Greeks

Greeks - Sunday, October 15, 2006 1 The Greek Letters FIN...

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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-Scholes Formulas 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 nondividend paying stock S 0 = 49, X = 50, r = 5%, σ σ = 20%, T = 20 weeks The Black-Scholes value 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 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
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Greeks - Sunday, October 15, 2006 1 The Greek Letters FIN...

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