008-04-20 03:09:29)

X'(y-Xβ)=0