MTH 4500: Introductory Financial Mathematics

# Final (Practice 4)

Problem 1

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 $$K=48$$.

Problem 2

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.

Problem 3

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.

Problem 4

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.

Problem 5

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 $$9$$.

Problem 6

• (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 $$\mathbb P\left( B_{5}+B_{7}\leq 6\right)$$.

Problem 7

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.

Problem 8

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?