# LO 11.2: Construct and apply an arbitrage argument to price a call option on a

LO 11.2: Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios.
The binomial interest rate model is used throughout this topic to illustrate the issues that must be considered when valuing bonds with embedded options. A binomial model is a model that assumes that interest rates can take only one of two possible values in the next period.
This interest rate model makes assumptions about interest rate volatility, along with a set of paths that interest rates may follow over time. This set of possible interest rate paths is referred to as an interest rate tree.
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Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7
Binomial Interest Rate Tree
The diagram in Figure 1 depicts a binomial interest rate tree.
Figure 1: 2-Period Binomial
Today
Period 1
Period 2
To understand this 2-period binomial tree, consider the nodes indicated with the boxes in Figure 1. A node is a point in time when interest rates can take one of two possible paths an upper path, U, or a lower path, L. Now consider the node on the right side of the diagram where the interest rate z2 rate, z’0, follows the lower path from node 0 to node 1 to become q L, then follows the upper of the two possible paths to node 2, where it takes on the value z’2 LU. At the risk of stating the obvious, the upper path from a given node leads to a higher rate than the lower path. Notice also that an upward move followed by a downward move gets us to the same place on the tree as a down-then-up move, so i2
appears. This is the rate that will occur if the initial
= i2 jjl *
The interest rates at each node in this interest rate tree are 1 -period forward rates corresponding to the nodal period. Beyond the root of the tree, there is more than one 1-period forward rate for each nodal period (i.e., at year 1, we have two 1-year forward rates, q ^ and q L). The relationship among the rates associated with each individual nodal period is a function of the interest rate volatility assumption of the model being employed to generate the tree.
Constructing the Binomial Interest Rate Tree
The construction of an interest rate tree, binomial or otherwise, is a tedious process. In practice, the interest rate tree is usually generated using specialized computer software. There is one underlying rule governing the construction of an interest rate tree: The values for on-the-run issues generated using an interest rate tree should prohibit arbitrage opportunities. This means that the value of an on-the-run issue produced by the interest rate tree must equal its market price. It should be noted that in accomplishing this, the interest rate tree must maintain the interest rate volatility assumption of the underlying model.
Valuing an Option-Free Bond W ith the Tree, Using Backward Induction
Backward induction refers to the process of valuing a bond using a binomial interest rate tree. The term backward is used because in order to determine the value of a bond at node 0, you need to know the values that the bond can take on at node 1. But to determine the values of the bond at node 1, you need to know the possible values of the bond at node 2,
2018 Kaplan, Inc.
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Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7
compounding periods, the current value of the bond and so on. Thus, for a bond that has is determined by computing the bonds possible values at period A^and working backward to node 0.
Consider the binomial tree shown in Figure 2 for a $100 face value, zero-coupon bond, with two years remaining until maturity, and a market price of$90,006. Starting on the top line, the blocks at each node include the value of the bond and the 1 -year forward rate at that node. For example, at the upper path of node 1, the price is $93,299, and the 1-year forward rate is 7.1826%. Figure 2: Valuing a 2-Year, Zero-Coupon, Option-Free Bond$90,006 4.5749%
$93,299 7.1826%$94,948 5.3210%
$100.00$ 100.00
$100.00$ 100.00
Today
End of 1 year
End of 2 years
Know that the value of a bond at a given node in a binomial tree is the average of the present values of the two possible values from the next period. The appropriate discount rate is the forward rate associated with the node under analysis.
Example: Valuing an option-free bond
Assuming the bonds market price is $90,006, demonstrate that the tree in Figure 2 is arbitrage free using backward induction. Answer: Consider the value of the bond at the upper node for period 1, Vi,u ($100×0.5)+ ($100×0.5) 1.071826$93,299
Similarly, the value of the bond at the lower node for period 1,
L is:
V
1+
($100×0.5) + ($100×0.5)
1.053210
$94,948 Page 134 2018 Kaplan, Inc. Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7 Now calculate Vq, the current value of the bond at node 0: = ($93.299×0.5) + ($94.948×0.5) = 1.045749 Since the computed value of the bond equals the market price, the binomial tree is arbitrage free. Professors Note: When valuing bonds with coupon payments, you need to add the coupons to the bond prices at each node. For example, with a$100 face value, 7% annual coupon bond, you would add the \$7 coupon to each price before computing present values. Valuing coupon-paying bonds with a binomial tree will be illustrated in LO 11.5.