Exam 12: Binomial Option Pricing

A stock is currently trading at $100. In each month, the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ . The risk-free rate of interest per month is 0.1668% in simple terms, i.e., an investment of $1 at the risk-free rate returns $1.001668 after one month. What is the early-exercise premium of a 100-strike, six-month American put option?

In order for a binomial tree to recombine, the following condition is necessary and sufficient:

Which of the following statements is valid?

The current price of a stock is $50. Every month, the stock price will rise by a factor of 1.2 or fall by a factor of 0.8; and a $1 risk-free investment will be worth $1.03. Dividends are paid as a proportion of the stock price. Which of the following $51-strike, six-month options has the highest value?

The current price of a stock is $50. This includes the value of a dividend of $2 that will be paid three months from now. In three months, the net of dividend stock price will rise by a factor of 1.2 or fall by a factor of 0.8. The risk-free rate of interest is 0.0834% per month (simple interest). Using the technique of Schroder (1988), what is the price of a six-month American call option at a strike of $51?

A stock is currently trading at $100. In each month, the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ . The risk-free rate of interest per month is 0.1668% in simple terms, i.e., an investment of $1 at the risk-free rate returns $1.001668 after one month. What is

The current price of a stock is $50. Every month, the stock price will rise by a factor of 1.2 or fall by a factor of 0.8; and a $1 risk-free investment will be worth $1.03. Dividends are paid as a proportion of 0.02 of the stock price. What can you say about the the 49-strike, European and American six-month calls if the risk-less investment grows to $1.05 (instead of $1.03) each period, and the dividend rate increases to 0.04?

A stock is currently trading at $100. In each month, the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ . The risk-free rate of interest per month is 0.1668% in simple terms, i.e., an investment of $1 at the risk-free rate returns $1.001668 after one month. What is the delta of a 100-strike, three-month European call option?

A stock is currently trading at $100. In each month, the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ . The risk-free rate of interest per month is 0.1668% in simple terms, i.e., an investment of $1 at the risk-free rate returns $1.001668 after one month. What is the delta of a 100-strike, three-month European put option?

Consider a stock index currently trading at 128. You have modeled the evolution of this index in a binomial tree and have come up with the following parameters: $\mathcal { L } = 1.08$ , $d = 0.93$ . The gross risk-free rate per step of the binomial tree is $R = 1.03$ and the dividend yield on the index is $\delta = 0.01$ . The risk-neutral probability of an up-move in this setting is

Consider a binomial tree setting in which in each period the price goes up by $u > 1$ (with probability $p$ ) or down by $d < 1$ (with probability $1 - p$ ). The risk-free interest rate per time step is zero, so a dollar invested at the beginning of the period returns $R = \$ 1$ at the end of the period. Let $Q$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there are no dividends; and let $Q ^ { D }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there is a dividend of size $D > 0$ between the first and second periods. Which of the following is most accurate?