MTH 4500: Introductory Financial Mathematics

Final (Practice 2)

Problem 1

Assume an one period binomial model in which the initial stock price is \( S=150 \) and in each period the stock price can go either up by a factor of \( u=\frac{7}{5} \) or down by a factor of \( d=\frac{2}{5} \). Assume that the simple interest rate over one time period is \( r=\frac{1}{5} \). Determine the price of the European call option with strike \( K=90 \).

Problem 2

Assume that the return \( K \) of a risky security depend on the market scenario in the following way: \[ \begin{array}{|c|c|c|}\hline \mbox{Scenario} & \mbox{Probability}& \mbox{Return }K \newline \hline \omega_1& 0.6& -20\% \newline \hline \omega_2&0.4 & 80\% \newline \hline\end{array} \] Determine the expected return and the risk of the given security.

Problem 3

Assume that there are three risky securities whose expected returns and covariance matrix are given by \[ m=\left[\begin{array}{ccc} 10\%&30\%&40\% \end{array}\right]\quad\quad\quad \mbox{and}\quad\quad\quad C=\left[\begin{array}{ccc}5 & 3&0\newline 3&2&0\newline 0&0&1 \end{array}\right].\] Assume that the rate of risk free return is \( R=10\% \). Determine the weights of the market portfolio.

Problem 4

Assume that the price of the stock follows the equation \( S(t)=S_0e^{7B_t} \), where \( B_t \) is the standard Brownian motion. Calculate the probability of the event \( \left\{S\left( 9\right)< 17 S\left(4\right)\right\} \).

Problem 5

Assume that the stock price is given by \(S(t)=S_0e^{\mu t+\sigma B_t}\), where \(S_0\), \(\mu\), and \(\sigma\) are given real numbers that satisfy \(S_0 > 0\) and \(\sigma > 0\). Assume that the risk-free interest rate is \(r\). Given positive real numbers \(T\), \(K_1\), and \(K_2\) that satisfy \(K_2 > K_1\), determine the price of the derived security that can be exercised only at time \(T\) and that pays \(K_2-K_1\), or \(S(T)-K_1\), or nothing, depending on whether the stock price \(S(T)\) is higher than \(K_2\), between \(K_1\) and \(K_2\), or below \(K_1\), respectively. In other words, the payoff function \(g\) for this security is given by \[g(x)=\left\{\begin{array}{ll} K_2-K_1, & \mbox{if } x > K_2,\newline x-K_1,& \mbox{if }x\in[K_1,K_2],\newline 0,&\mbox{if }x < K_1.\end{array}\right.\]

Problem 6

Given \( n \) risky securities, assume that the covariance matrix between their returns is \( C \). For each vector of weights \( w=[\begin{array}{cccc}w_1&w_2&\dots&w_n\end{array}] \) denote by \( \sigma(w) \) the risk of the portfolio whose weights are \( w_1 \), \( \dots \), \( w_n \). Find the vector of weights \( w \) that minimizes the function \( F(w)=\sigma^2(w) + 12\left(w_1^2+w_2^2+\cdots+w_n^2\right) \). We are assuming that all securities are risky (i.e. that all variances are non-zero) and that their covariance matrix is invertible.

Problem 7

Assume two period binomial model for bond prices. Assume that we know the values of \(B(0,2)\), \(B(0,1)\), \(B^u(1,2)\), \(B^u(1,3)\), \(B^d(1,2)\), \(B^d(1,3)\), \(B^{uu}(2,3)\), \(B^{ud}(2,3)\), \(B^{du}(2,3)\), and \(B^{dd}(2,3)\). Determine the value of \(B(0,3)\).

Problem 8

Assume that the risk free rate is \(r\) and that the stock price follows the equation \(S(t)=S_0e^{\mu t+\sigma B_t}\), where \(B_t\) is a standard Brownian motion and \(\mu\), \(S_0\), \(\sigma\), are constants (\(S_0 > 0\), \(\sigma > 0\)). Assume that \(T_1\), \(T_2\), and \(K\) are positive real numbers such that \(T_2 > T_1\). Determine the formula for the price of the Bermudan put option with strike \(K\) at time \(0\) that can be exercised either at time \(T_1\) or at time \(T_2\).