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 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.)

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

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 $K = 130$ ?

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 $u = 1.05$ 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?

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 American put option?

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 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 $50. In each month, the stock will either increase in price by a factor of $\mathcal { U } = 1.2$ 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.)

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

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

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?

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. 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?

Consider a two-period binomial tree setting in which the up and down moves are given, respectively, by $u > 1$ 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 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 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?

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

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.)

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?