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\noindent Math 236A, Fall 2012. 

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\centerline{\bf Homework Assignment 1}

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\noindent {\bf Due:} Oct.~12, 2012

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\noindent 1.  Assume $\Omega$ is a finite set. A {\it partition\/} of $\Omega$ comprises
a collection of nonempty disjoint sets whose union is $\Omega$. A partition $\Pi$ is a 
{\it refinement\/} of a partition $\Pi'$ if every set in $\Pi$ is a subset of a set in $\Pi'$. 

\noindent (a) Prove that the set of all $\sigma$-algebras on $\Omega$ is in 
natural one to one correspondence with a set of partitions of $\Omega$. Also, if the $\sigma$-algebras 
$\cF_0$ and $\cF_1$ correspond to respective partitions $\Pi_0$ and $\Pi_1$, then 
$\cF_0\subset \cF_1$ if and only if $\Pi_1$ is a refinement of $\Pi_0$. 

\noindent (b) Prove that a random variable is measurable with respect to 
a $\sigma$-algebra $\cF$ if and only if it is constant on every set of 
the corresponding partition. 

\noindent (c) Given a random variable on $(\Omega, 2^\Omega)$, and a sigma algebra $\cF$, 
find a formula for $E(X|\cF)$ in terms of the corresponding partition. 

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\noindent 2. Let $X$ be a standard Normal random variable. Compute $E(X|X_+)$. 

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\noindent 3. Assume that $P$ and $Q$ are equivalent probability measures on a space $(\Omega, \cF)$, 
and let $\Lambda=\frac{dQ}{dP}$. Let $X$ be a random variable with 
$E_P|X|<\infty$ and $E_Q|X|<\infty$. Prove a Bayes' formula: for a sigma algebra 
$\cG\subset \cF$, 
$$
E_Q(X|\cG)=\frac{E_P(X\Lambda|\cG)}{E_P(\Lambda|\cG)}, 
$$
$Q$-a.s. (First show that all conditional expectations exists and that the denominator is a.s.~nonzero.)

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\noindent 4. Assume that a discrete financial market is viable. The time interval is 
$t=0,\dots, T$. 

\noindent (a) Let $h$ be a contingent claim, for which a replicating strategy $\theta$ exists. 
Assume that $h$ is $\cF_t$ measurable for some $t<T$. Show that the discounted 
claim $e^{-r(T-t)}h$ is replicated by the same strategy at time $t$. 

\noindent (b) Assume that the market is also complete. 
Show that the filtration must be generated by the (vector of) prices alone, 
i.e., $\cF_t=\sigma(S_0, \dots, S_t)$. 

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\noindent 5. Assume that a discrete financial market is given by $S_t=(S_t^0, \dots, S_t^d)$, $t=0,\dots, T$. 
Assume also that the market is viable and complete. As in class, let 
$\tS_t=\frac{1}{S_t^0}S_t$. Moreover, let $\bS_t=\frac{1}{S_t^1}S_t$. 
Let $\tQ$ be the unique measure that makes 
$\tS_t$ a martingale. Denote
$$
\Lambda_t=\frac{S_t^1S_0^0}{S_t^0 S_0^1}.
$$
\noindent (a) Show that $\Lambda_t$ is a $Q$-martingale. 

\noindent (b)
Show that the unique measure $\bQ$ that makes $\bS_t$ a martingale is given by
$$
\frac{d\bQ}{d\tQ}=\Lambda_T.
$$

\noindent (c)
Show also that for any contingent claim $X$, 
$$
E_{\tQ}\left(X\cdot \frac {S_0^0}{S_T^0}\right)=E_{\bQ}\left(X\cdot \frac {S_0^1}{S_T^1}\right). 
$$
This is known as the {\it Principle of numeraire equivalence\/}. 

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\noindent 6. Assume a viable {\it binary\/} market (with one bond $\beta_t$ and one stock $S_t$, 
in which the conditional distribution of $S_{t+1}$ given $S_t$ is concentrated on two values), 
with natural filtration. Recall that such market is automatically complete. Let $Q$ be the risk-neutral 
measure. Assume that $Y_t$ is a $Q$-martingale. Prove that $Y_t$ is a martingale transform 
of $\tS_t$. 

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\noindent 7. Assume the CRR model. Every trader's dream is the {\it Buy Low Sell High\/} 
option 
$$
H=\max\{S_0,\dots, S_T\}-\min\{S_0,\dots, S_T\}. 
$$
Write a program that will compute the price of $H$ (which should be quite expensive)
for an arbitrary value of parameters $u$, $d$, $r$, $S_0$, and $T$. 

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\noindent 8. Assume the CRR model with parameters 
$r=\rho \frac sN $, $u=\exp(\rho\frac sN +\sigma \sqrt{\frac sN}$, 
$d=\exp(\rho\frac sN -\sigma \sqrt{\frac sN})$, and time interval $[0,N]$. Assume also $S_0=x$. 
Show that, as $N\to \infty$, the (time 0) price of a contingent claim $\Phi(S_N)$ converges to 
$$
E(\Phi(xe^Y))
$$
where $Y$ is a normal random variable. In the process, compute expectation and variance of $Y$. 

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\noindent 9. Assume that a discrete market has $r=1$ and $d=2$. 

\noindent (1) Assume that $S_0=(1,1,1)$ and that $T=1$, i.e., this is a one time-period market. 
Let $(S_1^1, S_1^2)$ be either $(2,4)$ or $(1/2,1/4)$. Determine the set of 
all arbitrage strategies. Is the market viable? 

\noindent (2) Assume that $S_0=(1,10,10)$ and that $T=2$. Assume that the pair $(S_1^1, S_1^2)$
of stocks at time 1 
is $(11,9)$, $(11,10)$, or $(8,11)$. From $t=1$ to $t=2$ the pair of stocks makes 
the following moves: from $(11,9)$ to $(14,8)$, $(10,13)$, or $(10,8)$; 
from $(11,10)$ to $(12,11)$, or $(10,9)$; and 
from $(8,11)$ to $(12,5)$, $(10,14)$, or $(6,11)$. Is the market viable and complete? If
so, determine the unique equivalent martingale measure. 







 

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