Consider two European call options with strike K and the same underlying non- dividend paying stock. The stock price is currently at So. Option 1 has price C₁ maturity T₁ and Option 2 has price C₂ and maturity T2, where T₂ > T₁. The risk-free interest rate is r. Use a no-arbitrage argument to prove that C₂ > C₁-

Consider two European call options with strike K and the same underlying non- dividend paying stock. The stock price is currently at So. Option 1 has price C₁ maturity T₁ and Option 2 has price C₂ and maturity T2, where T₂ > T₁. The risk-free interest rate is r. Use a no-arbitrage argument to prove that C₂ > C₁-

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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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.
Tick all those statements on options that are correct (and don't tick those statements that are incorrect). B a. The Black-Scholes formula is based on the assumption that the share price follows a geometric Brownian motion. b. If interest is compounded continuously then the put-call parity formula is P+ S(0) = C + Ker where T is the expiry time. An American put option should never be exercised before the expiry time. d. In general the equation S(T) +(K-S(T)) = (S(T)-K)+ +K is valid. e. The put-call parity formula necessarily requires the assumption that the share price follows a geometric Brownain motion. C.
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?
An up-and-out barrier call option with barrier B, strike price K and exercise time T has payoff H(T) = (S(T) − K) + if max {S(t)| 0 ≤ t ≤ T} < B, 0 otherwise, that is, the payoff is that of a call option if the underlying stock price does not reach or exceed the barrier B at any time up to and including time T, and 0 otherwise. For an up-and-out barrier call option with barrier B = 140, strike price K = 90 and exercise time T = 3 in the binomial model with parameters U = 0.2, D = −0.1, R = 0.1 and S(0) = 100 compute the following. (a) The option price at time 0;
Consider two put options on the same stock with the same time to maturity. The strike price of Put A is less than the strike price of Put B. Which of the following is true? O It is possible for Put A to be in the money and Put B to be out of the money. O It is possible for Put A to be out of the money and Put B to be in the money. One of the options must be in the money. All of the other answers are correct.
Consider a European call option for a non-dividend paying stock currently priced at S(0) with strike price K at maturity T. If the price of the call C(0) at time t = 0 is greater than the stock's price S(0), then which of the following trading strategies, if any, produces an arbitrage opportunity? Select one option. Short sell a call option and use the funds to buy the stock. Invest the O remaining cash at the risk-free rate. O Buy a call option by borrowing cash at the risk-free rate. O Short sell a call option. Invest the funds at the risk-free rate. O No arbitrage opportunities exist.
i need the answer quickly
D3) Finance Consider an option with α being a non-negative parameter and the option pays ((S(T))α − K)+ at maturity date T. Let Cα(S(0), σ, r) be the risk neutral price of the option (with interest rate r and volatility σ) when the initial price is S(0). Obviously, C1(S(0), σ, r) = C(S(0), σ, r) is the price of an ordinary call option. Show that, Cα(S(0), σ, r) = e(α−1)(r+ασ2/2)TC((S(0))α, ασ, rα), where rα = α(r − σ2/2) + α2σ2/2.
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 ?
3. A Consider two portfolios: Portfolio A consists of one European call option plus an zero which pays K at time T; Portfolio B consists of one European put option plus a share of the underlying stock. The stock pays no dividend. Both options have the same underlying stock, the same expiration date T and the strike price K. Graph the time-T value of both portfolios (in separate pictures) as functions of the stock price S = S(T).
Suppose that C is the price of a European call option to purchase a security whose present price is S. Show that if C>S then there is an opportunity for arbitrage (i.e. risk-less profit). You may assume the interest rate is r=0 so that the present value calculations are unnecessary.
Tick all those statements on options that are correct (and don't tick those statements that are incorrect). O a. The Black-Scholes formula is based on the assumption that the share price follows a geometric Brownian motion. Ob. The put-call parity formula necessarily requires the assumption that the share price follows a geometric Brownain motion. 0 C. If interest is compounded continuously then the put-call parity formula is P+ S(0) = C + Ke where I is the expiry time. Od. In general the equation S(T) + (K − S(T))+ (S(T) — K)† + K is valid. An American put option should never be exercised before the expiry time. e. =

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