Towards the end of his life, Fisher Black came up with a seminal idea about interest rates. He conjectured that interest rates – the short term interest rates only – can be viewed as a call option on a shadow interest rate, which is made up of a real interest rate and the rate of inflation. The call option was struck at a strike price of zero percent. He noted in this thesis that when the short rates were close to this strike price usual term structure relationships can be markedly altered by the value of the options embedded in the current and expected short rates. Here is a simple exposition of this theory:

Any (nominal) short term interest rate can be thought of as a sum of a real short term rate and an expected inflation. If is the real short term rate and is the expected inflation then we can write and expression for the nominal short term interest rate as:

Can the nominal rate be negative? Of course not! But real interest rates can be negative (translated in economic term this means that owing to falling demand for goods and services most projects are money losers). And the expected (as well as the actual inflation rate) is negative (as we saw in Japan in the late nineteen nineties). Therefore, at least mathematically speaking, if both the real short term rate and the expected inflation rate are negative then the nominal short term rate will also be negative. But actual nominal interest rates can never be negative; it can at best be zero. This is because then the investors will withdraw their money from the bank accounts and put the cash under their mattresses. Eventually, when all money is taken out of the savings accounts equilibrium is established and the nominal rates will return to zero value from a negative value. Hence, in a world where all financial transactions are done in cash, nominal interest rates cannot be negative.

Thus, in real world, the nominal interest can be either zero or a number greater than zero which is given by the sum of the real interest rate and the inflation rate. We can express this as:

can therefore be redefined as the shadow nominal short rate. One can see that the above formula represents a call option payoff formula with a strike price of zero and a spot rate equal to . The actual rate , therefore, is a call option on the shadow interest rate, with a strike rate of zero.

(Note: The above analysis borrows heavily from an excellent book titled *Financial Derivatives: Pricing, Applications and Mathematics * by Jamil Baz & George Chacko.)

Any comments and queries can
be sent through our
web-based form.

__
More on Quantitative Finance >>__

back to top