Assumptions of the BSM Model

• The underlying price follows a lognormal probability distribution as it evolves through time.
  A lognormal probability distribution is one in which the log return is normally distributed. For example, if a stock moves from 100 to 110, the return is 10% but the log return is ln(1.10) = 0.0953 or 9.53%. This assumption is reasonable for most assets that offer options. Additionally, the variance of the return is assumed to be constant for the life of the option.

• Interest rates remain constant and known.
  The model uses the risk-free rate to represent this constant and known rate. Borrowing and lending is allowed at the risk-free rate.
  During periods of rapidly changing interest rates, these 30-day rates are often subject to change, thereby violating one of the assumptions of the model. This assumption also becomes a problem for pricing options on bonds and interest rates.

• The underlying instrument is liquid. The price of the underlying instrument is continuous.

• The volatility of the underlying instrument is known and constant.
  The volatility of the underlying is specified in the form of the standard deviation of the log return. This is the most critical assumption.

• The market is "perfect." There are no transaction costs, regulatory constraints, or taxes. Continuous trading is available. Short selling of the underlying instrument with full use of the proceeds is permitted. There are no arbitrage opportunities in the marketplace.

• There are no cash flows on the underlying. If the underlying instrument pays a yield, it is expressed as a continuous known and constant yield at an annualized rate.

• European exercise terms are used.
  This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

BSM Model

The Black-Scholes-Merton formulas for the prices of call and put options are:

c = SN(d₁) - e^-rT XN(d₂)

p = e^-rTXN(-d₂) - SN(-d₁)

where:
d₁ = {ln(S/X) + [r + (σ²/2)] T} / [σ (T^1/2)]
d₂ = d₁ - σ (T^1/2)
σ = the annualized standard deviation of the continuously compounded return on the stock
r = the continuously compounded risk-free rate of return
N(.) = the cumulative normal distribution function

• N(-x) = 1 - N(x)
• N(d₁) is the basis for the number of units of the underlying instrument needed to replicate an option. It is the primary determinant of delta, and it answers the question of how much the option value will change with a small change in the underlying.
• N(d₂) is the probability that the call option expires in the money.

Example 1

Assume that a stock trades at $100 and the continuously compounding risk-free interest rate is 6%. A call option on the stock has an exercise price of $100 and expires in one year. The standard deviation of the stock's returns is 0.1 per year. We compute the values of d₁ and d₂ as follows:
d₁ = [ln(100/100) + (0.06 + (0.1²/2) 1] / (0.1 1^1/2) = 0.65
d₂ = 0.65 - 0.1 x 1^1/2 = 0.55

Next, we find the cumulative normal values associated with d₁ and d₂. These values are the probability that a normally distributed variable with a zero mean and a standard deviation of 1.0 will have a value equal to or less than the d₁ or d₂ term we are considering. Below is the relevant part of the cumulative probabilities table for a standard normal distribution.

For a value of 0.65 drawn from the border of the table, we find our probability in the interior: N(d₁) = N(0.65) = 0.7422. Similarly, N(d₂) = N(0.55) = 0.7088.

We now have:
c = $100 x 0.7422 - $100 x e ^{-0.06 x 1} x 0.7088 = $7.46
p = $100 x e ^{-0.06 x 1} x (1 - 0.7088) - $100 x (1 - 0.7422) = $1.64

The BSM model can be described as the present value of the expected option payoff at expiration. It can also be described as having two components: a stock component and a bond component.

• For call options, the stock component is SN(d₁) and the bond component is e^-rTXN(d₂). c = stock - bond = SN(d₁) - e^-rTXN(d₂)
• For put options, the stock component is SN(-d₁) and the bond component is e^-rTXN(-d₂). p = bond - stock = e^-rTXN(-d₂) - SN(-d₁)

Investors can replicate the stock option payoffs with stocks and bonds:

Replicating strategy cost = n_S S + n_B B Example 2

Assume d₁ = 0.3 and d₂ = 0.5. N(d₁) is 0.6179 and N(d₂) = 0.6915.

• To replicate a call option we should buy n_S = N(d₁) = 0.6179 shares, and buy -N(d₂) = -0.6915 zero-coupon bond (short the bond).
• To replicate a put option we should short n_S = -N(-d₁) = -(1 - 0.6179) = -0.3821 shares (short the stock), and buy N(-d₂) = (1 - 0.6915) = 0.3085 zero-coupon bond.
• The price of the zero-coupon bond is B = e^-rTX.

Note that for currency options, the stock component becomes the foreign exchange component. For example, the BSM call model applied to currencies is simply the foreign exchange component minus the bond component.

The BSM model can be adjusted to accommodate carry benefits.

c = Se^-γTN(d₁) - e^-rTXN(d₂)
p = e^-rTXN(-d₂) - Se^-γTN(-d₁)

where d₁= [ln(S/X) + (r-γ+σ²/2)T]/(σT,sup>1/2)

Note that d₁, d₂ and the stock component are all adjusted for carry benefits.

Carry benefits reduce the value of call option and raise the value of the put option. In replicating strategies, carry benefits lower the number of shares to buy (short) for calls (puts).

• For stock options, the carry benefits are dividends. γ is the continuously compounded dividend yield.
• For currency options, the carry benefits are the continuously compounded foreign interest risk-free rates. The domestic risk-free rate is the risk-free rate to be used in the model.