Copyright ⃝c Copyright University of Wales 2021. All rights reserved. Economics of Finance
Tutorial 4
1. Consider a three period binomial time-state model in which there are two securities, a bond and a stock. The payments made by these securities in each state are shown in the trees below:
gg 1 2.25
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Pq 1.20 1 1.20
gg 1 1.21
Pq 1.21 1 1.21
(i) Write down the payment matrix, Q, and corresponding price vector, pS, derived from the following elemental payment combinations:
B0: Buy a Bond at period 0, sell it at the end of the next period; S0: Buy a Stock at period 0, sell it at the end of the next period;
Bg: At period 1, if the state is g, buy a Bond, sell it at the end of the next period; Sg: At period 1, if the state is g, buy a Stock, sell it at the end of the next period; Bb: At period 1, if the state is b, buy a Bond, sell it at the end of the next period; Sb: At period 1, if the state is b, buy a Stock, sell it at the end of the next period.
(ii) Compute the atomic security prices (i.e., the price of one dollar in each of the six future time- states: g, b, gg, gb, bg, bb). Write down the formula you used to derive the atomic security price vector.
(iii) Write down the payment matrix, Q, and corresponding price vector, pS, derived from the following elemental payment combinations:
B0: Buy a Bond at period 0, sell it at the end of the next period; S0: Buy a Stock at period 0, sell it at the end of the next period;
Bb: At period 1, if the state is b, buy a Bond, sell it at the end of the next period; Sb: At period 1, if the state is b, buy a Stock, sell it at the end of the next period.
B02: Buy a Bond at period 0, sell it at the end of period 2; S02: Buy a Stock at period 0, sell it at the end of period 2;
(iv) Verify the atomic security prices computed using the payment matrix Q in part (iii) is the same as the one found using the payment matrix Q in part (i).
(v) Suppose an investor wants to obtain the following time-state payments: c=0 10 20 20 30 40′.
The vector of payment combination holdings, n, is calculated as follows: n = Q−1c. Calculate n for both the Q matrices considered in (i) and (iii) above.
(vi) Take each vector n from part (v) and calculate how much of the bond and stock the investor must buy or sell in aggregate in each state in period 1 to implement this dynamic strategy? Show your workings, and verify that both n vectors are describing the same overall strategy.
(vii) Compute the arbitrage-free price of the time-state payment vector c. Explain how you arrived at your answer.
2. Consider a three period binomial time-state model in which there are two securities, a bond and a stock. The payments made by the stock in each state are shown in the tree below. The bond pays 10 percent interest each period.
Figure 1: Lattice for the Stock
(i) Calculate the price in Period 0 of an American Call (buy) option that expires at the end of Period 2 with an exercise price of 1.60. Explain how you arrived at your answer.
(ii) Calculate the price in Period 0 of an American Put (sell) option that expires at the end of Period 2 with an exercise price of 1.60. Explain how you arrived at your answer.
3. An investor owns 2,000 shares of Walmart Inc, now selling at $110. The investor has a one-year investment horizon and is concerned of a downside risk. She considers buying a call or put option. A one-year call with a strike price of $110 is now selling at $1.50 whilst the put option with similar strike price and expiry date is selling at $0.41.
i) Given the above data, devise the protective strategy.
ii) Explain why in this situation a call option is more expensive than a put option.
iii) Under which conditions a call option price will be exactly equal to a put option price.
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