Time allowed: $2.5$ hours
Throughout this examination $W_{t}$ or $W(t)$ is a standard Brownian motion. You may assume (where appropriate) the following:
$$\begin{gathered} P(S, t)=-S e^{-D(T-t)} N\left(-d_{1}\right)+E e^{-r(T-t)} N\left(-d_{2}\right) \ d_{1,2}=\frac{\log (S / E)+\left(r-D \pm \frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}} \end{gathered}$$
$N(x)$ is the standard Normal Cumulative Distribution Function
\begin{aligned} N(x) &=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} e^{-s^{2} / 2} d s \ N^{\prime}(x) &=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \end{aligned}
SDE refers to Stochastic Differential Equation.
Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR questions will count towards the total mark. Each question is worth 25 marks. CALCULATORS ARE PERMITTED.

Problem 1.

1. a. [10 Marks] Suppose that $X$ is a normally distributed random variable, where $X \sim N\left(\mu, \sigma^{2}\right) .$ Show that
$$\mathbb{E}\left[e^{\theta X} f(X)\right]=e^{\theta \mu+\frac{1}{2} \theta^{2} \sigma^{2}} \mathbb{E}\left[f\left(X+\theta \sigma^{2}\right)\right]$$
where $f$ is a suitable function and $\theta \in \mathbb{R}$ is a scalar. Hint: Write $X=\mu+\sigma \phi ; \phi \sim N(0,1)$ and calculate the resulting integral.
b. [15 Marks] The ordinary differential equation
$$\mu S \frac{d u}{d S}+\frac{1}{2} \sigma^{2} S^{2} \frac{d^{2} u}{d S^{2}}=-1$$
for the function $u(S)$ is to be solved with boundary conditions
\begin{aligned} &u\left(S_{0}\right)=0 \ &u\left(S_{1}\right)=0 . \end{aligned}
$\mu$ and $\sigma$ are constants. Show that the solution is given by
$$u(S)=\frac{1}{\frac{1}{2} \sigma^{2}-\mu}\left(\log \left(S / S_{0}\right)-\frac{1-\left(S / S_{0}\right)^{1-2 \mu / \sigma^{2}}}{1-\left(S_{1} / S_{0}\right)^{1-2 \mu / \sigma^{2}}} \log \left(S_{1} / S_{0}\right)\right)$$
Hint: When solving for the particular integral, assume a solution of the form $C \log S$, where $C$ is a constant.

Problem 2.

1. a. [8 Marks] A spot rate $r_{t}$, evolves according to the popular form
$$d r_{t}=u\left(r_{t}\right) d t+\nu r_{t}^{\beta} d W_{t}$$
where $\nu$ and $\beta$ are constants. Suppose such a model has a steady state transition probability density function $p_{\infty}(r)$ that satisfies the forward Fokker Planck Equation. Show that this implies the drift structure of $(2.1)$ is given by
$$u\left(r_{t}\right)=\nu^{2} \beta r_{t}^{2 \beta-1}+\frac{1}{2} \nu^{2} r_{t}^{2 \beta} \frac{d}{d r}\left(\log p_{\infty}\right)$$
b. [12 Marks] The Ornstein-Uhlenbeck process satisfies the spot rate SDE and initial condition given by
$$d r_{t}=\kappa\left(\theta-r_{t}\right) d t+\sigma d W_{t}, r_{0}=x$$
where $\kappa, \theta$ and $\sigma$ are constants. Solve this SDE by setting $Y_{t}=$ $e^{\kappa t} r_{t}$ and using Itô’s lemma to show that
$$r_{t}=\theta+(x-\theta) e^{-\kappa t}+\sigma \int_{0}^{t} e^{-\kappa(t-s)} d W_{s}$$
c. [5 Marks] Calculate the mean of $r_{t}$ given by (2.2). You may use the result in your calculation
$$\mathbb{E}\left[\int_{0}^{t} f_{s} d W_{s}\right]=0$$
where $f_{t}$ is a time-dependent function.

Problem 3.

1. a. [9 Marks] An option $V(S, t)$ is to be written on a dividend paying stock $S_{t}$, that satisfies the $\mathrm{SDE}$
$$\frac{d S_{t}}{S_{t}}=\mu d t+\sigma d W_{t}$$
$\mu$ and $\sigma$ are constants. Assume that the asset receives a continuous and constant dividend yield, $D$, across each time-step. By constructing a hedged portfolio derive the partial differential equation
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+(r-D) S \frac{\partial V}{\partial S}-r V=0$$
for the fair price of an option with $r$ the risk-free interest rate.
b. [8 Marks] Separable solutions of (3.1) of the form $V(S, t)=$ $A(t) B(S)$ are sought. By substitution show that $(3.1)$ leads to a first order differential equation in $A(t)$ and a second order differential equation in $B(S)$. You are not required to solve this pair of differential equations.
c. [8 Marks] An At-The-Money-Forward (ATMF) option is struck when its strike price is $E=S e^{(r-D)(T-t)} .$ What is the BlackScholes pricing formula for an ATMF put option?

Problem 4.

a. [11 Marks] Consider the pricing equation for the value of a derivative security $V(S, t)$
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+(r-D) S \frac{\partial V}{\partial S}-r V=0$$
where $S \geq 0$ is the spot price of the underlying equity, $0<t \leq T$ is the time, $r \geq 0$ the constant rate of interest, and $\sigma$ is the constant volatility of $S$. The variables $(t, S)$ can be written as
$$t=m \delta t \quad 0 \leq m \leq M ; \quad S=n \delta S \quad 0 \leq n \leq N$$
where $(\delta t, \delta S)$ are fixed step sizes in turn. $V(S, t)$ is written discretely as $V_{n}^{m} .$ An Explicit Finite Difference Method is to be developed to solve (4.1) using a backward marching scheme.
i. Derive a difference equation in the form
$$V_{n}^{m-1}=a_{n} V_{n-1}^{m}+b_{n} V_{n}^{m}+c_{n} V_{n+1}^{m}$$
where $a_{n}, b_{n}, c_{n}$ should be defined; you may use the following as a starting point,
\begin{aligned} &\frac{\partial V}{\partial t} \sim \frac{V_{n}^{m}-V_{n}^{m-1}}{\delta t} ; \frac{\partial V}{\partial S} \sim \frac{V_{n+1}^{m}-V_{n-1}^{m}}{2 \delta S} \ &\frac{\partial^{2} V}{\partial S^{2}} \sim \frac{V_{n-1}^{m}-2 V_{n}^{m}+V_{n+1}^{m}}{\delta S^{2}} \end{aligned}
ii. Obtain expressions for the final and boundary conditions in finite difference form for a European Put Option.
b. [8 Marks] A binary put option is to be priced. Outline a. Monte Carlo method to do this.
c. [6 Marks] Give four advantages of the Monte Carlo methods for pricing derivatives