READING 56: THE BLACK-SCHOLES-MERTON MODEL

Stock Price And Return Distributions

        The Black-Scholes-Merton model suggests that stock prices are lognormal over longer periods of time, but suggests that stock returns are normally distributed.

Realized Return And Historical Volatility

        The realized return for a portfolio is computed using a geometric return.

Assumptions

        Assumptions underlying the BSM model:

• The price of the underlying asset follows a lognormal distribution.
  
• The (continuous) risk-free rate is constant and known.
  
• The volatility of the underlying asset is constant and known.
  
• Markets are “frictionless.”(There are no taxes, no transactions costs, and no restrictions on short sales or the use of short-sale proceeds.)
  
• The underlying asset generates no cash flows.
  
• The options are European.

Black-Scholes-Merton Formulas

        The formulas for the BSM model are:

        Cash flows on the underlying asset decrease call prices and increase put prices.

Valuation of Warrants

        Warrants are attachments to a bond issue that give the holder the right to purchase shares of a security at a stated price. Warrants can be valued as a separate call option on the firm’s shares.

Volatility Estimation

        Historical volatility is the standard deviation of a past series of continuously compounded returns for the underlying asset. Implied volatility is the volatility that, when used in the Black-Scholes-Merton formula, produces the current market price of the option.

American Options

        Dividends complicate the early exercise decision for American-style options because a dividend payment effectively decreases the price of the stock.

European Options

        To adjust the BSM model for assets with a continuously compounded rate of dividend payment equal to q, S₀e^−qT is substituted for S₀ in the formula.