Black-Scholes Formula

Contents (hide)

1. Assumptions
2. European call prices
3. Derivation

1. Assumptions

The Black-Scholes (BS) formula for the pricing of European options is one of the cornerstones of modern finance. The assumptions underlying the formula are the following

The asset price, $\fs4 S(t)$ follows a geometric Brownian motion with constant instantaneous drift and volatility ( $\fs4 \mu$ and $\fs4 \sigma$ , repsectively).
Short selling is permitted, with full use of the proceedings.
There are no dividends, transaction costs or taxes.
No arbitrage opportunities are present.
Trading is continuous in time.
There is a constant interest rate for all maturities, $\fs4 r$ .

2. European call prices

Based on these assumptions, the price of a European call $\fs4 C_{BMS}$ , written on an asset with current price $\fs4 S=S(0)$ , maturing at time $\fs4 T$ , with strike price $\fs4 K$ , is given by

$\fs4 C_{BMS}( S,K,T,r,\sigma ) =S\Phi( d_{1})-Ke^{-rT}\Phi( d_{2})$

where $\fs4 \mathcal{\Phi}(\cdot)$ is the cumulative normal distribution function, and

$\fs4 d_{1} =\frac{\ln (S/K) +\left( r+ \sigma ^{2}/2 \right) T}{\sigma \sqrt{T}}$ and

$\fs4 d_{2} = d_{1}-\sigma \sqrt{T}$

The price of the corresponding put $\fs4 P_{BMS}$ is given by

$\fs4 P_{BMS}( S,K,T,r,\sigma ) =-S\Phi( -d_{1})+Ke^{-rT}\Phi(-d_{2})$

3. Derivation

The BMS formula can be derived in a number of intuitive ways:

By solving the Black-Scholes partial differential equation, which is the original approach in the Black and Scholes (1973, Journal of Political Economy) paper. This approach illustrates that the market under the above assumptions is complete.
As the limit of a binomial model, as the number of nodes goes to infinity, and the discretization becomes finer.
By application of Girsanov's theorem, which allows one to construct expectations under different equivalent probability measures.
Using the martingale representation theorem to model the price of the asset under the risk adjusted probability measure.
By setting up agents that exhibit power utility in an equilibrium framework.

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Black-Scholes Formula

1. Assumptions

2. European call prices

3. Derivation