MTH 4500: Introductory Financial Mathematics
# Final (Practice 4)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
**Problem 7**
**Problem 8**

In one period binomial model in which the initial stock price is \( S=72 \) 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 \( r=\frac{1}{3} \). Find the price of the European call and European put options with strike \(K=48\).

Suppose that there are four risky securities \( S_1 \), \( S_2 \), \( S_3 \), \( S_4 \) and suppose that among all portfolios whose expected return is \( 4 \) the minimal risk has the portfolio with the weights \( (10\%, 20\%, 20\%, 50\%) \). Suppose that among all portfolios with expected return \( 9 \) the minimal risk has the portfolio with weights \( (25\%, 25\%, 35\%, 15\%) \). Among all portfolios with expected return \( 6 \), find the one with the minimal risk.

There are three risky securities whose expected returns are \( 0.9 \), \( 0.8 \), and \( 0.3 \). If \( \mathbb E\left[K_1^2\right]=\mathbb E\left[K_2^2\right]=\mathbb E\left[K_3^2\right]=1.8 \) and \( \mathbb E\left[K_1K_2\right]=\mathbb E\left[K_2K_3\right]=\mathbb E\left[K_3K_1\right]=0.4 \), determine the covariance matrix between the returns of these three securities.

Assume that the prices \( B(0,5) \), \( B(0,2) \), and \( B(2,5) \) of the zero coupon bonds with face value \( 1 \) are known at time \( 0 \) and are: \[ B(0,5)=0.19, \; B(0,2)=0.3,\; \mbox{and}\; B(2,5)=0.6.\] Prove that there is an arbitrage opportunity in this bond market and explain how to achieve this arbitrage.

Assume that the risk free rate is \( 0 \) and that the stock price is given by the equation \( S(t)=8e^{2 t + 2B(t)} \), where \( B(t) \) is the standard Brownian motion. Determine the price at time \( 0 \) of the European put option with strike \( K=8e^{18} \) and expiration \( 9 \).

**(a)**If \( B_t \) is a Brownian motion, calculate \( \mathbb P\left(B_{7} \geq B_{5}\;\mbox{and}\; B_{5} \geq 11\right) \).**(b)**If \( B_t \) is a Brownian motion, calculate \( \mathbb P\left( B_{5}+B_{7}\leq 6\right) \).

Suppose that the prices of the zero coupon bonds in two period binomial model are \[ B(0,1)=0.8,\; B(0,2)=0.55,\; B^u(1,2)=0.75,\; B^d(1,2)=0.6.\] The bank is offering a loan to its customer with the following terms:

**\( 1^{\circ} \)**The customer receives \( 300 \) at time \( 0 \);**\( 2^{\circ} \)**The customer must pay the interest at time \( 1 \), so that after this interest payment is completed, the customer still owes \( 300 \);**\( 3^{\circ} \)**The customer has to pay the interest and principal at time \( 2 \).

The bank is offering the customer an insurance product called *insurance against high interest* that costs USD\( 9 \). If this additional insurance is purchased, the customer is protected in the case that interest payment becomes \( 150 \) or higher. If the interest portion of the payment becomes higher than \( 150 \), the customer only needs to pay \( 150 \).

Should the customer accept this insurance or not? Provide a mathematically rigorous justification for your answer.

Let \( S(t) \) denote the price of stock at time \( t \). A European barrier call with barrier \( B= 50 \), expiration \( T=31 \), and strike \( K=33 \) costs \( 12 \). The investor is interested in a product that, unlike this barrier call, offers some protection for the case that the stock goes above the barrier \( 50 \). The investor wants to buy an investment product called *Secured Barrier Call* whose payoff structure is
\begin{eqnarray*}
\mbox{Payoff} &=&\left\{\begin{array}{rl} S(31)- 33,&\mbox{if }S(31) > 33\mbox{ and } S(t)< 50\mbox{ for all }t\leq 31,\newline
50,& \mbox{if }S(t)\geq 50\mbox{ for some }t\leq 31,\newline
0,&\mbox{otherwise.}\end{array}\right.
\end{eqnarray*}

The market also provides the quotes for American binary call options with strikes \( 33 \) and \( 50 \).

- An American binary call with strike \( 33 \) and expiration \( 31 \) costs \( 0.73 \).
- An American binary call with strike \( 50 \) and expiration \( 31 \) costs \( 0.70 \).

An American binary call option with strike \( K \) and expiration \( 31 \) is the derived security whose payoff at time \( 31 \) is either \( 0 \) or \( 1 \). Its payoff is \( 1 \) dollar if the stock price at any time gets greater than or equal to \( K \) dollars. Otherwise, if the stock price stays below \( K \), the payoff is \( 0 \).

What is the price of the described * Secured Barrier Call*?