Transcript Document

The Greek Letters
Chapter 14
1
The Greeks are coming!
The Greeks are coming!
Parameters of SENSITIVITY
Delta = 
Gamma = 
Theta = 
Vega = 
Rho = 
2
Example:
S=100; K = 100; r = 8%; T-t =180 days;
 = 30%.
Call
Put
Premium
10.3044
6.4360
Delta = 
Theta = 
Vega = 
Rho = 
Gamma = 
0.6151
-12.2070
26.8416
25.2515
0.0181
-0.3849
-4.5701
26.8416
-22.1559
0.0181
3
The GREEKS are measures of sensitivity.
The question is how sensitive the position’s
value is to changes in any of the variables
that contribute to the position’s market
value.These variables are S, X, T-t, r and
the return’s standard deviation, s.
DELTA
 = 0.6151
The Delta of any position measures the $
change/share in the position’s value that
ensues a “small” change in the value of the
underlying.
4
Delta (See Figure 14.2, page 302)
• Delta (D) is the rate of change of the
option price with respect to the underlying
Option
price
Slope = 
B
A
Stock price
5
In mathematical terms DELTA is the first
derivative of the option’s price with respect
to S. As such, Delta carries the units of the
option’s price; I.e., $ per share.
A Call: (c) = c/S
A Put:
The stock:
(p)= p/S
(S)= S/S = 1
6
THETA

Theta measures the sensitivity of the option’s
price to a “small” change in the time
remaining to expiration:
(c) = c/(T-t)
(p) = p/(T-t)
Theta is given in terms is $/1 year. Thus, if
(c) = - $12.2070/year, it means that if time to
expiration increases (decreases) by one year,
the call price will increase (decreases) by
$12.2070. Or, 12.2070/365 = 3.34 cent per day.
7
GAMMA 
Gamma measures the change in delta when
the market price of the undelrlying asset
changes.  = 0.0181
Gamma is the second derivative of the
option’s price with respect to the
underlying price.
(c) = (c)/S = 2c/ S2
(p) = (p)/S = 2p/ S2
(S) = (S)/S = 2S/ S2 = 0.
8
VEGA

Vega measures the sensitivity of the
option’s market price to “small” changes in
the volatility of the underlying asset’s
return.
Vega = 26.8416
(c) = c/s
(p) = p/s
Thus, Vega is in terms of $/1% change in
S.
9
RHO 
Rho measures the sensitivity of the option’s
price to “small changes in the rate of
interest. (c) = c/r
(p) = p/r
Rho = 
25.2515
-22.1559
Rho is in terms of $/%change of r.
10
DELTA-NEUTRAL POSITIONS
We just sold n(c) overvalued calls and we
wish to protect the profit against possible
adverse move of the underlying asset price.
To do so, we intend to purchase share of
the underlying in a quantity that
GUARANTIES that a small price change will
not have any impact on the short call and
long shares position.
Vposition = Sn(S) + cn(c)
(V) = n(S) + (c)n(c)
11
DELTA-NEUTRAL POSITIONS
Vposition = Sn(S) + cn(c)
(V) = n(S) + (c)n(c)
In order to have DELTA-neutral position we
solve the above equation for (V) = 0.
The solution is: n(S) = - n(c) (c)
The negative sign of the RHS of the
solution indicates that the calls and the
underlying asset must be held in opposite
direction.
12
EXAMPLE:
We just sold 10 CBOE calls whose delta is
$.50/shares. Each call covers 100 shares. How
many shares of the underlying stock we must
purchase in order to create a delta-neutral
position?
n(s) = - n(c)(c).
(c) = 0,50 and n(c) = 10 but every call covers
100 shares. Therfore: n(c) = - 1,000 shares.
n(s) = - [ - 1,000(0,50)] = 500.
The DELTA-neutral position consists of of 500
long shares and 10 short calls.
13
DELTA NEUTRAL POSITIONS
AGAIN, A DELTA-NEUTRAL PORTFOLIO IS
DEFINED BY ITS
DELTA(PORTFOLIO) = 0.
IN OUR CASE OF CALLS AND SHARES:
(V) = 0  n(S) + (c) n(c) = 0.
THE NUMBER OF SHARES OF THE
UNDERLYING TO BE PURCHASED IS:
n(S) = - n(c)(C).
14
EXAMPLE: We just sold 20 calls and 20 puts
whose deltas are $.7/share and -$.3/share,
respectively. Every call and every put covers
100 shares. How many shares of the
underlying stock we must purchase in order
to create a delta-neutral position?
the position’s delta is:
 = n(S) + (c)n(c) + (p)n(p).
n(S)+(.7)(-2,000)+(-.3)(-2,000)=0
n(S) = 800.
15
Portfolio: The portfolio consisting of 20
short calls, 20 short puts and 800 long
shares is
delta- neutral.
Price/share:
+$1
-$1
shares
+$800
-$800
calls
-$1,400
+$1,400
Puts
+$600
-$600
$0
$0
Portfolio
16
Results:
1. The deltas of a call and a put on the same
underlying asset, (with the same time to
expiration and the same exercise price)
must satisfy the following equality:
(c) = 1 + (p)
2. Using the Black and Scholes formula, it is
possible to show that:
(c) = n(d1) 
0 < (c) < 1
(p) = n(d1) - 1  -1 < (p) < 0
17
EXAMPLES
1. Long 100 shares of the underlying stock,
long one put and short one call on this
stock is always delta-neutral:
(position) =
100 + [n(d1) – 1](100)
+ n(d1)(-100) = 0.
2. The delta of a long STRADDLE long 15
puts and long 15 calls (same K and T-t)
with (c) = .64 and (p) = - .36 is:
15(100)[.64 + (- .36)] =$ 420/share.
18
3. A financial institution holds:
5,000 CBOE calls long;
delta .4,
6,000 CBOE puts long;
delta -.7,
10,000 CBOE puts short;
delta -.5,
Long 100,000 shares
(portfolio) = (.4)500,000 + (-.7)600,000
- (-.5)1,000,000 + 100,000
= $380,000/share
S  $1 => V(portfolio)  $380,000.
19
20
21
22
23
GAMMA 
Gamma measures the change in delta when
the market price of the undelrlying asset
changes.
In mathematical terms Gamma is the
second derivative of the option’s price with
respect to the underlying price.
(c) = (c)/S = 2c/ S2
(p) = (p)/S = 2p/ S2
(S) = (S)/S = 2S/ S2 = 0.
24
GAMMA 
In general, the Gamma of any portfolio is the
change of the portfolio’s delta due to a
“small” change in the underlying’s price.
As the second derivative of the option’s price
with respect to S, Gamma measures the
sensitivity of the option’s price to “large”
underlying asset’s price changes.
25
Interpretation of Gamma
• For a delta neutral portfolio,
dP  Q dt + ½dS 2
dP
dP
dS
dS
Positive Gamma
Negative Gamma
26
Result: The Gammas of a put and a call are
equal. Using the Black and Shcoles model:
(c) = n(d1)
(p) = n(d1) – 1.
Clearly, the derivatives of these deltas with
respect to S are equal.
EXAMPLE: (c) = .70, (p) = - .30 and let
gamma be .2345.
Holding a short call and a long put has:
 = - .70 + (- .30) = - 1.00 and
 = .2345 – .2345 = 0.
27
EXAMPLE:
(c) = .70, (p) = - .30 and let gamma be .2345.
Holding the underlying asset long, a long put
and a short call yields a portfolio with:
 = 1 - .70 + (- .30) = 0 and
 = 0 - 0,2345 + 0,2345 = 0,
simultaneously! This portfolio is
delta-gamma-neutral.
28
29
30
31
VEGA

Vega measures the sensitivity of the option’s
market price to “small” changes in the
volatility of the underlying asset’s return.
(c) = c/
(p) = p/
Thus, Vega is in terms of $/1% change in .
32
33
34
35
36
THETA

Theta measures the sensitivity of the
option’s price to a “small” change in the
time remaining to expiration:
(c) = c/(T-t)
(p) = p/(T-t)
Theta is given in terms is $/1 year. Thus, if
(c) = - $20/year, it means that if time to
expiration increases (decreases) by one
year, the call price will increase (decreases)
by $20. Or, 20/365 = 5.5 cent per day. 37
38
39
40
41
RHO

Rho measures the sensitivity of the option’s
price to “small changes in the rate of interest.
(c) = c/r
(p) = p/r
Rho is in terms of $/%change of r.
42
43
44
45
46
SUMMERY OF THE GREEKS
Position
Delta
Gamma
Vega
1
0
0
0
SHORT STOCK -1
0
0
0
LONG CALL
+
+
+
-
SHORT CALL
-
-
-
+
LONG STOCK
Theta
LONG PUT
-
+
+
-
SHORT PUT
+
-
-
+
47
The sensitivity of portfolios
1. A portfolio is a combination of securities
- assets and options.
2. All the sensitivity measures are
mathematical derivatives.
3. Theorem(Calculus): The derivative of a
combination of functions is the
combination of the derivatives of these
functions: The sensitivity measure of a
portfolio of securities is the portfolio of
these securities’ sensitivity measures.
48
Example:The DELTA of a portfolio of 5 long
calls, 5 short puts and 100 shares of the
stock long:
(portfolio) = (5c - 5p + 100S)
= (5c - 5p + 100S)/S
= 5c/S - 5p/S + 100
= 5c - 5p + 100
This delta reveals the $/share change in
the portfolio value as a function of a
“small” change in the underlying price
49
Example: the price of oil is S = $28.57/barrel.
Call
Delta
Gamma
A
$0.63/bbl
$0.22/bbl
B
$0.45/bbl
$0.34/bbl
C
$0.82/bbl
$0.18/bbl
Portfolio: Long:3 calls A; 2 calls C; 5 barrels
of oil.
Short: 10 calls B.
 = 3(0.63)+ 2(0.82) + 5(1) – 10(0.45)
= 3(0.22)+ 2(0.34) + 5(0) – 10(0.18)
 = $4.33/barrel and  = - $0.46/barrel.50
The portfolio’s Delta and Gamma are
 = 4.33 and  = - 0.46.
 = 4.33 means that a “small” change of the
oil price, say one cent per barrel, will change
the value of the above portfolio by 4.33
cents in the same direction.
 = - 0.46 means that a “small” change in the
oil price, say one cent per barrel, will change
the delta by about half a cent in the opposite
direction.
51
GREEKS BASED STRATEGIES
Greeks based strategies are opened and
maintained in order to attain a
specific level of sensitivity. Mostly,
these strategies are set to attain zero
sensitivity. What follows, is an
example of strategies that are:
1. Delta-neutral
2. Delta-Gamma-neutral
3. Delta-Gamma-Vega-Rho-neutral
52
EXAMPLE: The underlying asset is the
S&P100 stock index. The options on this
index are European.
S = $300; K = $300; T = 1yr;  = 18%;
r = 8%; q = 3%.
C = $28.25.
 = .6245
= .0067
= .0109
 = .0159
53
DELTA-NEUTRAL
Short the call. W0 = - 1. Long WS = .6245 of the index.
Case A1:
S increases from $300 to $301.
Portfolio
Initial Value
New value
Change
-Call
- $28.25
- $28.88
- $.63
$187.97
$.62
(.6245)S
$187.35
Error: - $.01
Case A2:
S decreases from $300 to $299.
Portfolio
Initial value
New value
Change
-Call
- $28.25
- $27.62
+ $.63
$186.73
- $.62
(.6245)S
$187.35
54
Error: + $.01
Case B1:
S increases from $300 to $310.
Portfolio
Initial Value
New value
Change
-Call
- $28.25
- $34.81
- $6.56
$197.59
$6.24
(.6245)S
$187.35
Error: - $.32
The point here is that Delta has changed significantly and
.6245 does not apply any more.
S = $300
$301
$310
=
.6311
.6879.
.6245
We conclude that the delta-neutral portfolio must be
adjusted for “large” changes of the underlying asset price.
55
Call #0
Call #1
S = $300
S = $300
K = $300
K = $305
T = 1yr
T = 90 days
= 18% r = 8% q = 3%
c = $28.25
c = $10.02
 = .6245
 = .4952
 = .0067
 = .0148
 = .0109
 = .0059
 = .0159
 = .0034
56
A DELTA-GAMMA-NEUTRAL PORTFOILO
 = 0: WS + W0(.6245) + W1(.4952) = 0
 = 0:
W0(.0067) + W1(.0148) = 0
Solution:
W0 = -1
W1 = - (.0067)(-1)/.0148 = .453
WS = - (.6245)(-1) – (.453)(.49520 = .4
Short the initial call :
W0 = -1.000
Long .453 of call #1
W1 = .453
Long .4 of the index
WS = .400
57
THE DELTA-GAMMA-NEUTRAL PORTFOLIO
Case A1:
S increases from $300 to $301.
Portfolio Initial value
New value
-1.0 #0
- $28.88
- $.63
$4.77
$.23
$120.4
$.40
(.453)#1
(.4)S
- $28.25
$4.54
$120
Change
Error:
Case B1:
Portfolio
-1.0 #0
0
S increases from $300 to $310.
Initial value
- $28.25
New value
Change
- $34.81
- $6.56
(.453)#1
$4.54
$7.11
$2.57
(.4)S
$120
$124
- $4.00
Error:
+ $.01
58
If we examine the exposure level to all parameters,
however, we observe that:
Portfolio
Delta
Gamma
Vega
Rho
-1.00(#0)
-.6245
- .0067
- .0109
- .0159
.453(#2)
.400S
.2245
.4000
.0067
0
.0027
0
.0015
0
Risk
0
0
- .0082
- .0144
The above numbers reveal that the DeltaGamma-neutral portfolio is exposed to risk
associated with
the volatility and the risk-free rate
59
Delta-neutral portfolio with volatility: 12%,
18% Y 24%,
12%
18%
24%
S

$270
$275
$280
$285
$290
$295
$300
$305
$310
$315
$320
$325
$330
$2.73
$4.05
$5.08
$5.82
$6.29
$6.47
$6.40
$6.09
$5.57
$4.84
$3.94
$2.89
$1.72
- $3,26
- $2,24
- $1,42
- $0,79
- $0,35
- $0,08
0.00
- $0.08
- $0.32
- $0.71
- $1.24
- $1.90
- $2.67
- $9.45
- $8.61
- $7.92
- $7.38
- $6.97
- $6.70
- $6.56
- $6.56
- $6.67
- $6.89
- $7.24
- $7.69
- $8.82
60
The Delta-Gamma-neutral portfoli0 Volatility
12%, 18% Y 24%,

S
$270
$275
$280
$285
$290
$295
$300
$305
$310
$315
$320
$325
$330
12%
$5.54
$6.04
$6.38
$6.57
$6.62
$6.55
$6.40
$6.17
$5.89
$5.56
$5.19
$4.80
$4.38
18%
- $0.45
- $0.25
- $0.12
- $0.04
- $0.01
0.00
0.00
0.00
$0.01
$0.01
$0.01
$0.01
- $0.01
24%
- $6.64
- $6.62
- $6.62
- $6.63
- $6.63
- $6.62
- $6.56
- $6.48
- $6.34
- $6.17
- $5.99
- $5.78
- $5.56
61
Case C1:
S increases from $300 to $310
and simultaneously,
r increases from 8% to 9%.
Portfolio Initial value
New value
Change
-1.0 (#0)
- $33.05
- $4.80
$4.5
$6.91
$2.37
$120
$124
- $4.00
(.453) #1
(.4)S
- $28.25
Error: - $1.57
62
Delta-Gamma-Vega-Rho-neutral portfolio
CALL
0
1
2
3
K
300
305
295
300
T(days)
365
90
90
180
Volatility
18%
18%
18%
18%
r
8%
8%
8%
8%
Dividends
3%
3%
3%
3%
c
$28.25
$10.02 $15.29
$18.59
63
Delta-Gamma-Vega-Rho-neutral portfolio
CALL




0
.6245
.0067
.0109
.0159
1
.4952
.0148
.0059
.0034
2
.6398
.0138
.0055
.0044
3
.5931
.0100
.0080
.0079
S
1.0
0.0
0.0
0.0
64
The DELTA-GAMMA-VEGA-RHO-NEUTRALPORTFOLIO
In order to neutralize the portfolio to all risk
exposures, following the sale of the initial call,
we determine the portfolio’s proportions such
that all the portfolio’s sensitivity parameters
are zero simultaneously.
Delta =  = 0 and Gamma = = 0 and
Theta =  = 0 and Vega =  = 0 and
Rho =  = 0
simultaneoulsy
65
Delta =  = 0
WS+W0(.6245)+W1(.4952)+W2(.6398)+W3(.5931) = 0
Gamma =  = 0
W0(.0067)+W1(.0148)+W2(.0138)+W3(.0100) = 0
Vega =  = 0
W0(.0109)+W1(.0059)+W2(.0055)+W3(.0080) = 0
Rho =  = 0
W0(.0159)+W1(.0034)+W2(.0044)+W3(.0079) = 0
66
We short the call 0, I.e.,
W0 = - 1 and we solve the
simultaneous equations. The solutions
is:
W0 = -1.000
short call #0
W1 =
long .840 call #1
.840
W2 = -1.900
short 1.900 call #2
W3 =
long 2.040 call #3
WS =
2.040
.2120
long .212 of the index
67
To see what this solution means in practical
terms, multiply all the weights by 10,000. The
portfolio becomes:
Short 100 CBOE calls #0;
Long 84 calls #1;
Short 190 calls #2;
Long 204 calls #3;
Long 2,120 units of the index.
Every index unit is $100, so buy $212,000
worth of the index.
68
THE DELTA-GAMMA-VEGA-RHO- NEUTRAL PORTFOLIO
Case D:
S increases from $300 to $310
r increases from 8% to 9%
 increases from 18% to 24%
Portfolio Initial value
New value
Change
-1.0(#0)
- $28.25
- $42.81
- $14.56
(.212)S
$63.60
$65.72
$2.12
(840)#1
$8.40
$16.42
$8.02
(-1.9)#2
- $29.05
- $48.97
- $19.92
(2.04)#3
$37.97
$62.20
- $24.25
Error:
- $.09
69