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 − − = •