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Deriving the Black-Scholes Equation

Assuming that the movement of a share price follows the geometric Brownian motion given by

The value V of an option or other derivatives contingent on S is a function of S and t, then from Itô's Lemma (see article), we get

Now consider a portfolio of value constructed by longing one option and shorting amount of shares, this gives

Differentiating this gives the change in portfolio as

Assuming that there are no arbitrage opportunities, the value of can be chosen such that the portfolio will earn the same rate of return as other risk-free securities,i.e.

Where r is the interest rate. Equating the change in portfolio to the risk-free return of , i.e. hedging the portfolio, gives

By equating the change in portfolio to the risk-free return, we have eliminated the randomness associated with the portfolio, this is only true if the coefficient of dX is equal to zero, therefore

So this gives

Rearranging gives

This is the Black-Scholes equation for pricing options.

In deriving this equation, we assumed that:
1. There are no arbitrage opportunities (no free lunch).
2. Short selling of shares is possible at all times.
3. No transaction costs or taxes in setting up a portfolio.
4. All securities are perfectly divisible.
5. Trading can take place continuously.
6. The underlying share pays no dividends during the lifetime of the option.
7. The risk-free rate r and the share volatility σ are known over the lifetime of the option.

Written by Henry Tang and Raffaello Vardavas

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