Business, Legal & Accounting Glossary
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:
Here, log denotes the natural logarithm, and:
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 , 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  and  are identical for puts and calls, as are vega formulas  and .
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.
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This glossary post was last updated: 3rd March, 2022 | 0 Views.