The Lognormal Interest Rate Model and Eurodollar Futures

Transcription

1 GLOBAL RESEARCH ANALYTICS The Lognormal Interest Rate Model and Eurodollar Futures CITICORP SECURITIES,INC. 399 Park Avenue New York, NY 143 Keith Weintraub Director, Analytics Michael Hogan Ex Citibank May 1998 The Eurodollar futures contract is basic to a large class of interest rate derivative instruments and failure to assign reasonable values to it is a serious flaw. We will use the lognormal short-term interest rate model as it is set up and explicitly solved in Dothan 1978 to show that the lognormal model assigns an infinite value to the Eurodollar futures contract. There are two reasons for this peculiar behavior. The first is the long positive tail of the lognormal distribution. Rate processes that are bounded by some absolute rate do not have this problem. The square root diffusion in which futures and forward prices can be calculated explicitly has a mild case of it. For normal values of the parameters, futures that expire within the first 1 years or so will have finite prices. The other reason is the design of the Eurodollar futures contract itself. At expiration, the Eurodollar futures contract settles at the prevailing value of the 9-day LIBOR rate. The LIBOR rate is quoted as an add-on or money market rate. A typical quote for a 9-day LIBOR rate is 4%; this means that $1 will get you $ in 9 days. If r L is the 9-day LIBOR rate and Z is a 9-day zero-coupon bond price on a nominal $1 of principal, then the following relationship exists: 1 r L 4 1 Z/Z. Eurodollar futures are quoted on price, that is a LIBOR rate of 4 would be quoted as a futures price of The T-bill Treasury bill futures contract is quoted on the same basis, although Treasury bill rates themselves are not quoted on a money market basis, so what we say about the Eurodollar futures contract applies to them as well. For analytical purposes, Eurodollar futures rates; are often used as forward rates, that is, if the Eurodollar futures rate for the contract maturing at time t is r, then it is treated as if it is the forward rate for purchase at time t of a zero-coupon bond maturing at time t + 9 days. This approach is wrong for two reasons. First, Cox, Ingersoll, and Ross 1981 discuss the difference between futures prices and forwards prices. They show that in the limit of instantaneous resettlement the futures price for the contract that settles at X at time t is E [X where E is expectation under the familiar pricing measure, while the value of the forward contract is t / [ t 3 E [X exp rsds E exp rsds

2 where rt is the path of instantaneous interest rates. Thus a technical adjustment is almost always made for the difference between forwards and futures due to the periodic resettlement feature. This adjustment is as large as possible when the terminal payoff is most highly correlated with the path of short-term rates, and Eurodollar futures are probably as extreme as possible in this respect. Second, Cox, Ingersoll, and Ross 1981 work out the case of instantaneously settled bill futures in detail. There is a key difference, however, between the Eurodollar futures contract and the contract that they very reasonably treat as a bill future. In effect, the Eurodollar futures pay at the wrong time. A forward contract on a zero-coupon bond expiring at time t would pay its rate at time t + 9 days, not at time t when the rate is set. The rate is earned over the period between time t and time t + 9 days. Meanwhile, the futures contract settles at the same rate at time t, which may be quite a strain. The degree to which it is a strain is reflected in the presence of Z in the denominator of 1. A forward rate agreement FRA also pays when the rate is set, but a 9-day FRA properly discounts the cash flows, and it would settle for an amount 1 Z, rather than 1 Z/Z. This discounting is entirely natural, and the price of such a product depends only on the yield curve, not on any assumptions about the rate process. STATEMENT AND PROOF OF RESULTS The analyses below closely follow Dothan It is useful to begin by considering the conditional discounting function. We will work with the following two instantaneous short-rate processes: 4 5 dr αrdt + σrdb t, d log r κθ log rdt + σdb t, where κ and θ are non-negative. Equation 5 is the model of Black and Karasinsky Proposition 1: If the short-term rate process r follows 4 we have [ cdr, r t, t E exp /π t rs ds r r, rt r t µsinhπµ K iµρ K iµ ρ t ρ ρ t exp 1 + µ t /4 dµ [ σ / logrt r t /σ /r σ t/ t φ [ / cd logrt /r σ t/ r, r t, t r t φ, where ρ s δr s, δx 8x/σ, K iµ is a parabolic cylinder function or Bessel function of the third kind, and φx 1/ π exp x / is the normal density function. Remark: The function name cd is a mnemonic for conditional discounting. Proof: Conditional on the terminal points, the process is a log Brownian bridge process between r and r t, and in particular, its distribution does not depend on α. We therefore compute Dothan s base case of α. Let f a,b u 1 if a < u < b and otherwise. Let [ t F a,b r, s E exp rx dx f a,b rt t s rt s r.

3 According to the Feynman-Kac formula Durrett 1984 this expectation solves the differential equation σ r F rr rf F s, subject to F r, f a,b r. This is equivalent to equation 5 in Dothan 1978 with a small change in notation. As in Dothan 1978, let z δr and τ σ s/. Let dz zσ /8 be the inverse transformation to δ. Then hz, τ F a,b dz, s satisfies 6 z h zz + zh z z + 1h 4h τ subject to the terminal boundary condition hz, f δa,δb z. According to sec 5.14 of Lebedev 197 hz, /π µsinhπµ K iµz hζ, K iµζ dµ dζ. z ζ Then following Dothan, we have hz, τ /π µsinhπµ K iµz z exp 1 + µ τ/4 hζ, K iµζ ζ dµ dζ. To obtain the conditional expectation, we divide by the probability that rt a, b and take the limit as a b, yielding /π µ sinhπµ K iµzk iµ β zβ exp 1 + µ t/4 dµ [ σ / r t σ logb/r σ t/ / t φ. σ t Substituting the definition of z in terms of r completes the calculation. We now show that the instantaneously resettled futures price is. Let Zt, τ, rt be the price at time t of a zero-coupon bond that pays $1 at time τ. Z is a function of the short-term rate rt at time t. From 1 and we see that it suffices to show Proposition : E[1/Zt, τ, rt. Remark: Because there is no restriction on α, Proposition holds even in the case of linear mean reversion as in the Courtadon model Hull Proof: The proof will proceed in two parts. First assume that the short-term rate process r follows 4. Let 7 η α σ //σ where α is defined in 4. By unconditioning we have Zt, τ, r cdr, ρ, τ t φ logρ/r α σ /τ t 1 σ τ t ρ σ τ t dρ 3

4 cd r, ρ, τ t exp η logρ/r η τ t/ 1 ρ σ τ t dρ cd r, ρ, τ tρ/r η exp η τ t/ 1 ρ σ τ t dρ. According to 8.43 of Gradshteyn and Ryzhik So, it is easy show that 9 K iµ z cosµt exp z coshtdt. K iµ z exp z /πz and that equality holds asymptotically. Substituting this into the formula for cd it is therefore easy to show that 1 Zt, τ, r C exp δrr η for some constant C >. Let r m be such that for r > r m. Then [ E 1/Zt, τ, rt r r 1. 1 C σ t Zt, τ, r Cr η exp δr 1/Zt, τ, u φ r m exp 8u/σ u η φ logu/r αt + σ t/ du u logu/r αt + σ t/ du u Now assume that the short-term rate process r follows 5. Without loss of generality we can assume θ. Then Z satisfies the differential equation σ Z t Z rr r Z r κ log r + σ r rz. Let then G satisfies G s G [ κ 1 κ log rt Z G exp σ, κ log r r σ σ r + G rr rσ G r, with boundary value Gτ, τ, rτ exp κ log rτ σ. 4

5 By the Feynman-Kac formula Gt, τ, r E r exp exp τ t [ κ 1 κ log rτ σ κ log rs σ rs ds where r evolves according to process 4 with α σ /. This implies that τ Gt, τ, r e κt/ E r κ log rτ exp rs ds exp t σ e κt/ logρ/r + σ τ t 1 cdr, ρ, τ t φ σ τ t ρ σ τ t κ log ρ exp σ dρ. Using an argument similar to that for process 4 we get and Gt, τ, r Cr exp δr, Zt, τ, r Cr exp δr exp which as shown above is sufficient for our result.. κ log r σ, When there is no resettlement we have a forward contract. For such contracts, if the short-rate follows process 4 with α >, Proposition still holds. Equations 1 and 3 imply that it suffices to show Proposition 3 for forward contracts to have infinite value. Proposition 3: If the short-rate process r follows 4, then [ t E exp rs ds 1/Zt, τ, t if and only if α >. Proof: Let η be as in 7. Then [ E exp t rs ds 1/Zt, τ, t r r cd r, r, t/zt, τ, r r/r η exp η t/ dr. As in Proposition we have as r, for some constant C >. cd r, r, t C expδr 5

6 Thus, cd r, r, t/zt, τ, r r/r η exp η t/ dr is finite if and only if r m r η dr is, hence, if and only if η > 1. Working through the definition yields the proposition. 6

7 REFERENCES 1. Black, F., and P. Karasinski, Bond and Option Pricing when Short Rates are Lognormal, Financial Analysts Journal, July/August 1991, pp Cox, J. C., J. E. Ingersoll Jr., and S. A. Ross, The Relation Between Forward Prices and Futures Prices, Journal of Financial Economics , pp Dothan, L. U., On the Term Structure of Interest Rates, Journal of Financial Economics , pp Durrett, R., 1984, Brownian Motion and Martingales in Analysis, Wadsworth Advanced Books and Software, Belmont, CA. 5. Gradshteyn, I.S., and I.S. Ryzhik, 198, Table of Integrals, Series, and Products, Corrected and Enlarged Edition, Academic Press, New York, NY. 6. Hull, J., 1989, Options, Futures, and Other Derivative Securities, Prentice-Hall, Englewood Cliffs, NJ. 7. Lebedev, N. N., 197, Special Functions and Their Applications, Dover Publications, Inc., New York, NY. 7