华东师范大学：《金融工程》英文版 Chapter 13

13.1 TThe greek letters Chapter 13 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.1 The Greek Letters Chapter 13

13.2 Example A FI has SOLD for $300,000 a European call on 100,000 shares of a non-dividend paying stock So=49X=50 5%=20% u =13% T =20 Weeks The Black-Scholes value of the option is $240,000 How does the fi hedge its risk? Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.2 Example • A FI has SOLD for $300,000 a European call on 100,000 shares of a non-dividend paying stock: S0 = 49 X = 50 r = 5%  = 20%  = 13% T = 20 weeks • The Black-Scholes value of the option is $240,000 • How does the FI hedge its risk?

13.3 Naked covered positions ° Naked position(裸期权头寸策略) Take No action Covered position(抵补期权头寸策略) Buy 100,000 shares today Both strategies leave the Fl exposed to significant risk Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.3 Naked & Covered Positions • Naked position (裸期权头寸策略) Take NO action • Covered position(抵补期权头寸策略) Buy 100,000 shares today Both strategies leave the FI exposed to significant risk

13.4 Stop-LosS Strategy This involves Fully covering the option as soon as it moves in-the-money Staying naked the rest of the time This deceptively simple hedging strategy does not work well !! Transactions costs, discontinuity of prices, and the bid-ask bounce kills it Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.4 Stop-Loss Strategy This involves – Fully covering the option as soon as it moves in-the-money – Staying naked the rest of the time • This deceptively simple hedging strategy does NOT work well !!! • Transactions costs, discontinuity of prices, and the bid-ask bounce kills it

13.5 Delt Delta(A)is the rate of change of the option price with respect f to the underlying △ Figure 13.2(p. 311) Option Price B ope A Stock Price Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.5 Delta • Delta () is the rate of change of the option price with respect to the underlying • Figure 13.2 (p. 311)  =   f S Option Price A B Stock Price Slope =  •

13.6 D elta Ledgin g This involves maintaining a delta neutral portfolio The delta of a European call on a stock paying dividends at a rate g is N(d,)e q The delta of a European put is [N(d,)-1]e 9 The hedge position must be frequently rebalanced Delta hedging a written option involves a BUYhigh, SELL low?' trading rule Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.6 Delta Hedging • This involves maintaining a delta neutral portfolio • The delta of a European call on a stock paying dividends at a rate q is • The delta of a European put is • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “BUY high, SELL low” trading rule qT N d − ( ) e 1 qT N d − [ ( ) −1]e 1 •

13.7 Delta Neutral Portfolio Example (in-the-money) Table132(p.314) um ost of tock Shares Shares n Week Price Delta Purch Purch. Interest Cost 049.000.52252.2002,557825578 08.0 800 19798 1854.6200.990 200 65.55.197.35.0 20 2501.000 0.05,2633 Options, Futures, and Other Derivatives, 4th edition o 2000 by John C Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.7 Delta Neutral Portfolio Example (in-the-money) Cum. Cost of Cost Stock Shares Shares Incl. Int. Week Price Delta Purch. Purch. Interest Cost 0 49.000 0.522 52,200 2,557.8 2,557.8 2.5 1 48.120 0.458 (6,400) (308.0) 2,252.3 2.2 2 47.370 0.400 (5,800) (274.7) 1,979.8 1.9 18 54.620 0.990 1,200 65.5 5,197.3 5.0 19 55.870 1.000 1,000 55.9 5,258.2 5.1 20 57.250 1.000 0 0.0 5,263.3 … … … … … … … Table 13.2 (p. 314) •

138 Delta Neutral Portfolio Example (out-of-the-money) Table 13. 3(p. 315 um ost of tock Shares Shares n Week Price Delta Purch Purch. Interest Cost 049.0000.52252.200 82 84.600 252.0000.70513.700 805504 7124 18481300.18312.100582.41.109.6 600 290.0 2048.1200.000 Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.8 Delta Neutral Portfolio Example (out-of-the-money) Cum. Cost of Cost Stock Shares Shares Incl. Int. Week Price Delta Purch. Purch. Interest Cost 0 49.000 0.522 52,200 2,557.8 2,557.8 2.5 1 49.750 0.568 4,600 228.0 2,789.2 2.7 2 52.000 0.705 13,700 712.4 3,504.3 3.4 18 48.130 0.183 12,100 582.4 1,109.6 1.1 19 46.630 0.007 (17,600) (820.7) 290.0 0.3 20 48.120 0.000 (700) (33.7) 256.6 … … … … … … … Table 13.3 (p. 315) •

13.9 Delta for futures From Chapter 3, we have F=S where T is the maturity of futures contract Thus, the delta of a futures contract is aF a(")T e as aS So, if Ha is the required position in the asset for delta hedging and he is the required position in futures for the same delta hedging H H rt* H Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.9 Delta for Futures • From Chapter 3, we have where T* is the maturity of futures contract • Thus, the delta of a futures contract is • So, if HA is the required position in the asset for delta hedging and HF is the required position in futures for the same delta hedging, * 0 0 e rT F = S * * e ( e ) rT rT S S S F =   =   A r T HF r T HA H * * e e 1 − = = •

13.10 Delta for other futures For a stock or stock index paying a continuous dividend F-e(g)7* H H For a currency H (r-r;) H Option s RAGHelativoMasketsiVejnapGRi6 5 2SdRriygoAE3Hull Tang Yincai, C 203 iShprBhdjCthal University

Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 13.10 Delta for other Futures • For a stock or stock index paying a continuous dividend, • For a currency, Speculative Markets, Finance 665 Spring 2003 Brian Balyeat A r q T HF H ( ) * e − − = A r r T HF H f ( ) * e − − = •