Suppose you can trade a stock (S), and three-month European calls on the stock struck at...
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Suppose you can trade a stock (S), and three-month European calls on the stock struck at 250, 300, and 350. The stock pays no dividends, is trading at $312.50, and has a volatility of 64%. Three-month zero-coupon bonds with faces of $100 are trading at $99.50 (a continuously compounded yield of r = 2%). The Greeks of each of the calls are given below: K A Г e P 250 0.809 2.725 x10-3 -58.04 44.14 42.58 300 0.619 3.810 x10-3 -79.15 36.81 59.53 350 0.429 3.926 × 10-³ -80.68 26.91 61.35 (a) (5 point) Use the Black-Scholes PDE (our pricing formula in terms of Greeks) to calculate the price of the call struck at 300. • That is, figure out the call price without using the Black-Scholes formula. (b) (5 points) Suppose you own the call struck at 250. What positions in the call struck at 350 and the underlying do you need to take to simultaneously gamma and delta hedge your position? (c) (5 points) Use put-call parity to calculate the Delta, the Gamma, and the Theta of a three-month European put on the stock struck at 250. • Remember, you can think about the forward as its replicating portfolio. - And the Greeks of a portfolio are the sum of the Greeks of the portfo- lio's holdings. (d) (5 point) Suppose you bought a butterfly on 1,000 shares of the stock (using the same options provided in the start of the problem). If over the next five days the underlying falls $30, what will be the approximate change in the value of your position? • You can assume interest rates and implied volatility are unchanged. Suppose you can trade a stock (S), and three-month European calls on the stock struck at 250, 300, and 350. The stock pays no dividends, is trading at $312.50, and has a volatility of 64%. Three-month zero-coupon bonds with faces of $100 are trading at $99.50 (a continuously compounded yield of r = 2%). The Greeks of each of the calls are given below: K A Г e P 250 0.809 2.725 x10-3 -58.04 44.14 42.58 300 0.619 3.810 x10-3 -79.15 36.81 59.53 350 0.429 3.926 × 10-³ -80.68 26.91 61.35 (a) (5 point) Use the Black-Scholes PDE (our pricing formula in terms of Greeks) to calculate the price of the call struck at 300. • That is, figure out the call price without using the Black-Scholes formula. (b) (5 points) Suppose you own the call struck at 250. What positions in the call struck at 350 and the underlying do you need to take to simultaneously gamma and delta hedge your position? (c) (5 points) Use put-call parity to calculate the Delta, the Gamma, and the Theta of a three-month European put on the stock struck at 250. • Remember, you can think about the forward as its replicating portfolio. - And the Greeks of a portfolio are the sum of the Greeks of the portfo- lio's holdings. (d) (5 point) Suppose you bought a butterfly on 1,000 shares of the stock (using the same options provided in the start of the problem). If over the next five days the underlying falls $30, what will be the approximate change in the value of your position? • You can assume interest rates and implied volatility are unchanged.
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Related Book For
Financial Reporting and Analysis
ISBN: 978-0078025679
6th edition
Authors: Flawrence Revsine, Daniel Collins, Bruce, Mittelstaedt, Leon
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