# L2.T5.43 Multi-period binomial interest rate tree (Tuckman)

#### kausthub

##### New Member
@David Harper CFA FRM
Hi,
Firstly I am posting this here because whenever I click on the links to threads in the study notes, I get the message "Oops, you do not have permission to access this thread. ". I am unsure what to do about that.

Anyways, my question is in Page 25 of the study notes Tuckman chapter-7 in market risk management. You mentioned the below -
"We already inferred the risk-neutral probabilities in the move from date 0 to date 1: the risk-neutral probability of an up move on date 0 is 80.24%; the risk-neutral probability of a down move on date 0 is 19.76%."
However this risk neutral probability is calculated for the previous example where the market value at time 0 is 950.423. In this example since there are three 6 month periods, as opposed to 2 in the previous example, we calculate the market value at time 0 to be 925.21. How are we using the same risk neutral probabilities as before since we calculate those probabilities by equating the market value to the value obtained from the time 1 values.

My second query is on how to intuitively think about a call option on a bond being replicated by a long position in 1 year bonds, and short position in 6 month bond. How are the cash flows same?

Thanks, and sorry for posting it here. (Kindly help me with these issues)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
@David Harper CFA FRM
Hi,
Firstly I am posting this here because whenever I click on the links to threads in the study notes, I get the message "Oops, you do not have permission to access this thread. ". I am unsure what to do about that.

Anyways, my question is in Page 25 of the study notes Tuckman chapter-7 in market risk management. You mentioned the below -
"We already inferred the risk-neutral probabilities in the move from date 0 to date 1: the risk-neutral probability of an up move on date 0 is 80.24%; the risk-neutral probability of a down move on date 0 is 19.76%."
However this risk neutral probability is calculated for the previous example where the market value at time 0 is 950.423. In this example since there are three 6 month periods, as opposed to 2 in the previous example, we calculate the market value at time 0 to be 925.21. How are we using the same risk neutral probabilities as before since we calculate those probabilities by equating the market value to the value obtained from the time 1 values.

My second query is on how to intuitively think about a call option on a bond being replicated by a long position in 1 year bonds, and short position in 6 month bond. How are the cash flows same?

Thanks, and sorry for posting it here. (Kindly help me with these issues)
Hello @kausthub

Regarding the issue that you were having with the forum threads, I have fixed your forum permissions so you should no longer have this issue.

Thank you,

Nicole

#### kausthub

##### New Member
Hello @kausthub

Regarding the issue that you were having with the forum threads, I have fixed your forum permissions so you should no longer have this issue.

Thank you,

Nicole
Thanks Nicole!

#### kausthub

##### New Member
@David Harper CFA FRM
Sorry if I posted my question in the wrong thread.
my question is in Page 25 of the study notes Tuckman chapter-7 in market risk management. You mentioned the below -
"We already inferred the risk-neutral probabilities in the move from date 0 to date 1: the risk-neutral probability of an up move on date 0 is 80.24%; the risk-neutral probability of a down move on date 0 is 19.76%."
However this risk neutral probability is calculated for the previous example where the market value at time 0 is 950.423. In this example since there are three 6 month periods, as opposed to 2 in the previous example, we calculate the market value at time 0 to be 925.21. How are we using the same risk neutral probabilities as before since we calculate those probabilities by equating the market value to the value obtained from the time 1 values.

My second query is on how to intuitively think about a call option on a bond being replicated by a long position in 1 year bonds, and short position in 6 month bond. How are the cash flows same?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @kausthub

When Tuckman (Chapter 7) extends the risk-neutral probability example from 1.0 year to 1.5 years (three six month periods, as you say), he only adds to the scenario: the 1.0 year assumptions that informed the risk-neutral (RN) p = 80.24% remain. That is to say, the assumption in the multi-period example is a 1.5 years spot rate term structure of {six-month spot rate = 5.00%, 1.0 year spot rate = 5.15%, and 1.5 year spot rate of 5.25%}. The 1-year spot rate of 5.15% informs the observed (traded) 1.0 year bond price of $1,000/(1+5.15%/2)^2 =$950.42; and this 950.42 is effectively in input into solving for the RN probabilities; i.e.,. the p = 80.1% is solved-for in order to generate a expected discount value that equals the 950.42. When Tuckman extends to 1.5 years, he adds the spot rate, S(1.0) = 5.25% but does not change the prior 1.0 year spot rate assumption. Just as there is a 1.5 year term structure of spot rates, there is a trading 1.0 year bond price ($950.42) and a trading 1.5 year bond price ($925.21) which effectively becomes an input so that the RN probabilities can be solved. The RN probabilities match the expected discounted value (EDV) to the traded price, so effectively the traded prices (here given as a function of the spot rate) become "inputs" in order to solve for the probabilities.

Re the options, I don't have an easy intuition. It's a simple one-step binomial with only two possible future states (i.e., rate goes up to 5.5% or down to 4.5%) so the bond price in six months is either $973.24 or$978.00; given a strike price of $975.00, this option under this extremely simple tree either pays$3.00 or nothing. The short six-month bond is simply a promise to pay the 612.50 in six months, so it gets subtracted from the positive value of the one-year bond (which is six months will be a six month bond). Under the replication, it is either the case that the positive value of the long one year bond (i.e., six months remaining) either equals the short repayment (612.50), or its value is $3.00 greater than the repayment. So, I guess I just look at it as: in six months, we'll have PV(long bond with six months to mature) - repayment due on short bond position = either zero or$3.00, in order to replicate. I hope that's helpful,

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
moved duplicate question to where I answered

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @frogs This illustrates Tuckman's example, which he steps through in detail (I do not have time to re-draft his explainer). Briefly:
• Under this simple binomial, the call option either pays zero or $3.00. The replication goal is to find the bond portfolio with the exactly same payoff profile {i.e., zero or$3.00). That solution is the the portfolio of two bonds: long face value of $629.34 of the 1.0 year bond (his number is 630.2521, but mine is an exact solution in the XLS) plus short face value of$612.50 of the 0.5 year bond. In the forward 0.5 year nodes, the 0.5 year bond matures such that the 612.50 face = 612.50. In regard to the long bond (which is then only 0.5 remining maturity): $973.24 * 629.34 face / 1000 =$612.50; i.e., is the value at that point. Same in the lower node.
• The replicating portfolio has a net cost of $0.58 because 629.34 is purchased (the long) offset by$612.50 received (the short). Please note the face values can get confusing: 629.34*950.42/1000 - 612.50*975.61/1000 = 598.14 - 597.56 = $0.58; i.e., to purchase today 629.34 face value of the one-year bond (that has a price of$950.42) costs \$598.14. Hope that's helpful,

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