How Black-Scholes Began Part Final

We are almost done!! The equation you see below is from the previous post equation 10 but the only thing changed was that u=a and x=0 (on maturity) to y(u,x).

If we replace the equation 2 in the form of (a) we get the 2nd equation in 2. With the same concept equation 1 becomes 3 with its bottom ranged infinity changed to -(u/sigma*root(x)). If we write equation 3 into two parts we get equation 4 down below divided into parts a and b.

If we take the residue equation from a) and b) and replace it back into equation 4 we will get equation 5. If you look closely into that function it is the Probability Distribution Function. The graph of such function can be represented below.

We are now approaching the final step which is to find f(S,t). This can be done by subing. equation 5 into 6. This equation then forms 7, our ultimate answer of B-S model.

With this formula not only can we calculate the theoretical option price with respect to time, but also calculate each individual option's volatility. The ultimate stock pricing tool. If you were able to follow all the steps then you should be proud!

-Charles Sin