Download Lecture 11: The Greeks and Risk Management This lecture studies

Lecture 11:
The Greeks and Risk Management
This lecture studies market risk management from
the perspective of an options trader. First, we show how
to describe the risk characteristics of derivatives. Then,
we construct portfolios that eliminate these risks.
I. Motivation
II. Partial Derivatives of Simple Securities
III. Partial Derivatives of European Options
IV. The Gamma-Theta Relationship
V. Popular Options Strategies and the Greeks
VI. Risk Management
A.
B.
C.
D.
E.
Portfolio Hedging
Delta Hedging
Gamma Hedging
Simultaneous Delta and Gamma Hedging
Theta, Vega, and Rho Hedging
VII. The Cost of Greeks
VIII. Other Risk Management Approaches
Risk Management with Options
I. Motivation
Traders who write derivatives must hedge their risk
exposure.
We’d like to simply characterize the main risks
associated with a complicated portfolio of positions
on an underlying.
– Ultimately, that’s where we’re headed.
Example: Suppose you’re trading options for
Goldman-Sachs, and you just wrote, for $5, a 10week, ATM European call.
The underlying’s trading at $50, and D 50%.
The risk-free rate is 3.0%.
Black-Scholes tells you that the call option is worth
$4.50. How can you make the profit of $0.50 per
option without risk?
Buy the same option for $4.50 elsewhere.
Spend $4.50 on a replicating portfolio (i.e., buy a
synthetic option) that has the same payoff.
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Robert Novy-Marx
Risk Management with Options
Question 1 : can you really perfectly replicate the
option’s payoff?
That is, can you perfectly hedge away all of the risk
associated with the call you wrote?
You can if both:
– The binomial tree model perfectly describes the
stock price dynamics.
– You can trade without transaction costs.
You can if both:
– The log-normal model perfectly describes the
stock price dynamics.
– You can trade continuously and without transaction
costs.
But in the real world:
We can’t trade continuously.
Transaction costs can be substantial.
The volatility of the underlying and the risk-free
rate aren’t constant.
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Risk Management with Options
Question 2: So what should you do, if you can’t
perfectly hedge the risk of the call you’ve written?
1. Identify the different sources of risk.
The value of the call changes if any of the
following factors changes:
S D Stock Price
t D Time
D Volatility
r D Interest Rate
2. Form an approximate replicating portfolio for the
written call option.
The value of this portfolio should change by
about the same amount as that of the option.
– At least for small changes in the factors.
But how do we figure out how sensitive the option is
to the factors?
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Robert Novy-Marx
Risk Management with Options
The “Greeks”
To construct the approximate replicating portfolio, we
have to know how much the value of the option
changes as the various factors change.
That is, the sensitivity of the call to each factor.
Using calculus (i.e., a linear approximation), for
small changes in the factors, the value of the call
option changes by:
1 @2 C
@C
@C
@C
@C
2
(dS
)
dS C
C
dt
C
d
C
dr,
dC D
2
@S
2 @S
@t
@
@r
„ƒ‚…
„ƒ‚…
„ƒ‚…
„ƒ‚…
„ƒ‚…
Delta
Theta
Gamma
or, using the symbols ,
dC
D c dS C 12
2
c (dS )
Vega
Rho
, , and ,
C c dt C c d C c dr.
– These “Greeks” characterize the market risk
associated with the option.
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Robert Novy-Marx
Risk Management with Options
II. Example: Stocks, Bonds, and Forwards
Before we consider the sensitivity of an option’s
price to each of the factors that determine an
option’s value, we’ll do some simpler securities.
– To get a feel.
First, what are ,
D
@S t
D 1
@S t
S
D
@S
D 0
@S t
S
D
@S t
D 0
@t
D
@S t
D 0
@
D
@S t
D 0.
@r
S
S
S
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, , and for a stock?
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Robert Novy-Marx
Risk Management with Options
What about a bond?
– Bt,T D e
r (T t)
, so
D
@Bt
D 0
@S t
B
D
@B
D 0
@S t
B
D
@Bt
D rBt,T
@t
D
@Bt
D 0
@
D
@Bt
D
@r
B
B
B
(T
t )Bt,T .
B D rBt,T > 0 )
Bond becomes more valuable as time passes.
B D
(T
t )Bt,T < 0 )
Bond looses value when interest rates rise.
Note: B D
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DB Bt,T .
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Risk Management with Options
What about a forward contract?
– Replication ) f t D S t
Ke
f
D
@f t
D 1
@S t
f
D
@f
@S t
f
D
@f t
D
@t
f
D
@f t
D 0
@
D
@f t
D (T
@r
f
r (T t)
, so
D 0
rKe
r (T t)
t )Ke
r (T t)
.
A long forward position is worth more (ceteris
paribus)
– If the underlying goes up.
– If interest rates rise.
– With more time to maturity.
Note: f D 1 and f D 0 explains why we can
replicate a forward statically.
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Risk Management with Options
III. The Greeks for European Options
How does the price of a derivative change when we
vary one factor and hold all others fixed?
Delta ()
Delta measures a derivative’s sensitivity to the
price of the underlying security.
c
D
@C
@S
p
D
@P
D
@S
D N(d1) > 0
N( d1 ) < 0.
– Note that:
c ! 0 as S ! 0
c ! 1 as S ! 1
Important: In the Black-Scholes model, delta tells
us how many shares of the stock to buy for the
replicating portfolio.
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Risk Management with Options
What does c look like?
Delta vs. Underlying price
T = 1 week, 1 month, and 1 quarter
Delta HDL
1
0.8
0.6
0.4
0.2
Spot HSL
60
80
100
120
140
160
Delta as a function of the spot price of the underlying, for
three different time-to-expirations [T D 0.02 (solid line),
T D 0.0833 (dashed line) and T D 0.25 (dotted line)].
Figure depicts the case when K D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
How about c and “moneyness”?
Delta vs. Time-to-Expiration
ITM, ATM, OTM
Delta HDL
1
0.8
0.6
0.4
0.2
Time HTL
0.1
0.2
0.3
0.4
0.5
Delta as a function of time-to-maturity, for three different
levels of “moneyness” [K D 100 (solid line, ATM), K D
90 (dashed line, ITM) and K D 110 (dotted line, OTM)].
Figure depicts the case when S D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
Gamma ( )
Gamma measures a derivative’s convexity.
c
D
D
p
D
@c
@S
@d1
@S
N 0 (d1)
N (d1) D
p
> 0
S T t
0
@( d1)
@S
N 0 ( d1) D
c.
– Note that:
! 0 as S ! 0
! 0 as S ! 1
is high when S K.
– What does gamma tell us?
It tells us how much we gain as the
underlying rises.
It also tells us how quickly a delta-hedged
derivative becomes unhedged.
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Risk Management with Options
What does
c
look like?
Gamma vs. Underlying price
T = 1 week, 1 month, and 1 quarter
Gamma HGL
0.05
0.04
0.03
0.02
0.01
Spot HSL
80
100
120
140
Gamma as a function of the spot price of the underlying,
for three different time-to-expirations [T D 0.02 (solid
line), T D 0.0833 (dashed line) and T D 0.25 (dotted
line)]. Figure depicts the case when K D 100, D 0.56
and r D 0.05.
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Risk Management with Options
How about
c
and “moneyness”?
Gamma vs. Time-to-Expiration
ATM, OTM, ITM
Gamma HGL
0.035
0.03
0.025
0.02
0.015
0.01
0.005
Time HTL
0.1
0.2
0.3
0.4
0.5
Gamma as a function of time-to-maturity, for three
different levels of “moneyness” [K D 100 (solid line, ATM),
K D 80 (dashed line, ITM) and K D 120 (dotted line,
OTM)]. Figure depicts the case when S D 100, D 0.56
and r D 0.05.
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Risk Management with Options
Theta ()
Theta measures the derivative’s sensitivity to the
passage of time. It captures time-decay.
@C
D
@t
c
@
D
@(T t)
SN(d1)
Ke
@N(d1 )
S @(T
C Ke
t)
D
r (T t)
N(d2)
r (T t) @N(d2 )
@(T t)
rKe
r (T t)
N(d2)
This can be simplified using
0
SN (d1) D
p
(d2 C T t)2 =2
Se
p
2
0
D SN (d2)e
D Ke
r (T t)
p
( T t )d2 2 (T t)=2
N 0 (d2)
and
@ (d1
@(T
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d2)
t)
D
p
@ T t
D p
@(T t )
2 T
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Robert Novy-Marx
Risk Management with Options
Taken together, these imply
c
D
N 0(d1 )S
p
2 (T t )
rKe
r (T t)
N(d2)
< 0.
c < 0 ) the value of a call decreases as time
goes by, ceteris paribus.
As time-to-expiration decreases:
– The variance of the stock price at maturity
decreases.
Less value in the right to not exercise.
– The time discounting of the exercise price
decreases.
Expected cost of exercise is higher.
Important: the fact that c is negative does not
imply that the call price is expected to fall.
– Remember, the stock price, on average, rises
over time.
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By put-call parity,
P
D C
f,
we have
p
D
@(C
f)
@t
@f
D c C
@(T t )
D
N 0 (d1) S
p
C rKe
2 (T t )
r (T t)
N( d2 ).
The first term is again because the variance of
the stock price at maturity decreases as time-tomaturity decreases.
The second term is positive.
– The PV of the strike grows over time.
I.e., with less time-to-maturity.
– Put receive the strike, so this tends to make the
put more valuable as time passes.
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Risk Management with Options
What does c look like?
Theta vs. Underlying price
T = 1 week, 1 month, and 1 quarter
Theta HQL
Spot HSL
80
100
120
140
160
180
-20
-40
-60
-80
Theta as a function of the spot price of the underlying, for
three different time-to-expirations [T D 0.02 (solid line),
T D 0.0833 (dashed line) and T D 0.25 (dotted line)].
Figure depicts the case when K D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
How about c and “moneyness”?
Theta vs. Time-to-Expiration
ATM, OTM, ITM
Theta HQL
Time HTL
0.1
0.2
0.3
0.4
0.5
-10
-20
-30
-40
-50
Theta as a function of time-to-maturity, for three different
levels of “moneyness” [K D 100 (solid line, ATM), K D
80 (dashed line, ITM) and K D 120 (dotted line, OTM)].
Figure depicts the case when S D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
Vega ()
Vega measures the derivative’s sensitivity to the
volatility of the underlying security.
– You’ll occasionally see it called “Kappa” ().
p
@C
c D
DS T
@
p D
t N 0 (d1) > 0
@P
D c
@
Note that:
0 for S K
is largest for S K e r (T
t)
0 for S K
Important: Vega is important to traders who worry
about changes in the volatility of the underlying
security.
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Risk Management with Options
What does c look like?
Vega vs. Underlying price
T = 1 week, 1 month, and 1 quarter
Vega HΝL
20
15
10
5
Spot HSL
80
100
120
140
160
180
200
Vega as a function of the spot price of the underlying, for
three different time-to-expirations [T D 0.02 (solid line),
T D 0.0833 (dashed line) and T D 0.25 (dotted line)].
Figure depicts the case when K D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
How about c and “moneyness”?
Vega vs. Time-to-Expiration
ATM, OTM, ITM
Vega HΝL
17.5
15
12.5
10
7.5
5
2.5
Time HTL
0.05
0.1
0.15
0.2
Vega as a function of time-to-maturity, for three different
levels of “moneyness” [K D 100 (solid line, ATM), K D
80 (dashed line, ITM) and K D 120 (dotted line, OTM)].
Figure depicts the case when S D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
The Gamma-Vega Relationship
Remember:
c
D
N 0 (d1)
p
S T t
So:
c
p
D S T
t N 0 (d1)
D S 2(T
t)
c
or
c
c
D S 2(T
t)
– They always come together
Closely related; sensitivities to expected and
realized volatilities
– Come in fixed proportion, for a given series
– Calender spreads allow you to bet more on one
than the other
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Risk Management with Options
Rho (c )
Rho measures the derivative’s sensitivity to the
risk-free interest rate.
c
D
@
@r
SN(d1)
1)
D S @N(d
@r
C (T
Ke
Ke
r (T t)
N(d2)
r (T t) @N(d2 )
@r
t )Ke
r (T t)
N(d2).
Now
@N(d1 )
@r
@N(d2 )
@r
D
D
D
@d1 0
N (d1)
@r
p
@(d1 T t) 0
N (d2 )
@r
@d1 0
N (d2)
@r
and
SN 0(d1) D Ke
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r (T t)
N 0 (d2).
Robert Novy-Marx
Risk Management with Options
So
c
D (T
t )Ke
r (T t)
N(d2) > 0.
Put-call parity gives
p
D
@(C
D c
D
(T
S C Ke
@r
(T
t )Ke
t )Ke
r (T t)
)
r (T t)
r (T t)
N( d2 ) < 0.
The value of the call always goes up with the
interest rate.
– The P V of S (T ) is always S (t ).
– The P V of K drops.
The opposite is true for puts.
– The value of a put falls with the interest rate.
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Risk Management with Options
What does c look like?
Rho vs. Underlying price
T = 1 week, 1 month, and 1 quarter
Rho HΡL
20
15
10
5
Spot HSL
80
100
120
140
160
180
Rho as a function of the spot price of the underlying, for
three different time-to-expirations [T D 0.02 (solid line),
T D 0.0833 (dashed line) and T D 0.25 (dotted line)].
Figure depicts the case when K D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
How about c and “moneyness”?
Rho vs. Time-to-Expiration
ATM, OTM, ITM
Rho HΡL
25
20
15
10
5
Time HTL
0.1
0.2
0.3
0.4
0.5
Rho as a function of time-to-maturity, for three different
levels of “moneyness” [K D 100 (solid line, ATM), K D
80 (dashed line, ITM) and K D 120 (dotted line, OTM)].
Figure depicts the case when S D 100, D 0.56 and
r D 0.05.
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Risk Management with Options
Other “Greeks”
Much less common; just mentioning their existence
Lambda
– Delta per dollar invested
c
D
c
C
Volga (or Vega-Gamma)
– Second-order sensitivity to volatility
@2 C
@ 2
D
@c
@
Vanna
– Sensitivity of Delta to volatility
@2 C
@ @S
D
@c
@
– Moneyness changes with underlying price, and
implied volatilities change with moneyness
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Risk Management with Options
IV. The Gamma-Theta Relationship
Remember the Black-Scholes partial differential
equation:
1 2 2 @2 C
S @S 2
2
C rS
@C
@S
C
@C
@t
rC
D 0.
Using the Greeks we can rewrite this as
1 2 2
S c
2
C rSc C c
rC
D 0.
That is, the Black-Scholes PDE implies a relation
between C , , , and for a European call
option.
– They are not determined independently.
– This is true in general, not just for calls.
How can we interpret this constraint?
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Risk Management with Options
First, rewrite the constraint as
rC
D rSc C 12 2S 2
c
C c .
Now remember: r C is the expected, risk-neutral
yield to the call.
– Here yield = price rate of return.
The equation says that yield comes from three
sources.
If you own a call you’re:
– Long the underlying stock.
And earning a return on that.
– You’re also “Earning” c .
Because you’re “long volatility.”
– And you’re “Paying” c .
Remember: c < 0.
Time-to-expiration “runs backwards.”
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Risk Management with Options
rC
D rSc C 12 2S 2
c
C c .
This equation also says that if two calls are:
1. Priced the same, and
2. Have the same sensitivity to the underlying,
– I.e., if C D C 0 and C D C 0 , then:
1 2 2
S c
2
C c
D
1 2 2
S c0
2
C c 0 .
That is, the call with the higher Gamma also has
a higher Theta.
– You “pay for” Gamma by taking on Theta.
Traders care about this!
– A lot.
– Having this answer in an interview is the kind of
thing that can get you a job.
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Risk Management with Options
It also gives us another way to understand this:
Theta vs. Time-to-Expiration
ATM, OTM, ITM
Theta HQL
Time HTL
0.1
0.2
0.3
0.4
0.5
-10
-20
-30
-40
-50
Theta as a function of time-to-maturity, for three
different levels of “moneyness.” Solid line, ATM: K=S D
1; dashed line, ITM: K=S D 0.8; dotted line, OTM:
K=S D 1.2. Other parameters: D 0.56 and r D 0.05.
How so?
– Well what happens to Gamma ATM, ITM, and
OTM as T ! 0?
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Risk Management with Options
V. Popular Options Strategies and the Greeks
We can also recast option portfolio strategies in
terms of the Greeks.
– Traders tend to think about them this way.
For example, what are you “buying” when you buy
a straddle?
– That is, when you buy both a put and a call with
the same strike.
py g
(
) y
Straddle:
C=f(S,t)
ATM CALL + ATM PUT
S = 100
K = 100
Profit
(BUY ATM CALL @ $18.84)
(BUY ATM PUT @ $5.80)
t =1
r = 1.15
25
d = 1.00
V = . 3
50
75
125
150
Future Asset Price
-25
Straddle Value = $18.84 + $5.80 = $24.64
Loss
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Strategy:
Strategy: Believe
Believevolatility
volatilityof
of
asset
price
will
be
high,
asset price will be high,but
buthave
have
no
noclue
clueabout
aboutdirection.
direction.
Robert Novy-Marx
Risk Management with Options
You’re not buying Delta.
– At least not very much:
pCc
D c C p
D N(d1 )
N( d1) 0.
pCc D 0 if d1 D 0 , K D Se (r C
2=2)T
.
But you’re definitely buying Gamma.
pCc
D
c
C
p
D 2 c.
– And the Gamma of the call is high, if it’s near the
money.
You’re paying for it with Theta.
– Strike-discounting isn’t the problem.
You’re paying on the put, earning on the call.
– But the premia on both options shrinks with time.
Each moment, your exposure to Gamma costs
you Theta.
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Risk Management with Options
What happens to the straddle price as (T t ) ! 0?
– Assuming everything else is unchanged.
Straddle Price
T = 1 day, 1 week, and 1 month
Price H$L
20
15
10
5
Spot HSL
90
100
110
120
Straddle prices (P100 CC100) as a function of the spot price
of the underlying, for three different time-to-expirations
[T D 0.004 (solid line), T D 0.02 (dashed line) and
T D 0.0833 (dotted line)]. Figure depicts the case when
D 0.56 and r D 0.05.
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Risk Management with Options
Butterfly spreads do more or less the opposite.
– BS: Buy one ITM, sell two ATM, buy one OTM.
But near the money:
– Delta is linear w.r.t. moneyness (more or less).
– Gamma is convex w.r.t. moneyness.
Delta HDL
1
Gamma HGL
0.8
0.04
0.6
0.03
0.4
0.02
0.2
0.01
0.05
Spot HSL
Spot HSL
60
80
100
120
140
160
80
That is,
K
K
1
2
1
2
>
(K
(
ı
K ı
100
120
140
C KCı )
C
KCı ) ,
so
BFS
BFS
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D K
D
ı
2K C KCı 0
K ı
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2
K
C
KCı
< 0.
Robert Novy-Marx
Risk Management with Options
What happens to the Butterfly price at the central
strike as (T t ) ! 0?
– Assuming everything else is unchanged.
Butterfly Price
T = 1 day, 1 week, and 1 month
Price H$L
10
8
6
4
2
Spot HSL
85
90
95
100
105
110
115
120
Butterfly prices (C90 2C100 C C110) as a function of the
spot price of the underlying, for three different time-toexpirations [T D 0.004 (solid line), T D 0.02 (dashed line)
and T D 0.0833 (dotted line)]. Figure depicts the case
when D 0.56 and r D 0.05.
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Robert Novy-Marx
Risk Management with Options
VI. Risk Management
A. Portfolio Hedging
The basic idea of portfolio hedging is that the
value of a portfolio can be made invariant to the
factors affecting it, such as S , and r .
For example, suppose a portfolio consists of
three assets:
V
D n1 A1 C n2 A2 C n3 A3
where:
V is the value of the portfolio,
ni is the number of shares of asset i, and
Ai is the market value of one share of asset i.
Then the sensitivity of the portfolio to some
arbitrary factor, x, is
@V
@x
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D n1
@A1
@A2
@A3
C n2
C n3
.
@x
@x
@x
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Robert Novy-Marx
Risk Management with Options
The objective of x-hedging is to pick the ni such
that the value of the portfolio stays constant
when x changes.
– That is, pick n1, n2, and n3 so that:
@V
@x
D n1
@A1
@A2
@A3
C n2
C n3
D 0.
@x
@x
@x
Then the value of the portfolio stays approximately
constant when x changes by a small amount:
dV
D
@V
dx 0.
@x
Important: It takes n securities to hedge against
n 1 sources of uncertainty.
For example, with two assets you can only
hedge one risk.
– E.g., you could pick the relative weights so that
the portfolio is Delta-neutral.
– All the other exposures are determined by
these weights.
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B. Delta Hedging
A portfolio is Delta neutral (i.e., Delta hedged) if
the of the portfolio is zero.
For example, take our portfolio of three assets,
and let x D S :
portfolio D
@V
@S
@A1
@A2
@A3
D n1
C n2
C n3
@S
@S
@S
D n1 1 C n2 2 C n3 3.
The portfolio will be Delta-hedged if we pick the
ns so that this is zero.
Then the portfolio value will be insensitive to
small changes in S :
dV
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portfolio dS D 0.
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Risk Management with Options
More concretely:
Remember the call you wrote for GoldmanSachs:
T
S
D 50
K
D 50
10
52
t
D
D 0.50
r
D 0.03.
Questions: how many share of the stock should
we buy to Delta hedge the option?
We’re short the call, and c D 0.554.
S of a share is one, so we buy nS shares such
that:
nS 1
0.554 D 0.
So, we buy nS D S D 0.554 shares of the stock.
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C. Gamma Hedging
A portfolio is Gamma neutral (i.e., Gamma
hedged) if the of the portfolio is zero.
Take the portfolio of three assets and let x D S :
portfolio
D
@2 V
@S 2
D
@portfolio
@S
@1
@2
@3
D n1
C n2
C n3
@S
@S
@S
D n1
1
C n2
2
C n3
3.
Question:
If a portfolio is already Delta hedged, so its value
stays approximately constant for small changes in
S , why do we want to Gamma hedge it?
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Example continued ...
We just learned that the portfolio
1. Short the ATM call, and
2. Long 0.554 shares of the stock,
is Delta hedged.
How stable is the value of this portfolio if S
changes?
Small change in S:
– Suppose S increases from 50 to 51.
C (50, 50, 10
, 0.50, 0.03) D 4.498
52
C (51, 50, 10
, 0.50, 0.03) D 5.070.
52
– Then 0.554(51 50) (5.070 4.498) D
0.018.
A loss of less than 2 cents for a $1 increase in
the stock price.
Not too bad.
– But what about bigger moves in the underlying?
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Large change in S :
– Suppose S increases from 50 to 60.
C (50, 50, 10
, 0.50, 0.03) D 4.498
52
C (60, 50, 10
, 0.50, 0.03) D 11.541.
52
– Then 0.554(60 50) (11.541 4.498) D
1.54.
A loss of $1.54 for a $10 increase in the stock
price.
Not so good. Our “hedged” position still had
an effective 15% exposure the large move in
the underlying.
Lesson from the Example:
Delta hedging works well for small changes in S
only.
Gamma hedging can improve the quality of the
hedge.
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To understand why, look at the Taylor series
expansion for the change in the call price
C (S C dS )
C (S )
c dS C 12
2
c (dS )
D 0.554 dS C 0.018 (dS )2.
Buying 0.554 shares hedges the first term.
We’re still exposed to the second term.
– We’d need to Gamma hedge as well to
eliminate that exposure.
For the large change in S (dS D 10), the second
term is $1.80, which explains our $1.55 loss.
– Any discrepancy is due to the missing 3rd-,
4th-, and higher- order terms.
Questions:
Is Gamma hedging alone more effective than
Delta hedging alone?
Can we use stock to Gamma hedge an option?
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D. Simultaneous Delta and Gamma Hedging
What if we wanted to hedge our writhen call such
that our net position was both Delta-neutral and
Gamma-neutral?
– For that ATM call 1 D 0.554 and 1 D 0.0361.
We need another asset
– One that has Gamma.
So any call should do
We’ll consider the call struck at 55.
Delta-hedging requires that
nS S
C nC55 C55
C nC50 C50
D 0.
Gamma-hedging requires that
nS
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S
C
nC55
C55
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C
nC50
C50
D
0.
Robert Novy-Marx
Risk Management with Options
We already know
C50
C50
And S D 1,
S
D 0.554
D 0.0361.
D 0.
Black-Scholes gives
C55
C55
D 0.382
D 0.0348.
Finally, nC50 D 1, so need to buy nS shares of
the stock and nC55 calls at 55 such that:
nS
C
0.382 nC55
0.554
D
0
0
C
0.0348 nC55
0.0361
D
0.
Solving these yields nS D 0.158 and nC55 D 1.037.
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How stable is the value of this portfolio to changes
in S ?
Small change in S:
– Suppose S increases from 50 to 51.
C50(51)
C50(50) D 5.067
4.498
D 0.569
C55(51)
C55(50) D 3.002
2.602
D 0.400
and
0.158 1 C 1.037 0.400
1 0.569 D 0.001.
– The value of the portfolio changes (increases)
by less than 0.1 cent.
That’s pretty good hedging.
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Large change in S :
– Suppose S increases from 50 to 60.
C50(60)
C50(50) D 11.581
4.498
D 7.084
C55(60)
C55(50) D 8.104
2.602
D 5.501
and
0.158 10 C 1.037 5.501
1 7.084 D 0.201.
– The value of the portfolio changes (increases)
by 20 cents.
That’s much less than a change of $1.55
change for the Delta hedged portfolio.
Important: We needed three securities to form a
portfolio hedged in both Delta and Gamma.
In general we need n securities to form a
portfolio that is insensitive to small variations in
n 1 factors.
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E. Theta, Vega, and Rho Hedging
These don’t get worried about as much
– They’re still important, but not as important.
The mechanics of these hedging strategies are
similar to Delta or Gamma hedging.
– For example, a portfolio is Theta hedged, or is
Theta neutral, if its is zero.
Some questions:
How does the of a portfolio relate to the ’s of
the securities that form the portfolio?
Can a bond be used for Theta hedging?
Can a stock be used to Theta hedge an option?
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VII. The Cost of Greeks
We can construct pure exposures to individual
Greeks
– By hedging all the other risks away
This allows us to “price” the exposures
– Figure what it costs to take on a pure exposure
to a given Greek
The easiest Greek to price is delta
– How do you get a pure exposure to ?
– What’s the cost of a unit exposure to ?
The underlying is a pure exposure to delta
– So cost of a unit exposure:
P D S
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Rho is also easy to price
– What does “duration” cost?
Nothing!
Remember, for a bond B D
B
B
D
@Bt
D rBt,T
@t
B
D
@Bt
D
@r
(T
D B D 0, and
t )Bt,T
So buy $1 of a long bond, sell $1 of a short bond
– It’s free
– l=s D l=s D l=s D 0, and
l=s
D r
r D 0
l=s
D
(Tl
t ) C (Ts
D
(Tl
Ts )
t)
So P D 0
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Now it’s easy to price theta
Again, for a bond B D
B
D B D 0, and
B
D
@Bt
D rBt,T
@t
B
D
@Bt
D
@r
(T
t )Bt,T
Rho is free
So buy a bond, at a cost of Bt,T
– Hedge the Rho risk
using a zero-cost portfolio of bonds
– Hedging Rho is free
Your pure theta exposure of rBt,T cost you Bt,T
Unit price is (price paid) / (total exposure), so
P
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D
Bt,T
rBt,T
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D
r
Robert Novy-Marx
Risk Management with Options
Gamma and Vega are slightly harder
Let’s start by summarizing what we know
– Cost of , , and :
P D S
– C D SN(d1)
c
c
P
D 0
P
D 1=r
r (T t)
Ke
N(d2 ) and
D N(d1)
D
N 0(d1)
p
S T t
N 0 (d1)S
p
2 (T t )
c
D
c
D S 2(T
c
D (T
t)
t )Ke
rKe
r (T t)
N(d2)
c
r (T t)
N(d2)
Cost of a delta/rho/theta-hedged call?
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Cost and Greeks of the hedged call:
P C
D
P C
SN(d1)
1
r
D
P C
r (T t)
Ke
0
N (d1)S
p
2 (T t )
P C
N(d2)
rKe
SN(d1)
r (T t)
!
N(d2)
N 0(d1)S
p
2r (T t )
and
P P
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D 0
D
N 0 (d1)
p
S T t
P D 0
P D S 2(T
P D 0
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c
Robert Novy-Marx
Risk Management with Options
Now note that
P
D
P
N 0(d1)S
2r
i.e., P
p
(T t)
N 0(d1 )
p
S T t
D
D
2S 2
2r
2S 2
2r
P
This is independent of all contractual parameters
– $1 of any delta/rho/theta-hedged call gives you
the same amount of gamma: P =P D 2r= 2S 2
So a long/short portfolio of delta/rho/theta-hedged
calls is gamma-neutral
– It’s also delta/rho/theta-neutral
– It’s not generally Vega-neutral
Unless same maturity on both sides
I.e., pure Vega-exposure is free ) P D 0
So ame equation gives us the price of gamma:
P
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D
P
P
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D
2S 2
2r
Robert Novy-Marx
Risk Management with Options
Summarizing:
P D S
P
P
2S 2
D
2r
D 0
P
D 0
P
D 1=r
So the value of any derivative V on S , in terms of
its exposures to the risk factors, is
V
C P C P C P D P C P
2 2
S
1
D S C 2r
C r – We can also write this as
rV
D rS CS C 12 2S 2CS S C C t
The Black-Scholes PDE!
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VIII. Other Risk Management Approaches
Stop-Loss Rules.
Scenario Analysis:
1. Monte Carlo Simulations.
2. Stress Testing.
3. Value at Risk (VAR).
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