Binomial interest rate tree to calculate market price

[monsieuruzairo3]

Helllo @David Harper CFA FRM CIPM

I quote your question from the reading P2.T5.40. Replicating callable bond

40.1 Assume the market six-month and one-year spot rates are 2.0% and 2.2%,
respectively. Assume, per Tuckman's two-step binomial interest rate tree (i.e., each step is
six months), that the six months from now the six-month rate will be either 2.5% (+0.5%) or
1.5% (-0.5%) with equal probability. If a bond's face value is $1,000, what is the market
price of the bond (note: Tuckman assumes semi-annual compounding)?
a) $968.45
b) $964.63
c) $978.36
d) $982.12

My approach (immediately after going through the reading and attempting this as first question)

Calculate PV at upper node & lower node
UN: FV =1000, RATE = 2.5%/2, N=1 ----> PV =987.65432
LN: FV=1000, RATE =1.5%/2 , N=1 -----> PV=992.55583

Now using "True probabilities" and discounting them with 2%
(0.5*987.65 + 0.5*992.555)/[1 + 0.02/2] ~ 980.3

But the answer is using 1 year discount rate 1000/ [1 + 0.022/2]^2

On reflection, I feel that this mismatch between my answer and BT answer is that I have used True probabilities. Maybe I need to use Risk-neutral probabilities to arrive at 978.36.

Am I right? And what more information do i need to calculate Risk neutral probabilities

Many thanks
Uzi

 

[David Harper CFA FRM]

Hi @monsieuruzairo3 I just noticed this question does not specify zero-coupon bond, fwiw. This is a query of Tuckman and your calculation is valid, but your value is the answer to the question "What is the expected discounted value of the bond?" (answer = $980.30) rather than, as asked, "What is the market price of the bond?" (answer = $978.36). Emphasis mine. On a superficial level, $978.36 is the answer we should want to see (right?) because we are accustomed to retrieving the price of a one year zero as 1000/(1+2.2%/2)^2 = $978.36; or for that matter, it's not necessary but out of habit: N = 2, I/Y = 2.20/2 = 1.1, PMT = 0, FV = 1000 and CPT PV = 978.36

Tuckman's emphasis is the distinction between expected discounted value and market price hinges on risk-aversion. If you have any risk aversion, the expected discounted value is more than you will pay for the bond (risk is here the dispersion or "volatility" around the expected value). In the same way that I will offer to pay you under two scenarios in one year:

• Either I will pay you $100, or
• I will pay you $70 with 50% probability or $130 with 50% probability (expected future value = $100).

Say risk free rate is 3%. You should be willing to pay $100*exp(-3%*1) = $97.05 today for the first riskless scenario
But it's an open question what you would pay for the second scenario

• If you are risk-seeking, you'd pay more than $97.05!
• But Tuckman's (classic) argument is that you are risk-averse and you are only willing to pay something less than 97.05 due to the dispersion around the mean. Hence the market price. To me, the confusion can arises due to the label of "risk neutral probabilities" ... those are the probability that equate the exp discounted value to the market price, but the market price presumes risk aversion. I hope that explains