Stochastic Processes and Financial Mathematics
(part one)
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1.5 Exercises on Chapter 1
On the one-period market
All these questions refer to the market defined in Section 1.2 and use notation \(u,d,p_u,p_d,r,s\) from that section.
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1.1 Suppose that our portfolio at time \(0\) has \(10\) units of cash and \(5\) units of stock. What is the value of this portfolio at time
\(1\)?
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1.2 Suppose that \(0<d<1+r<u\). Our portfolio at time \(0\) has \(x\geq 0\) units of cash and \(y\geq 0\) units of
stock, but we will have a debt to pay at time \(1\) of \(K>0\) units of cash.
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(a) Assuming that we don’t buy or sell anything at time \(0\), under what conditions on \(x,y,K\) can we be certain of paying off our
debt?
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(b) Suppose that do allow ourselves to trade cash and stocks at time \(0\). What strategy gives us the best chance of being able to pay
off our debt?
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1.3
Revision of probability and analysis
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1.4 Let \(Y\) be an exponential random variable with parameter \(\lambda >0\). That is, the probability density function of \(Y\) is
\[ f_X(x)= \begin {cases} \lambda e^{-\lambda x} & \text { for }x>0\\ 0 & \text { otherwise.}\\ \end {cases} \]
Calculate \(\E [X]\) and \(\E [X^2]\). Hence, show that \(\var (X)=\frac {1}{\lambda ^2}\).
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1.5 Let \((X_n)\) be a sequence of independent random variables such that
\[ \P [X_n=x]= \begin {cases} \frac {1}{n} & \text { if }x=n^2\\ 1-\frac {1}{n}& \text { if }x=0. \end {cases} \]
Show that \(\P [|X_n|>0]\to 0\) and \(\E [X_n]\to \infty \), as \(n\to \infty \).
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1.6 Let \(X\) be a normal random variable with mean \(\mu \) and variance \(\sigma ^2>0\). By calculating \(\P
[Y\leq y]\) (or otherwise) show that \(Y=\frac {X-\mu }{\sigma }\) is a normal random variable with mean \(0\) and variance \(1\).
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1.7 For which values of \(p\in (0,\infty )\) is \(\int _1^\infty x^{-p}\,dx\) finite?
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1.8 Which of the following sequences converge as \(n\to \infty \)? What do they converge too?
\(\seteqnumber{0}{1.}{2}\)
\begin{align*}
e^{-n} \quad \quad \quad \sin \l (\frac {n\pi }{2}\r ) \quad \quad \quad \frac {\cos (n\pi )}{n} \quad \quad \quad \sum \limits _{i=1}^n 2^{-i} \quad \quad \quad \sum \limits
_{i=1}^n\frac {1}{i}.
\end{align*}
Give brief reasons for your answers.
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1.9 Let \((x_n)\) be a sequence of real numbers such that \(\lim _{n\to \infty } x_n=0\). Show that \((x_n)\) has a subsequence
\((x_{n_r})\) such that \(\sum _{r=1}^\infty |x_{n_r}|<\infty \).