Ch13_The Greek Letters(金融工程学,华东师大).
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– 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, © 2003, Shanghai Normal University
•
13.10
Delta for other Futures
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?
…
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
•
13.8
Delta Neutral Portfolio Example
…
48.130 0.183 12,100 46.630 0.007 (17,600) 48.120 0.000 (700)
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
• 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
•
13.6
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 N (d1 ) e qT • The delta of a European put is [ N (d1 ) 1] e qT • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “BUY high, SELL low” trading rule
•
13.9
Delta for Futures
• From Chapter 3, we have F0 S0erT * where T* is the maturity of futures contract • Thus, the delta of a futures contract is F ( SerT * ) rT * e S S • 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, 1 H F rT * H A e rT * H A e
0
Theta
Out-of-the-Money
Time to Maturity
At-the-Money In-the-Money
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 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)
13.1
The Greek Letters
Chapter 13
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
• For a stock or stock index paying a continuous dividend,
H F e ( r q )T * H A
• For a currency,
HF e
( r r f )T *
HA
Speculative Markets, Finance 665 Spring 2003 Options, Futures, and Other Derivatives , 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Brian Balyeat
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
13.3
Naked & Covered Positions
(in-the-money)
Table 13.2 wk.baidu.comp. 314)
Week 0 1 2 18 19 20 Stock Price Delta 49.000 0.522 48.120 0.458 47.370 0.400 … … 54.620 0.990 55.870 1.000 57.250 1.000 Shares Purch. 52,200 (6,400) (5,800) … 1,200 1,000 0 Cum. Cost of Cost Shares Incl. Purch. Interest 2,557.8 2,557.8 (308.0) 2,252.3 (274.7) 1,979.8 … … 65.5 5,197.3 55.9 5,258.2 0.0 5,263.3 Int. Cost 2.5 2.2 1.9 5.0 5.1 …
• The Gamma () for a European call or put paying a continuous dividend q is
N ' (d1 )e qT S 0 T
where
1 d12 / 2 N ' (d1 ) n(d1 ) e 2
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
Gamma
X
Stock Price •
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
13.12
Equation for Gamma
Option Price
f S
B A
Slope =
Stock Price
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
•
13.14
Theta
• Theta () of a derivative (or a portfolio of derivatives) is the rate of change of the value with respect to the passage of time f • Figure 13.6 (p. 321) t
•
13.11
Gamma
• Gamma () is the rate of change of delta () with respect to the price of the underlying 2 f 2 S S • Figure 13.9 (p. 325) [for a call or put]
(out-of-the-money)
Table 13.3 (p. 315)
Week 0 1 2 18 19 20 Stock Price Delta 49.000 0.522 49.750 0.568 52.000 0.705 … … Shares Purch. 52,200 4,600 13,700 … Cum. Cost of Cost Shares Incl. Purch. Interest 2,557.8 2,557.8 228.0 2,789.2 712.4 3,504.3 … … 582.4 1,109.6 (820.7) 290.0 (33.7) 256.6 Int. Cost 2.5 2.7 3.4 1.1 0.3 …
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
•
13.13
Gamma Addresses Delta Hedging Errors Caused By Curvature
• Figure 13.7 (p. 322)
C'' C' C
Call Price
S
S' Stock Price
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 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
• 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, © 2003, Shanghai Normal University
•
13.10
Delta for other Futures
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?
…
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
•
13.8
Delta Neutral Portfolio Example
…
48.130 0.183 12,100 46.630 0.007 (17,600) 48.120 0.000 (700)
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
• 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
•
13.6
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 N (d1 ) e qT • The delta of a European put is [ N (d1 ) 1] e qT • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “BUY high, SELL low” trading rule
•
13.9
Delta for Futures
• From Chapter 3, we have F0 S0erT * where T* is the maturity of futures contract • Thus, the delta of a futures contract is F ( SerT * ) rT * e S S • 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, 1 H F rT * H A e rT * H A e
0
Theta
Out-of-the-Money
Time to Maturity
At-the-Money In-the-Money
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 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)
13.1
The Greek Letters
Chapter 13
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
• For a stock or stock index paying a continuous dividend,
H F e ( r q )T * H A
• For a currency,
HF e
( r r f )T *
HA
Speculative Markets, Finance 665 Spring 2003 Options, Futures, and Other Derivatives , 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Brian Balyeat
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
13.3
Naked & Covered Positions
(in-the-money)
Table 13.2 wk.baidu.comp. 314)
Week 0 1 2 18 19 20 Stock Price Delta 49.000 0.522 48.120 0.458 47.370 0.400 … … 54.620 0.990 55.870 1.000 57.250 1.000 Shares Purch. 52,200 (6,400) (5,800) … 1,200 1,000 0 Cum. Cost of Cost Shares Incl. Purch. Interest 2,557.8 2,557.8 (308.0) 2,252.3 (274.7) 1,979.8 … … 65.5 5,197.3 55.9 5,258.2 0.0 5,263.3 Int. Cost 2.5 2.2 1.9 5.0 5.1 …
• The Gamma () for a European call or put paying a continuous dividend q is
N ' (d1 )e qT S 0 T
where
1 d12 / 2 N ' (d1 ) n(d1 ) e 2
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
Gamma
X
Stock Price •
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
13.12
Equation for Gamma
Option Price
f S
B A
Slope =
Stock Price
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University
•
13.14
Theta
• Theta () of a derivative (or a portfolio of derivatives) is the rate of change of the value with respect to the passage of time f • Figure 13.6 (p. 321) t
•
13.11
Gamma
• Gamma () is the rate of change of delta () with respect to the price of the underlying 2 f 2 S S • Figure 13.9 (p. 325) [for a call or put]
(out-of-the-money)
Table 13.3 (p. 315)
Week 0 1 2 18 19 20 Stock Price Delta 49.000 0.522 49.750 0.568 52.000 0.705 … … Shares Purch. 52,200 4,600 13,700 … Cum. Cost of Cost Shares Incl. Purch. Interest 2,557.8 2,557.8 228.0 2,789.2 712.4 3,504.3 … … 582.4 1,109.6 (820.7) 290.0 (33.7) 256.6 Int. Cost 2.5 2.7 3.4 1.1 0.3 …
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
•
13.13
Gamma Addresses Delta Hedging Errors Caused By Curvature
• Figure 13.7 (p. 322)
C'' C' C
Call Price
S
S' Stock Price
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 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