MTH 4500: Introductory Financial Mathematics
# Midterm 2 (Practice 3)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**

Assume an one period binomial model in which the initial stock price is \( S=72 \) and in each period the stock price can go either up by a factor of \( u=\frac{7}{3} \) or down by a factor of \( d=\frac{1}{3} \). Assume that the simple interest rate over one time period is \( r=\frac{1}{3} \). Determine the price of the European put option with strike \( K=48 \).

Assume that the price of the stock is governed by the binomial model. The initial price of the stock is \( S \) and in each step the price can go up by factor \( u \) or down by factor \( d \). Assume that \( u\cdot d=1 \). Consider the barrier European call option on the stock with strike \( K=S \), expiration \( 4 \), and barrier \( B=Su^3 \). This means that the option becomes knocked-out (worthless) if at any moment prior to its expiration \( 4 \), the stock price reaches \( Su^3 \) or raises above \( Su^3 \).

It turns out that in the first time period the stock price went up, in the second period it went up, in the third period it went down, and in the fourth period it went up. Determine the payoff of this barrier call option.

Consider the \( n \)-period binomial model for the price of the stock. Assume that the interest is simple and that the rate is \( r \). Let us denote by \( C_E(S,K) \) the price of the European call option with strike \( K \) and expiration \( n \) if the initial price of the stock is \( S \). Assume that \( S_2> S_1 \) and \( K_2< K_1 \).

**(a)**Prove that \( C_E\left( S_2,K_2\right)\geq C_E\left( S_1, K_1\right) \).**(b)**Prove that \( C_E\left( S_2,K_2\right)-C_E\left( S_1, K_1\right)\leq \left( S_2- S_1\right)+ \frac{ K_1-K_2}{\left(1+r\right)^n} \).

Assume that the investor holds a portfolio that consists of one European call option with strike 70, three European call options with strike \( 80 \), and nine European put options with strike \( 100 \). All the options have the same expiration. What is the payoff on this portfolio if the price of the underlying asset at the expiration becomes \( 90 \)?

Assume that the stock price is governed by a binomial model. The initial price of the stock is \( S=100 \), and in each period the stock can go either up by a factor of \( u \) or down by a factor of \( d \). The numbers \( u \) and \( d \) are not known. The interest rate is also not known. It is known that the price of European call option with strike \( 210 \) and expiration \( n=100 \) is \( 17 \) and the price of European call option with strike \( 350 \) and expiration \( n=100 \) is \( 11 \). Determine the price of the derived security whose payoff at time \( 100 \) is \( 140 \) if the stock price \( S_{100} \) is above \( 350 \); \( S_{100}-210 \) if the stock price \( S_{100} \) at time \( 100 \) is between \( 210 \) and \( 350 \); or nothing if the stock price \( S_{100} \) at time \( 100 \) is smaller than \( 210 \).

* Remark.*
In other words, the payoff function depends on the stock price \( S_{100} \) in the following way:
\[ \mbox{Payoff}\left(S_{100}\right)=\left\{\begin{array}{rl}140, &\mbox{ if } S_{100}> 350,\newline
S_{100}-210, &\mbox{ if } 210\leq S_{100}\leq 350,\newline
0, &\mbox{ if } S_{100}< 210.\end{array}\right. \]

Determine the number of sequences \( \left(X_1, X_2, \dots, X_{100}\right) \) with terms in \( \{-1,1\} \) such that \begin{eqnarray*}X_1+X_2+\cdots+ X_{100}&=&50\quad\mbox{and} \newline X_1+X_2+\cdots+X_k&\geq& 0\quad \mbox{for each }k\in\{1,2,\dots, 48\}.\end{eqnarray*}