Assume the price, S, of a non-dividend paying stock follows geometric Brownian motion withdrift r and volatility o. Consider a perpetual call option on S. The option is exercised whenS = Ŝ, and the payoff on the option is Ŝ– X, where X is the exercise price of the option.Assume all investors are risk-neutral and the instantaneous risk-free rate of interest is aconstant, r. Note that the drift of the stock the instantaneous rate of return on the stock-isequal to r. Of course, this makes sense, since the return on all traded assets in a risk-neutralworld must be the risk-free rate of interest. Compute the value of the call option C(S;$) andthe optimal exercise policy S = arg max[C(S; $)].

A call option gives the right to the buyer to buy a stock at an exercise price X. An option is a…

Answered: Assume the price, S, of a non-dividend…

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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The buyer of a call option on stock benefits if the underlying stock price rises or if the volatility of the stock's price increases. Select one: True False
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.
4. Valuation of a Derivative Consider a derivative on a stock with the time to expiration T and the following payoff: 0 K₁ 0 if ST K₁. What is the present value of the derivative? Provide an analytic expression of the price using N(), the cumulative probability distribution function of a standard normal random variable.
Assume that the CAPM assumptions hold. Consider the following statements:i. A stock with a beta below zero will tend to move in the same direction as the market but will tend to move less aggressively in that direction than the market does.ii. Alpha measures the additional risk we take on top of the risk of the market portfolio.
Both call and put options are affected by the following five factors: the exercise price, the underlying stock price, the time to expiration, the stock’s standard deviation, and the risk-free rate. However, the direction of the effects on call and put options could be different. Use the following table to identify whether each statement describes put options or call options. Statement Put Option Call Option 1. An option is more valuable the longer the maturity. 2. A longer maturity in-the-money option on a risky stock is more valuable than the same shorter maturity option. 3. When the exercise price increases, option prices increase. 4. As the risk-free rate increases, the value of the option increases.
The price S(u), 0 ≤ u ≤ t, of the share is driven by a geometric Brownian motion: S(u) = Seμu+σW(u).A proportional dividend on this share is paid continuously at rate q > 0 and is reinvested in the share. The continuously compounded interest rate is r. Compute the no-arbitrage price of a derivative with expiration time t and payoff function R(t) = [S(t/3)S(2t/3)S(t)]1/3 .
Assume now that in our usual setting the stock has continuous dividends. If the interest rate is r = 0.06 and the dividends rate is q = 0.03, will the option A be greater or smaller than 0.5?
Stock HB has a beta of 1.5 and Stock LB has a beta of 0.5. The market is in equilibrium, with required returns equaling expected returns. Which of the following statements is CORRECT? a. If expected inflation remains constant but the market risk premium (rM – rRF) declines, the required return of Stock LB will decline but the required return of Stock HB will increase. b. If both expected inflation and the market risk premium (rM – rRF) increase, the required returns of both stocks will increase by the same amount. c. Since the market is in equilibrium, the required returns of the two stocks should be the same. d. If expected inflation remains constant but the market risk premium (rM – rRF) declines, the required return of Stock HB will decline but the required return of Stock LB will increase. e. If both expected inflation and the market risk premium (rM – rRF) increase, the required return on Stock HB will increase by…
Stocks A and B have the same price and are in equilibrium, but Stock A has the higher required rate of return. Which of the following statements is CORRECT? a. Stock B must have a higher dividend yield than Stock A. b. Stock A must have a higher dividend yield than Stock B. c. If Stock A has a higher dividend yield than Stock B, its expected capital gains yield must be lower than Stock B's. d. Stock A must have both a higher dividend yield and a higher capital gains yield than Stock B. e. If Stock A has a lower dividend yield than Stock B, its expected capital gains yield must be higher than Stock B's.
The premium on a call option is primarily a function of the difference in spot price S relative to the strike price X, the length of time until expiration T, and the volatility of the currency o. C = f(S-X, T, o) For each characteristic of a call option, use the table to indicate whether that would lead to a higher call option premium or a low call option premium (all else equal). Characteristic A lower spot price relative to the strike price A shorter time before expiration A higher level of volatility for the currency Higher Call Option Premium O Lower Call Option Premium When using a call option to hedge payables in an international currency, a U.S. based MNC can lock in the to obtain the needed foreign currency. maximum minimum amount of dollars needed
Question 1. Let St be the current price of a stock that pays no dividends. a)Let rbid be the interest rate at which one can invest/lend money, and roff be theinterest rate at which one can borrow money, rbid≤roff. Both rates are continuously compounded. Using arbitrage arguments, find upper and lower bounds for the forwardprice of the stock for a forward contract with maturity T > t. b)How does your answer change if the stock itself has bid price St,bid and offer price St,off?
The security market line (SML) is an equation that shows the relationship between risk as measured by beta and the required rates of return on individual securities. The SML equation is given below: Required return on Stock = Risk-free return + (Market risk premium) (Stock's beta) If a stock's expected return plots on or above the SML, then the stock's return is sufficient ✓to compensate the investor for risk. If a stock's expected return plots below the SML, the stock's return is insufficient to compensate the investor for risk. The SML line can change due to expected inflation and risk aversion. If inflation changes, then the SML plotted on a graph will shift up or down parallel to the old SML. If risk aversion changes, then the SML plotted on a graph will rotate up or down becoming more or less steep if investors become more or less risk averse. A firm can influence market risk (hence its beta coefficient) through changes in the composition of its assets and through changes in the…

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