# P2.T5.23.1. Risk-neutral interest rate term structure

#### Nicole Seaman

##### Director of CFA & FRM Operations
Staff member
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Learning objectives: Calculate the expected discounted value of a zero-coupon security using a binomial tree. Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios. Define risk-neutral pricing and apply it to option pricing. Distinguish between true and risk-neutral probabilities and apply this difference to interest rate drift.

Questions:

23.1.1. An option pays either $6.10 or nothing (aka, zero) in six months when it matures. The riskfree rate is 3.0% per annum with semiannual compounding. The option's expected discounted value (EDV) with real-world probabilities is$3.00, but the market's price is $1.80. What are the implied risk-neutral probabilities? a. 45.0% and 55.0% b. 37.5% and 62.5% c. 30.0% and 70.0% d. 22.5% and 77.5% 23.1.2. In Tuckman's (4th edition) illustration, the current six-month spot rate is 2.00%, and the one-year spot rate is 2.15% per annum with semi-annual compounding. The featured bond is a one-year zero-coupon bond with face value of$1,000.00. The binomial assumption is that six months from today, the six-month rate will jump 50 basis points to either 2.50% or 1.50%. The real-world probabilities are p(w) = 50.0% and 50.0%, while the risk-neutral probabilities are p(n) = 80.1% and 19.9%. The derivative is a call option on the zero-coupon bond that expires in six months and gives the right to purchase $1,000 face value of the (then six-month) zero at$990.00; i.e., the option's strike price is $990.00. Which of the following statements is TRUE? a. The arbitrage price of the option is$0.504
b. The replicating portfolio demonstrates that the expected return on all assets in the risk-neutral world is zero
c. The bond's market price of $980.30 is given by its expected discounted value (EDV) conditional on the assumption of real-world probabilities d. In the recombining multi-period binomial interest rate tree, risk-neutral (aka, arbitrage) pricing solves for the drift (aka, expected change) that is constant from each date to the next; i.e., the drift from date zero to date one equals the drift from date one to date two 23.1.3. The spot rate term structure is a current six-month rate of 4.00% and a one-year rate of 4.15% per annum with semi-annual compound frequency. In a binomial interest rate tree, the jump is 1.00%; in six months, the six-month rate jumps to either 3.00% or 5.00%. The real-world probabilities are 50% and 50%. The real-neutral probabilities are 65.2% and 34.8%. The bond is a zero-coupon bond that matures in one year. The derivative is a call option that expires in six months and has a strike price of$980.00.

Each of the following is true EXCEPT which is false?

a. The arbitrage price of the option is about $1.78 b. The cost of portfolio that replicates the call option's cash flows (i.e., zero or$5.22) is $2.56 c. The implied drift (aka, expected change) in the six-month interest rate is about 30.4 basis points d. The expected discounted value (EDV) of the bond using the risk-neutral probabilities returns the bond's market price of$959.76