数学代写|Financial Engineering代写

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.

2. 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.

3. 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?

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