03-PS5-OP_sol

# 5401 32341 67912 142625 1 at time 0 you want to short

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1.5401 0 3.2341 0 6.7912 14.2625 1

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At time 0, you want to short some stock so that you have a payoff of 0 if stock goes up and 3.234 if stock goes down. You can achieve that by shorting 3 . 2341 115 - 95 = 0 . 1617 units of stock, and putting \$17.7107 in the bank. This will costs you \$1.5401 today. Similary, the replicating portfolio in the other nodes: Time 1:Stock price goes up, unwind all transactions. Stock price goes up, short -0.3396 stock and put \$37.1925 in the bank (in total). Time 2: unwind all transactions if stock goes up once or twice during the past two years. If stock went down twice, short -0.7131 stock and put \$77.1042 in the bank. (c) Notice that the standard deviation of return is 10%, so we use σ = 10% instead of 5%. The black-scholes price is 1.7129, which is quite close. Use the Black-Scholes formula to price the option, using a standard deviation of 5 percent per year. Contrast the result to your answer above. (Note: the standard deviation of stock return is 5 percent in this case) (d) Just build the tree considering early exercise also. You will exercise early if stock price goes down, so the value of the put is now \$2.3801 (e) Adjust the tree manually such that the stock price goes down by an additional 20 at time 2. Note that the tree does not re-combine like it normally does (at time 3) because we are subtracting a constant amount from the nodes, not a percentage. Now the european put is worth \$8.9515 and the american put is worth \$11.6213. See the attached Excel Spreadsheet for calculations (f) Using risk-neutral pricing gives you the ”fair” price of the option, given the market prices. If the stock is correctly priced, then the option price must be such as to avoid any arbitrage. However, it is apparent that your broker tells you that the stock is overpriced. If you agree with him, then you agree that the put is underpriced so you should buy the put. However, this is a subjective judgement, and you are not making risk-free
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• Fall '03
• Wang
• Options, Strike price

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