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.If there are no dividends,what is the early-exercise premium of a 100-strike,six-month American call 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: $u = 1.08$ , $d = 0.93$ .The gross risk-free rate per step of the binomial tree is $R = 1.03$ and the dividend yield on the index is $\delta = 0.01$ .What is the price of a one-period put option with a strike of $K = 130$ ?

Suppose you were replicating a two-period put option in a binomial tree.Then,the replicating strategy involves an initial short position in the underlying stock that

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 early-exercise premium of a 100-strike,six-month American put option?

Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } > 1$ (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 a dollar at the end of the period. In this setting,the risk-neutral probability of an at-the-money two-period put finishing in-the-money is _____________ as that of a one-period at-the-money put finishing in-the-money.

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?

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.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.A dividend as a proportion 0.01 of the stock price is paid each period.What is the price of a six-month American put option at a strike of $51?

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 $u = 1.05$ 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 price of a 100-strike,two-month European put option?

Schroder's (1988)approach to binomial option pricing offers a way of

A binomial tree setting has an up-move of $u > 1$ (with probability $p = 0.75$ )and a down move of $d < 1$ (with probability $1 - p = 0.25$ ),with $u - 1 = 1 - d$ .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.The risk-neutral probability of an up-move in this setting

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 price 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?

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 call option when a dividend of $1 is paid at the end of each month? (Assume that if the option is exercised,it is done just before the dividends are paid. )

Consider a binomial tree setting in which in each period the price goes up by $u = 1.10$ (with probability $p = 0.60$ )or down by $d = 0.90$ (with probability $1 - p = 0.40$ ).The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,the risk-neutral probability of a two-period call with strike $K = 95$ finishing in-the-money is

For a call option on a stock that pays dividends,which of the following statements is valid?

A binomial tree setting has an up-move of $u > 1$ (with probability $p = 0.75$ )and a down move of $d < 1$ (with probability $1 - p = 0.25$ ),with $u d = 1$ .The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,the risk-neutral probability of an up-move

A stock is currently trading at $50.In each month,the stock will either increase in price by a factor of $u = 1.2$ or fall by a factor of $d = 0.9$ .The risk-free rate of interest is 0.0834% per month in simple terms,i.e. ,an investment of $1 today returns $1.00834 after one period.Consider a 52-strike,three-month European put option when a dividend of $0.5 is paid at the end of each month.Assume that the company just announces a cancellation of future dividends.Ceteris paribus,by how much does the option price change on this announcement? (Assume that if the option is exercised,it is done before dividends are paid. )