MTH 4500: Introductory Financial Mathematics
# Midterm 2 (Practice 1)

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
The initial price of the stock is \(S(0)=100\) and the return on the risk-less security is \(R=0\). In each period the stock price either increases by the factor of \(u=1.25\), or decreases by the factor of \(d=0.75\). Consider the option that at time \(100\) pays \(1\) dollar if for every \(k\in\{1,2,\dots, 100\}\) the price of the stock at time \(k\) is smaller than or equal to \( B_k=100\cdot \sqrt{ud}^k \). Determine the price of this option.

What is a forward contract? What is the formula for the forward price?

Consider a two-period binomial model. Let the current stock price be \(100\) and the risk-free rate is assumed to be \(0\). In each period the stock can go either up by \(10\%\) or down by \(10\%\). The strike price of a European call option is \(100\), and the option expires at the end of the second period.

**(a)**Find the stock price tree.**(b)**Determine the call price tree.**(c)**Determine the hedge ratio tree.

Let \(S(0)=50\), \(R=5\%\), \(U=30\%\), and \(D=-10\%\). Find the price of a European put option with strike price \(X=60\) to be exercised after \(n=3\) time steps.

Assume that the stock price is governed by a binomial model. The initial price of the stock is \(S(0)\), the return on a risk-less security over one period of time is \(R\) and in each period the stock price can either increase by the factor of \(u\) or decrease by the factor of \(d\), where \(u\) and \(d\) are positive real numbers such that \(u > d\). Using a no-arbitrage principle prove that \[d < 1+R < u.\]

Assume that the stock price is governed by a binomial model. The initial price of the stock is \(S(0)=100\). The return on a risk-less security over one period of time is \(R=10\%\). The parameters \(u\) and \(d\) are not known.

**(a)**Which of the following two options is more expensive: A European put option with strike \(120\) and expiration \(1\) or a European call option with strike \(120\) and expiration \(1\)?**(b)**Which of the following two options is more expensive: A European put option with strike \(110\) and expiration \(2\) or a European call option with strike \(120\) and expiration \(2\)?