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.

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