Chapter 10, Term-Structure Models Video Solutions, Stochastic Calculus for Finance II : Continuous-Time Models

Problem 1

Exercise 10.1 (Statistics in the two-factor Vasicek model). According to Example 4.7.3, $Y_1(t)$ and $Y_2(t)$ in $(10.2 .43)-(10.2 .46)$ are Gaussian processes.
(i) Show that
$$
\widetilde{\mathbb{E}} Y_1(t)=e^{-\lambda_1 t} Y_1(0),
$$
that when $\lambda_1 \neq \lambda_2$, then
$$
\tilde{\mathbb{E}} Y_2(t)=\frac{\lambda_{21}}{\lambda_1-\lambda_2}\left(e^{-\lambda_1 t}-e^{-\lambda_2 t}\right) Y_1(0)+e^{-\lambda_2 t} Y_2(0),
$$
and when $\lambda_1=\lambda_2$, then
$$
\widetilde{\mathbb{E}} Y_2(t)=-\lambda_{21} t e^{-\lambda_1 t} Y_1(0)+e^{-\lambda_1 t} Y_2(0) .
$$

We can write
$$
Y_1(t)-\widetilde{\mathbb{E}} Y_1(t)=e^{-\lambda_1 t} I_1(t),
$$
when $\lambda_1 \neq \lambda_2$,
$$
Y_2(t)-\mathbb{E} Y_2(t)=\frac{\lambda_{21}}{\lambda_1-\lambda_2}\left(e^{-\lambda_1 t} I_1(t)-e^{-\lambda_2 t} I_2(t)\right)-e^{-\lambda_2 t} I_3(t),
$$
and when $\lambda_1=\lambda_2$,
$$
Y_2(t)-\tilde{\mathbb{E}} Y_2(t)=-\lambda_{21} t e^{-\lambda_1 t} I_1(t)+\lambda_{21} e^{-\lambda_1 t} I_4(t)+e^{-\lambda_1 t} I_3(t),
$$
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10 Term-Structure Models
where the Itô integrals
$$
\begin{aligned}
& I_1(t)=\int_0^t e^{\lambda_1 u} d \widetilde{W}_1(u), I_2(t)=\int_0^t e^{\lambda_2 u} d \widetilde{W}_1(u), \\
& I_3(t)=\int_0^t e^{\lambda_2 u} d \widetilde{W}_2(u), I_4(t)=\int_0^t u e^{\lambda_1 u} d \widetilde{W}_1(u),
\end{aligned}
$$
all have expectation zero under the risk-neutral measure $\widetilde{\mathbb{P}}$. Consequently, we can determine the variances of $Y_1(t)$ and $Y_2(t)$ and the covariance of $Y_1(t)$ and $Y_2(t)$ under the risk-neutral measure from the variances and covariances of $I_j(t)$ and $I_k(t)$. For example, if $\lambda_1=\lambda_2$, then

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07:49

Problem 2

(Ordinary differential equations for the mixed affineyield model). In the mixed model of Subsection 10.2.3, as in the two-factor Vasicek model and the two-factor Cox-Ingersoll-Ross model, zero-coupon bond prices have the affine-yield form
$$
f\left(t, y_1, y_2\right)=e^{-y_1 C_1(T-t)-y_2 C_2(T-t)-A(T-t)},
$$
where $C_1(0)=C_2(0)=A(0)=0$.
(i) Find the partial differential equation satisfied by $f\left(t, y_1, y_2\right)$.
(ii) Show that $C_1, C_2$, and $A$ satisfy the system of ordinary differential equations
$$
\begin{aligned}
C_1^{\prime} & =-\lambda_1 C_1-\frac{1}{2} C_1^2-\sigma_{21} C_1 C_2-(1+\beta) C_2^2+\delta_1, \\
C_2^{\prime} & =-\lambda_2 C_2+\delta_2, \\
A^{\prime} & =\mu C_1-\frac{1}{2} \alpha C_2^2+\delta_0 .
\end{aligned}
$$

Vipender Yadav

Numerade Educator

Problem 3

(Calibration of the two-factor Vasicek model). Consider the canonical two-factor Vasicek model (10.2.4), (10.2.5), but replace the interest rate equation $(10.2 .6)$ by
$$
R(t)=\delta_0(t)+\delta_1 Y_1(t)+\delta_2 Y_2(t)
$$
where $\delta_1$ and $\delta_2$ are constant but $\delta_0(t)$ is a nonrandom function of time. Assume that for each $T$ there is a zero-coupon bond maturing at time $T$. The price of this bond at time $t \in[0, T]$ is
$$
B(t, T)=\tilde{\mathbb{E}}\left[e^{-\int_t^T R(u) d u} \mid \mathcal{F}(t)\right] .
$$

Because the pair of processes $\left(Y_1(t), Y_2(t)\right)$ is Markov, there must exist some function $f\left(t, T, y_1, y_2\right)$ such that $B(t, T)=f\left(t, T, Y_1(t), Y_2(t)\right)$. (We indicate the dependence of $f$ on the maturity $T$ because, unlike in Subsection 10.2.1, here we shall consider more than one value of $T$.)
(i) The function $f\left(t, T, y_1, y_2\right)$ is of the affine-yield form
$$
f\left(t, T, y_1, y_2\right)=e^{-y_1 C_1(t, T)-y_2 C_2(t, T)-A(t, T)} .
$$

Holding $T$ fixed, derive a system of ordinary differential equations for $\frac{d}{d t} C_1(t, T), \frac{d}{d t} C_2(t, T)$, and $\frac{d}{d t} A(t, T)$.

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Problem 4

4. Hull and White [89] propose the two-factor model
$$
\begin{aligned}
& d U(t)=-\lambda_1 U(t) d t+\sigma_1 d \widetilde{B}_2(t), \\
& d R(t)=\left[\theta(t)+U(t)-\lambda_2 R(t)\right] d t+\sigma_2 d \widetilde{B}_1(t),
\end{aligned}
$$
where $\lambda_1, \lambda_2, \sigma_1$, and $\sigma_2$ are positive constants, $\theta(t)$ is a nonrandom function, and $\widetilde{B}_1(t)$ and $\widetilde{B}_2(t)$ are correlated Brownian motions with $d \widetilde{B}_1(t) d \widetilde{B}_2(t)=$ $\rho d t$ for some $\rho \in(-1,1)$. In this exercise, we discuss how to reduce this to the two-factor Vasicek model of Subsection 10.2.1, except that, instead of (10.2.6), the interest rate is given by $(10.7 .7)$, in which $\delta_0(t)$ is a nonrandom function of time.
10.7 Exercises
455
(i) Define
$$
\begin{gathered}
X(t)=\left[\begin{array}{l}
U(t) \\
R(t)
\end{array}\right], \quad K=\left[\begin{array}{cc}
\lambda_1 & 0 \\
-1 & \lambda_2
\end{array}\right], \quad \Sigma=\left[\begin{array}{cc}
\sigma_1 & 0 \\
0 & \sigma_2
\end{array}\right] \\
\Theta(t)=\left[\begin{array}{c}
0 \\
\theta(t)
\end{array}\right], \quad \widetilde{B}(t)=\left[\begin{array}{c}
\widetilde{B}_1(t) \\
\widetilde{B}_2(t)
\end{array}\right],
\end{gathered}
$$
so that $(10.7 .10)$ and $(10.7 .11)$ can be written in vector notation as
$$
d X(t)=\Theta(t) d t-K X(t) d t+\Sigma d \widetilde{B}(t) .
$$

Now set
$$
\widehat{X}(t)=X(t)-e^{-K t} \int_0^t e^{K u} \theta(u) d u .
$$

Show that
$$
d \widehat{X}(t)=-K \widehat{X}(t) d t+\Sigma d \widetilde{B}(t)
$$
(ii) With
$$
C=\left[\begin{array}{cc}
\frac{1}{\sigma_1} & 0 \\
-\frac{\rho}{\sigma_1 \sqrt{1-\rho^2}} & \frac{1}{\sigma_2 \sqrt{1-\rho^2}}
\end{array}\right],
$$
define $Y(t)=C \widehat{X}(t), \widetilde{W}(t)=C \Sigma \widetilde{B}(t)$. Show that the components of $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are independent Brownian motions and
$$
d Y(t)=-\Lambda Y(t)+d \widetilde{W}(t),
$$
where
$$
\Lambda=C K C^{-1}=\left[\begin{array}{cc}
\lambda_1 & 0 \\
\frac{\rho \sigma_2\left(\lambda_2-\lambda_1\right)-\sigma_1}{\sigma_2 \sqrt{1-\rho^2}} & \lambda_2 .
\end{array}\right] .
$$

Equation (10.7.14) is the vector form of the canonical two-factor Vasicek equations (10.2.4) and (10.2.5).
(iii) Obtain a formula for $R(t)$ of the form (10.7.7). What are $\delta_0(t), \delta_1$, and $\delta_2$ ?

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Problem 5

(Correlation between long rate and short rate in the one-factor Vasicek model). The one-factor Vasicek model is the one-factor Hull-White model of Example 6.5.1 with constant parameters,
$$
d R(t)=(a-b R(t)) d t+\sigma d \widetilde{W}(t),
$$
where $a, b$, and $\sigma$ are positive constants and $\widetilde{W}(t)$ is a one-dimensional Brownian motion. In this model, the price at time $t \in[0, T]$ of the zero-coupon bond maturing at time $T$ is
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10 Term-Structure Models
$$
B(t, T)=e^{-C(t, T) R(t)-A(t, T)},
$$
where $C(t, T)$ and $A(t, T)$ are given by (6.5.10) and (6.5.11):
$$
\begin{aligned}
C(t, T) & =\int_t^T e^{-\int_t^s b d v} d s=\frac{1}{b}\left(1-e^{-b(T-t)}\right), \\
A(t, T) & =\int_t^T\left(a C(s, T)-\frac{1}{2} \sigma^2 C^2(s, T)\right) d s \\
& =\frac{2 a b-\sigma^2}{2 b^2}(T-t)+\frac{\sigma^2-a b}{b^3}\left(1-e^{-b(T-t)}\right)-\frac{\sigma^2}{4 b^3}\left(1-e^{-2 b(T-t)}\right) .
\end{aligned}
$$

In the spirit of the discussion of the short rate and the long rate in Subsection 10.2.1, we fix a positive relative maturity $\bar{\tau}$ and define the long rate $L(t)$ at time $t$ by $(10.2 .30)$ :
$$
L(t)=-\frac{1}{\bar{\tau}} \log B(t, t+\bar{\tau}) .
$$

Show that changes in $L(t)$ and $R(t)$ are perfectly correlated (i.e., for any $0 \leq t_1<t_2$, the correlation coefficient between $L\left(t_2\right)-L\left(t_1\right)$ and $R\left(t_2\right)-R\left(t_1\right)$ is one). This characteristic of one-factor models caused the development of models with more than one factor.

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Problem 6

(Degenerate two-factor Vasicek model). In the discussion of short rates and long rates in the two-factor Vasicek model of Subsection 10.2.1, we made the assumptions that $\delta_2 \neq 0$ and $\left(\lambda_1-\lambda_2\right) \delta_1+\lambda_{21} \delta_2 \neq 0$ (see Lemma 10.2.2). In this exercise, we show that if either of these conditions is violated, the two-factor Vasicek model reduces to a one-factor model, for which long rates and short rates are perfectly correlated (see Exercise 10.5).
(i) Show that if $\delta_2=0$ (and $\delta_0>0, \delta_1>0$ ), then the short rate $R(t)$ given by the system of equations (10.2.4)-(10.2.6) satisfies the one-dimensional stochastic differential equation
$$
d R(t)=(a-b R(t)) d t+d \widetilde{W}_1(t) .
$$

Define $a$ and $b$ in terms of the parameters in (10.2.4)-(10.2.6).
(ii) Show that if $\left(\lambda_1-\lambda_2\right) \delta_1+\lambda_{21} \delta_2=0$ (and $\delta_0>0, \delta_1^2+\delta_2^2 \neq 0$ ), then the short rate $R(t)$ given by the system of equations (10.2.4)-(10.2.6) satisfies the one-dimensional stochastic differential equation
$$
d R(t)=(a-b R(t)) d t+\sigma d \tilde{B}(t) .
$$

Define $a$ and $b$ in terms of the parameters in (10.2.4)-(10.2.6) and define the Brownian motion $\widetilde{B}(t)$ in terms of the independent Brownian motions $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ in $(10.2 .4)$ and $(10.2 .5)$.
10.7 Exercises
457

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Problem 7

Forward measure in the two-factor Vasicek model). Fix a maturity $T>0$. In the two-factor Vasicek model of Subsection 10.2.1, consider the $T$-forward measure $\widetilde{\mathbb{P}}^T$ of Definition 9.4.1:
$$
\widetilde{\mathbb{P}}^T(A)=\frac{1}{B(0, T)} \int_A D(T) d \tilde{\mathbb{P}} \text { for all } A \in \mathcal{F} .
$$
(i) Show that the two-dimensional $\widetilde{\mathbf{P}}^T$-Brownian motions $\widetilde{W}_1^T(t), \widetilde{W}_2^T(t)$ of (9.2.5) are
$$
\widetilde{W}_j^T(t)=\int_0^t C_1(T-u) d u+\widetilde{W}_j(t), \quad j=1,2,
$$
where $C_1(\tau)$ and $C_2(\tau)$ are given by (10.2.26)-(10.2.28).
(ii) Consider a call option on a bond maturing at time $\bar{T}>T$. The call expires at time $T$ and has strike price $K$. Show that at time zero the risk-neutral price of this option is
$$
B(0, T) \tilde{\mathbb{E}}^T\left[\left(e^{-C_1(\bar{T}-T) Y_1(T)-C_2(\bar{T}-T) Y_2(T)-A(\bar{T}-T)}-K\right)^{+}\right]
$$

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Problem 8

(Reversal of order of integration in forward rates). The forward rate formula (10.3.5) with $v$ replacing $T$ states that
$$
f(t, v)=f(0, v)+\int_0^t \alpha(u, v) d u+\int_0^t \sigma(u, v) d W(u)
$$

Therefore,
$$
-\int_t^T f(t, v) d v=-\int_t^T\left[f(0, v)+\int_0^t \alpha(u, v) d u+\int_0^t \sigma(u, v) d W(u)\right] d v .
$$
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10 Term-Structure Models
(i) Define
$$
\widehat{\alpha}(u, t, T)=\int_t^T \alpha(u, v) d v, \quad \widehat{\sigma}(u, t, T)=\int_t^T \sigma(u, v) d v .
$$

Show that if we reverse the order of integration in (10.7.20), we obtain the equation
$$
\begin{aligned}
& -\int_t^T f(t, v) d v \\
& =-\int_t^T f(0, v) d v-\int_0^t \widehat{\alpha}(u, t, T) d u-\int_0^t \widehat{\sigma}(u, t, T) d W(u) .
\end{aligned}
$$
(In one case, this is a reversal of the order of two Riemann integrals, a step that uses only the theory of ordinary calculus. In the other case, the order of a Riemann and an Itô integral are being reversed. This step is justified in the appendix of [83]. You may assume without proof that this step is legitimate.)
(ii) Take the differential with respect to $t$ in (10.7.21), remembering to get two terms from each of the integrals $\int_0^t \widehat{\alpha}(u, t, T) d u$ and $\int_0^t \widehat{\sigma}(u, t, T) d W(u)$ because one must differentiate with respect to each of the two $t s$ appearing in these integrals.
(iii) Check that your formula in (ii) agrees with (10.3.10).

Victor Salazar

Numerade Educator

Problem 9

(Multifactor HJM model). Suppose the Heath-JarrowMorton model is driven by a $d$-dimensional Brownian motion, so that $\sigma(t, T)$ is also a $d$-dimensional vector and the forward rate dynamics are given by
$$
d f(t, T)=\alpha(t, T) d t+\sum_{j=1}^d \sigma_j(t, T) d W_j(t) .
$$
(i) Show that (10.3.16) becomes
$$
\alpha(t, T)=\sum_{i=1}^d \sigma_j(t, T)\left[\sigma_j^*(t, T)+\Theta_j(t)\right] .
$$

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Problem 10

. (i) Use the ordinary differential equations (6.5.8) and (6.5.9) satisfied by the functions $A(t, T)$ and $C(t, T)$ in the one-factor Hull-White model to show that this model satisfies the HJM no-arbitrage condition (10.3.27).
(ii) Use the ordinary differential equations (6.5.14) and (6.5.15) satisfied by the functions $A(t, T)$ and $C(t, T)$ in the one-factor Cox-Ingersoll-Ross model to show that this model satisfies the HJM no-arbitrage condition $(10.3 .27)$.

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Problem 11

. Let $\delta>0$ be given. Consider an interest rate swap paying a fixed interest rate $K$ and receiving backset LIBOR $L\left(T_{j-1}, T_{j-1}\right)$ on a principal of 1 at each of the payment dates $T_j=\delta j, j=1,2, \ldots, n+1$. Show that the value of the swap is
$$
\delta K \sum_{j=1}^{n+1} B\left(0, T_j\right)-\delta \sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right) .
$$

Remark 10.7.1. The swap rate is defined to be the value of $K$ that makes the initial value of the swap equal to zero. Thus, the swap rate is
$$
K=\frac{\sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right)}{\sum_{j=1}^{n+1} B\left(0, T_j\right)}
$$

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Problem 12

In the proof of Theorem 10.4.1, we showed by an arbitrage argument that the value at time 0 of a payment of backset LIBOR $L(T, T)$ at time $T+\delta$ is $B(0, T+\delta) L(0, T)$. The risk-neutral price of this payment, computed at time zero, is
$$
\widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]
$$

Use the definitions
$$
\begin{aligned}
L(T, T) & =\frac{1-B(T, T+\delta)}{\delta B(T, T+\delta)}, \\
B(0, T+\delta) & =\widetilde{\mathbb{E}}[D(T+\delta)],
\end{aligned}
$$
and the properties of conditional expectations to show that
$$
\widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]=B(0, T+\delta) L(0, T) .
$$

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