Essentials Of Investments
11th Edition
ISBN: 9781260013924
Author: Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher: Mcgraw-hill Education,
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Chapter 16, Problem 6PS
Reconsider the determination of the hedge ratio in the two-state model (Section
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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.
We will derive a two-state call option value in this problem. Data: S0 = $190; X = $200; 1 + r = 1.10. The two possibilities for ST are $220 and $120. The portfolio consists of 1 share of stock and 5 calls short.
Required:
a. The range of S is $100 while that of C is $20 across the two states. What is the hedge ratio of the call? (Round your answer to 2 decimal places.)
b. Calculate the value of a call option on the stock with an exercise price of $200. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.) (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Data: S0 = 102; X = 115; 1 + r = 1.1. The two possibilities for ST are 146 and 84.
Required:
The range of S is 62 while that of P is 31 across the two states. What is the hedge ratio of the put?
Form a portfolio of one share of stock and two puts. What is the (nonrandom) payoff to this portfolio?
What is the present value of the portfolio?
Given that the stock currently is selling at 102, calculate the put value.
Chapter 16 Solutions
Essentials Of Investments
Ch. 16 - Prob. 1PSCh. 16 - A put option on a stock with a current price of 33...Ch. 16 - Prob. 3PSCh. 16 - Prob. 4PSCh. 16 - In each of the following questions, you are asked...Ch. 16 - Reconsider the determination of the hedge ratio in...Ch. 16 - Show that Black-Scholes call option hedge ratios...Ch. 16 - We will derive a two-State put option value in...Ch. 16 - a. Calculate the value of a call option on the...Ch. 16 - Prob. 10PS
Ch. 16 - Prob. 11PSCh. 16 - Prob. 12PSCh. 16 - Prob. 13PSCh. 16 - Prob. 14PSCh. 16 - Prob. 15PSCh. 16 - Prob. 16PSCh. 16 - 17. Find the Black-Scholes value of a put option...Ch. 16 - Prob. 18PSCh. 16 - What would be the Excel formula in Spreadsheet...Ch. 16 - Prob. 20PSCh. 16 - Prob. 21PSCh. 16 - Prob. 22PSCh. 16 - Prob. 23PSCh. 16 - Prob. 24PSCh. 16 - Prob. 25PSCh. 16 - Prob. 26PSCh. 16 - Prob. 27PSCh. 16 - Prob. 28PSCh. 16 - Prob. 29PSCh. 16 - Prob. 30PSCh. 16 - Prob. 31PSCh. 16 - Prob. 32PSCh. 16 - Prob. 33PSCh. 16 - Prob. 34PSCh. 16 - Prob. 35PSCh. 16 - Prob. 36PSCh. 16 - Prob. 38CCh. 16 - Prob. 39CCh. 16 - Prob. 40CCh. 16 - Prob. 41CCh. 16 - Prob. 42CCh. 16 - Prob. 43CCh. 16 - Prob. 44CCh. 16 - Prob. 2CP
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- Data: So=102; X = 115; 1 + r=1.1. The two possibilities for ST are 146 and 84. Required: a. The range of S is 62 while that of P is 31 across the two states. What is the hedge ratio of the put? b. Form a portfolio of one share of stock and two puts. What is the (nonrandom) payoff to this portfolio? c. What is the present value of the portfolio? d. Given that the stock currently is selling at 102, calculate the put value. × Answer is not complete. Complete this question by entering your answers in the tabs below. Required A Required B Required C Required D What is the present value of the portfolio? Note: Round your answer to 2 decimal places. Present value $ 146.00 xarrow_forward1. An option is trading at $5.26, has a delta of .52, and a gamma of .11. what would the delta of the option be if the underlying increases by $.75? What would the delta of the option be if the underlying decreases by $1.05? Explain.arrow_forwardMf2. Assume a one-period binomial model in which the initial stock price is S = 60 and in each period the stock price can go either up by a factor of u = 7 3 or down by a factor of d = 2 3 . Assume that the simple interest rate over one time period is r = 1 3 . (a) Determine the fair price of the European put option with strike K = 60. (b) Assume that instead of the price determined in part (a), the European call option is trading at 11. Prove that there is an arbitrage and explain how the arbitrage can be achievedarrow_forward
- 4. Consider an exchange option. Suppose the initial prices (time 0) of the two stocks are S =S2 = 100 and a =0.40,. Suppose also that the returns on the stocks are uncorrelated. Assume no dividends and final maturity of the option is T = 2 year: (a) Using the closed-form expressions for the price of these options, identify the price of the exchange option when o = 0, a2 =0.20, ag =0.40, and @2 =0.60. (b) Is there a trend in the price? Intuitively, why is this the case?arrow_forwardA share price is modelled via a two-period binomial model with initial stock price S=40, up/down multiplication factors u = per time period r = 4%. and d = and interest rate Explain Does this model satisfy the no-arbitrage assumption? Write 0 if your answer is 'no' and 1 if your answer is 'yes'. Answer: your argument in your hand-written answer (b) Calculate the risk-neutral probabilities of up and down movements in the share price. State your answer to three valid digits. Answer: P = Pd= (c) to three valid digits. Answer: Determine the no-arbitrage price of a European call option on the share with strike price K = 70 and expiry time T = 2. State your answer Explain your calculation steps in your hand-written answer.arrow_forwardWe will derive a two-state put option value in this problem. Data: S0 = 100; X = 110; 1 + r = 1.10. The two possibilities for ST are 130 and 80.a. Show that the range of S is 50, whereas that of P is 30 across the two states. What is the hedge ratio of the put?b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to this portfolio?c. What is the present value of the portfolio?d. Given that the stock currently is selling at 100, solve for the value of the put.arrow_forward
- 1. Consider a family of European call options on a non - dividend - paying stock, with maturity T, each option being identical except for its strike price. The current value of the call with strike price K is denoted by C(K) . There is a risk - free asset with interest rate r >= 0 (b) If you observe that the prices of the two options C( K 1) and C( K 2) satisfy K2 K 1<C(K1)-C(K2), construct a zero - cost strategy that corresponds to an arbitrage opportunity, and explain why this strategy leads to arbitrage.arrow_forwardV3. By looking at the sensivities of your portfolio to δs = -$2 and δσ = -1%, you decide to hedge delta, gamma and Vega risk of your portfolio with the underlying stock and two different options on the same asset with below data. Calculate the units of stock you need to trade to hedge away all delta, gamma and Vega risks of your portfolio.(Note that here you have to calculate the units of stock, Option A and Option B, but you will only submit the units of stock.)arrow_forwardSuppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rƒ. The characteristics of two of the stocks are as follows: Correlation = -1 Stock A B Rate of return Required: a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be formed to create a "synthetic" risk-free asset?) (Round your answer to 2 decimal places.) Yes No X Answer is not complete. Expected Return 9% 13% % Standard Deviation 45% 55% b. Could the equilibrium rƒ be greater than rate of return?arrow_forward
- b) Suppose you are given the following information: Current market price of a share= R200 000 Option’s exercise price = R300 000 Time until the option expires= 5 yrs Risk free rate =4% Standard deviation of the returns on the share= 0.35 Required: i. Calculate d1 and d2 ii. Suppose N(d1) =0.7517 and N(d2) =0.4602; calculate the price of the call option on the sharearrow_forwardA. An option is trading at $5.03. If it has a delta of -.56, what would the price of the option be if the underlying increases by $.75? What would the price of the option be if the underlying decreases by $.55? B. What type of option is this and how? C. With a delta of -.56, is this option ITM, ATM or OTM and how?arrow_forward4. Consider a stock with a current price of S0 = $60. The value of the stock at time t = 1 can take one of two values: S1,u = $100, S1,d = $40. The price of a risk-free bond that pays out $1 in period t = 1 is $0.90. (a) Using a one-step binomial tree, write down the possible payoffs of a put option on stock S with strike K = $60 and maturity t = 1. (b) What is the price of this put option? (c) What is the price of a call option with strike K = $60 and maturity t = 1? Please use put-call parity to find the call price.arrow_forward
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