GENERAL TREE MODELS Sample Clauses

GENERAL TREE MODELS. In this lesson, we will use binomial trees to model changing short-term interest rates. We will also use these interest rate models to price bonds. We begin with a discussion of notation. We will use the following notation when referring to the prices of zero-coupon bonds. • Let P (t ,T ) denote the price determined at time t, and paid at time t, of a bond maturing for $1 at time T. • Let P0(t , T ) denote the price determined at time 0, but paid at time t, of a bond maturing for $1 at time T. We can make the following observations about these bond prices. • Notice that P0(t , T ) is the forward price of P (t ,T ) . As such, we also denote P0(t , T ) by F 0,T (t , T ) . • Interest theory concepts tell us that P (t , T ) = P ( 0,T ) . 0 P (0,t ) • If risk free rate is constant, then P (0,T ) = e−rT . Even if the risk free rate varies, we can use P (0,T ) as a present value factor. The present value of a payment of K occurring at time T would be P (0,T )⋅ K . • These bond prices are related to spot rates as follows: P (0,T ) = 1 . (1 + sT ) Binomial trees can be used to model changes in short term interest rates over time. The details of how tree rate models work are provided in the following comments. • Each node in the tree will represent the interest rate during a period of length h. Typically, h will be 1 year. It is important to remember that in interest rate models nodes represent an entire period, not a particular moment in time. • The process used for pricing bonds using tree rate models will be path-dependent. As such, our binomial trees must not be allowed to recombine. • Some method will be provided to determine the magnitude of an up-move or down-move in the rates when moving from one period to the next. Risk-neutral probabilities of an up-move or a down-move will also be provided. • Unless otherwise specified, rates will be continuously compounded. There are exceptions to this rule, however. For example, the Black-Derman-Toy model that we will study in the next section makes use of annual effective rates rather than continuously compounded rates.
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