How Black-Scholes Began Part 2

The B-S partial differential equation that we saw from the previous post is the same as 1) below. Now we are going to use classical partial differential equation method to solve for f(t,s) for Call option.

The boundaries of equation 1 is the same as equation 2. Keeping that in mind, Option's Time Value (T) is equally proportionate to underlying stock price (S) on maturity and strike price (X). If stock price (s) is bigger than strike price (x) on maturity then the value of this option should be that (ST-X). If S is smaller than X then this particular option will expire worthless.

Partial Differential Equation Process

To make our lives more simple we will replace the varibles in equation 1 and 2 to like that of the equations below in 3 and 4.

x= T-t will represent time value of the option and therefore on maturity T=t and therefore T-t=0. If that is the case then eqation 2's boundaries will be the same with that of equation 4.

From here let us try to show that equation 1 could look like 5

The above equation tells us that when days left until maturity is x and option's value is y(u,x) then this option's current price denoted f(S,t) is then also y(u,x). Therefore we can show that f(S,t) like that of the formula below

equation 10's 5th step is just chain rule. f, s, and t are variables and S, t are same as u and x variables and therefore with those subsitution we have the 5th step in equation 10.  Now that we have equation 7, 9 and 11 lets subsitute this into equation 3 and see what we get.

lets make equation 13 alittle bit more clean and we get.

Basically what we have done until now is to simplify equation 1 with known variables.
stay tuned.


-Charles Sin