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).
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- Stocks A and B have the following data. Assuming the stock marketyls efficient and the stocks are in equilibrium, which of the following statements is CORRECT? \table[[,A,BIn the Black-Scholes option pricing model, the value of a call is inversely related to: a. the risk-free interest stock b. the volatility of the stock c. its time to expiration date d. its stock price e. its strike priceQUESTION 2. Consider a stock valued So at time t = 0, and taking only two possible values S1 = S, or Si = S1 at time t = 1, with S, < I1. a) Compute the initial portfolio allocation (a, 3) of a portfolio made of a units of stock and $B in cash, hedging the call option with strike price K e [S,,3¡] and claim payoff Si – K if Si = 3ı C = (Si – K)* $0 if S = S1, at time t = 1. b) Show that the risky asset allocation a satisfies the condition a € [0, 1]. c) Compute the Superhedging Risk Measure SRMc' of the claim C = (S1 – K)+.
- Consider two binomial trees, one for the spot price of an asset and the other for the futures price on the same asset. Let the up factor, u, and the down factor, d, be the same for both trees. The difference in the risk-neutral probabilities for the up movement on spot tree and on the futures tree is: Oa (1-d)/e Ob. (u-1)/et Oc ert /u Od (e".1)/ (u-d) Oo (u-et)/ (u-d)Describe the effect of a change in each of the following factors on the value of a calloption:1. Stock price2. Exercise price3. Option life4. Risk-free rateConsider the three stocks in the following table. Pt represents price at time t, and Qt represents shares outstanding at time t. Stock C splits two for one in the last period. a. A market-value-weighted index. rate of returnb. An equally weighted index. rate of return
- Which of the following statements true? A call option price is increasing in stock return volatility A put option price is decreasing in stock return volatility I. II. A) I. and II. are true B) I. is true and II. is false C) II. is true and I. is false D) I. and II. are false |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 the stock’s future prices of stock A and stock B as the following distribution State Future Price Stock A Future price Stock B 1 $10 $7 2 $8 $9 If the time 1 price of stock A is $6, and the time 1 price of stock B is $5. And C1 represents the time 1 price of claim on state 1, C2 represents the time 1 price of claim on state 2 Use the information about stock prices and payoffs to Find the time 1 price C1 and C2. Find the risk–free rate of return, obtained in this market.
- Refer to the graph below, what is the beta of portfolio X under CAPM? E(r) A 0.14 0.10 0.06 N Beta of portfolio X = place) M X o(r) (to the nearest 1 decimalConsider two portfolios A and B. At the expiration date, t, both portfolios have identical payoffs. Portfolio A consists of a put option and one share of stock. Portfolio B has a call option (with the same strike price and expiration date as the put option) and cash in the amount equal to the present value (PV) of the strike price discounted at the continuously compounded risk-free rate, which is Xe−rRFtXe−rRFt. At expiration, the stock price is Ptt, and the value of this cash will equal the strike price, X. As defined in the Black–Scholes model, let N(didi) stand for probability that a deviation less than didi will occur in a standard normal distribution and N(d₁) and N(d₂) represent areas under a standard normal distribution function. Complete the equation for the value of a put option. Put Option = – Now consider the stock of Grotesque Enterprises (GE) traded at the price P = $35 per share. A put option written on GE’s stock has an exercise price…Assume the price, S, of a non-dividend paying stock follows geometric Brownian motion with drift r and volatility o. Consider a perpetual call option on S. The option is exercised when S = Ŝ, and the payoff on the option is Ŝ– X, where X is the exercise price of the option. Assume all investors are risk-neutral and the instantaneous risk-free rate of interest is a constant, r. Note that the drift of the stock the instantaneous rate of return on the stock-is equal to r. Of course, this makes sense, since the return on all traded assets in a risk-neutral world must be the risk-free rate of interest. Compute the value of the call option C(S;$) and the optimal exercise policy S = arg max[C(S; $)].