Merton (1973) Option Pricing Formula

The Merton (1973) option pricing formula generalization the Black-Scholes (1973) formula so it can price European options on stocks or stock indices paying a known dividend yield. The yield is expressed as an annual continuously compounded rate q. Values for a call price c or put price p are:

where:

Here, log denotes the natural logarithm, and:

• s = the price of the underlying stock
• x = the strike price
• r = the continuously compounded risk free interest rate
• q = the continuously compounded annual dividend yield
• t = the time in years until the expiration of the option
• σ = the implied volatility for the underlying stock
• Φ = the standard normal cumulative distribution function.

Consider a call option on a stock index. The option is struck at EUR 8000 and expires in .18 years. The index is trading at EUR 7986 and has 24% (that is .24) implied volatility. The continuously compounded risk free interest rate is .0293. Based upon recent dividends, assume an annual dividend yield of q = .0254. Applying formula [1], the option has market value EUR 319. Because the option is out-of-the-money, that value is entirely time value.

The Greeks—delta, gamma, vega, theta and rho—for a call are:

where ϕ denotes the standard normal probability density function. For a put, the Greeks are:

Note that gamma formulas [6] and [11] are identical for puts and calls, as are vega formulas [7] and [12].

A shortcoming of the Merton formula is its assumption that dividends are paid out continuously. For a stock index, this is an imperfect but often reasonable approximation. For individual stocks, which typically distribute dividends in two payments each year, it is more problematic. The stock’s annual yield is immaterial. The quantity q needs to reflect the dividends that will be earned prior to the option’s expiration. If the stock has no dividend record date prior to the option’s expiration, set q = 0. Otherwise, calculate the stock’s dividend yield through expiration and annualize. Another problem is the fact that the model assumes that the dividend yield is a known constant. Often a dividend payment will be scheduled during the life of an option, but the amount of the payment has not yet been announced. This is an additional source of uncertainty the Merton model cannot reflect.