Module 40.6: The Black Scholes Merton Model

The BSM model allows us to value options continuously in real time, as long as the no-arbitrage condition holds. The no-arbitrage option price guarantees that the hedge portfolio will yield the risk free rate. There are six underlying assumptions of the BSM Model:

1. The underlying asset price follows a geometric Brownian motion process. The return on the underlying asset follows a lognormal distribution. In other words, the logarithmic continuously compounded return is normally distributed.
2. The (continuously compounded) risk-free rate is constant and known. Borrowing and lending are both at the risk-free rate.
3. The volatility of the returns on the underlying asset is constant and known. The price of the underlying changes smoothly (i.e., does not jump abruptly).
4. Markets are “frictionless.” There are no taxes, no transactions costs, and no restrictions on short sales or the use of short-sale proceeds. Continuous trading is possible, and there are no arbitrage opportunities in the marketplace.
5. The (continuously compounded) yield on the underlying asset is constant.
6. The options are European options (i.e., they can only be exercised at expiration).

The formula for valuing a European option using the BSM model is:

C₀ = S₀N(d₁) − e^–rTXN(d₂)

and

P₀ = e^–rTXN(–d₂) − S₀N(–d₁)

where:

C₀ and P₀ = values of call and put option

T = time to option expiration

r = continuously compounded risk-free rate

S₀ = current asset price

X = exercise price

σ = annual volatility of asset returns

N(*) = cumulative standard normal probability

N(–x) = 1 – N(x)

The BSM value can be thought of as the present value of the expected option payoff at expiration. Calls can be thought of as a leveraged stock investment where N(d₁) units of stock are purchased using e^–rTXN(d₂) of borrowed funds. A portfolio that replicates a put option consists of a long position in N(–d₂) bonds and a short position in N(–d₁) stocks. N(d₂) is interpreted as the risk-neutral probability that a call option will expire in the money. Similarly, N(–d₂) or 1 − N(d₂) is the risk-neutral probability that a put option will expire in the money.

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