Lattice Term Structure Model
IEOR E4706: Financial Engineering: Discrete-Time Models c 2010 by Martin Haugh
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.
Arbitrage
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.
t=0
t=1
t=2
t=3
t=4
Term Structure Lattice Models We use risk-neutral pricing on this lattice to
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.
Arbitrage
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.
t=0
t=1
t=2
t=3
t=4
Term Structure Lattice Models We use risk-neutral pricing on this lattice to
References: Luenberger, D.G. 1998. Investment Science, Oxford University Press, New York.