P2.T5.304. Ho-Lee and arbitrage-free interest rate models

[Suzanne Evans]

AIMs: Describe the process of and construct a tree for a short-term rate under the Ho-Lee Model with time dependent drift. Describe uses and benefits of the arbitrage-free models and assess the issue of fitting models to market prices.

Questions:

304.1. The current short-term rate, r(0) is 4.00%. Under a Ho-Lee Model with time-dependent drift, the time step is monthly and the annualized drifts are as follows: +100 basis points in the first month and +80 basis points in the second month. The annual basis point volatility (sigma) is 200 bps.

What is the value of the missing node [2,0] in this Ho-Lee interest rate tree?

a. 2.447%
b. 2.677%
c. 2.995%
d. 3.256%

304.2. Analyst Susan wants to employ the Ho-Lee short-term interest rate model, which contains a time-dependent drift and is described by the following process:

(Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011))

Her model assumes the following:

• The time step is monthly; i.e., dt = 1/12
• the current rate, r(0) = 4.000%
• The annual basis point volatility = 300 basis points
• Annualized drift in the first month, lamba(1) = 110 basis points
• Annualized drift in the second month, lambda(2) = 70 basis points

Her simulation requires a random number each month. In the first month, the uniform random number is 0.30 such that (via inverse transformation) the random standard normal is -0.52. In the second month, the uniform random number is 0.80 such that the random standard normal is 0.84; i.e., NORM.S.INV(30%) ~= 0.520 and NORM.S.INV(80%) ~= 0.84. What short-term rate does this Ho-Lee model generate in the second month, r(2/12)?

a. 3.930%
b. 4.425%
c. 5.006%
d. 5.327%

304.3. Tuckman's short-term interest rate models classify into two categories: equilibrium models and arbitrage-free models. Consider the following statements about no-arbitrage interest rate models: (Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011))

I. Model 1 (Normally distributed rates and no drift) and Model 2 (Constant drift and risk premium) are classified as arbitrage-free models​

II. The Ho-Lee Model, due to time-dependent drift, maybe be used to match the observed prices of securities and therefore may be classified as a arbitrage-free model​

III. Important uses of arbitrage-free models include quoting non-actively traded securities (based on prices of liquid securities) and to value derivatives for purposes of market-making or proprietary trading​

IV. Because an arbitrage-free model matches market prices does not necessarily imply that it provides fair values and accurate hedges: the model itself could be bad; and/or market prices may not be "fair" in the context of the model.​

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Which of the above statements are TRUE according to Tuckman?

a. None
b. I. and II. only
c. II., III. and IV.
d. All are true

Answers:

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