Term Structure Lattice Models
1 Binomial-Lattice Models
In these lecture notes1 we introduce binomial-lattice models for modeling the “short-rate”, i.e. the one-period spot interest rate. We will also use these models to introduce various interest rate derivatives that are commonly traded in the ﬁnancial markets. First we deﬁne what an arbitrage means.
A type A arbitrage is an investment that produces immediate positive reward at t = 0 and has no future cost at t = 1. An example of a type A arbitrage would be somebody walking up to you on the street, giving you a positive amount of cash, and asking for nothing in return, either then or in the future. A type B arbitrage is an investment that has a non-positive cost at t = 0 but has a positive probability of yielding a positive payoﬀ at t = 1 and zero probability of producing a negative payoﬀ then. An example of a type B arbitrage would be a stock that costs nothing, but that will possibly generate dividend income in the future. In ﬁnance we always assume that arbitrage opportunities do not exist since if they did, market forces would quickly act to dispel them.
Constructing an Arbitrage-Free Lattice
Consider the binomial lattice below where we specify the short rate, ri,j , that will apply for the single period beginning at node N (i, j). This means for example that if $1 is deposited in the cash account at t = i, state j, (i.e. node N (i, j), then this deposit will be worth $(1 + ri,j ) at time t + 1 regardless of the successor node to N (i, j). ¨ ¨¨ ¨
r3,3 ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ r2,2 ¨¨ r3,2 ¨¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ r1,1 ¨¨ r2,1 ¨¨ r3,1 ¨¨ ¨ ¨¨ ¨¨ ¨¨ ¨ ¨¨ ¨¨ ¨¨ ¨¨ r0,0 ¨¨ r1,0 ¨¨ r2,0 ¨¨ r3,0 ¨¨ ¨ ¨ ¨ ¨ 1 Many
of our examples are drawn from Investment Science (1998) by David G. Luenberger.
Term Structure Lattice Models We use risk-neutral pricing on this lattice to compute security prices. For example, if Si (j) is the value of a non-dividend / coupon2 paying security at time i and state j, then we insist that Si (j) = 1 [qu Si+1 (j + 1) + qd Si+1 (j)] 1 + ri,j
where qu and qd are the probabilities of up- and down-moves, respectively. If the security pays a coupon then it should be included in the right-hand side of (1). Such a model is arbitrage-free by construction. Exercise 1 Can you see why (1) implies that the lattice is arbitrage-free?
Computing the Term-Structure from the Lattice
It is easy to compute the price of a zero-coupon bond once the risk-neutral probability distribution, Q say, and the short-rate lattice are speciﬁed. In the short rate-lattice below (where the short rate increases by a factor of u = 1.25 or decreases by a factor of d = .9 in each period), we assume that the Q-probability of each branch is .5 and node-independent. We can then use risk-neutral-pricing to compute the prices of zero-coupon bonds.
Short Rate Lattice 0.117 0.084 0.061 0.044 t=3 0.146 0.105 0.076 0.055 0.039 t=4
0.075 0.054 t=1
0.094 0.068 0.049 t=2
0.183 0.132 0.095 0.068 0.049 0.035 t=5
Example 1 (Pricing a Zero-Coupon Bond) We compute the price of a 4-period zero-coupon bond with face value 100 that expires at t = 4. Assuming the short-rate lattice is as given above, we see, for example, that the bond price at node (2, 2) is given by 83.08 = 1 1 1 89.51 + 92.22 . 1 + .094 2 2
Iterating backwards, we ﬁnd that the zero-coupon bond is worth 77.22 at t = 0. 4-Year Zero 83.08 87.35 90.64 t=2 89.51 92.22 94.27 95.81 t=3 100.00 100.00 100.00 100.00 100.00 t=4
79.27 84.43 t=1
these notes focus on ﬁxed-income securities, we will henceforth refer to any intermediate cash-ﬂow as a coupon.
Term Structure Lattice Models
Note that given the price of the 4-period zero-coupon bond, we can now ﬁnd the 4-period spot rate, s4 . It satisﬁes 77.22 = 1/(1 + s4 )4 if...