1. Suppose there are two basis assets on the market - a stock and a risk-free...
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1. Suppose there are two basis assets on the market - a stock and a risk-free zero-coupon bond with face value $100 and time-to-maturity of one month. The current price of the bond and the stock are $98 and $100, respectively. The stock may increase to $105 or decrease to $90 in one month, with probabilites 0.25 and 0.75, respectively. (a) What is the monthly risk-free interest rate r? (b) Suppose there is a put option with time-to-maturity of one month and strike price $110. What is the payoff of this option? What is the fair price of this option? If one buys one share of this put option, what should he/she do to hedge the risk. (c) Suppose one constructs a portfolio by longing one share of the above put option and one share of the bond. How to hedge this portfolio? (d) Suppose there is a future contract, in which one must buy one share of the stock at $100 in one month. What is the payoff this future contract? What is the fair price of this contact? 2. Suppose there is a risky asset S and a risk-free asset B. On the second day, the risky asset will be worth S₁ = uSo with probability p and S₁ = d.So with probability 1- p. Let r be the constant risk-free interest rate. (a) Let Po be the "no-arbitrage" (fair) price of a put option with strike K; and let Co be the "no-arbitrage" (fair) price of a call option with strike K. Prove the put-call parity: Co-Po= So K 1+r (b) Suppose you observe that the market trading price of a call option with strike K is less than the "no-arbitrage" price Co. If you believe the model, how do you arbitrage from this? 3. Consider a one-period market with three states 1, 2, and 3, as well as three assets - stock A, stock B, and riskless bond C. In state 1, the returns of stocks A and B are uA = 1.2 and uB = 1.1, respectively. In state 2, the returns of stocks A and B are both 1. In state 3, the returns of stocks A and B are dд = 0.7 and dB = 0.9, respectively. The return of bond C is always 1 + r. (a) Is the market complete? (b) To avoid any arbitrage opportunity, what condition should r satisfy? (c) Suppose r= 1%. What is a fair price of a security with payoff vector (2, 1,0.5)¹? 1. Suppose there are two basis assets on the market - a stock and a risk-free zero-coupon bond with face value $100 and time-to-maturity of one month. The current price of the bond and the stock are $98 and $100, respectively. The stock may increase to $105 or decrease to $90 in one month, with probabilites 0.25 and 0.75, respectively. (a) What is the monthly risk-free interest rate r? (b) Suppose there is a put option with time-to-maturity of one month and strike price $110. What is the payoff of this option? What is the fair price of this option? If one buys one share of this put option, what should he/she do to hedge the risk. (c) Suppose one constructs a portfolio by longing one share of the above put option and one share of the bond. How to hedge this portfolio? (d) Suppose there is a future contract, in which one must buy one share of the stock at $100 in one month. What is the payoff this future contract? What is the fair price of this contact? 2. Suppose there is a risky asset S and a risk-free asset B. On the second day, the risky asset will be worth S₁ = uSo with probability p and S₁ = d.So with probability 1- p. Let r be the constant risk-free interest rate. (a) Let Po be the "no-arbitrage" (fair) price of a put option with strike K; and let Co be the "no-arbitrage" (fair) price of a call option with strike K. Prove the put-call parity: Co-Po= So K 1+r (b) Suppose you observe that the market trading price of a call option with strike K is less than the "no-arbitrage" price Co. If you believe the model, how do you arbitrage from this? 3. Consider a one-period market with three states 1, 2, and 3, as well as three assets - stock A, stock B, and riskless bond C. In state 1, the returns of stocks A and B are uA = 1.2 and uB = 1.1, respectively. In state 2, the returns of stocks A and B are both 1. In state 3, the returns of stocks A and B are dд = 0.7 and dB = 0.9, respectively. The return of bond C is always 1 + r. (a) Is the market complete? (b) To avoid any arbitrage opportunity, what condition should r satisfy? (c) Suppose r= 1%. What is a fair price of a security with payoff vector (2, 1,0.5)¹?
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1 a The monthly riskfree rate r is 2 implied by the bond price of 98 for a face value of 100 maturin... View the full answer
Related Book For
Introduction to Operations Research
ISBN: 978-1259162985
10th edition
Authors: Frederick S. Hillier, Gerald J. Lieberman
Posted Date:
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