Merton (1973) Option Pricing Formula

Business, Legal & Accounting Glossary

Definition: Merton (1973) Option Pricing Formula


Merton (1973) Option Pricing Formula


Full Definition of 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:

[1]

[2]

where:

[3]

[4]

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:

[5]

[6]

[7]

[8]

[9]

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

[10]

[11]

[12]

[13]

[14]

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.


Cite Term


To help you cite our definitions in your bibliography, here is the proper citation layout for the three major formatting styles, with all of the relevant information filled in.

Page URL
https://payrollheaven.com/define/merton-1973-option-pricing-formula/
Modern Language Association (MLA):
Merton (1973) Option Pricing Formula. PayrollHeaven.com. Payroll & Accounting Heaven Ltd.
October 09, 2024 https://payrollheaven.com/define/merton-1973-option-pricing-formula/.
Chicago Manual of Style (CMS):
Merton (1973) Option Pricing Formula. PayrollHeaven.com. Payroll & Accounting Heaven Ltd.
https://payrollheaven.com/define/merton-1973-option-pricing-formula/ (accessed: October 09, 2024).
American Psychological Association (APA):
Merton (1973) Option Pricing Formula. PayrollHeaven.com. Retrieved October 09, 2024
, from PayrollHeaven.com website: https://payrollheaven.com/define/merton-1973-option-pricing-formula/

Definition Sources


Definitions for Merton (1973) Option Pricing Formula are sourced/syndicated and enhanced from:

  • A Dictionary of Economics (Oxford Quick Reference)
  • Oxford Dictionary Of Accounting
  • Oxford Dictionary Of Business & Management

This glossary post was last updated: 3rd March, 2022 | 0 Views.