MTH 4500: Introductory Financial Mathematics

Final (Practice 5)

Problem 1

The initial price of the stock is \( S=140 \). The price is assumed to follow one-period binomial model. At time one the price can be either \( Su=224 \) or \( Sd=112 \). Assume that the simple interest rate over one time period is \( r=\frac{2}{5} \).

  • (a) What is the price of the European call option with strike \( K=140 \)?

  • (b) What is the price of the European put option with strike \( K=140 \)?

Problem 2

Assume that the risk free rate is \( 0 \) and that the stock price is given by the equation \[ S(t)=10e^{2 t + 2B(t)},\] where \( B(t) \) is the standard Brownian motion. Determine the price at time \( 0 \) of the European put option with strike \( K=10e^{18} \) and expiration \( 9 \).

Problem 3

There are only three stocks in the stock market called \( X \), \( Y \), and \( Z \). Their current prices are \( 10 \), \( 20 \), and \( 30 \), respectively. The covariance matrix between the returns on the stocks \( X \), \( Y \), and \( Z \) is \[ C=\left[\begin{array}{ccc}1 & 1&0\newline 1&2&0\newline 0&0&1 \end{array}\right].\] What is the risk of the portfolio that consists of \( 2 \) shares of \( X \), \( 3 \) shares of \( Y \), and \( 4 \) shares of \( Z \)?

Problem 4

The prices \( B(0,9) \), \( B(0,5) \), and \( B(5,9) \) of the zero coupon bonds with face value \( 1 \) are known at time \( 0 \) and are: \[ B(0,9)=0.22, \; B(0,5)=0.3,\; \mbox{and}\; B(5,9)=0.7.\] Prove that there is an arbitrage opportunity and explain how to achieve this arbitrage.

Problem 5

Assume that the price of a security follows Black-Scholes model and that its price at time \( 0 \) is \( 20 \). Consider a portfolio \( M \) that is long three European put options with strike \( 25 \) and expiration \( 1 \) and short three European call options with strike \( 25 \) and expiration \( 1 \) written on the given security. Sketch the graph of the price of the portfolio \( M \) at time \( 0 \) as the function of the interest rate \( r \).

Problem 6

A \( 5\times 5 \) matrix \( C \) is the covariance matrix between the returns of \( 5 \) risky securities. It is known that \[ \left[9,3,9,5,6\right] \cdot C=\left[8,8,8,8,8\right].\] Determine \( \overrightarrow u C^{-1}\overrightarrow u^T \), where \( \overrightarrow u\in\mathbb R^5 \) is the vector whose all components are \( 1 \).

Problem 7

Assume that \(X\) is a standard normal random variable and that \(Y\) is independent from \(X\) and satisfies \(\mathbb P(Y=1)=\mathbb P(Y=-1)=\frac12\). Assume that the return on the security \(A\) is \(K_A=e^X-1\) and the return on the security \(B\) is \(K_B=e^{XY}-1\). Determine the covariance matrix between the returns on portfolios \(A\) and \(B\) and the minimal possible variance for the return on a portfolio that consists entirely of shares of \(A\) and \(B\)?

Problem 8

Assume that the numbers \(\mu\), \(r\), \(\sigma_1\), \(\sigma_2\), \(T_1\), \(T_2\), \(K\), and \(S_0\) are given and that \(0 < T_1 < T_2\). The function \(\sigma(t)\) is defined as \begin{eqnarray*} \sigma(t)&=&\left\{\begin{array}{ll}\sigma_1, & \mbox{ if } t\in[0,T_1),\newline \sigma_2,&\mbox{ if }t\in[T_1,T_2].\end{array}\right. \end{eqnarray*} What is the price of European call option with strike \(K\) whose underlying security has the price that satisfies \(S(t)=S_0e^{\mu t+\sigma(t)B(t)}\), where \(B(t)\) is a standard Brownian motion?