MTH 4500: Introductory Financial Mathematics
# Midterm 2 (Practice 5)

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

Assume that the investor holds a portfolio that consists of one European call option with strike 20, two European call options with strike \( 60 \), and six European put options with strike \( 90 \). 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 \( 80 \)?

Assume an one period binomial model in which the initial stock price is \( S=60 \) 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 \( d=\frac{2}{3} \). Assume that the simple interest rate over one time period is \( r=\frac{1}{3} \).

**(a)**Determine the fair price of the European put option with strike \( K=60 \).**(b)**Assume that instead of the price determined in part (a), the European put option is trading at 11. Prove that there is an arbitrage and explain how the arbitrage can be achieved.

Calculate the price of binary call and put options with strike \(K\). Assume that the stock price follows the binomial model with parameters \(u\), \(d\), \(r\), \(K\). Binary call pays \(1\) if \(S(n)\geq K\) and 0 if \(S(n) < K\). Binary put pays \(1\) if \(S(n) < K\) and \(0\) if \(S(n)\geq K\). What is the price of a portfolio which consists of 1 binary call option and 1 binary put option with the same strike?

**(a)**Assume a two period binomial model in which the initial stock price is \( S=9 \) 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 \( r=100\% \). Determine the price of the put option of strike \( K=15 \) with expiration \( T=2 \) that can be exercised either at time \( 1 \) or at time \( 2 \).**(b)**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 down, in the third period it went down, and in the fourth period it went down. Determine the payoff of this barrier call option.

Determine the number of sequences \( \left(X_1, X_2, \dots, X_{100}\right) \) with terms in \( \{-1,1\} \) such that the following two conditions are satisfied \begin{eqnarray*}X_1+X_2+\cdots+ X_{100}=0\quad\mbox{ and}&& \newline \left(X_1+X_2+\cdots+X_{k}\right)^2< 30^2 \mbox{ for all }k\in\{1,2,\dots, 100\} .&&\end{eqnarray*}

Assume that instead of binomial model the stock price \( S_n \) evolves in discrete time according to the formula \( S_n=S_0+\sigma W_n \), where \( W_n \) is the standard random walk and \( S_0 \) and \( \sigma \) are given positive real numbers. In other words, the initial price is \( S_0 \) and in each step the stock price increases by \( \sigma \) or decreases by \( \sigma \). Assume that the interest is \(r\) in each step.

Determine the formulas for risk-neutral probabilities for this model and using these risk-neutral probabilities determine the formula for the price of European call option with strike \( K=S_0 \) and expiration 3 under the assumption that \( S_0 > 3\sigma \).