Consider the SDE
$$
d X_{t}=\left(t+X_{t}\right) d t+2 t d B_{t} .
$$
(a) Write this SDE in integral form, and show that $f(t)=\mathbb{E}\left[X_{t}\right]$ satisfies the differential equation
$$
f^{\prime}(t)=t+f(t)
$$
Show that this equation is satisfied by $f(t)=C e^{t}-t-1$.
(b) Let $Y_{t}=X_{t}^{2}$. Show that
$$
d Y_{t}=2\left(2 t^{2}+t X_{t}+X_{t}^{2}\right) d t+4 t X_{t} d B_{t}
$$
(

a. [6 Marks] A stock price $S(t)$ evolves according to Geometric Brownian Motion
$$
\frac{d S(t)}{S(t)}=\mu d t+\sigma d W
$$
where $\mu$ and $\sigma$ are constants. Using Itô’s lemma show that
$$
S(t)=S_{0} e^{\left(\mu-\sigma^{2} / 2\right) t+\sigma W}
$$
where $S_{0}=S(0)$.
b. [12 Marks] Consider the following SDE
$$
d \sigma=a d t+b d W
$$
where both $a$ and $b$ are functions of $\sigma$ and $t$. The Forward Kolmogorov Equation, for the transition density function $p\left(\sigma, t ; \sigma^{\prime}, t^{\prime}\right)$ is
$$
\frac{\partial p}{\partial t^{\prime}}=\frac{1}{2} \frac{\partial^{2}}{\partial \sigma^{\prime 2}}\left(b^{2} p\right)-\frac{\partial}{\partial \sigma^{\prime}}(a p)
$$
where the primed variables refer to future states. The steady state solution is given by setting $\frac{\partial p}{\partial t^{\prime}}=0$. Considering the boundary conditions that as $\sigma^{\prime} \rightarrow \pm \infty, p \rightarrow 0$ and $\frac{\partial p}{\partial \sigma^{\prime}} \rightarrow 0$, show that the steady state solution is given by
$$
p\left(\sigma^{\prime}\right)=\frac{A}{b^{2}} e^{\int \frac{2 \alpha}{b^{2}} d \sigma^{\prime}}
$$
where $A$ is a constant. (During your working you may drop the primed notation).
c. [7 Marks] Consider the process
$$
d \sqrt{v}=(\alpha-\beta \sqrt{v}) d t+\delta d W
$$
The parameters $\alpha, \beta, \delta$ are constant. Using Itô’s lemma show that
$$
d v=\left(\delta^{2}+2 \alpha \sqrt{v}-2 \beta v\right) d t+2 \delta \sqrt{v} d W
$$

a. [7 Marks] Consider the process
$$
Z_{t}=\left(W_{t}+t\right) \exp \left(-W_{t}+k t\right), t \geq 0
$$
where $k$ is a constant. For what value of $k$ is $Z_{t}$ driftless (i.e. has zero drift)? Justify your answer.
b. [6 Marks] Use Itô’s lemma to show that
$$
d\left(\cos W_{t}\right)=\alpha\left(\cos W_{t}\right) d t+\beta\left(\sin W_{t}\right) d W_{t}
$$
and
$$
d\left(\sin W_{t}\right)=\alpha\left(\sin W_{t}\right) d t-\beta\left(\cos W_{t}\right) d W_{t}
$$
and determine the constants $\alpha$ and $\beta$.
c. [12 Marks] The dynamics of a state variable $Y_{t}$ can be modelled by the SDE
$$
\frac{d Y_{t}}{Y_{t}}=\alpha\left(\theta-\log Y_{t}\right) d t+\sigma d W_{t}
$$
where $\theta, \sigma, \alpha>0$ are constant. Consider a time $T>t .$ By performing Itô on $f\left(Y_{t}\right)=\log Y_{t}$ and then setting
$$
d f=-\alpha(f-\mu) d t+\sigma d W_{t}
$$
for a suitable choice of constant $\mu$, show that
$$
\log Y_{T}=e^{-\alpha(T-t)} \log Y_{t}+\left(\theta-\frac{1}{2 \alpha} \sigma^{2}\right)\left(1-e^{-\alpha(T-t)}\right)+\sigma \int_{t}^{T} e^{-\alpha(T-s)} d W_{s}
$$

3. An asset $S$ follows Geometric Brownian Motion $d S=\mu S d t+\sigma S d W$, where $\mu$ and $\sigma$ are constants. We wish to value an option that pays off at expiry $T$ an amount which is a function of the path taken by the asset between time zero and expiry.
   a. [10 Marks] Assuming that an option value $V$ depends on $S, t$ and a quantity
   $$
   I(t)=\int_{0}^{t} f(S, \tau) d \tau
   $$
   where $f$ is a specified function and $r$ the risk free interest rate, derive the pricing equation
   $$
   \frac{\partial V}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+f(S, t) \frac{\partial V}{\partial I}+r S \frac{\partial V}{\partial S}-r V=0
   $$
   for the function $V(S, I, t)$
   b. [12 Marks] For an arithmetic strike Asian call option the payoff at time $T$ is
   $$
   \max \left(S-\frac{1}{T} \int_{0}^{T} S(\tau) d \tau, 0\right)
   $$
   By writing the value of this option as
   $$
   V(S, I, t)=S F(R, t)
   $$
   where $R=I / S$, show that the partial differential equation for $F(R, t)$ is given by
   $$
   \frac{\partial F}{\partial t}+\frac{1}{2} \sigma^{2} R^{2} \frac{\partial^{2} F}{\partial R^{2}}+(1-r R) \frac{\partial F}{\partial R}=0
   $$
   c. [3 Marks] Show that the payoff can now be written as
   $$
   F(R, T)=\max \left(1-\frac{R}{T}, 0\right)
   $$

4. A two factor interest rate model depends on the spot rate $r$ and another quantity $l$. The state variables $r$ and $l$ follow the SDEs, in turn,
   $$
   \begin{aligned}
   d r &=u(r, t) d t+w(r, t) d W_{1} \
   d l &=p(r, t) d t+q(r, t) d W_{2}
   \end{aligned}
   $$
   where the Brownian motions are correlated with $\mathbb{E}\left[d W_{1} d W_{2}\right]=\rho d t$. A bond with maturity $T$ has value $V(r, l, t ; T)$.
   a. [ 10 Marks] Consider a portfolio where one bond of maturity $T$ is hedged with two others of maturities $T_{1}$ and $T_{2}$ which is given by
   $$
   \Pi=V(r, l, t ; T)-\Delta_{1} V_{1}\left(r, l, t ; T_{1}\right)-\Delta_{2} V_{2}\left(r, l, t ; T_{2}\right)
   $$
   Use Itô’s lemma to show that in one time step the change in the portfolio value is given by
   $$
   \begin{aligned}
   d \Pi=& \mathcal{L}(V) d t+\frac{\partial V}{\partial r} d r+\frac{\partial V}{\partial l} d l \
   &-\Delta_{1}\left(\mathcal{L}\left(V_{1}\right) d t+\frac{\partial V_{1}}{\partial r} d r+\frac{\partial V_{1}}{\partial l} d l\right) \
   &-\Delta_{2}\left(\mathcal{L}\left(V_{2}\right) d t+\frac{\partial V_{2}}{\partial r} d r+\frac{\partial V_{2}}{\partial l} d l\right)
   \end{aligned}
   $$
   where the operator
   $$
   \mathcal{L} \equiv \frac{\partial}{\partial t}+\frac{1}{2} w^{2} \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{2} q^{2} \frac{\partial^{2}}{\partial l^{2}}+\rho w q \frac{\partial^{2}}{\partial r \partial l}
   $$
   b. [7 Marks] Using the No Arbitrage Principle show that the problem reduces to the inconsistent system
   $$
   \begin{aligned}
   \frac{\partial V}{\partial r}-\Delta_{1} \frac{\partial V_{1}}{\partial r}-\Delta_{2} \frac{\partial V_{2}}{\partial r} &=0 \
   \frac{\partial V}{\partial l}-\Delta_{1} \frac{\partial V_{1}}{\partial l}-\Delta_{2} \frac{\partial V_{2}}{\partial l} &=0 \
   \mathcal{L}^{\prime}(V)-\Delta_{1} \mathcal{L}^{\prime}\left(V_{1}\right)-\Delta_{2} \mathcal{L}^{\prime}\left(V_{2}\right) &=0
   \end{aligned}
   $$
   where
   $$
   \mathcal{L}^{\prime}(V)=\mathcal{L}(V)-r V
   $$

Hence obtain the two factor interest rate model
$$
\frac{\partial V}{\partial t}+\frac{1}{2} w^{2} \frac{\partial^{2} V}{\partial r^{2}}+\rho w q \frac{\partial^{2} V}{\partial r \partial l}+\frac{1}{2} q^{2} \frac{\partial^{2} V}{\partial l^{2}}+\left(u-\lambda_{r} w\right) \frac{\partial V}{\partial r}+\left(p-\lambda_{l} q\right) \frac{\partial V}{\partial l}-r V=0
$$
where the arbitrary functions $\lambda_{r}$ and $\lambda_{l}$ represent the market price of risk associated with $r$ and $l$ respectively.
c. [8 Marks] Given that
$$
\begin{aligned}
u-\lambda_{r} w &=0=p-\lambda_{l} q \
w &=q=\sqrt{a+b r+c l}
\end{aligned}
$$
where $a, b$ and $c$ are constants, derive a set of first order equations and boundary conditions for $A, B$ and $C$ such that a bond $V$ is of the form
$$
V=\exp (A(t ; T)-r B(t ; T)-l C(t ; T))
$$
is a solution of the Bond Pricing Equation with redemption value
$$
V(r, l, T ; T)=1
$$
You should not solve these resulting equations.

.