MTH 4500: Introductory Financial Mathematics

Final (Practice 3)

Problem 1

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 \( r=\frac{1}{4} \). Determine the price of the European call option with strike \( K=80 \).

Problem 2

Assume that there are only three stocks in the stock market called \( P \), \( Q \), and \( R \) and that their prices at time \( 0 \) are \( 10 \), \( 10 \), and \( 20 \), respectively. The covariance matrix between the returns on the stocks \( P \), \( Q \), and \( R \) is given by \[ C=\left[\begin{array}{ccc}2 & 1&0\newline 1&1&0\newline 0&0&1 \end{array}\right].\] Calculate the risk of the portfolio that consists of \( 13 \) shares of \( P \), \( 3 \) shares of \( Q \), and \( 2 \) shares of \( R \).

Problem 3

The bond with face value \( F \) and maturity \( T=26 \) years has two coupon payments in the amount \( C \) at the end of years \( 7 \) and \( 8 \). Determine the formula for the price of this bond if the continuous compounding rate is \( r \)?

Problem 4

Assume that the risk free rate is \( 0 \) and that the stock price is given by the equation \( S(t)=6e^{2 t + 2B(t)} \), where \( B(t) \) is the standard Brownian motion. Determine the price at time \( 0 \) of the European call option with strike \( K=6e^{18} \) and expiration \( 9 \).

Problem 5

Assume that \( B(t) \) is a Brownian motions and that \( S(t) \) is a random variable defined as \( S(t)=7 \cdot 2^{B(t)} \). Determine \( \mathbb P\left(S(11)> 4S(5)\right) \).

Problem 6

Assume that the covariance matrix between the returns of \( 100 \) given securities is \( C \). For each vector of weights \( w=[\begin{array}{cccc}w_1&w_2&\dots&w_{100}\end{array}] \) denote by \( \sigma(w) \) the risk of the portfolio whose weights are \( w_1 \), \( \dots \), \( w_{100} \). Find the vector of weights \( w \) that minimizes the function \[ F(w)=\sigma^2(w) + 9\left(w_1^2+w_2^2+\cdots+w_{50}^2\right)+ 2\left(w_{51}^2+w_{52}^2+\cdots+w_{100}^2\right).\] 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 that bond prices are following a binomial model and assume that we know the values of the zero-coupon bond with maturity \(4\) in all possible scenarios, i.e. assume that the values \(B(0,4)\), \(B^u(1,4)\), \(B^d(1,4)\), \(B^{uu}(2,4)\), \(B^{ud}(2,4)\), \(B^{du}(2,4)\) , \(B^{dd}(2,4)\), \(B^{uuu}(3,4)\), \(\dots\), \(B^{ddu}(3,4)\), and \(B^{ddd}(3,4)\) are known. Assume also that the short rates \(r(0)\), \(r^u(1)\), and \(r^d(1)\) are known. Determine the price of the zero-coupon bond with maturity \(2\).

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 \(K\) is a positive real number. Determine the formula for the price of the derived security whose payoff at time \(T\) depends on the stock price \(S(T)\) in the following way \[g(S(T))=\left(\mbox{max }\left\{S(T)-K, 0\right\}\right)^2.\]