How Black-Scholes Began Part 3

In this section we will take equation 1 do partial differentials and solve for y(u,x). When we obtain
y(u,x) we will be able to get f(S,t)

 

y(u,x) can also be written as F(u) and T(x), which comes from a physics equation when finding temperature with time.

After doing partial differentials for u and x we will substitue those numbers back into equation 1 so that we can end up with equation 3. What k represents in equation 3 is called Separation constant. We can re write equation 3 into 2 different equations: 4 and 6.



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If we take equation 5,7 and T and substitute into equation 2 then we will be able to get the equation 8. Since we know that e function is an infinity function we can rewrite equation 8 in a more basic form as that of 9

More simplifing begins...

After simplifying equation 12 and using 13's equation method we obtain equation 14. If we switch up the integral then we will get equation 15

We now need to substitute equation 16 into 15, but in order to do so we need to simplify our equation which takes place from 17 to 20.

Sub. equation 20 to 15 to get 22. We do this so that we have an equation that has a function of y(u,x).

In equation we can see a function g(a). This function show that y(u,0) is on the day of maturity. We are now almost at our goal!