91Ó°ÊÓ

Compute \(E\left[B\left(t_{1}\right) B\left(t_{2}\right) B\left(t_{3}\right)\right]\) for \(t_{1}

Consider the random walk that in each \(\Delta t\) time unit either goes up or down the amount \(\sqrt{\Delta t}\) with respective probabilities \(p\) and \(1-p\), where \(p=\frac{1}{2}(1+\mu \sqrt{\Delta t})\). (a) Argue that as \(\Delta t \rightarrow 0\) the resulting limiting process is a Brownian motion process with drift rate \(\mu\). (b) Using part (a) and the results of the gambler's ruin problem (Section 4.5.1), compute the probability that a Brownian motion process with drift rate \(\mu\) goes up \(A\) before going down \(B, A>0, B>0\)

A stock is presently selling at a price of $$\$ 50$$ per share. After one time period, its selling price will (in present value dollars) be either $$\$ 150$$ or $$\$ 25 .$$ An option to purchase \(y\) units of the stock at time 1 can be purchased at cost \(c y\). (a) What should \(c\) be in order for there to be no sure win? (b) If \(c=4\), explain how you could guarantee a sure win. (c) If \(c=10\), explain how you could guarantee a sure win. (d) Use the arbitrage theorem to verify your answer to part (a).

The present price of a stock is 100 . The price at time 1 will be either 50,100 , or 200\. An option to purchase \(y\) shares of the stock at time 1 for the (present value) price \(k y\) costs \(c y\). (a) If \(k=120\), show that an arbitrage opportunity occurs if and only if \(c>80 / 3\). (b) If \(k=80\), show that there is not an arbitrage opportunity if and only if \(20 \leqslant\) \(c \leqslant 40\).

Let \(\\{X(t), t \geqslant 0\\}\) be Brownian motion with drift coefficient \(\mu\) and variance parameter \(\sigma^{2}\). That is, $$ X(t)=\sigma B(t)+\mu t $$ Let \(\mu>0\), and for a positive constant \(x\) let $$ \begin{aligned} T &=\operatorname{Min}\\{t: X(t)=x\\} \\ &=\operatorname{Min}\left\\{t: B(t)=\frac{x-\mu t}{\sigma}\right\\} \end{aligned} $$ That is, \(T\) is the first time the process \(\\{X(t), t \geqslant 0\\}\) hits \(x .\) Use the Martingale stopping theorem to show that $$ E[T]=x / \mu $$