How Black-Scholes Began part1

When it comes to algorithm trading there isn't a day where you will miss out on OPT. With the devlopement in technology we can simply open up our trading systems and BAM there is our theoretical option prices and its greeks.

 

From my personal experience it is not optimal to just simply DDE those numbers to your excel spreadsheet. It is crucial that you how these numbers were calculated and when you do master it, you can use it to develop more sophisticated strategies.

There are two ways in which we can derive BS formula. One, Fischer Black and Myron Scholes way (1973)  or two Stochastic Differential equation way. I shall introduce both.

If you were ever interested in finance you probably have seen the equation down below, it is the Black-Scholes option pricing model. We are now going to attempt to build this model!

 

To get the above formulas we need to start with Brownian Motion, Wiener Process and Ito Calculus. For our insanity I am not going to explain them but these are the formulas that we need.

 

1) With the notion of Wiener Process we can show the change in stock price, with change with time. Basically, during our given dt, the stock price change is the same as risk free price change (u) + sigma. Simply put its risk free price + premium for taking on risk.  If Wiener Process value gets bigger then so is our premium. 

What we want at the end of the day is to find f(s,t) [Futures Price]  from our underlying Security "S". If what we are trying to find is the value of the option, then it would consists of time (t), underlying price (s), risk free rate (r),  Volatility (Sigma) and its strike price X. Therefore, an option price has

f(S,t,r,sigma,x) as variables.

2) This formula is Ito-Doublin formula revised from Stochastic Process f(s(t),t).

From the equations 1) and 2)  we want to try to find the unknow value "f" however we cannot do this beucase of dw. We need a way to get rid of dw! and this is how.

1. Portfolio

Lets think about a scenario where we buy the underlying "s" and short sell an option with price of "f".

If that is the case then through formula 3 and 4 we get formula 5. We tweak the formula to show us changes in portfolio values then we get formula 6, with the position being 100% hedged: Delta-hedge with value of 0.

2.If risk-free

if we created a portfolio like that of 5 which is completely risk-free then it should have a value of which is equivalent to risk-free rate. Remember, there is no free lunch!

 

Formula 7 is basically what i just said about no arbitrage rule and that the portfolio that is completely hedged should have the same return as that of rf investment. If we add 7) and 6) together we will get formula 9)

Formula 9) is the first key to solving B-S formula stay tuned!

-Charles Sin