Exam 12: Binomial Option Pricing

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,let $Q _ { 1 }$ be the risk-neutral probability of a one-period at-the-money call finishing in-the-money.and let $Q _ { 2 }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money.Which of the following is true?

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 is the early-exercise premium on the 49-strike,six-month American call?

Which of the following statements is valid?

A stock is currently trading at $100.In each period of a binomial tree,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 per period of the binomial tree is 0.1668%,i.e. ,an investment of a dollar at the risk-free rate returns $1.001668 after one period.What is the risk-neutral probability of the stock finishing in the money on a 100-strike,two-month 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(a)the price of a 100-strike,three-month European put option,and(b)the price of a 100-strike,two-month European 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 $R = \$ 1$ 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?

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

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 American put option?

Consider a two-period binomial tree setting in which the up and down moves are given,respectively,by $\mathcal { U } > 1$ and $d < 1$ .The risk-free gross rate of interest per time step is $R$ .Suppose there is a dividend $D > 0$ paid after the first period of the tree. Let $C _ { 1 }$ denote the value of a one-period call that is initially at-the-money.Let $C _ { 2 } ^ { A } , C _ { 2 } ^ { E }$ denote the values of two-period American and European calls,respectively,that too are initially at-the-money. Which of the following statements is always true regardless of the specific parameter values?

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 $\delta = 0.01$ .The risk-neutral probability of an up-move in this setting is

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,six-month European 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 a dividend payment. )