• Paper 1, Section II, 30K

(a) What does it mean to say that a stochastic process $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale with respect to a filtration $\left(\mathcal{F}_{n}\right)_{n \geqslant 0}$ ?

(b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a martingale, and let $\xi_{n}=X_{n}-X_{n-1}$ for $n \geqslant 1$. Suppose $\xi_{n}$ takes values in the set $\{-1,+1\}$ almost surely for all $n \geqslant 1$. Show that $\left(X_{n}\right)_{n \geqslant 0}$ is a simple symmetric random walk, i.e. that the sequence $\left(\xi_{n}\right)_{n \geqslant 1}$ is $\operatorname{IID}$ with $\mathbb{P}\left(\xi_{1}=1\right)=1 / 2=$ $\mathbb{P}\left(\xi_{1}=-1\right) .$

(c) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a martingale and let the bounded process $\left(H_{n}\right)_{n \geqslant 1}$ be previsible.

Let $\hat{X}_{0}=0$ and

$\hat{X}_{n}=\sum_{k=1}^{n} H_{k}\left(X_{k}-X_{k-1}\right) \text { for } n \geqslant 1$

Show that $\left(\hat{X}_{n}\right)_{n \geqslant 0}$ is a martingale.

(d) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk with $X_{0}=0$, and let

$T_{a}=\inf \left\{n \geqslant 0: X_{n}=a\right\}$

where $a$ is a positive integer. Let

$\hat{X}_{n}= \begin{cases}X_{n} & \text { if } n \leqslant T_{a} \\ 2 a-X_{n} & \text { if } n>T_{a}\end{cases}$

Show that $\left(\hat{X}_{n}\right)_{n \geqslant 0}$ is a simple symmetric random walk.

(e) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk with $X_{0}=0$, and let $M_{n}=\max _{0 \leqslant k \leqslant n} X_{k}$. Compute $\mathbb{P}\left(M_{n}=a\right)$ for a positive integer $a$.

comment
• Paper 2, Section II, 30K

Consider a one-period market model with $d$ risky assets and one risk-free asset. Let $S_{t}$ denote the vector of prices of the risky assets at time $t \in\{0,1\}$ and let $r$ be the interest rate.

(a) What does it mean to say a portfolio $\varphi \in \mathbb{R}^{d}$ is an arbitrage for this market?

(b) An investor wishes to maximise their expected utility of time-1 wealth $X_{1}$ attainable by investing in the market with their time- 0 wealth $X_{0}=x$. The investor's utility function $U$ is increasing and concave. Show that, if there exists an optimal solution $X_{1}^{*}$ to the investor's expected utility maximisation problem, then the market has no arbitrage. [Assume that $U\left(X_{1}\right)$ is integrable for any attainable time-1 wealth $X_{1}$.]

(c) Now introduce a contingent claim with time-1 bounded payout $Y$. How does the investor in part (b) calculate an indifference bid price $\pi(Y)$ for the claim? Assuming each such claim has a unique indifference price, show that the map $Y \mapsto \pi(Y)$ is concave. [Assume that any relevant utility maximisation problem that you consider has an optimal solution.]

(d) Consider a contingent claim with time-1 bounded payout $Y$. Let $I \subseteq \mathbb{R}$ be the set of initial no-arbitrage prices for the claim; that is, the set $I$ consists of all $p$ such that the market augmented with the contingent claim with time- 0 price $p$ has no arbitrage. Show that $\pi(Y) \leqslant \sup \{p \in I\}$. [Assume that any relevant utility maximisation problem that you consider has an optimal solution. You may use results from lectures without proof, such as the fundamental theorem of asset pricing or the existence of marginal utility prices, as long as they are clearly stated.]

comment
• Paper 3, Section II, 29K

(a) Let $M=\left(M_{n}\right)_{n \geqslant 0}$ be a martingale and $\hat{M}=\left(\hat{M}_{n}\right)_{n \geqslant 0}$ a supermartingale. If $M_{0}=\hat{M}_{0}$, show that $\mathbb{E}\left(M_{T}\right) \geqslant \mathbb{E}\left(\hat{M}_{T}\right)$ for any bounded stopping time $T$. [If you use a general result about supermartingales, you must prove it.]

(b) Consider a market with one stock with time- $n$ price $S_{n}$ and constant interest rate $r$. Explain why a self-financing investor's wealth process $\left(X_{n}\right)_{n \geqslant 0}$ satisfies

$X_{n}=(1+r) X_{n-1}+\theta_{n}\left[S_{n}-(1+r) S_{n-1}\right]$

where $\theta_{n}$ is the number of shares of the stock held during the $n$th period.

(c) Given an initial wealth $X_{0}$, an investor seeks to maximize $\mathbb{E}\left[U\left(X_{N}\right)\right]$, where $U$ is a given utility function. Suppose the stock price is such that $S_{n}=S_{n-1} \xi_{n}$, where $\left(\xi_{n}\right)_{n \geqslant 1}$ is a sequence of independent copies of a random variable $\xi$. Let $V$ be defined inductively by

$V(n-1, x)=\sup _{t \in \mathbb{R}} \mathbb{E}[V(n,(1+r) x+t(1+r-\xi))]$

with terminal condition $V(N, x)=U(x)$ for all $x \in \mathbb{R}$.

Show that the process $\left(V\left(n, X_{n}\right)\right)_{0 \leqslant n \leqslant N}$ is a supermartingale for any trading strategy $\left(\theta_{n}\right)_{1 \leqslant n \leqslant N}$. Suppose that the trading strategy $\left(\theta_{n}^{*}\right)_{1 \leqslant n \leqslant N}$ with corresponding wealth process $\left(X_{n}^{*}\right)_{0 \leqslant n \leqslant N}$ are such that the process $\left(V\left(n, X_{n}^{*}\right)\right)_{0 \leqslant n \leqslant N}$ is a martingale. Show that $\left(\theta_{n}^{*}\right)_{1 \leqslant n \leqslant N}$ is optimal.

comment
• Paper 4, Section II, 29K

(a) What does it mean to say that a stochastic process is a Brownian motion? Show that, if $\left(W_{t}\right)_{t \geqslant 0}$ is a continuous Gaussian process such that $\mathbb{E}\left(W_{t}\right)=0$ and $\mathbb{E}\left(W_{s} W_{t}\right)=s$ for all $0 \leqslant s \leqslant t$, then $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion.

For the rest of the question, let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion.

(b) Let $\widehat{W}_{0}=0$ and $\widehat{W}_{t}=t W_{1 / t}$ for $t>0$. Show that $\left(\widehat{W}_{t}\right)_{t \geqslant 0}$ is a Brownian motion. [You may use without proof the Brownian strong law of large numbers: $W_{t} / t \rightarrow 0$ almost surely as $t \rightarrow \infty$.]

(c) Fix constants $c \in \mathbb{R}$ and $T>0$. Show that

$\mathbb{E}\left[f\left(\left(W_{t}+c t\right)_{0 \leqslant t \leqslant T}\right)\right]=\mathbb{E}\left[\exp \left(c W_{T}-\frac{1}{2} c^{2} T\right) f\left(\left(W_{t}\right)_{0 \leqslant t \leqslant T}\right)\right]$

for any bounded function $f: C[0, T] \rightarrow \mathbb{R}$ of the form

$f(\omega)=g\left(\omega\left(t_{1}\right), \ldots, \omega\left(t_{n}\right)\right),$

for some fixed $g$ and fixed $0, where $C[0, T]$ is the space of continuous functions on $[0, T]$. [If you use a general theorem from the lectures, you should prove it.]

(d) Fix constants $x \in \mathbb{R}$ and $T>0$. Show that

$\mathbb{E}\left[f\left(\left(W_{t}+x\right)_{t \geqslant T}\right)\right]=\mathbb{E}\left[\exp \left((x / T) W_{T}-\frac{1}{2}\left(x^{2} / T\right)\right) f\left(\left(W_{t}\right)_{t \geqslant T}\right)\right]$

for any bounded function $f: C[T, \infty) \rightarrow \mathbb{R}$. [In this part you may use the Cameron-Martin theorem without proof.]

comment

• Paper 1, Section II, 30K

Consider a single-period asset price model $\left(\bar{S}_{0}, \bar{S}_{1}\right)$ in $\mathbb{R}^{d+1}$ where, for $n=0,1$,

$\bar{S}_{n}=\left(S_{n}^{0}, S_{n}\right)=\left(S_{n}^{0}, S_{n}^{1}, \ldots, S_{n}^{d}\right)$

with $S_{0}$ a non-random vector in $\mathbb{R}^{d}$ and

$S_{0}^{0}=1, \quad S_{1}^{0}=1+r, \quad S_{1} \sim N(\mu, V) .$

Assume that $V$ is invertible. An investor has initial wealth $w_{0}$ which is invested in the market at time 0 , to hold $\theta^{0}$ units of the riskless asset $S^{0}$ and $\theta^{i}$ units of risky asset $i$, for $i=1, \ldots, d$.

(a) Show that in order to minimize the variance of the wealth $\bar{\theta} \cdot \bar{S}_{1}$ held at time 1 , subject to the constraint

$\mathbb{E}\left(\bar{\theta} \cdot \bar{S}_{1}\right)=w_{1}$

with $w_{1}$ given, the investor should choose a portfolio of the form

$\theta=\lambda \theta_{m}, \quad \theta_{m}=V^{-1}\left(\mu-(1+r) S_{0}\right)$

where $\lambda$ is to be determined.

(b) Show that the same portfolio is optimal for a utility-maximizing investor with CARA utility function

$U(x)=-\exp \{-\gamma x\}$

for a unique choice of $\gamma$, also to be determined.

comment
• Paper 2, Section II, 29K

Let $\left(S_{n}^{0}, S_{n}\right)_{0 \leqslant n \leqslant T}$ be a discrete-time asset price model in $\mathbb{R}^{d+1}$ with numéraire.

(i) What is meant by an arbitrage for such a model?

(ii) What does it mean to say that the model is complete?

Consider now the case where $d=1$ and where

$S_{n}^{0}=(1+r)^{n}, \quad S_{n}=S_{0} \prod_{k=1}^{n} Z_{k}$

for some $r>0$ and some independent positive random variables $Z_{1}, \ldots, Z_{T}$ with $\log Z_{n} \sim N\left(\mu, \sigma^{2}\right)$ for all $n$.

(iii) Find an equivalent probability measure $\mathbb{P}^{*}$ such that the discounted asset price $\left(S_{n} / S_{n}^{0}\right)_{0 \leqslant n \leqslant T}$ is a martingale.

(v) By considering the contingent claim $\left(S_{1}\right)^{2}$ or otherwise, show that this model is not complete.

comment
• Paper 3, Section II, 29K

(a) Let $\left(B_{t}\right)_{t \geqslant 0}$ be a real-valued random process.

(i) What does it mean to say that $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion?

(ii) State the reflection principle for Brownian motion.

(b) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is a Brownian motion and set $M_{t}=\sup _{s \leqslant t} B_{s}$ and $Z_{t}=M_{t}-B_{t}$.

(i) Find the joint distribution function of $B_{t}$ and $M_{t}$.

(ii) Show that $\left(M_{t}, Z_{t}\right)$ has a joint density function on $[0, \infty)^{2}$ given by

$\mathbb{P}\left(M_{t} \in d y \text { and } Z_{t} \in d z\right)=\frac{2}{\sqrt{2 \pi t}} \frac{(y+z)}{t} e^{-(y+z)^{2} /(2 t)} d y d z$

(iii) You are given that two of the three processes $\left(\left|B_{t}\right|\right)_{t \geqslant 0},\left(M_{t}\right)_{t \geqslant 0}$ and $\left(Z_{t}\right)_{t \geqslant 0}$ have the same distribution. Identify which two, justifying your answer.

comment
• Paper 4, Section II, K

(i) What does it mean to say that $\left(S_{t}^{0}, S_{t}\right)_{0 \leqslant t \leqslant T}$ is a Black-Scholes model with interest rate $r$, drift $\mu$ and volatility $\sigma$ ?

(ii) Write down the Black-Scholes pricing formula for the time- 0 value $V_{0}$ of a time- $T$ contingent claim $C$.

(iii) Show that if $C$ is a European call of strike $K$ and maturity $T$ then

$V_{0} \geqslant S_{0}-e^{-r T} K$

(iv) For the European call, derive the Black-Scholes pricing formula

$V_{0}=S_{0} \Phi\left(d^{+}\right)-e^{-r T} K \Phi\left(d^{-}\right)$

where $\Phi$ is the standard normal distribution function and $d^{+}$and $d^{-}$are to be determined.

(v) Fix $t \in(0, T)$ and consider a modified contract which gives the investor the right but not the obligation to buy one unit of the risky asset at price $K$, either at time $t$ or time $T$ but not both, where the choice of exercise time is to be made by the investor at time $t$. Determine whether the investor should exercise the contract at time $t$.

comment

• Paper 1, Section II, 30K

(a) What does it mean to say that $\left(M_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale? (b) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain defined by $X_{0}=0$ and

\begin{aligned} & \mathbb{P}\left[X_{n}=0 \mid X_{n-1}=0\right]=1-\frac{1}{n} \end{aligned}

\begin{aligned} &\mathbb{P}\left[X_{n}=1 \mid X_{n-1}=0\right]=\mathbb{P}\left[X_{n}=-1 \mid X_{n-1}=0\right]=\frac{1}{2 n} \\ &\mathbb{P}\left[X_{n}=0 \mid X_{n-1}=0\right]=1-\frac{1}{n} \end{aligned}

and

$\mathbb{P}\left[X_{n}=n X_{n-1} \mid X_{n-1} \neq 0\right]=\frac{1}{n}, \quad \mathbb{P}\left[X_{n}=0 \mid X_{n-1} \neq 0\right]=1-\frac{1}{n}$

for $n \geqslant 1$. Show that $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale with respect to the filtration $\left(\mathcal{F}_{n}\right)_{n} \geqslant 0$ where $\mathcal{F}_{0}$ is trivial and $\mathcal{F}_{n}=\sigma\left(X_{1}, \ldots, X_{n}\right)$ for $n \geqslant 1$.

(c) Let $M=\left(M_{n}\right)_{n \geqslant 0}$ be adapted with respect to a filtration $\left(\mathcal{F}_{n}\right)_{n \geqslant 0}$ with $\mathbb{E}\left[\left|M_{n}\right|\right]<\infty$ for all $n$. Show that the following are equivalent:

(i) $M$ is a martingale.

(ii) For every stopping time $\tau$, the stopped process $M^{\tau}$ defined by $M_{n}^{\tau}:=M_{n \wedge \tau}$, $n \geqslant 0$, is a martingale.

(iii) $\mathbb{E}\left[M_{n \wedge \tau}\right]=\mathbb{E}\left[M_{0}\right]$ for all $n \geqslant 0$ and every stopping time $\tau$.

[Hint: To show that (iii) implies (i) you might find it useful to consider the stopping time

$T(\omega):= \begin{cases}n & \text { if } \omega \in A, \\ n+1 & \text { if } \omega \notin A,\end{cases}$

for any $\left.A \in \mathcal{F}_{n .} .\right]$

comment
• Paper 2, Section II, $29 K$

(a) In the context of a multi-period model in discrete time, what does it mean to say that a probability measure is an equivalent martingale measure?

(b) State the fundamental theorem of asset pricing.

(c) Consider a single-period model with one risky asset $S^{1}$ having initial price $S_{0}^{1}=1$. At time 1 its value $S_{1}^{1}$ is a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ of the form

$S_{1}^{1}=\exp (\sigma Z+m), \quad m \in \mathbb{R}, \sigma>0,$

where $Z \sim \mathcal{N}(0,1)$. Assume that there is a riskless numéraire $S^{0}$ with $S_{0}^{0}=S_{1}^{0}=1$. Show that there is no arbitrage in this model.

[Hint: You may find it useful to consider a density of the form $\exp (\tilde{\sigma} Z+\tilde{m})$ and find suitable $\tilde{m}$ and $\tilde{\sigma}$. You may use without proof that if $X$ is a normal random variable then $\mathbb{E}\left(e^{X}\right)=\exp \left(\mathbb{E}(X)+\frac{1}{2} \operatorname{Var}(X)\right)$.]

(d) Now consider a multi-period model with one risky asset $S^{1}$ having a non-random initial price $S_{0}^{1}=1$ and a price process $\left(S_{t}^{1}\right)_{t \in\{0, \ldots, T\}}$ of the form

$S_{t}^{1}=\prod_{i=1}^{t} \exp \left(\sigma_{i} Z_{i}+m_{i}\right), \quad m_{i} \in \mathbb{R}, \sigma_{i}>0$

where $Z_{i}$ are i.i.d. $\mathcal{N}(0,1)$-distributed random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that there is a constant riskless numéraire $S^{0}$ with $S_{t}^{0}=1$ for all $t \in\{0, \ldots, T\}$. Show that there exists no arbitrage in this model.

comment
• Paper 3, Section II, K

In the Black-Scholes model the price $\pi(C)$ at time 0 for a European option of the form $C=f\left(S_{T}\right)$ with maturity $T>0$ is given by

$\pi(C)=e^{-r T} \int_{-\infty}^{\infty} f\left(S_{0} \exp \left(\sigma \sqrt{T} y+\left(r-\frac{1}{2} \sigma^{2}\right) T\right)\right) \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y$

(a) Find the price at time 0 of a European call option with maturity $T>0$ and strike price $K>0$ in terms of the standard normal distribution function. Derive the put-call parity to find the price of the corresponding European put option.

(b) The digital call option with maturity $T>0$ and strike price $K>0$ has payoff given by

$C_{\mathrm{digCall}}= \begin{cases}1 & \text { if } S_{T} \geqslant K \\ 0 & \text { otherwise }\end{cases}$

What is the value of the option at any time $t \in[0, T]$ ? Determine the number of units of the risky asset that are held in the hedging strategy at time $t$.

(c) The digital put option with maturity $T>0$ and strike price $K>0$ has payoff

$C_{\text {digPut }}= \begin{cases}1 & \text { if } S_{T}

Find the put-call parity for digital options and deduce the Black-Scholes price at time 0 for a digital put.

comment
• Paper 4, Section II, K

(a) Describe the (Cox-Ross-Rubinstein) binomial model. What are the necessary and sufficient conditions on the model parameters for it to be arbitrage-free? How is the equivalent martingale measure $\mathbb{Q}$ characterised in this case?

(b) Consider a discounted claim $H$ of the form $H=h\left(S_{0}^{1}, S_{1}^{1}, \ldots, S_{T}^{1}\right)$ for some function $h$. Show that the value process of $H$ is of the form

$V_{t}(\omega)=v_{t}\left(S_{0}^{1}, S_{1}^{1}(\omega), \ldots, S_{t}^{1}(\omega)\right)$

for $t \in\{0, \ldots, T\}$, where the function $v_{t}$ is given by

$v_{t}\left(x_{0}, \ldots, x_{t}\right)=\mathbb{E}_{\mathbb{Q}}\left[h\left(x_{0}, \ldots, x_{t}, x_{t} \cdot \frac{S_{1}^{1}}{S_{0}^{1}}, \ldots, x_{t} \cdot \frac{S_{T-t}^{1}}{S_{0}^{1}}\right)\right]$

You may use any property of conditional expectations without proof.

(c) Suppose that $H=h\left(S_{T}^{1}\right)$ only depends on the terminal value $S_{T}^{1}$ of the stock price. Derive an explicit formula for the value of $H$ at time $t \in\{0, \ldots, T\}$.

(d) Suppose that $H$ is of the form $H=h\left(S_{T}^{1}, M_{T}\right)$, where $M_{t}:=\max _{s \in\{0, \ldots, t\}} S_{s}^{1}$. Show that the value process of $H$ is of the form

$V_{t}(\omega)=v_{t}\left(S_{t}^{1}(\omega), M_{t}(\omega)\right)$

for $t \in\{0, \ldots, T\}$, where the function $v_{t}$ is given by

$v_{t}(x, m)=\mathbb{E}_{\mathbb{Q}}\left[g\left(x, m, S_{0}^{1}, S_{T-t}^{1}, M_{T-t}\right)\right]$

for a function $g$ to be determined.

comment

• Paper 1, Section II, K

(a) What does it mean to say that $\left(M_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

(b) Let $Y_{1}, Y_{2}, \ldots$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ with $Y_{i}>0 \mathbb{P}$-a.s. and $\mathbb{E}\left[Y_{i}\right]=1, i \geqslant 1$. Further, let

$M_{0}=1 \quad \text { and } \quad M_{n}=\prod_{i=1}^{n} Y_{i}, \quad n \geqslant 1$

Show that $\left(M_{n}\right)_{n \geqslant 0}$ is a martingale with respect to the filtration $\mathcal{F}_{n}=\sigma\left(Y_{1}, \ldots, Y_{n}\right)$.

(c) Let $X=\left(X_{n}\right)_{n \geqslant 0}$ be an adapted process with respect to a filtration $\left(\mathcal{F}_{n}\right)_{n \geqslant 0}$ such that $\mathbb{E}\left[\left|X_{n}\right|\right]<\infty$ for every $n$. Show that $X$ admits a unique decomposition

$X_{n}=M_{n}+A_{n}, \quad n \geqslant 0,$

where $M=\left(M_{n}\right)_{n \geqslant 0}$ is a martingale and $A=\left(A_{n}\right)_{n \geqslant 0}$ is a previsible process with $A_{0}=0$, which can recursively be constructed from $X$ as follows,

$A_{0}:=0, \quad A_{n+1}-A_{n}:=\mathbb{E}\left[X_{n+1}-X_{n} \mid \mathcal{F}_{n}\right]$

(d) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a super-martingale. Show that the following are equivalent:

(i) $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale.

(ii) $\mathbb{E}\left[X_{n}\right]=\mathbb{E}\left[X_{0}\right]$ for all $n \in \mathbb{N}$.

comment
• Paper 2, Section II, K

Consider the Black-Scholes model, i.e. a market model with one risky asset with price $S_{t}$ at time $t$ given by

$S_{t}=S_{0} \exp \left(\sigma B_{t}+\mu t\right),$

where $\left(B_{t}\right)_{t \geqslant 0}$ denotes a Brownian motion on $(\Omega, \mathcal{F}, \mathbb{P}), \mu>0$ the constant growth rate, $\sigma>0$ the constant volatility and $S_{0}>0$ the initial price of the asset. Assume that the riskless rate of interest is $r \geqslant 0$.

(a) Consider a European option $C=f\left(S_{T}\right)$ with expiry $T>0$ for any bounded, continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$. Use the Cameron-Martin theorem to characterize the equivalent martingale measure and deduce the following formula for the price $\pi_{C}$ of $C$ at time 0 :

$\pi_{C}=e^{-r T} \int_{-\infty}^{\infty} f\left(S_{0} \exp \left(\sigma \sqrt{T} y+\left(r-\frac{1}{2} \sigma^{2}\right) T\right)\right) \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y$

(b) Find the price at time 0 of a European option with maturity $T>0$ and payoff $C=\left(S_{T}\right)^{\gamma}$ for some $\gamma>1$. What is the value of the option at any time $t \in[0, T] ?$ Determine a hedging strategy (you only need to specify how many units of the risky asset are held at any time $t$ ).

comment
• Paper 3, Section II, $29 K$

Consider a multi-period model with asset prices $\bar{S}_{t}=\left(S_{t}^{0}, \ldots, S_{t}^{d}\right), t \in\{0, \ldots, T\}$, modelled on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and adapted to a filtration $\left(\mathcal{F}_{t}\right)_{t \in\{0, \ldots, T\}}$. Assume that $\mathcal{F}_{0}$ is $\mathbb{P}$-trivial, i.e. $\mathbb{P}[A] \in\{0,1\}$ for all $A \in \mathcal{F}_{0}$, and assume that $S^{0}$ is a $\mathbb{P}$-a.s. strictly positive numéraire, i.e. $S_{t}^{0}>0 \mathbb{P}$-a.s. for all $t \in\{0, \ldots, T\}$. Further, let $X_{t}=\left(X_{t}^{1}, \ldots, X_{t}^{d}\right)$ denote the discounted price process defined by $X_{t}^{i}:=S_{t}^{i} / S_{t}^{0}, t \in\{0, \ldots, T\}, i \in\{1, \ldots, d\}$.

(a) What does it mean to say that a self-financing strategy $\bar{\theta}$ is an arbitrage?

(b) State the fundamental theorem of asset pricing.

(c) Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ which is equivalent to $\mathbb{P}$ and for which $\mathbb{E}_{\mathbb{Q}}\left[\left|X_{t}\right|\right]<\infty$ for all $t$. Show that the following are equivalent:

(i) $\mathbb{Q}$ is a martingale measure.

(ii) If $\bar{\theta}=\left(\theta^{0}, \theta\right)$ is self-financing and $\theta$ is bounded, i.e. $\max _{t=1, \ldots, T}\left|\theta_{t}\right| \leqslant c<\infty$ for a suitable $c>0$, then the value process $V$ of $\bar{\theta}$ is a $\mathbb{Q}$-martingale.

(iii) If $\bar{\theta}=\left(\theta^{0}, \theta\right)$ is self-financing and $\theta$ is bounded, then the value process $V$ of $\bar{\theta}$ satisfies

$\mathbb{E}_{\mathbb{Q}}\left[V_{T}\right]=V_{0}$

[Hint: To show that (iii) implies (i) you might find it useful to consider self-financing strategies $\bar{\theta}=\left(\theta^{0}, \theta\right)$ with $\theta$ of the form

$\theta_{s}:= \begin{cases}\mathbf{1}_{A} & \text { if } s=t \\ 0 & \text { otherwise }\end{cases}$

for any $A \in \mathcal{F}_{t-1}$ and any $t \in\{1, \ldots, T\}$.]

(d) Prove that if there exists a martingale measure $\mathbb{Q}$ satisfying the conditions in (c) then there is no arbitrage.

comment
• Paper 4, Section II, K

Consider a utility function $U: \mathbb{R} \rightarrow \mathbb{R}$, which is assumed to be concave, strictly increasing and twice differentiable. Further, $U$ satisfies

$\left|U^{\prime}(x)\right| \leqslant c|x|^{\alpha}, \quad \forall x \in \mathbb{R},$

for some positive constants $c$ and $\alpha$. Let $X$ be an $\mathcal{N}\left(\mu, \sigma^{2}\right)$-distributed random variable and set $f(\mu, \sigma):=\mathbb{E}[U(X)]$.

(a) Show that

$\mathbb{E}\left[U^{\prime}(X)(X-\mu)\right]=\sigma^{2} \mathbb{E}\left[U^{\prime \prime}(X)\right]$

(b) Show that $\frac{\partial f}{\partial \mu}>0$ and $\frac{\partial f}{\partial \sigma} \leqslant 0$. Discuss this result in the context of meanvariance analysis.

(c) Show that $f$ is concave in $\mu$ and $\sigma$, i.e. check that the matrix of second derivatives is negative semi-definite. [You may use without proof the fact that if a $2 \times 2$ matrix has nonpositive diagonal entries and a non-negative determinant, then it is negative semi-definite.]

comment

• Paper 1, Section II, J

(a) What does it mean to say that $\left(X_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

(b) Let $\Delta_{0}, \Delta_{1}, \ldots$ be independent random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{E}\left[\left|\Delta_{i}\right|\right]<\infty$ and $\mathbb{E}\left[\Delta_{i}\right]=0, i \geqslant 0$. Further, let

$X_{0}=\Delta_{0} \quad \text { and } \quad X_{n+1}=X_{n}+\Delta_{n+1} f_{n}\left(X_{0}, \ldots, X_{n}\right), \quad n \geqslant 0$

where

$f_{n}\left(x_{0}, \ldots, x_{n}\right)=\frac{1}{n+1} \sum_{i=0}^{n} x_{i}$

Show that $\left(X_{n}\right)_{n \geqslant 0}$ is a martingale with respect to the natural filtration $\mathcal{F}_{n}=$ $\sigma\left(X_{0}, \ldots, X_{n}\right)$.

(c) State and prove the optional stopping theorem for a bounded stopping time $\tau$.

comment
• Paper 2, Section II, $27 \mathrm{~J}$

(a) What is a Brownian motion?

(b) Let $\left(B_{t}, t \geqslant 0\right)$ be a Brownian motion. Show that the process $\tilde{B}_{t}:=\frac{1}{c} B_{c^{2} t}$, $c \in \mathbb{R} \backslash\{0\}$, is also a Brownian motion.

(c) Let $Z:=\sup _{t \geqslant 0} B_{t}$. Show that $c Z \stackrel{(d)}{=} Z$ for all $c>0$ (i.e. $c Z$ and $Z$ have the same laws). Conclude that $Z \in\{0,+\infty\}$ a.s.

(d) Show that $\mathbb{P}[Z=+\infty]=1$.

comment
• Paper 3, Section II, J

(a) State the fundamental theorem of asset pricing for a multi-period model.

Consider a market model in which there is no arbitrage, the prices for all European put and call options are already known and there is a riskless asset $S^{0}=\left(S_{t}^{0}\right)_{t \in\{0, \ldots, T\}}$ with $S_{t}^{0}=(1+r)^{t}$ for some $r \geqslant 0$. The holder of a so-called 'chooser option' $C\left(K, t_{0}, T\right)$ has the right to choose at a preassigned time $t_{0} \in\{0,1, \ldots, T\}$ between a European call and a European put option on the same asset $S^{1}$, both with the same strike price $K$ and the same maturity $T$. [We assume that at time $t_{0}$ the holder will take the option having the higher price at that time.]

(b) Show that the payoff function of the chooser option is given by

$C\left(K, t_{0}, T\right)= \begin{cases}\left(S_{T}^{1}-K\right)^{+} & \text {if } S_{t_{0}}^{1}>K(1+r)^{t_{0}-T} \\ \left(K-S_{T}^{1}\right)^{+} & \text {otherwise }\end{cases}$

(c) Show that the price $\pi\left(C\left(K, t_{0}, T\right)\right)$ of the chooser option $C\left(K, t_{0}, T\right)$ is given by

$\pi\left(C\left(K, t_{0}, T\right)\right)=\pi(E C(K, T))+\pi\left(E P\left(K(1+r)^{t_{0}-T}, t_{0}\right)\right),$

where $\pi(E C(K, T))$ and $\pi(E P(K, T))$ denote the price of a European call and put option, respectively, with strike $K$ and maturity $T$.

comment
• Paper 4, Section II, J

(a) Describe the (Cox-Ross-Rubinstein) binomial model. When is the model arbitragefree? How is the equivalent martingale measure characterised in this case?

(b) What is the price and the hedging strategy for any given contingent claim $C$ in the binomial model?

(c) For any fixed $0 and $K>0$, the payoff function of a forward-start-option is given by

$\left(\frac{S_{T}^{1}}{S_{t}^{1}}-K\right)^{+}$

Find a formula for the price of the forward-start-option in the binomial model.

comment

• Paper 1, Section II, 28K

(a) What is a Brownian motion?

(b) State the Brownian reflection principle. State the Cameron-Martin theorem for Brownian motion with constant drift.

(c) Let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Show that

$\mathbb{P}\left(\max _{0 \leqslant s \leqslant t}\left(W_{s}+a s\right) \leqslant b\right)=\Phi\left(\frac{b-a t}{\sqrt{t}}\right)-e^{2 a b} \Phi\left(\frac{-b-a t}{\sqrt{t}}\right)$

where $\Phi$ is the standard normal distribution function.

(d) Find

$\mathbb{P}\left(\max _{u \geqslant t}\left(W_{u}+a u\right) \leqslant b\right)$

comment
• Paper 2, Section II, K

In the context of the Black-Scholes model, let $S_{0}$ be the initial price of the stock, and let $\sigma$ be its volatility. Assume that the risk-free interest rate is zero and the stock pays no dividends. Let $\operatorname{EC}\left(S_{0}, K, \sigma, T\right)$ denote the initial price of a European call option with strike $K$ and maturity date $T$.

(a) Show that the Black-Scholes formula can be written in the form

$\mathrm{EC}\left(S_{0}, K, \sigma, T\right)=S_{0} \Phi\left(d_{1}\right)-K \Phi\left(d_{2}\right)$

where $d_{1}$ and $d_{2}$ depend on $S_{0}, K, \sigma$ and $T$, and $\Phi$ is the standard normal distribution function.

(b) Let $\operatorname{EP}\left(S_{0}, K, \sigma, T\right)$ be the initial price of a put option with strike $K$ and maturity $T$. Show that

$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(S_{0}, K, \sigma, T\right)+K-S_{0}$

(c) Show that

$\operatorname{EP}\left(S_{0}, K, \sigma, T\right)=\operatorname{EC}\left(K, S_{0}, \sigma, T\right)$

(d) Consider a European contingent claim with maturity $T$ and payout

$S_{T} I_{\left\{S_{T} \leqslant K\right\}}-K I_{\left\{S_{T}>K\right\}}$

Assuming $K>S_{0}$, show that its initial price can be written as $\mathrm{EC}\left(S_{0}, K, \hat{\sigma}, T\right)$ for a volatility parameter $\hat{\sigma}$ which you should express in terms of $S_{0}, K, \sigma$ and $T$.

comment
• Paper 3, Section II, K

Consider the following two-period market model. There is a risk-free asset which pays interest at rate $r=1 / 4$. There is also a risky stock with prices $\left(S_{t}\right)_{t \in\{0,1,2\}}$ given by

The above diagram should be read as

$\mathbb{P}\left(S_{1}=10 \mid S_{0}=7\right)=2 / 3, \quad \mathbb{P}\left(S_{2}=14 \mid S_{1}=10\right)=1 / 2$

and so forth.

(a) Find the risk-neutral probabilities.

(b) Consider a European put option with strike $K=10$ expiring at time $T=2$. What is the initial no-arbitrage price of the option? How many shares should be held in each period to replicate the payout?

(c) Now consider an American put option with the same strike and expiration date. Find the optimal exercise policy, assuming immediate exercise is not allowed. Would your answer change if you were allowed to exercise the option at time 0 ?

comment
• Paper 4, Section II, K

Let $U$ be concave and strictly increasing, and let $\mathcal{A}$ be a vector space of random variables. For every random variable $Z$ let

$F(Z)=\sup _{X \in \mathcal{A}} \mathbb{E}[U(X+Z)]$

and suppose there exists a random variable $X_{Z} \in \mathcal{A}$ such that

$F(Z)=\mathbb{E}\left[U\left(X_{Z}+Z\right)\right]$

For a random variable $Y$, let $\pi(Y)$ be such that $F(Y-\pi(Y))=F(0)$.

(a) Show that for every constant $a$ we have $\pi(Y+a)=\pi(Y)+a$, and that if $\mathbb{P}\left(Y_{1} \leqslant Y_{2}\right)=1$, then $\pi\left(Y_{1}\right) \leqslant \pi\left(Y_{2}\right)$. Hence show that if $\mathbb{P}(a \leqslant Y \leqslant b)=1$ for constants $a \leqslant b$, then $a \leqslant \pi(Y) \leqslant b .$

(b) Show that $Y \mapsto \pi(Y)$ is concave, and hence show $t \mapsto \pi(t Y) / t$ is decreasing for $t>0$.

(c) Assuming $U$ is continuously differentiable, show that $\pi(t Y) / t$ converges as $t \rightarrow 0$, and that there exists a random variable $X_{0}$ such that

$\lim _{t \rightarrow 0} \frac{\pi(t Y)}{t}=\frac{\mathbb{E}\left[U^{\prime}\left(X_{0}\right) Y\right]}{\mathbb{E}\left[U^{\prime}\left(X_{0}\right)\right]}$

comment

• Paper 1, Section II, $26 K$

(i) What does it mean to say that $\left(X_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

(ii) If $Y$ is an integrable random variable and $Y_{n}=E\left[Y \mid \mathcal{F}_{n}\right]$, prove that $\left(Y_{n}, \mathcal{F}_{n}\right)$ is a martingale. [Standard facts about conditional expectation may be used without proof provided they are clearly stated.] When is it the case that the limit $\lim _{n \rightarrow \infty} Y_{n}$ exists almost surely?

(iii) An urn contains initially one red ball and one blue ball. A ball is drawn at random and then returned to the urn with a new ball of the other colour. This process is repeated, adding one ball at each stage to the urn. If the number of red balls after $n$ draws and replacements is $X_{n}$, and the number of blue balls is $Y_{n}$, show that $M_{n}=h\left(X_{n}, Y_{n}\right)$ is a martingale, where

$h(x, y)=(x-y)(x+y-1)$

Does this martingale converge almost surely?

comment
• Paper 2, Section II, $27 \mathrm{~K}$

(i) What is Brownian motion?

(ii) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is Brownian motion, and the price $S_{t}$ at time $t$ of a risky asset is given by

$S_{t}=S_{0} \exp \left\{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t\right\}$

where $\mu>0$ is the constant growth rate, and $\sigma>0$ is the constant volatility of the asset. Assuming that the riskless rate of interest is $r>0$, derive an expression for the price at time 0 of a European call option with strike $K$ and expiry $T$, explaining briefly the basis for your calculation.

(iii) With the same notation, derive the time-0 price of a European option with expiry $T$ which at expiry pays

$\left\{\left(S_{T}-K\right)^{+}\right\}^{2} / S_{T}$

comment
• Paper 3, Section II, $26 \mathrm{~K}$

A single-period market consists of $n$ assets whose prices at time $t$ are denoted by $S_{t}=\left(S_{t}^{1}, \ldots, S_{t}^{n}\right)^{T}, t=0,1$, and a riskless bank account bearing interest rate $r$. The value of $S_{0}$ is given, and $S_{1} \sim N(\mu, V)$. An investor with utility $U(x)=-\exp (-\gamma x)$ wishes to choose a portfolio of the available assets so as to maximize the expected utility of her wealth at time 1. Find her optimal investment.

What is the market portfolio for this problem? What is the beta of asset $i$ ? Derive the Capital Asset Pricing Model, that

Excess return of asset $i=$ Excess return of market portfolio $\times \beta_{i}$.

The Sharpe ratio of a portfolio $\theta$ is defined to be the excess return of the portfolio $\theta$ divided by the standard deviation of the portfolio $\theta$. If $\rho_{i}$ is the correlation of the return on asset $i$ with the return on the market portfolio, prove that

Sharpe ratio of asset $i=$ Sharpe ratio of market portfolio $\times \rho_{i}$.

comment
• Paper 4, Section II, $26 \mathrm{~K}$

(i) An investor in a single-period market with time- 0 wealth $w_{0}$ may generate any time-1 wealth $w_{1}$ of the form $w_{1}=w_{0}+X$, where $X$ is any element of a vector space $V$ of random variables. The investor's objective is to maximize $E\left[U\left(w_{1}\right)\right]$, where $U$ is strictly increasing, concave and $C^{2}$. Define the utility indifference price $\pi(Y)$ of a random variable $Y$.

Prove that the map $Y \mapsto \pi(Y)$ is concave. [You may assume that any supremum is attained.]

(ii) Agent $j$ has utility $U_{j}(x)=-\exp \left(-\gamma_{j} x\right), j=1, \ldots, J$. The agents may buy for time- 0 price $p$ a risky asset which will be worth $X$ at time 1 , where $X$ is random and has density

$f(x)=\frac{1}{2} \alpha e^{-\alpha|x|}, \quad-\infty

Assuming zero interest, prove that agent $j$ will optimally choose to buy

$\theta_{j}=-\frac{\sqrt{1+p^{2} \alpha^{2}}-1}{\gamma_{j} p}$

units of the risky asset at time 0 .

If the asset is in unit net supply, if $\Gamma^{-1} \equiv \sum_{j} \gamma_{j}^{-1}$, and if $\alpha>\Gamma$, prove that the market for the risky asset will clear at price

$p=-\frac{2 \Gamma}{\alpha^{2}-\Gamma^{2}}$

What happens if $\alpha \leqslant \Gamma ?$

comment

• Paper 1, Section II, K

Suppose that $\bar{S}_{t} \equiv\left(S_{t}^{0}, \ldots, S_{t}^{d}\right)^{T}$ denotes the vector of prices of $d+1$ assets at times $t=0,1, \ldots$, and that $\bar{\theta}_{t} \equiv\left(\theta_{t}^{0}, \ldots, \theta_{t}^{d}\right)^{T}$ denotes the vector of the numbers of the $d+1$ different assets held by an investor from time $t-1$ to time $t$. Assuming that asset 0 is a bank account paying zero interest, that is, $S_{t}^{0}=1$ for all $t \geqslant 0$, explain what is meant by the statement that the portfolio process $\left(\bar{\theta}_{t}\right)_{t \geqslant 0}$ is self-financing. If the portfolio process is self-financing, prove that for any $t>0$

$\bar{\theta}_{t} \cdot \bar{S}_{t}-\bar{\theta}_{0} \cdot \bar{S}_{0}=\sum_{j=1}^{t} \theta_{j} \cdot \Delta S_{j}$

where $S_{j} \equiv\left(S_{j}^{1}, \ldots, S_{j}^{d}\right)^{T}, \Delta S_{j}=S_{j}-S_{j-1}$, and $\theta_{j} \equiv\left(\theta_{j}^{1}, \ldots, \theta_{j}^{d}\right)^{T}$.

Suppose now that the $\Delta S_{t}$ are independent with common $N(0, V)$ distribution. Let

$F(z)=\inf E\left[\sum_{t \geqslant 1}(1-\beta) \beta^{t}\left\{\left(\bar{\theta}_{t} \cdot \bar{S}_{t}-\bar{\theta}_{0} \cdot \bar{S}_{0}\right)^{2}+\sum_{j=1}^{t}\left|\Delta \theta_{j}\right|^{2}\right\} \mid \theta_{0}=z\right]$

where $\beta \in(0,1)$ and the infimum is taken over all self-financing portfolio processes $\left(\bar{\theta}_{t}\right)_{t \geqslant 0}$ with $\theta_{0}^{0}=0$. Explain why $F$ should satisfy the equation

$F(z)=\beta \inf _{y}\left[y \cdot V y+|y-z|^{2}+F(y)\right]$