Backward Induction for Callable Bonds on an Interest Rate Tree
Summary
The document explains how an issuer’s call right is incorporated when valuing a callable bond on a binomial interest-rate tree. Valuation proceeds backward from later dates: at each node, the model discounts the possible next-period bond values using the tree’s interest rates and probabilities. At a date when exercise is permitted, compare the continuation value with the call price. If the continuation value exceeds the amount paid on exercise, cap the bond’s value at the call price at that node.
This node-by-node replacement resolves the concern that a call at a later date might imply the bond is called repeatedly. Once exercise is optimal at a node, later values along that branch no longer determine the value carried backward from it; the capped value feeds into its predecessor. The responses illustrate this logic with a small tree and callable dates, but do not provide a full derivation or discuss variations in call terms, accrued interest, or exercise conventions. The method depends on the specified rate tree and its assumptions.
Key ideas
- Value the bond by backward induction through the interest-rate tree.
- At an eligible call date, compare the continuation value with the contractual call price.
- When exercise is optimal, replace the node’s continuation value with the call price.
- The capped node value is passed backward, so later branch values beyond exercise do not affect that path’s earlier value.
- A bond is not called repeatedly along one path; the tree represents exercise decisions at separate possible nodes.
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# Question about using binomial interest rate tree to value callable bonds # Question about using binomial interest rate tree to value callable bonds When we use a binomial interest rate tree to value callable bond, we work backward, right? If any computed bond value is larger than the call price, the bond will be called. The call price will replace the computed bond value and we go on to calculate the bond value in previous year. My question is: we cannot call the bond twice. If the bond is call at year 2 on some node, we use the call price to calculate the bond price at year 1 and the results shows that the bond will be called in year 1 too. So confused! ## Answer by Lennart_R (score 3) https://quant.stackexchange.com/a/37133 we basically make assumptions about the possible values of future interest rates, i.e. we jump forward in time and then gradually work our way back to the presence by checking if the bond surpasses a value of 100 in any time where it may be called by the issuer. The bond value in any given node of the tree will then depend only on the two previous values (from right to left side) that "lead" into the node, as from the given node the interest rate can only change according to the parameters that we assumed earlier (e.g. go up 30 % with a probability of 50 % or go down 4 % with a probability of 50 % - see the example below). Obviously, the bond cannot be called twice, so in a node where the bond may be called but its value is > 100, we have to set the value to 100. If the bond is callable in a node that is closer to t=0 and the bond value in that node is >100 than we will set that node to 100. The bond value now does NOT depend on the values that follow from that node in later periods as the bond would get called anyways and therefore has a value (to the buyer) of 100. Look at the following (quick Excel) example: The bond is callable in t=1 and t=1.5; the value of the callable bond does however not depend on the value of the circled node on the right, but on the node to the left of it as the bond value does exceed 100 (as calculated in the circled node on the left) as well - remember, that the bond is callable in t=1 as well). If you want to try and recreate this example: - up-factor: 1.3 - down-factor: 0.96 - Interest in t=0: 0.07 - bond is callable in t=1, t=1.5 ## Answer by R. Steigmeier (score 0) https://quant.stackexchange.com/a/37130 Well, the final price at t0 is only dependent on the two nodes on date t1. Those nodes (t1) are dependent on the next (t2) and so on. So if the bond is going to be called at one of the three nodes in t2 (let's Call this node A), the price is set to 100 (instead of let's say 102) and this will partially (50%) affect one of the nodes in t1. Because of this reset, all the nodes coming after node A in t2 (A t3u and At3d) are no longer used for this path.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.