Options, Futures, and Other Derivatives

John C. Hull

Chapter 31 Equilibrium models of the short rate - all with Video Answers

Educators

Chapter Questions

Problem 1

Suppose that the short rate is currently $4 \%$ and its standard deviation in a short period of time $\Delta t$ is $0.01 \sqrt{\Delta t}$. What happens to this standard deviation when the short rate increases to $8 \%$ in (a) Vasicek's model, (b) Rendleman and Bartter's model, and (c) the Cox, Ingersoll, and Ross model?

Gus Steppen

Numerade Educator

01:24

Problem 2

If a stock price were mean reverting or followed a path-dependent process there would be market inefficiency. Why is there not a market inefficiency when the short-term interest rate does so?

Rashmi Sinha

Numerade Educator

00:55

Problem 3

Explain the difference between a one-factor and a two-factor model.

Matt Gibson

Numerade Educator

Problem 4

Suppose that in a risk-neutral world the Vasicek parameters are $a=0.1, b=0.03$, and $\sigma=0.01$. What is the price of a 5 -year zero-coupon bond with a principal of $$\$ 1$$ when the short rate is $2 \%$.

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Problem 5

Suppose that in a risk-neutral world the CIR parameters are $a=0.1, b=0.03$, and $\sigma=0.07$. The market price of interest rate risk is -1 times the square root of the short rate. What are the risk-neutral and real-world processes for (a) the short rate and (b) a zerocoupon bond with a current maturity of 4 years.

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Problem 6

Suppose that the market price of risk of the short rate is $\lambda_1+\lambda_2 r$ (with $\lambda_1$ and $\lambda_2$ negative). Show that if the real-world process for the short rate is the one assumed by Vasicek, the risk-neutral process has the same functional form as the real-world process. Derive the relationship between (a) the real-world reversion rate and the risk-neutral reversion rate and (b) the real-world reversion level and the risk-neutral reversion level.

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Problem 7

Observations spaced at intervals $\Delta t$ are taken on the short rate. The $i$ th observation is $r_i$ $(0 \leqslant i \leqslant m)$. Show that the maximum-likelihood estimates of $a, b^*$, and $\sigma$ in Vasicek's model are given by maximizing
$$
\sum_{i=1}^m\left(-\ln \left(\sigma^2 \Delta t\right)-\frac{\left[r_i-r_{i-1}-a\left(b^*-r_{i-1}\right) \Delta t\right]^2}{\sigma^2 \Delta t}\right)
$$

Victor Salazar

Numerade Educator

02:40

Problem 8

Calculate the alternative duration measure explained in Section 31.2 for a 2-year bond with a principal of $$\$ 100$$ paying coupons semiannually at the rate of $$\$ 3$$ per year when Vasicek's model is used with $a=0.13, b=0.012, \sigma=0.01$, and $r=1 \%$. Show that it correctly predicts the effect of an increase in $r$ to $1.05 \%$.

Anand Jangid

Numerade Educator

Problem 9

Suppose that $a=0.1$ and $b=0.1$ and in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is $10 \%$ and the initial standard deviation of the short-rate change in a short time $\Delta t$ is $0.02 \sqrt{\Delta t}$. Compare the prices given by the models for a zero-coupon bond that matures in year 10 .

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Problem 10

Suppose the short rate $r$ is $4 \%$ and its real-world process is $d r=0.1(0.05-r) d t+0.01 d z$, while the risk-neutral process is $d r=0.1(0.11-r) d t+0.01 d z$ :
(a) What is the market price of interest rate risk?
(b) What is the expected return and volatility for a 5 -year zero-coupon bond in the riskneutral world?
(c) What is the expected return and volatility for a 5 -year zero-coupon bond in the real world?

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04:33

Problem 11

(a) What is the second partial derivative of $P(t, T)$ with respect to $r$ in the Vasicek and CIR models?
(b) In Section $31.2, \hat{D}$ is presented as an alternative to the usual duration measure, $D$. What is a similar alternative, $\hat{C}$, to the convexity measure in Section 4.11 ?
(c) What is $\hat{C}$ for $P(t, T)$ ? How would you calculate $\hat{C}$ for a coupon-bearing bond?
(d) Give a Taylor series expansion for $\Delta P(t, T)$ in terms of $\Delta r$ annd $(\Delta r)^2$ for Vasicek and CIR.

Breanna Ollech

Numerade Educator

Problem 12

Suppose that in the risk-neutral Vasicek process $a=0.15, b=0.025$, and $\sigma=0.012$. The market price of interest rate risk is -0.2 . What are the risk-neutral and real-world processes for (a) the short rate and (b) a zero-coupon bond with a current maturity of 3 years.

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Problem 13

Suppose that in a risk-neutral world the CIR parameters are $a=0.15, b=0.025$, and $\sigma=0.075$. What is the price of a 5 -year zero-coupon bond with a principal of $$\$ 1$$ when the short rate is $2.5 \%$ ?

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Problem 14

Suppose that the market price of risk of the short rate is $\lambda_1 / \sqrt{r}+\lambda_2 \sqrt{r}$. Show that if the real world process for the short rate is the one assumed by $\mathrm{CIR}$, the risk-neutral process has the same functional form. Derive the relationship between (a) the real-world reversion rate and the risk-neutral reversion rate and (b) the real-world reversion level and the risk-neutral reversion level.

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08:31

Problem 15

In the two-factor extension of Vasicek given in Section 31.5, derive the differential equations which must be satified by a bond price, $P(t, T)$. Use this to derive differential equations that must be satisfied by $A(t, T), B(t, T)$, and $C(t, T)$ in $P(t, T)=$ $A(t, T) e^{-B(t, T) r-C(t, T) u}$. Show that the expressions given for $B(t, T)$ in equation (31.7) and $C(t, T)$ in equation (31.14) satisfy these equations. [Hint: Use equation (14A.10) to obtain the drift of $P(t, T)$ and set this drift equal to $r P(t, T)$.]

Stanley Enemuo

Numerade Educator

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Problem 16

Use the result in Problem 31.7 to determine the best fit parameters for the Vasicek model using the same data as in Section 31.4 (see www-2.rotman.utoronto.ca/ hull/ VasicekCIR). Verify that the regression approach in Section 31.4 and the maximumlikelihood approach give the same answer.

Rashmi Sinha

Numerade Educator

Problem 17

What is the result corresponding to that given in Problem 31.7. for the CIR model. Use maximum likelhood methods to estimate the $a, b$, and $\sigma$ parameters for the CIR model using the same data as that used for the Vasicek model in Section 31.4 (see www-2.rotman.utoronto.ca/ hull/VasicekCIR). Setting the market price of risk equal to $\kappa \sqrt{r}$ use the market data in Table 31.1 to estimate the best fit $\kappa$.

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