Hi Can someone tell me the difference between the Black-scholes formula in Chapter 12 and Garman-Kohlhagen formula on pg 47 in the formula book? Why is the r and q 'missing' from Black-scholes formula? In what case do I use Black-scholes or Garman-Kohlhagen? Thank you very much.
They are two different but related beasts. Assume for a moment that there is a function that gives you the price of a call option. The B-S equation is simply an equation that any such function must satisfy (or bad things, such as arbitrage, happen). The G-K function is a function that can be shown to satisfy the B-S equation, and therefore it can be used to give the price of a call option. I don't have notes in front of me, but I think q is left out of the derivation for simplicity, but r should be used in B-S unless the derivation is assuming a zero risk rate.
The Garman-Kohlhagen formula is an extension of the Black Scholes model to allow it to cope with two different interest rates, one domestic and one foreign. This allows you to value options on a foreign exchange rate. In which case would you use either in real life? You wouldn't. In which case would you use either in an exam? If you are valuing an option on a share (or really any asset where you are assuming lognormal returns), use Black-Scholes. If you are valuing an option on a foreign exchange rate, use Garman-Kohlhagen. However I doubt they would ever examine this formula since it's not that well known, or in the core reading.
Thanks cjno1. So r is the domestic interest rate and q is not the dividend yield as defined in pg 45 but the foreign interest rate?
Correct. r is the domestic continously compounded risk-free rate. q is the foreign continously compounded risk-free rate.
