Stochastic Processes and

Advanced Mathematical Finance

Homework

Due Wed, November 10, 2010

- If you buy a lottery ticket in 50 independent lotteries,
and in each lottery your chance of winning a prize is
$1\u2215100$,
write down and evaluate the probability of winning and also approximate
the probability using the Central Limit Theorem.
- exactly one prize,
- at least one prize,
- at least two prizes.

Explain with a reason whether or not you expect the approximation to be a good approximation.

- A bank has $1,000,000 available to make for car loans. The loans are in random amounts uniformly distributed from $5,000 to $20,000. Make a model for the total amount that the bank loans out. How many loans can the bank make with 99% confidence that it will have enough money available?
- Let $W\left(t\right)$
be standard Brownian motion.
- Find the probability that $0<W\left(1\right)<1$.
- Find the probability that $0<W\left(1\right)<1$ and $1<W\left(2\right)-W\left(1\right)<3$.
- Find the probability that $0<W\left(1\right)<1$ and $1<W\left(2\right)-W\left(1\right)<3$ and $0<W\left(3\right)-W\left(2\right)<1\u22152$.

- Let $W\left(t\right)$
be standard Brownian motion.
- Evaluate the probability that $W\left(5\right)\le 3$ given that $W\left(1\right)=1$.
- Find the number $c$ such that $Pr\left[W\left(9\right)>c|W\left(1\right)=1\right]=0.10$.

- Use your Homework 5 record of a $100$-flip coin flip sequence. Scoring ${Y}_{i}=+1$ for each Head and ${Y}_{i}=-1$ for each Tail on each flip, keep track of the accumulated sum ${T}_{n}={\sum}_{i=1}^{n}{Y}_{i}$ for $n=1,\dots ,100$ representing the net fortune at any time. Plot the resulting ${T}_{n}$ versus $n$ on the interval $\left[0,100\right]$. Finally, using $N=10$ plot the rescaled approximation ${W}_{10}\left(t\right)=\left(1\u2215\sqrt{10}\right)S\left(10t\right)$ on the interval $\left[0,10\right]$ on the same set of axes.
- Let $Z$
be a single normally distributed random variable, with mean
$0$ and
variance $1$,
i.e. $Z\sim N\left(0,1\right)$.
Then consider the continuous time stochastic process
$X\left(t\right)=\sqrt{t}Z$.
- Using a normal random variable generator (from Excel, Maple, Mathematica, Octave, MATLAB, R etc. , all have one and probably the TI-89 or equivalent has one too), find sample values of $X\left(1\right)$, $X\left(2\right)$, $X\left(4\right)$ and $X\left(9\right)$.
- Explain why the distribution of $X\left(t\right)$ is normal with mean $0$ with variance $t$.
- Is $X\left(t\right)$ a Brownian motion? Explain why or why not.