The Black-Scholes Model

In this lesson, we learned about the Black-Scholes model and how it related to the binomial model that we have been using in previous lessons.

The Black-Scholes model has four main assumptions:

1. A continuously compounding interest rate, r.
2. Geometric Brownian motion dynamic for stock prices, so that:
   

where W(t) is a standard Brownian motion.

     3.  The stock pays a dividend yield of c.

     4.  Continuous trading with no transaction costs and short-selling allowed.

Here are some sample paths of Geometric Brownian Motion:

sdf

The Black-Scholes formula for pricing European call options with strike price K and maturity at time T is:

where d(1) and d(2) are as follows:

and N(d)  P(N(0,1) ≤ d). 

Note that mu (the drift of the Brownian motion) does not appear in the formula, just as p and (1-p) do not appear in the binomial model.

Once we have the call option price C(0), we can calculate the put option price P(0) from the put-call parity:

We can show that under the Black-Scholes model uses a similar replicating strategy to the one we used for the binomial model:

We can convert the Black-Scholes parameters of r and sigma into binomial parameters.

The Black-Scholes parameters:

1. r, the continuously compounding interest rate
2. sigma, the annualized volatility of the stock

can be converted into binomial model parameters:

and we can redefine the risk-neutral probabilities as:

The Black-Scholes model is actually used in industry to quote option prices and the binomial model is often used as an approximation to the Black-Scholes model.

The binomial model also converges to the Black-Scholes model as the number of time periods goes to infinity.  A longer proof was included in the lesson that I have omitted here for brevity.