Suppose that a one-year zero sells for $93.30, and a two-year zero sells for $80

Question

Suppose that a one-year zero sells for $93.30, and a two-year zero sells for $80.32. Further, you
have strong reason to believe that the one-year rate next year will be 13%.

1. Suppose that you could borrow up to $1,000,000 at current market rates (i.e., you can
borrow and lend at the same rates). What transactions would you undertake to profit
from your information? How much profit could you make if you utilized the entire line
of credit?
2.What is the maximum spread between the borrowing and lending rates that would still
allow you to profit?

Explanation / Answer

A model of the nominal term structure of interest rates is developed that has a positive and stationary process for the interest rate and delivers closed form expressions for the prices of discount bonds and European options on bonds. Unlike the one-state-variable version of the Cox, Ingersoll and Ross (1985) model this model - even in its one-state-variable version - allows the term premium to change sign as a function of the state and the term to maturity, and also allows for shapes of the yield curve that are observed in the U.S. data but that are disallowed in the Cox, Ingersoll and Ross model The paper derives a general form of the termstructure of interest rates. The following assumptions are made: (A.1) The instantaneous (spot) interest rate follows a diffusion process; (A.2) the price of a discount bond depends only on the spot rate over its term; and (A.3) the market is efficient. Under these assumptions, it is shown by means of an arbitrage argument that the expected rate of return on any bond in excess of the spot rate is proportional to its standard deviation. This property is then used to derive a partial differential equation for bond prices. The solution to that equation is given in the form of a stochastic integral representation. An interpretation of the bond pricing formula is provided. The model is illustrated on a specific case.

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