Candidates should attempt ALL questions.

MODULE CODE : COMP0048
ASSESSMENT : COMP0048A7PC
PATTERN
MODULE NAME : COMP0048 – Financial Engineering
LEVEL: : Postgraduate
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
Additional material N/A
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.

(a) Consider a $T$-period market model with $n+1$ assets (including a bank account) defined on a probability space with our usual assumptions. Let $\left{\boldsymbol{R}{t}^{i}\right}{i=0, \ldots, n ; t=1, \ldots, T}$ be the one-period gross returns (all well-defined). Show that if $Z$ is an adapted process such that $Z_{0}=1$ and
$$\mathbb{E}{t}\left[Z{t+1} R_{t+1}^{i}\right]=1 ; \quad \text { for all } i=0, \ldots, n ; t=0, \ldots, T-1 \text {, }$$
then $M_{t}=\prod_{t=0}^{t} Z_{\ell}$ is an SDF. [9 marks]

Consider a probability space with $\Omega={1,2}^{2}$, the power sigma algebra $\mathcal{F}=\mathcal{P}(\Omega)$ and uniform probability. On this space, we define a market model in one period composed of two assets: a riskfree asset with return $R_{1}^{0}=R_{2}^{0}=3 / 2$, and a risky asset with $S_{0}^{1}=8$ and
$$R_{1}^{1}\left(\left(\omega_{1}, \omega_{2}\right)\right)=\left{\begin{array}{ll} 1 / 2 & \text { if } \omega_{1}=1 \ 2 & \text { if } \omega_{1}=2 \end{array} ; \quad R_{2}^{1}\left(\left(\omega_{1}, \omega_{2}\right)\right)= \begin{cases}1 / 2 & \text { if } \omega_{2}=1 \ 2 & \text { if } \omega_{2}=2\end{cases}\right.$$

(b) Find all SDFs (if any) that can be defined on this market model. Conclude on whether the market is arbitrage-free and/or complete. [9 marks]
(c) A butterfly option on the risky asset with maturity $T=2$ wants to be introduced into the market: this is the option with payoff $\phi\left(S_{2}^{1}\right)$ with
$$\phi(x):=\left(1-\frac{1}{4}|x-8|\right)^{+} .$$
Find the set of arbitrage-free prices for this option. [7 marks]

Proof .

Problem 2.

Q2 (25 points)
A pure investor has utility function
$$u(x)= \begin{cases}x^{1 / 2} & \text { if } x \geq 0 \ -|x|^{1 / 2} & \text { if } x<0\end{cases}$$
(a) Find the coefficients of relative risk aversion of this investor as a function of their wealth. How does the investor behave facing risk? [8 marks]
(b) Let $W$ be a continuous random variable with even density $f$ (i.e. $f(x)=f(-x)$ for all $x \in \mathbb{R}$ ), and so that $\mathbb{E}[|u(W)|]<\infty$. Find the certainty equivalent of a wealth $W$ for this investor. [8 marks]
(c) Suppose that our investor has initial wealth $w_{0}$. They will act on a one-period market with two assets: one risk-free with return $R^{0}$ and one risky with a binomial return
$$R^{1}= \begin{cases}h & \text { with probability } p \ \ell & \text { with probability } 1-p\end{cases}$$
with $\ell<R^{0}<h$. Find the optimal investment strategy with no short positions for this agent, at time 0 , in terms of the amount to invest on each asset. [9 marks]

Proof .

Problem 3.

Q3 (25 points)
Let $g:[0,1] \rightarrow \mathbb{R}{+}$be a positive function such that $$\int{0}^{1} g(u) \mathrm{d} u=1$$
For $X \in L^{1}$, define the risk measure
$$\rho_{g}(X):=\int_{0}^{1} g(u) q_{-X}(u) \mathrm{d} u$$
where $q_{-X}$ is the quantile function of $-X$.
(a) Find an example of a function $g$ satisfying the assumptions above and such that the measure $\rho_{g}$ is coherent. [8 marks]
(b) Show that $\rho_{g}$ is a monetary risk measure.
Remark: You can use properties of risk measures studied in class. [8 marks]

(c) Take a sample of size $T$ of results of one-trading periods. Consider the excess indicators $\left(I_{t, u}\right){t=1, \ldots, T ; u \in(0,1)}$ where Assume that $I{t, u} \Perp I_{s, v}$ for all $t \neq s$ and that $\mathbb{P}\left[I_{t, u}=1\right]=1-u$ for all $t=1, \ldots, T ; u \in(0,1)$. Let
$$Y_{g}:=\frac{1}{T} \sum_{t=1}^{T} \int_{0}^{1} g(u) I_{t, u} \mathrm{~d} u$$
Show that
$$\mathbb{E}\left[Y_{g}\right]=\int_{0}^{1} g(u)(1-u) \mathrm{d} u .$$
Then, using in addition the (given) fact that
$$\operatorname{var}\left(Y_{g}\right)=\frac{2}{T} \int_{0}^{1} \int_{v}^{1} g(u) g(v)(1-u) \mathrm{d} u \mathrm{~d} v-\mathbb{E}\left[Y_{g}\right]^{2}<\infty$$
propose a coverage backtest for the measure $\rho_{g}$. [9 marks]

Proof .

Problem 4.

Q4 (25 points)
A one-period market model contains a risk-free asset with return $R^{0}$ and $n$ risky assets.
The mean-variance frontier (excluding the risk-free asset) of returns in this market contains two portfolios, $\pi_{1}, \pi_{2}$ such that:

• $\mathbb{E}\left(R^{\pi_{1}}\right)=R^{0}+a$ for $a>0, \mathbb{E}\left(R^{\pi_{2}}\right)=R^{0}$;
• $\mathrm{sd}\left(\mathrm{R}^{x_{1}}\right)=\mathrm{sd}\left(\mathrm{R}^{\pi_{2}}\right)=\sigma>0$;
• $\operatorname{corr}\left(R^{\pi_{1}}, R^{\pi 2}\right)=1 / 2$.
(a) Find an expression that describes all portfolios in the mean-variance frontier (without risk-free asset), and give their corresponding mean and variance. [9 marks]
(b) Find the tangency portfolio and the maximal Sharpe ratio $\left(S_{\max }\right)$ in this market. [9 marks]
(c) Assume that an investor would like to choose a portfolio with maximal Sharpe ratio, mean return larger than the risk-free rate, and no short position on the risk-free asset. Describe the set of portfolios that satisfy these properties. [7 marks]

Proof .

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# 概率论代考

## MATH0094 Market Risk and PortfolioTheory

Year:2021-2022
Code:MATH0094

Value:15 UCL credits (= 7.5 ECTS)

Assessment:In-class test with programming component (20%) and a final examination (80%). To pass the course, students must
obtain an overall weighted mark of 50%.

Lecturers:Dr C Garcia Trillos

#### Course description and objectives

Risk is an intrinsic element in financial markets. Its quantitative modelling and understanding is a cornerstone of modern financial theory, as it is essential to many activities like choosing investment strategies, calculating capital requirements and creating new financial products.

This module aims to study quantitatively (by using several mathematical tools from probability, optimisation, linear algebra,…) the effects of market risk under some modelling assumptions. We will pay particular attention to the effects associated to decision making for investors and regulators. Important aspects related to the implementation of these concepts will be highlighted.