MTH 4500: Introductory Financial Mathematics

# Midterm 2 (Practice 4)

Problem 1

Consider the path of the random walk that starts from the point $$(0,1)$$ and corresponds to the sequence $\omega=\left(+1,+1,+1,-1,-1, -1,-1, -1, +1,+1,+1,-1,-1,+1,+1,+1\right).$

• (a) Sketch the described path.

• (b) Determine the path $$\hat\omega$$ that is obtained from the path $$\omega$$ using the reflection principle.

• (c) Draw the path that corresponds to $$\hat\omega$$.

Problem 2

Assume an one period binomial model in which the initial stock price is $$S=80$$ and in each period the stock price can go either up by a factor of $$u=\frac{7}{4}$$ or down by a factor of $$d=\frac{3}{4}$$. Assume that the simple interest rate over one time period is $$r=\frac{1}{4}$$.

• (a) Determine the fair price of the European call option with strike $$K=80$$.

• (b) Assume that instead of the price determined in part (a), the European call option is trading at $$18$$. Prove that there is an arbitrage and explain how the arbitrage can be achieved.

Problem 3

There are two portfolios $$A$$ and $$B$$ consisting of European put and call options written on the same underlying asset and with the same expiration.

• The portfolio $$A$$ consists of $$5$$ European put options with strike $${USD} 145$$, $$9$$ European call options with strike $${USD} 145$$, and 7 European put options with strike $${USD} 125$$.

• The portfolio $$B$$ consists of $$7$$ European put options with strike $${USD} 135$$, $$9$$ European call options with strike $${USD} 135$$, and $$5$$ European put options with strike $${USD} 155$$.

Which portfolio is more expensive: $$A$$ or $$B$$?

Problem 4

For each $$k\in\{0,1,2,\dots, 30\}$$ the symbol $$S(k)$$ denotes the price of the stock at time $$k$$. A European call option with strike $${USD} 90$$ and expiration $$n=30$$ costs $${USD} 15$$. A European put option with strike $${USD} 100$$ and expiration $$30$$ costs $${USD} 11$$. Both options have the same stock as their underlying security. What is the price of the security whose payoff structure is \begin{eqnarray*} \mbox{Payoff} &=&\left\{\begin{array}{rl} 7 S(30)-{USD} 630,&\mbox{if }S(30) >{USD} 100,\newline \;S(30)-{USD} 30,& \mbox{if }{USD} 90\leq S(30)\leq {USD} 100,\newline {USD} 600-6 S(30),&\mbox{if }S(30)< {USD} 90?\end{array}\right. \end{eqnarray*}

Problem 5

Assume that the stock price follows a binomial model with $$n=30$$ steps. The initial price of the stock at time $$0$$ is $$S_0={USD} 8$$ and in each step the price of the stock goes up by the factor $$u=\frac{5}{2}$$ or down by the factor $$d=\frac{2}{5}$$. The interest rate is assumed to be $$0$$. Consider the derived security that pays $${USD} 1$$ if the price of the stock at time $$n$$ is exactly $${USD} 50$$ and during the time interval $$[0,30]$$ the price has reached the level $$B={USD} 125$$ or went above $$B={USD} 125$$. Determine the price of the described security.

Problem 6

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}=0&&\quad\mbox{and} \newline -32 < X_1+X_2+\cdots+X_k < 18& & \quad \mbox{for each }k\in\{1,2,\dots, 100\}.\end{eqnarray*}