Use the binomial method to determine the value of an American Put optionat time t = 0. The option expires at time t = T = 1/2 and has exercise priceE = 55. The current value of the underlying is S(0) = 50 with the underlyingpaying continuous dividends at the rate D = 0.05. The interest rate is r = 0.3.Use a time step of St = 1/6. Consider the case of p = 1/2 and suppose thevolatility is σ = 0.3. Perform all calculations using a minimum of 4 decimalplaces of accuracy.=

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Answered: Use the binomial method to determine…

Essentials Of Investments

11th Edition

ISBN:9781260013924

Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.

Chapter1: Investments: Background And Issues

Section: Chapter Questions

Problem 1PS

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Assume that K=61, St =65, t = 0.25 (i.e. time to expiry is 3 months), and the risk-free rate is 0.04. The current price of the put option is p = 4. What would the price of the call option ‘c’ need to be for put-call parity to hold?
Assume that K=61, St =65, t = 0.25 (i.e. time to expiry is 3 months), and the risk-free rate is 0.04. The current price of the put option is p = 4. If the price of the call option is 7.17, describe the arbitrage that would be possible, and calculate the profit that would result.
Assume an interest rate of zero. A Call option and a Put option with the same exercise price, X = 100p are priced at 9p for the Call and 4p for the Put. By completing the table below (attached) show that the net position at expiry is zero.
2. Derive the single - period binomial model for a put option. Include a single - period example where: u = 1.10, d = 0.95, Rf = 0.05, SO = $100, X = $100. 3. Assume ABC stock's price follows a binomial process, is trading at SO = $100, has u 1.10, d = 0.95, and probability of its price increasing in one period is 0.5 (q = 0.5). a. Show with a binomial tree ABC's possible stock prices, logarithmic returns, and probabilities after one period and two periods. . b. What are the stock's expected logarithmic return and variance for 2 periods and 3 periods? c. Define the properties of a binomial distribution.
Assume a security follows a geometric Brownian motion with volatility parameter = 0.2. Assume the initial price of the security is 21 and the interest rate is 0. It is known that the price of a down-and-in barrier option and a down-and-out barrier option with strike price 19 and expiration 30 days have equal risk-neutral prices. Compute this common risk-neutral price.
Let C be the price of a call option that enables its holder to buy one share of a stock at an exercise price K at time t; also, let P be the price of a European put option that enables its holder to sale one share or the stock for the amount K at time t. Let S be the price of the stock at time 0. Then, assuming that interest is continuously discounted at a nominal rate r, either S+P-C=Ke-rt or there is an arbitrage opportunity. Question: How do I verify that the strategy of selling one share of stock, selling one put option, and buying one call option always results in a positive win if S+P-C>Ke-rt ?
A Vanilla American put and European put options with the same underlying, time to expiry and strike price $26.00, the underlying asset S (0) is $26 and the return over each period R=1.06. CRR notation d=0.8 and u=1.25 Construct a three-step binomial pricing tree for both the Vanilla American and European put options and calculate the premiums. Please Show all working.
Suppose that, in each period, the cost of a security either goes up by a factor of 2 or goes down by a factor of 1/2 (i.e. u=2, d=1/2). If the initial price of the security is 100, determine the no-arbitrage cost of a call option to purchase the security at the end of two periods for a price of 150.
With all other variables being equal (the same excerise price, underlying asset, implied volatility, interest rate, etc.), an at-the-money option with 30 days to expiration will tpyically have a gamma that is higher than an at-the-moeny option with 180 days to expiration (hint: think of the different shapes of the associated probability distribution and the change in delta) True or False?
Consider the following data (interest rate is per period): S = 100; K = 75;R = 1.20 ; u = 1.5 ; d = .5. 1. What is the binomial price of a European call option with two periods until expiration? What is the price an American option with the same strike price and same time to expiration. Is there ever early exercise? 2. Show that the binomial option price for a European put option with two periods to go until expiration is 3.125. Show that the binomial price for an American put is 6.25. Can you conclude from the difference in prices alone that early exercise may be optimal? When is it optimal? 3. Use your answers to (i) and (ii) to verify that put-call parity holds for European options, but not for American options. Explain all answers in detail or step by step solution.
Assume that the risk free rate is currently 3% and that the market retunr is currently 11%. Calculate the market risk premium Give the previous datea, calculate the required retun on asset A having a beta of 0.3 and asset B having a beta of 1.5.
Assume a security follows a geometric Brownian motion with volatility parameter sigma=0.2. Assume the initial price of the security is $25 and the interest rate is 0. It is known that the price of a down-and-in barrier option and a down-and-out barrier option with strike price $22 and expiration 30 days have equal risk-neutral prices. Compute this common risk-neutral price. (I attempted this problem and got a final answer of $1.50. Not sure if that is right.)

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