Candidates should attempt ALL questions.

MODULE CODE : COMP0048
ASSESSMENT : COMP0048A7PC
PATTERN
MODULE NAME : COMP0048 – Financial Engineering
DATE : 24/08/2021
TIME : 14:30

This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
2020/21
Special instructions Answers must be hand-written and uploaded as a single PDF
Exam paper word
count
10 pages
TURN

Department of Computer Science
University College London
COMP0048 Financial Engineering
MSc Examination (LSA)
2021
Throughout this examination W or Wt is a standard Brownian motion:
You may assume (where appropriate) the following: SDE refers to
Stochastic Di¤erential Equation.
There are four questions in this examination. You are required to answer all
questions.

Problem 1.

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
$$d v=\left(\delta^{2}+2 \alpha \sqrt{v}-2 \beta v\right) d t+2 \delta \sqrt{v} d W$$

Proof .

Problem 2.

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
$$\log Y_{T}=e^{-\alpha(T-t)} \log Y_{t}+\left(\theta-\frac{1}{2 \alpha} \sigma^{2}\right)\left(1-e^{-\alpha(T-t)}\right)+\sigma \int_{t}^{T} e^{-\alpha(T-s)} d W_{s}$$

Proof .

Problem 3.

1. 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
$$F(R, T)=\max \left(1-\frac{R}{T}, 0\right)$$

Proof .

Problem 4.

1. 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.

.

Proof .

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