# Some history on the Option pricing with Black-Scholes for quants

(Last Updated On: October 14, 2010)

## Some history on the Option pricing with Black-Scholes for quants

Some highlights from this quick and dirty article:

If there was no inventory at hand, a market maker knew that they should hedge their position using the underlying asset. Nelson describes this concept already in 1904. Later the concept is generalised and extended by Thorp from ‘at the money hedge’ to ‘off the money hedging’, essentially formulating the concept of ?-hedging.

Thorp went on to make a small fortune by discovering risk neutral probabilities and putting the concept to practice in the Las Vegas casinos. He also went on to discover the formula now know as the Black-Scholes equation. Thorp formulated the equation, but didn’t have a derivation at first. It was Black and Scholes who added the dynamic hedging concept used to make a portfolio of a derivative and its underlying asset risk free, allowing for the derivation of the equation.

Later Bachelier (1900), Einstein (1905) and finally Wiener (1923) developed a rigorous theory and the mathematics to describe Brownian Motion. The resulting theory is essential for the derivation of the Black and Scholes formula, which uses Brownian Motion to model asset prices as stochastic differential equations. To work with these equations we need Ito calculus. Ito (1951) showed how given a stochastic differential equation for some independent random variable, one can derive the stochastic differential equation of a function of that variable.

Some quant algo highlights:

Can we find a choice of ? that eliminates all the dX terms? Yes, choose:

?

Something amazing happens using this choice of ?, the ?-term also disappears! Implying that
we no longer face the difficult task of coming up with a number for stock return (?).

. We are left with a risk free portfolio, since the equation no longer contains a dX term. 