Financial mathematics

Let

the cumulative normal distribution function. Rewrite the Black-Scholes formula for the value of a European call in terms of $\Phi$. This is the way the Black-Scholes formula is written in finance books.

A European put that gives one the option to sell a share of stock at price $K$ at time $t_{E}$ has value $\left(K-S_{t_{E}}\right)^{+}$at time $t_{E}$. Find the present-day value of the European put at time 0 .

Carry out the details of the proof of Theorem 28.1 for Case 2.

If the utility function in Section 28.4 is $U(x)=\log x$ instead of $U(x)=x^{p}$, what is the optimal choice for $u$ ?

Let $a, b>0$, let $Y_{i}$ be i.i.d. random variables that take only the values $b$ and $-a$, and let $S_{n}=\sum_{i=1}^{n} Y_{i}$. Show that if $\mathbb {P}\left(Y_{1}=b\right)>0$ and $\mathbb {P}\left(Y_{1}=-a\right)>0$, there exists a probability measure $\mathbb {Q}$ equivalent to $\mathbb {P}$ under which $S_{n}$ is a martingale. Describe the Radon-Nikodym derivative of $\mathbb {Q}$ with respect to $\mathbb {P}$.

Suppose the interest rate $r$ is equal to 0 and an option $V$ has payoff

at time $t_{e}$. What is the price of $V$ at time 0 ?

Suppose the interest rate $r$ is equal to 0 . Let $U$ be the option that pays off $-\inf_{s \leq t_{e}} S_{s}$ at time $t_{e}$. What is the price of $U$ at time 0 ?

If $V$ is as in Exercise 28.6, then $U+V$ is the option that pays off the maximum of the stock price minus the minimum of the stock price, in other words, "buy low, sell high." Naturally such an option would be expensive. It is remarkable that there exists a trading strategy that can duplicate this payoff, even though the times when the maximum and minimum occur are not stopping times.