** CHAPTER 6**

** Black′s World**

** T**he Black-Scholes-Merton formula, and its Black variant for futures was historically derived for non-interest-rate-related underlying assets (equities, FX, commodities), and under the assumption that interest rates are nonrandom. When interest rate options (cap/floors, swaptions) were introduced, traders co-opted these formulas and applied them to interest rates. While everyone recognized that the formulae need to be adjusted since interest rates are not traded assets, and *are* random, nevertheless, in the absence of any other simple alternatives, Black′s formula became (and still continues to be) the standard option pricing formula for interest-rate flow products.

Therefore, we will suspend disbelief for a while and derive the Black-Scholes-Merton formula in a world where interest rates are deterministic, and then turn around and apply these to interest rate options!

Before we get there, however, a bit of probability review is in order.

#
** A LITTLE BIT OF RANDOMNESS**

A random variable (r.v.) *X* is said to have a *Normal* distribution with mean *µ* and standard deviation *σ* , if the probability that it lies in some region [*x, x* + *dx*] is approximately

We will use the shorthand

*X* ~

*N*(

*µ, σ* ^{2} ). More precisely, the

*cumulative distribution function* (CDF) of an

*N*(

*µ, σ* ^{2} ) random variable

*X* is

**FIGURE 6.1** Distribution Function of a Normal

*N*(

*µ, σ* ^{2}) Random Variable