# LO 11.5: Explain how the principles o f arbitrage pricing o f derivatives on fixed

LO 11.5: Explain how the principles o f arbitrage pricing o f derivatives on fixed income securities can be extended over multiple periods.
There are three basic steps to valuing an option on a fixed-income instrument using a binomial tree:
Step 1: Price the bond value at each node using the projected interest rates. Step 2: Calculate the intrinsic value of the derivative at each node at maturity. Step 3: Calculate the expected discounted value of the derivative at each node using the
risk-neutral probabilities and working backward through the tree.
Note that the option cannot be properly priced using expected discounted values because the call option value depends on the path of interest rates over the life of the option. Incorporating the various interest rate paths will prohibit arbitrage from occurring.
Example: Call option
Assume that you want to value a European call option with two years to expiration and a strike price of \$100.00. The underlying is a 7%, annual-coupon bond with three years to maturity. Figure 3 represents the first two years of the binomial tree for valuing the underlying bond. Assume that the risk-neutral probability of an up move is 0.76 in year 1 and 0.60 in year 2.
Fill in the missing data in the binomial tree, and calculate the value of the European call option.
Professor’s Note: Since the option is European, it can only he exercised at maturity.
Figure 3: Incomplete Binomial Tree for European Call Option on 3-Year, 7% Bond
Int. rate = 3.00% Bond price = ? Option value = ?
0.76
Int. rate = 5.99% Bond price = ? Option value = ?
0.24
Int. rate = 4.44% Bond price = ? Option value = ?
Int. rate = 8.56% Bond price = \$98.56 Option value = ?
0.60
Int. rate = 6.34% Bond price = ? Option value = ?
0.40 0.60
Int. rate = 4.70% Bond price = ? Option value = ?
0.40
Today
End of 1 year
End of 2 years
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2018 Kaplan, Inc.
Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7
Step 1: Calculate the bond prices at each node using the backward induction methodology.
At the middle node in year 2, the price is \$100.62. You can calculate this by noting that at the end of year 2 the bond has one year left to maturity:
N = 1; I / Y = 6.34; PMT = 7; FV = 100; CPT – PV = 100.62
At the bottom node in year 2, the price is \$102.20:
N = 1; I / Y = 4.70; PMT = 7; FV = 100; CPT > PV = 102.20
At the top node in year 1, the price is \$100.37:
(\$103.36 x 0.6) + (\$107.62 x 0.4)
1.0599
= \$100.37
At the bottom node in year 1, the price is \$103.65:
(\$107.62 x 0.6) + (\$109.20 x 0.4)
1.0444
= \$103.65
Today, the price is \$105.01:
(\$107.37 x 0.76) + (\$110.65 x 0.24)
1.03
\$105.01
As shown here, the price at a given node is the expected discounted value of the cash flows associated with the two nodes that feed into that node. The discount rate that is applied is the prevailing interest rate at the given node. Note that since this is a European option, you really only need the bond prices at the maturity date of the option (end of year 2) if you are given the arbitrage-free interest rate tree. However, its good practice to compute all the bond prices.
Step 2: Determine the intrinsic value of the option at maturity in each node. For example, the intrinsic value of the option at the bottom node at the end of year 2 is \$2.20 = \$102.20 – \$100.00. At the top node in year 2, the intrinsic value of the option is zero since the bond price is less than the call price.
Step 3: Using the backward induction methodology, calculate the option value at each
node prior to expiration. For example, at the top node for year 1, the option price is \$0.23:
(\$0.00×0.6)+ (\$0.62×0.4) ——————————– = \$0.23
1.0599
2018 Kaplan, Inc.
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Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7
Figure 4 shows the binomial tree with all values included.
Figure 4: Completed Binomial Tree for European Call Option on 3-Year, 7% Bond
Int. rate = 3.00% Bond price = \$105.01 Option value = \$0.45
0.76
Int. rate = 5.99% Bond price = \$100.37 Option value = \$0.23
0.24
Int. rate = 4.44% Bond price = \$103.65 Option value = \$1.20
Int. rate = 8.56% Bond price = \$98.56 Option value = \$0.00
0.60
Int. rate = 6.34% Bond price = \$100.62 Option value = \$0.62
Int. rate = 4.70% Bond price = \$102.20 Option value = \$2.20
0.40
Today
End of 1 year
End of 2 years
The option value today is computed as:
(\$0.23 x 0.76) + (\$ 1.20 x 0.24)
1.03
= \$0.45
Recombining and Nonrecombining