(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]

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]

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]

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]