Black-Scholes Model

The Black Scholes model is the most famous formula in finance. Published in 1973 by Fischer Black and Myron Scholes, with key contributions from Robert Merton, it gives a price for a European option from five inputs. It transformed options trading by giving traders a common language for value and risk. Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997; Black had died in 1995. Today the model is used less to find "the" price and more to translate prices into implied volatility and to calculate the Greeks.

The five inputs#

Because volatility is the only input you cannot see, traders run the formula backwards: given the market price, they solve for the volatility that produces it. That is implied volatility. See Implied Volatility (IV).

The formula#

For a European call on a non dividend paying stock:

C = S × N(d1) - K × e^(-rT) × N(d2)
d1 = [ln(S / K) + (r + σ² / 2) × T] / (σ × √T)
d2 = d1 - σ × √T

For a European put:

P = K × e^(-rT) × N(-d2) - S × N(-d1)

N(x) is the cumulative standard normal distribution: the probability that a standard normal variable is below x. See Normal Distribution.

Reading the formula#

• S × N(d1): the expected value of receiving the stock if the option finishes in the money, in present value terms. N(d1) is also the call's delta. See Delta.
• K × e^(minus rT) × N(d2): the present value of paying the strike, weighted by N(d2), the risk neutral probability that the call finishes in the money.

Worked example#

The assumptions#

See Lognormal Distribution.

Where the model breaks down#

The clearest sign is the volatility smile: if Black Scholes were exactly right, every option on the same underlying and expiry would have the same implied volatility. In reality, out of the money puts on equity indices trade at higher implied volatilities, especially since the 1987 crash. Traders handle this by using different volatilities for different strikes, or by using models such as local volatility, Heston and SABR. See Local Volatility, Stochastic Volatility and the Heston Model and SABR Model.

The replication idea#

The deep insight of Black, Scholes and Merton was that an option's payoff can be replicated by continuously adjusting a position in the stock and a risk free bond. If replication is possible, the option's price must equal the cost of the replicating portfolio, regardless of investors' views on expected returns. This is why the stock's expected return does not appear in the formula. It is also the basis of Delta Hedging.

Extensions#

• Merton (1973): continuous dividend yield, by replacing S with S × e^(minus qT).
• Black 76: options on futures. See Black-76 and Bachelier Models.
• Garman and Kohlhagen (1983): currency options, using domestic and foreign interest rates. See FX Options.

Frequently asked questions#

What is the Black Scholes model?#

A mathematical model, published in 1973, that prices European options from the underlying price, strike, time to expiry, interest rate and volatility.

Why is Black Scholes still used if its assumptions are wrong?#

It provides a standard way to quote options in implied volatility terms and to compute Greeks, and traders adjust for its flaws with volatility smiles and better models.

What does N(d2) mean in Black Scholes?#

It is the risk neutral probability that a call option finishes in the money at expiration.

Next, see versions for futures and negative prices in Black-76 and Bachelier Models.

Sources#

• Wikipedia, Black Scholes model