Stochastic Exponential and Girsanov - Exercises

Exercise 1

probability space and $\mathcal{G}$ be a $\sigma$-algebra such that $\mathcal{G}\subseteq \mathcal{F}$. Let also $Z_{\infty} \in L^1$ be a random variable such that $Z_{\infty}>0$ almost surely and $E[Z_{\infty}]=1$. 1. Show that $Q:\mathcal{F}\to [0,1]$, defined as $Q[A]=E[1_A Z_\infty]$ defines a probability measure which is equivalent to $P$ and for which holds $E^{Q}[ X ]=E[ Z_\infty X ]$ for every positive random variable $X$, 2. Show that for every $\mathcal{G}$-measurable positive random variable $X$, it holds $E^{Q}[ X ]=E[ E[ Z_\infty|\mathcal{G} ] X]$. 3. Let $\mathcal{H}$ be a $\sigma$-algebra such that $\mathcal{H}\subseteq \mathcal{G}\subseteq \mathcal{F}$ and $X$ be a $\mathcal{G}$-measurable positive random variable. Show that it holds

$$E^Q\left[ X|\mathcal{H} \right]=\frac{1}{E\left[ Z_\infty|\mathcal{H} \right]}E\left[ Z_\infty X |\mathcal{H} \right]$$

Let $\mathbb{F}=(\mathcal{F}_t)$ be a filtration. Define the process $Z$ by $Z_t=E[Z_{\infty}|\mathcal{F}_t]$ for every $t=1,\ldots$. Show that a process $X$ is a martingale with respect to the measure $Q$ if and only if the process $ZX$ is a martingale with respect to the measure $P$.

Exercise 2

and $\mathbb{F}=(\mathcal{F}_t)_{0\leq t\leq T}$ a right-continuous filtration and $B=(B_t)_{0\leq t\leq T}$ be a Brownian motion adapted to $\mathbb{F}$. We define the process

\[X_t = 1+\int_{0}^{t}Z_s dB_s, \quad 0\leq t\leq T\]

for some progressive process $Z=(Z_t)_{0\leq t\leq T}$ such that

\[E\left[ \int_{0}^{T}Z_s^2 d\langle B\rangle_s \right]=E\left[ \int_{0}^{T}Z_s^2 ds \right]<\infty\]

We know that $X$ is then a continuous square integrable martingale. We suppose that

\[0<\varepsilon < X_t, \quad \text{almost surely}\]

for every $0\leq t\leq T$ and some $\varepsilon>0$. 1. Define the process $q=(q_t)_{0\leq t\leq T}$ as

$$q_t=\frac{Z_t}{X_t},\quad 0\leq t\leq T$$

Show that

$$E\left[ \int_{0}^{T}q_s^2 ds  \right]<\infty$$

Applying Ito's formula to the process $Y=(Y_t)_{0\leq t\leq T}$ given by

\[Y_t=\ln\left( X_t \right), \quad 0\leq t\leq T\]

show that

\[X_t=\exp\left( \int_{0}^{t}q_s dB_s -\frac{1}{2}\int_{0}^{t}q_s^2 ds \right)\]

Exercise 3

The financial markets admits an arbitrage if an only if there exists a strategy $\alpha$ such that

\[P\left[ \sum_{t=1}^T \alpha_s\cdot (S_s - S_{s-1}) \geq 0 \right] = 1 \quad \text{and}\quad P\left[ \sum_{t=1}^T \alpha_s\cdot (S_s - S_{s-1}) > 0 \right] >0\]
If there exists a probability measure $Q\sim P$ such that each stock $S^k$ is a martingale under $Q$, then the financial market does not admit an arbitrage. (Hint: Assume that there exists such a martingale measure $Q$ and an arbitrage strategy $\alpha$ and show that it is a contradiction.) The reciprocal is also true and the proof relies on Hahn Banach separation theorem, and is called the fundamental theorem of asset pricing. Fundamental Theorem of Asset Pricing: The financial market does not admit an arbitrage strategy if and only if there exists a probability measure $Q\sim P$ such that each stock $S^k$ is a martingale under $Q$.
Assume the fundamental theorem of asset pricing and consider the following market with a single stock $S=S^1$. We consider a binomial model with iid random variables $Y_1, \ldots Y_T$ with $P[Y_1 = 1] = p$ and $P[Y_1 = -1] = 1-p$ for $0<p<1$. We specify the stock price evolution as follows

\[ S_0>0 \quad \text{and} \quad S_t = \begin{cases} S_{t-1}(1+u) & \text{if }Y_t = 1\\ S_{t-1}(1+d) & \text{if }Y_t = -1 \end{cases} \]

where $-1<d <u$ so that the stock price is always positive. In other terms the stock increases by $u/100$ or decreases by $d/100$ percent each time according to $Y$. - Show that the market is arbitrage free if and only if $d<0<u$ (assume the fundamental theorem of asset pricing above). - Furthermore, show that if it is arbitrage free, there exists a unique measure $Q \sim P$ for which $S$ is a martingale measure, and under this measure the returns of the stock
```
$$R_{t+1} = \frac{S_{t+1} - S_t}{S_t}, \quad t=0, \ldots, T-1$$

are iid with $Q[R_t = u] = -\frac{d}{u-d}$
```

Exercise 4 (BSDE)

the auxiliary function $g$ is affine, that is

\[g(Y,Z) = bY + cZ - a\]

where $a, b$ and $c$ are progressively measurable processes such that each of them are uniformly bounded. 1. Show that if $\xi$ is in $L^2$ and $g =0$, then the BSDE [eq:01] has a solution. 2. Consider that $c= a = 0$, proceeding to the variable change $\tilde{Y} = e^{\int b ds}Y$, write the resulting BSDE for $\tilde{Y}$, solve it and derive the explicit solution for $Y$. 3. Consider that $b = a = 0$, make the measure change with the stochastic exponential $\tilde{Y} = \mathcal{E}(\int c dW)Y$, write the resulting BSDE for $\tilde{Y}$, solve it and derive the explicit solution for $Y$. 4. Consider that $b= c = 0$, proceed to the variable change $\tilde{Y} = \tilde{Y} + \int ads$, write the resulting BSDE for $\tilde{Y}$, solve it and derive the explicit solution for $Y$. 5. From these three steps derive what is the explicit solution of the BSDE when $g(Y,Z) = bY + cZ - a$.