Greeks - Sunday 1 The Greek Letters FIN 4934 2006 Dr James...

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FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 1 The Greek Letters FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 2 The Black-ScholesFormulas c S N d X e N d p X e N d S N d d S X r T T d S X r T T d T rT rT = - = - - - = + + = + - = - - - 0 1 2 2 0 1 1 0 2 0 1 2 2 2 2 where ( ) ( ) ( ) ( ) ln( / ) ( / ) ln( / ) ( / ) σ σ σ σ σ FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 3 Example A bank has sold for $300,000 a European call option on 100,000 shares of a nondividendpaying stock S 0 = 49, X = 50, r = 5%, σ = 20%, T = 20 weeks The Black-Scholesvalue of the option is $240,000 How does the bank hedge its risk?
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FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 4 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 5 Stop-Loss Strategy This involves: Buying 100,000 shares as soon as price reaches $50 Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well Why? FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 6 Delta Delta ( Δ ) is the rate of change of the option price with respect to the underlying Option price A B Slope = Δ= Stock price S C
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FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 7 Delta Hedging This involves maintaining a delta neutral portfolio The delta of a European call on a stock paying dividends at rate q is N ( d 1 ) e qT The delta of a European put is e qT [ N ( d 1 ) –1] What is the delta of a forward contract? FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 8 Delta Hedging continued The hedge position must be frequently rebalanced Delta hedging a written option involves a “buy high, sell low”trading rule FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 9 Option Closes in the Money: Cost to the writer=$258,110 5,258.11 0.17 3 1.000 57.25 20 5.05 5,252.89 55.99 1,003 1.000 55.83 19 4.99 5,191.91 63.34 1,159 0.990 54.63 18 4.93 5,123.65 617.71 11,257 0.978 54.88 17 4.33 4,501.61 563.66 10,659 0.866 52.88 16 3.78 3,934.16 (44.56) (859) 0.759 51.88 15 3.82 3,974.90 921.20 17,673 0.768 52.13 14 2.93 3,050.77 (1,340.32) (26,607) 0.591 50.38 13 4.22 4,386.87 139.68 2,597 0.857 53.78 12 4.08 4,243.12 484.06 9,048 0.831 53.50 11 3.61 3,755.45 (104.50) (2,002) 0.741 52.20 10 3.71 3,856.25 11.02 210 0.761 52.58 9 3.69 3,841.53 (307.87) (5,850) 0.759 52.63 8 3.99 4,145.42 66.86 1,243 0.817 53.78 7 3.92 4,074.64 97.55 1,819 0.805 53.64 6 3.82 3,973.27 499.06 9,352 0.786 53.36 5 3.34 3,470.88 500.27 9,667 0.693 51.75 4 2.85 2,967.76 984.44 19,591 0.596 50.25 3 1.91 1,981.41 (274.67) (5,798) 0.400 47.38 2 2.17 2,253.91 (304.41) (6,325) 0.458 48.13 1 2.46 2,555.86 2,555.86 52,160 0.522 49.00 0 Interests Cost Cumulative Costs (000) Cost (000) Shares Purchased Delta Stock Price Week
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FIN 4934 2006, Dr. James S. Doran, Department of Finance, Florida State University garnet.acns.fsu.edu/~jsdoran Sunday, October 15, 2006 10
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