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
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?
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
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
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 =?
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
qTdN?e)( 1
qTdN e]1)([ 1
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 2.51 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 5.019 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)
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 2.51 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 1.119 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)
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,
*00 e rTSF?
*
*
e)e( rT
rT
S
S
S
F?
A
rT
ArTF HHH
*
* ee
1
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
ATqrF HH *)(e
ATrrF HH f *)(e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.11
Gamma
Gamma (?) is the rate of change of delta (?)
with respect to the price of the underlying
Figure 13.9 (p,325) [for a call or put]
2
2
S
f
S?
Gamma
Stock PriceX
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.12
Equation for Gamma
The Gamma (?) for a European call or put
paying a continuous dividend q is
where
TS
dN qT
0
1 e)('
2/
11
2
1e
2
1)()(' ddndN
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.13
Gamma Addresses Delta Hedging
Errors Caused By Curvature
Figure 13.7 (p,322)
Call
Price
S
C
Stock PriceS'
C''
C'
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
Figure 13.6 (p,321) ft
0 Theta Time to Maturity
At-the-Money
In-the-Money
Out-of-the-Money
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.15
Equations for Theta
The Theta (?) of an European call option
paying a dividend at rate q is
The Theta (?) of an European put option
paying a dividend at rate q is
)(ee)(
2
e)('
210
10 dNrKdNqS
T
dNS rTqTqT
c
)(ee)(
2
e)('
210
10 dNrKdNqS
T
dNS rTqTqT
p
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.16
Relationship Among
Delta,Gamma,and,Theta
For a non-dividend paying stock
This follows from the
Black-Scholes differential equation
221
2 ( 1 3,7 )r S S r f
2
221
2 2 ( 1 1,1 5 )
f f fr S S r f
t S S
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.17
Vega
Vega (?) is the rate of change of a
derivatives portfolio with respect to volatility
Figure 14.11 (p,317) [for a call or put]
Vega
Stock PriceX
f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.18
Equation for Vega
The Vega (?) for a European call or put
paying a continuous dividend q is
qTdNTS e)('
10?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.19
Managing Delta,Gamma,and Vega
can be changed by taking a position in the
underlying
To adjust? and? it is necessary to take a position
in an option or other derivative
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.20
Hedging Example(ref,p.324,p327)
Assume that a company has a portfolio of the
following S&P100 stock options
Type Position Delta Gamma Vega
Call 2000 0.6 2.2 1.8
Call -500 0.1 0.6 0.2
Put 1000 -0.2 1.3 0.7
Put -1500 -0.7 1.8 1.4
An option is available which has a delta of 0.6,a gamma of 1.8,
and a vega of 0.1.
What position in the traded option and the S&P100 would make
the portfolio both gamma and delta neutral?
Both vega and delta neutral?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.21
Hedging Example
(continued)
First,calculate the delta,gamma,and vega of the portfolio.
deltap = 2000*0.6 - 500*0.1 +1000*(-0.2) -1500*(-0.7)
= +2000
gammap = 2000*2.2 - 500*0.6 +1000* 1.3 -1500* 1.8
= +2700
vegap = 2000*1.8 - 500*0.2 +1000* 0.7 -1500* 1.4
= +2100
To be gamma neutral,we need to add -2700/1.8 = -1500
traded options ( )
This changes the delta of the new portfolio to be
-1500*0.6 + 2000 = 1100
In addition to selling 1500 traded options,we would need a
short position of 1100 shares in the index
*0pT O P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.22
Hedging Example
(continued)
To be vega neutral,we need to add -2100/0.1 = -21000
traded options (i.e,short 21000 options)( )
This changes the delta of the new portfolio to be
-21000*0.6 + 2000 = -10600
In addition to shorting the 21000 traded options,we would need
a long position of 10600 shares in the index
To be delta,gamma,and vega neutral we would need a second
(independent) option,We would then solve a system of
two equations in 2 unknowns to determine how many of
each type of option needs to be purchased to be both
gamma and vega neutral,Then,we take a position in the
underlying to assure delta neutrality.
*0pT O P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.23Hedging Example
(continued)
Assume that a second option is available which has a delta of
0.2,a gamma of 0.9,and a vega of 0.8.
Solving 2 equations with 2 unknowns,we have
The solution to this system is OPT1=-200 and OPT2= -2600
This gives a new? of
Thus,1,360 shares must be shorted to become delta neutral
02*8.01*1.02 1 0 0
02*9.01*8.12 7 0 0
O P TO P T
O P TO P T
1360
2.0*)2600(6.0*)200(2000
2.0*26.0*12000
O P TO P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.24
Rho
Rho is the rate of change of the value of a
derivative with respect to the interest rate
For currency options there are 2 rhos
r h o fr
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.25
Equations for Rho
The Rho (?) of an European call option paying a
dividend at rate q is
The Rho (?) of an European put option paying a
dividend at rate q is
The same formulas apply to European call and put
options on non-dividend stock
2r h o e ( )
rTX T N d
2r h o e ( )
rTX T N d
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.26
Equations for Rho in
Currency Options
In addition to the two previous formulas,which
correspond to the domestic interest rate r,we have
those rhos correspond to rf
The Rho (?f) of an European call currency option is
The Rho (?f) of an European put currency option is
)(eohr 10 dNTS Trf f
)(eohr 10 dNTS Trf f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.27
Hedging in Practice
Traders usually ensure that their portfolios are
delta-neutral at least once a day
Whenever the opportunity arises,they improve
gamma and vega
As portfolio becomes larger hedging becomes
less expensive
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.28
Scenario Analysis
Scenario analysis and the calculation of
value at risk (VaR) is an alternative to
relying exclusively on?,?,?,etc.
Typical VaR question,
What loss level are we 99% certain will not be
exceeded over the next 10 days?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.29
Hedging vs,
Creation of an Option Synthetically
When we are hedging,
we take positions that offset?,?,?,etc.
When we create an option synthetically,
we take positions that match?,?,and?
Thus,the procedure for creating an
option position synthetically is the reverse of
the procedure for hedging the option
position.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.30
Portfolio Insurance
In October 1987,many portfolio managers attempted
to create put options on their portfolios by matching?
This involves initially SELLING enough of the portfolio
(or of index futures) to match the? of the put option
As the value of the portfolio increases,the? of the put
becomes less negative and the position in the portfolio
is increased
As the value of the portfolio decreases,the? of the
put becomes more negative and more of the portfolio
must be SOLD
This strategy did NOT work well on October 19,1987
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.31
Portfolio Insurance Example
A fund manager has a well-diversified portfolio that
mirrors the performance of the S&P500 and is worth
$90 million,The value of the S&P500 is 300 and the
portfolio manager would like to insure against a
reduction of more than 5% in the value of the portfolio
over the next six months,The risk-free rate is 6% per
annum,The dividend yield on both the portfolio and
the S&P500 is 3% and the volatility of the index is
30% per annum.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.32Portfolio Insurance Example
(continued)
If the fund manager buys traded European options,how
much would the insurance cost?
If the value of the portfolio falls by 5%,so does the index as
Return from Change in Portfolio -5.0% in 6 mths
Dividends from Portfolio 1.5% per 6 mths
Total Portfolio Return -3.5% per 6 mths
Risk-free rate 3.0% per 6 mths
Excess Portfolio Return -6.5% per 6 mths
Excess Index Return -6.5% per 6 mths
Total Index Return -3.5% per 6 mths
Dividends from Index 1.5% per 6 mths
Increase in Value of Index -5.0% in 6 mths
Thus,we need to evaluate a put option on the S&P500 with a strike of
300*(1.0-0.05) = 300*0.95 = 285
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.33Portfolio Insurance Example
(continued)
Using
So,we have the total cost of the hedge being
4 1 8 2.0)(2 0 6 4.0
3 3 7 8.0)(4 1 8 6.0
22
11
dNd
dNd
000,755,4$300 000,000,90*85.15?
2 0 1e ( ) e ( )r T q Tp X N d S N d
2 0 1
0,0 6 * ( 6 / 1 2 ) 0,0 3 * ( 6 / 1 2 )
e ( ) e ( )
2 8 5 * 0,4 1 8 2 3 0 0 * 0,3 3 7 8
1 5,8 5
r T q Tp X N d S N d
ee
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.34Portfolio Insurance Example
(continued)
Explain carefully alternative strategies open to the fund
manager involving traded European call options,and show
that they lead to the same result
From the put-call parity
This shows that a put option can be created by buying a call
option,selling (or shorting) e-qT of the index,and lending
the net present value of the strike at the risk-free rate of
interest.
0
0
ee
ee
q T r T
q T r T
S p c X
p c S X
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.35Portfolio Insurance Example
(continued)
Applying this to this situation,the fund manager could,
1,Sell 90e-0.03*6/12 = $88.66 million of stock
2,Buy 300,000 call options on the S&P500 with exercise
price = 285 and 6 months to maturity
3,Invest remaining cash at the risk-free rate of 6%
Thus,$1.34 million of stock is retained
The value of one call is
The total cost of the call options is 300,000*34.80 = $10.44 mill
80.34
5 8 1 8.0*e2856 6 2 2.0*e300
)(e)(e
12/6*06.012/6*03.0
210
dNKdNSc rTqT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.36Portfolio Insurance Example
(continued)
The value of the portfolio at the end of the six months is
pay-off of put dividends
future value of price of put
Note that
12/6*06.012/6*06.0
12/6*06.0
12/6*06.0
12/6*06.012/6*06.0
e*03.0*12/6e75.4)5.85,m a x (
e34.190.4)5.85,m a x (
e34.160.80)0,5.85m a x (
e)44.1066.88(e34.1)0,5.85m a x (
T
T
T
T
S
S
S
S
75.444.1066.8897.82e5.85 12/6*06.0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.37Portfolio Insurance Example
(continued)
If the fund manager decides to provide
insurance by keeping part of the portfolio in
risk-free securities,what should the initial
position be?
The delta of one put option is
This indicates that 33.27% of the portfolio (i.e,
$29.94 million) should be initially sold and
invested in risk-free securities
3327.0
]16622.0[*e
]1)([*e
12/6*03.0
1
dNqT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.38Portfolio Insurance Example
(continued)
If the fund manager decides to provide insurance by using
nine-month index futures,what should the initial position be?
The delta of a nine-month index futures contract is
From before,the spot position is 29,940,000,so we need
contracts.
023.1
e
e
12/9*)03.006.0(
)(
Tqr
3 9 00 2 3.1 1*2 5 0*3 0 0 0 0 0,9 4 0,29?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.39
Assignments
13.2,13.8,13.10,13.12,13.14,13.15,
13.18,13.19,13.22
Assignment Questions
Tang Yincai,? 2003,Shanghai Normal University
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
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
13.3
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
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
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 =?
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
qTdN?e)( 1
qTdN e]1)([ 1
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 2.51 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 5.019 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)
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 2.51 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 1.119 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)
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,
*00 e rTSF?
*
*
e)e( rT
rT
S
S
S
F?
A
rT
ArTF HHH
*
* ee
1
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
ATqrF HH *)(e
ATrrF HH f *)(e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.11
Gamma
Gamma (?) is the rate of change of delta (?)
with respect to the price of the underlying
Figure 13.9 (p,325) [for a call or put]
2
2
S
f
S?
Gamma
Stock PriceX
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.12
Equation for Gamma
The Gamma (?) for a European call or put
paying a continuous dividend q is
where
TS
dN qT
0
1 e)('
2/
11
2
1e
2
1)()(' ddndN
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.13
Gamma Addresses Delta Hedging
Errors Caused By Curvature
Figure 13.7 (p,322)
Call
Price
S
C
Stock PriceS'
C''
C'
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
Figure 13.6 (p,321) ft
0 Theta Time to Maturity
At-the-Money
In-the-Money
Out-of-the-Money
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.15
Equations for Theta
The Theta (?) of an European call option
paying a dividend at rate q is
The Theta (?) of an European put option
paying a dividend at rate q is
)(ee)(
2
e)('
210
10 dNrKdNqS
T
dNS rTqTqT
c
)(ee)(
2
e)('
210
10 dNrKdNqS
T
dNS rTqTqT
p
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.16
Relationship Among
Delta,Gamma,and,Theta
For a non-dividend paying stock
This follows from the
Black-Scholes differential equation
221
2 ( 1 3,7 )r S S r f
2
221
2 2 ( 1 1,1 5 )
f f fr S S r f
t S S
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.17
Vega
Vega (?) is the rate of change of a
derivatives portfolio with respect to volatility
Figure 14.11 (p,317) [for a call or put]
Vega
Stock PriceX
f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.18
Equation for Vega
The Vega (?) for a European call or put
paying a continuous dividend q is
qTdNTS e)('
10?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.19
Managing Delta,Gamma,and Vega
can be changed by taking a position in the
underlying
To adjust? and? it is necessary to take a position
in an option or other derivative
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.20
Hedging Example(ref,p.324,p327)
Assume that a company has a portfolio of the
following S&P100 stock options
Type Position Delta Gamma Vega
Call 2000 0.6 2.2 1.8
Call -500 0.1 0.6 0.2
Put 1000 -0.2 1.3 0.7
Put -1500 -0.7 1.8 1.4
An option is available which has a delta of 0.6,a gamma of 1.8,
and a vega of 0.1.
What position in the traded option and the S&P100 would make
the portfolio both gamma and delta neutral?
Both vega and delta neutral?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.21
Hedging Example
(continued)
First,calculate the delta,gamma,and vega of the portfolio.
deltap = 2000*0.6 - 500*0.1 +1000*(-0.2) -1500*(-0.7)
= +2000
gammap = 2000*2.2 - 500*0.6 +1000* 1.3 -1500* 1.8
= +2700
vegap = 2000*1.8 - 500*0.2 +1000* 0.7 -1500* 1.4
= +2100
To be gamma neutral,we need to add -2700/1.8 = -1500
traded options ( )
This changes the delta of the new portfolio to be
-1500*0.6 + 2000 = 1100
In addition to selling 1500 traded options,we would need a
short position of 1100 shares in the index
*0pT O P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.22
Hedging Example
(continued)
To be vega neutral,we need to add -2100/0.1 = -21000
traded options (i.e,short 21000 options)( )
This changes the delta of the new portfolio to be
-21000*0.6 + 2000 = -10600
In addition to shorting the 21000 traded options,we would need
a long position of 10600 shares in the index
To be delta,gamma,and vega neutral we would need a second
(independent) option,We would then solve a system of
two equations in 2 unknowns to determine how many of
each type of option needs to be purchased to be both
gamma and vega neutral,Then,we take a position in the
underlying to assure delta neutrality.
*0pT O P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.23Hedging Example
(continued)
Assume that a second option is available which has a delta of
0.2,a gamma of 0.9,and a vega of 0.8.
Solving 2 equations with 2 unknowns,we have
The solution to this system is OPT1=-200 and OPT2= -2600
This gives a new? of
Thus,1,360 shares must be shorted to become delta neutral
02*8.01*1.02 1 0 0
02*9.01*8.12 7 0 0
O P TO P T
O P TO P T
1360
2.0*)2600(6.0*)200(2000
2.0*26.0*12000
O P TO P T
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.24
Rho
Rho is the rate of change of the value of a
derivative with respect to the interest rate
For currency options there are 2 rhos
r h o fr
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.25
Equations for Rho
The Rho (?) of an European call option paying a
dividend at rate q is
The Rho (?) of an European put option paying a
dividend at rate q is
The same formulas apply to European call and put
options on non-dividend stock
2r h o e ( )
rTX T N d
2r h o e ( )
rTX T N d
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.26
Equations for Rho in
Currency Options
In addition to the two previous formulas,which
correspond to the domestic interest rate r,we have
those rhos correspond to rf
The Rho (?f) of an European call currency option is
The Rho (?f) of an European put currency option is
)(eohr 10 dNTS Trf f
)(eohr 10 dNTS Trf f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.27
Hedging in Practice
Traders usually ensure that their portfolios are
delta-neutral at least once a day
Whenever the opportunity arises,they improve
gamma and vega
As portfolio becomes larger hedging becomes
less expensive
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.28
Scenario Analysis
Scenario analysis and the calculation of
value at risk (VaR) is an alternative to
relying exclusively on?,?,?,etc.
Typical VaR question,
What loss level are we 99% certain will not be
exceeded over the next 10 days?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.29
Hedging vs,
Creation of an Option Synthetically
When we are hedging,
we take positions that offset?,?,?,etc.
When we create an option synthetically,
we take positions that match?,?,and?
Thus,the procedure for creating an
option position synthetically is the reverse of
the procedure for hedging the option
position.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.30
Portfolio Insurance
In October 1987,many portfolio managers attempted
to create put options on their portfolios by matching?
This involves initially SELLING enough of the portfolio
(or of index futures) to match the? of the put option
As the value of the portfolio increases,the? of the put
becomes less negative and the position in the portfolio
is increased
As the value of the portfolio decreases,the? of the
put becomes more negative and more of the portfolio
must be SOLD
This strategy did NOT work well on October 19,1987
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.31
Portfolio Insurance Example
A fund manager has a well-diversified portfolio that
mirrors the performance of the S&P500 and is worth
$90 million,The value of the S&P500 is 300 and the
portfolio manager would like to insure against a
reduction of more than 5% in the value of the portfolio
over the next six months,The risk-free rate is 6% per
annum,The dividend yield on both the portfolio and
the S&P500 is 3% and the volatility of the index is
30% per annum.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.32Portfolio Insurance Example
(continued)
If the fund manager buys traded European options,how
much would the insurance cost?
If the value of the portfolio falls by 5%,so does the index as
Return from Change in Portfolio -5.0% in 6 mths
Dividends from Portfolio 1.5% per 6 mths
Total Portfolio Return -3.5% per 6 mths
Risk-free rate 3.0% per 6 mths
Excess Portfolio Return -6.5% per 6 mths
Excess Index Return -6.5% per 6 mths
Total Index Return -3.5% per 6 mths
Dividends from Index 1.5% per 6 mths
Increase in Value of Index -5.0% in 6 mths
Thus,we need to evaluate a put option on the S&P500 with a strike of
300*(1.0-0.05) = 300*0.95 = 285
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.33Portfolio Insurance Example
(continued)
Using
So,we have the total cost of the hedge being
4 1 8 2.0)(2 0 6 4.0
3 3 7 8.0)(4 1 8 6.0
22
11
dNd
dNd
000,755,4$300 000,000,90*85.15?
2 0 1e ( ) e ( )r T q Tp X N d S N d
2 0 1
0,0 6 * ( 6 / 1 2 ) 0,0 3 * ( 6 / 1 2 )
e ( ) e ( )
2 8 5 * 0,4 1 8 2 3 0 0 * 0,3 3 7 8
1 5,8 5
r T q Tp X N d S N d
ee
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.34Portfolio Insurance Example
(continued)
Explain carefully alternative strategies open to the fund
manager involving traded European call options,and show
that they lead to the same result
From the put-call parity
This shows that a put option can be created by buying a call
option,selling (or shorting) e-qT of the index,and lending
the net present value of the strike at the risk-free rate of
interest.
0
0
ee
ee
q T r T
q T r T
S p c X
p c S X
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.35Portfolio Insurance Example
(continued)
Applying this to this situation,the fund manager could,
1,Sell 90e-0.03*6/12 = $88.66 million of stock
2,Buy 300,000 call options on the S&P500 with exercise
price = 285 and 6 months to maturity
3,Invest remaining cash at the risk-free rate of 6%
Thus,$1.34 million of stock is retained
The value of one call is
The total cost of the call options is 300,000*34.80 = $10.44 mill
80.34
5 8 1 8.0*e2856 6 2 2.0*e300
)(e)(e
12/6*06.012/6*03.0
210
dNKdNSc rTqT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.36Portfolio Insurance Example
(continued)
The value of the portfolio at the end of the six months is
pay-off of put dividends
future value of price of put
Note that
12/6*06.012/6*06.0
12/6*06.0
12/6*06.0
12/6*06.012/6*06.0
e*03.0*12/6e75.4)5.85,m a x (
e34.190.4)5.85,m a x (
e34.160.80)0,5.85m a x (
e)44.1066.88(e34.1)0,5.85m a x (
T
T
T
T
S
S
S
S
75.444.1066.8897.82e5.85 12/6*06.0
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.37Portfolio Insurance Example
(continued)
If the fund manager decides to provide
insurance by keeping part of the portfolio in
risk-free securities,what should the initial
position be?
The delta of one put option is
This indicates that 33.27% of the portfolio (i.e,
$29.94 million) should be initially sold and
invested in risk-free securities
3327.0
]16622.0[*e
]1)([*e
12/6*03.0
1
dNqT
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.38Portfolio Insurance Example
(continued)
If the fund manager decides to provide insurance by using
nine-month index futures,what should the initial position be?
The delta of a nine-month index futures contract is
From before,the spot position is 29,940,000,so we need
contracts.
023.1
e
e
12/9*)03.006.0(
)(
Tqr
3 9 00 2 3.1 1*2 5 0*3 0 0 0 0 0,9 4 0,29?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
13.39
Assignments
13.2,13.8,13.10,13.12,13.14,13.15,
13.18,13.19,13.22
Assignment Questions
