1
The Greek Letters
Chapter 17
2
The Greeks are coming!
The Greeks are coming!
Parameters of SENSITIVITY
Delta =
Theta =
Gamma =
Vega =
Rho =
3
c = SN(d1) – Ke–r(T – t)N(d2)
p = Ke–r)T –t)N(-d2) – SN(-d1)
Notationally:
c = c(S; K; T-t; r; σ)
p = p(S; K; T-t; r; σ)
Once c and p are calculated, WHAT IF?
4
The GREEKS are measures of sensitivity. The question is how sensitive a position’s value is to
changes in any of the variables that contribute to the position’s market
value.These variables are:
S, K, T-t, r and .
Each one of the Greek measures indicates the change in the value of the position as a result of a “small”
change in the corresponding variable.
Formally, the Greeks are partial derivatives.
5
Delta =
In mathematical terms DELTA is the first derivative of the option’s
premium with respect to S. As such, Delta carries the units of the option’s
price; I.e., $ per share.
For a Call: (c)= c/S
For a Put: (p)= p/S
Results: (p) = (c) - 1
For the (S) = S/S = 1
6
THETA
Theta measures are given by:
(c)= c/(T-t) (p)= p/(T-t)
s are positive but the they are reported as negative values. The
negative sign only indicates that as time passes, t increases, time to
expiration, T – t, diminishes and so does the option’s value, ceteris
paribus. This loss of value is labeled the option’s “time decay.”
Also, (S) = 0.
7
GAMMA
Gamma measures the change in delta when the price of the underlying asset
changes.
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
Results: (c) = (p)
(S) = 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. (c)= c/
(p)= p/
Thus, Vega is in terms of
$/1% change in .
(S)= 0.
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 is in terms of $/%change of r.
(S) = 0.
10
Example:
S=100; K = 100; r = 8%; T-t =180 days;
= 30%. Call Put
Premium $10.3044 $6.4360
The Greeks:
Delta = 0.6151 -0.3849
Theta = -0.03359 -0.01252Gamma = 0.0181
0.0181Vega = .268416
.268416Rho = .252515
-.221559
11
Again. 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.
(c)= 0.6151
(p) = - 0.3849
12
Call Delta (See Figure 15.2, page 345)
• Delta is the rate of change of the call price with respect to the underlying
Call price
A
CSlope =
Stock price
13
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.
(c) = - $12.2607/year if time to expiration increases (decreases) by one year, the call price will increase (decrease) by $12.2607. Or, 12.2607/365 = 3.35 cent per day.
14
GAMMA
Gamma measures the change in delta when the price of the underlying asset
changes.
= 0.0181. (c) = .6151; (p) = -.3849.
If the stock price increases to $101:(c) increases
to .6332 (p) increases to -.3668.
If the stock price decreases to $99:(c) decreases to .5970(p) decreases to -.4030.
15
VEGA
Vega measures the sensitivity of the option’s market price to
“small” changes in the volatility of the underlying asset’s return.
= .268416
(Check on Computer)
16
RHO Rho measures the sensitivity of
the option’s price to “small changes in the rate of interest.
Rho = Call Put.252515
-.221559
Rho is in terms of $/%change of r.
(check on computer)
17
DELTA-NEUTRAL POSITIONS
A market maker wrote n(c) calls and wishes to protect the revenue against
possible adverse move of the underlying asset price. To do so,
he/she uses shares of the underlying asset in a quantity that GUARANTEES
that a small price change will not have any impact on the call-shares position.
Definition: A portfolio is Delta-neutral if
(portfolio) = 0
18
DELTA neutral position in the simple case of call-stock portfolios.
Vportfolio = Sn(S) + cn(c;S)
(portfolio) = (S)n(S) + (c)n(c;S)
(portfolio) = 0 n(S) + (c)n(c;S) = 0.
n(S) = - n(c;S)(c).
The call delta is positive. Thus, the negative sign indicates that the calls and the shares of the underlying asset must be held in opposite direction.
19
EXAMPLE: call - stock portfolio
We just sold 10 CBOE calls whose delta is $.54/shares. Each call covers 100 shares.
n(S) = - n(c;S)(c).
(c) = 0.54 and n(c) = -10.
n(c;S) = - 1,000 shares.
n(s) = - [ - 1,000(0.54)] = 540.
The DELTA-neutral position consists of the 10 short calls and 540 long shares.
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The Hedge Ratio c
Definition: Hedge ratio.
options by the covered shares ofnumber The
portfolio theneutralize torequired shares ofnumber The
Ratio Hedge
In the example:
Hedge ratio = 540/1,000 = .54
Notice that this is nothing other than (c).
21
In the numerical example, Slide 9:
The hedge ratio: (c) = 0.6151
With 100 CBOE short calls:
n(S) = -(c)n(c;S).
n(c;S) = -10,000.
n(S) = -(.6151)[-10,000] = +6,151 shares The value of this portfolio is:
V = -10,000($10.3044) + 6,151($100)
V = $512,056
22
Suppose that the stock price rises by $1.
SNEW = 100 + 1 = $101/share.
V = - 10,000($10.3044 + $.6151)
+6,151($101)
V = - 10,000($10.3044) + 6,151($100) - 10,000($.6151) + $1(6,151)
V = $512,056 - $6,151 + $6,151
V = $512,056.
23
Suppose that the stock price falls by $1.
SNEW = 100 - 1 = $99/share.
V = - 10,000($10.3044 - $.6151)+6,151($99)
V = - 10,000($10.3044) + 6,151($100) - 10,000( - $.6151) - $1(6,151)
V = $512,056 + $6,151 – $6,151
V = $512,056.
24
In summary:
The portfolio consisting of 100 short calls and 6,151 long shares is
delta- neutral.Price/share: +$1 -$1
shares +$6,151 -$6,151
calls +(-$6,151) -(-$6,151)
Portfolio $0 $0
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DELTA neutral position in the simple case of put-stock portfolios.
Vportfolio = Sn(S) + pn(p;S)
(portfolio) = (S)n(S) + (p)n(p;S)
(portfolio) = 0 n(S) + (p)n(p;S) = 0.
n(S) = - n(p;S)(p)
Since the put delta is negative, then the negative sign indicates that the puts and the underlying asset must be held in the same direction.
26
EXAMPLE: put – stock portfolio.
We just bought 10 CBOE puts whose delta is -$.70/share. Each put covers 100 shares. n(S) = - n(p;S)(p).
(p) = -.70 and n(p) = 10.
n(p;S) = 1,000 shares.
n(S) = - 1,000(-.70) = 700.
The DELTA-neutral position consists of the 10 long puts and 700 long shares.
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Portfolio: The portfolio consisting of 10 long puts and 700 long shares is
delta- neutral.Price/share: +$1 -$1
shares +$700 -$700
puts -$700 $700
Portfolio $0 $0
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In the numerical example, Slide 9:
The hedge ratio: (p) = -0.3849
The Delta neutral position with 100 CBOE long puts requires the holding
of:
n(S) = -(p)n(p;S)
n(S) = -(-.3849)[10,000] = +3,849shares The value of this
portfolio is:
V = 10,000($6.4360) + 3,849($100)
V = $449,260.
29
Suppose that the stock price rises by $1.
SNEW = 100 + 1 = $101/share.
V = 10,000($6.4360 - .3849)
+3,849($101)
V = 10,000( $6.4360) – 3,849($100) - 10,000(.3849) + $1(3,849)
V = $449,260- $3,849 + $3,849
V = $449,260.
30
Suppose that the stock price falls by $1.
SNEW = 100 - 1 = $99/share.
V = 10,000($6.4360 + $.3849)+3,849($99)
V = 10,000( $6.4360) + 3,849($100) 10,000($.3849) - $1(3,849)
V = $449,260+ $3,849 - $3,849
V = $449,260.
31
In summary:
The portfolio consisting of 100 long puts and 6,151 long shares is
delta- neutral.Price/share: +$1 -$1
shares +$3,849 -$3,849
calls +(-$3,849) -(-$3,849)
Portfolio $0 $0
32
An extension:
calls, puts and the stock position
Vportfolio = Sn(S) + cn(c;S) + pn(p;S)
portfolio = (S)n(S) + (c)n(c;S) + (p)n(p;S).
But S = 1.
Thus, for delta-neutral portfolio:
portfolio = 0 and
n(S) = -(c)n(c;S) -(p)n(p;S).
33
EXAMPLE: We short 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?
n(S) = -(c)n(c;S) + [-(p)]n(p;S).
n(S) = -(.7)(-2,000) – [-.3](-2,000)
n(S) = 800.
34
Example continued
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
Portfolio $0 $0
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EXAMPLES
The put-call parity:
Long 100 shares of the underlying stock,
long one put and short one call on this
stock is always delta-neutral:
(position)
= 100 + (p)n(p;S) + (c)n(c;S)
= 100 + [(c) – 1](100) + (c)(-100)
= 0.
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EXAMPLES
A long STRADDLE:
Long 15 puts and long 15 calls
(same underlying asset, K and T-t),
with: (c) = .64; (p) = - .36.
(straddle) = 15(100)[.64 + (- .36)]
=$420/share.
Long 420 shares to delta neutralize this straddle.
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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:
(p) = (c) - 1
2. Using the Black and Scholes formula:
(c) = N(d1) 0 < (c) < 1
(p) = N(d1) – 1 -1 < (p) < 0
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39
40
41
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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.
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44
45
46
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GAMMA
Gamma measures the change in delta when the market price of the underlying asset changes.
(c) = (c)/S = 2c/ S2
(p) = (p)/S = 2p/ S2
Results:Results:
(c) = (p)
(S) = 0.
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GAMMA
In general, the Gamma of any portfolio is the change of the portfolio’s delta
due to a “small” change in the underlying asset 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.
May be positive or negative.
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Interpretation of GammaThe delta neutral position with 100
short calls and 6,151 long shares has Γ= -$181
S
Negative Gamma means that the position loses value when the stock price moves more and more away from it initial value.
75 100 125
$512,056Position value
More negative Γ
50
Interpretation of Gamma
S
Negative Gamma
S
Positive Gamma
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Result: The Gammas of a put and a call are equal. Using the Black and
Shcoles model:
(c) = n(d1) and (p) = n(d1) – 1.
Clearly, the derivatives of these deltas with respect to S are equal.
EXAMPLE: (c) = .70; (p) = - .30;
= .2345.
Holding a long call and a short put has: = .70 - (- .30) = 1.00.
= .2345 – .2345 = 0.
52
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.
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54
55
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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
57
58
59
60
61
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.
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63
64
65
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SUMMERY OF THE GREEKS
Position Delta Gamma Vega Theta Rho
LONG STOCK 1 0 0 00
SHORT STOCK -1 0 0 00
LONG CALL + + + -+
SHORT CALL - - - +-
LONG PUT - + + --
SHORT PUT + - - + +
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The sensitivity of portfolios, a summary.
1.A portfolio is a combination of securities and options.
2.All the sensitivity measures are partial derivatives.
3.Theorem(Calculus): The derivative of a linear combination of functions is the combination of the derivatives of these functions. Thus, the sensitivity measure of a portfolio of securities is the portfolio of these securities’ sensitivity measures.
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Example:The DELTA of a portfolio of 5 long CBOE calls, 5 short puts and 100
shares of the stock long:
(portfolio) = (500c - 500p + 100S)
= (500c - 500p + 100S)/S
= 500c/S - 500p/S + 100
= 500c - 500p + 100
This delta reveals the $/share change in the portfolio value as a function of a “small” change in the underlying
price
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Example: S = $48.57/barrel.
1 call = 1,000bbls.
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; 5,000 barrels. Short: 10 calls B.
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Example:
= (0.63)3,000+ (0.82)2,000
+ (1)5,000 + (0.45)(-10,000)
= (0.22)3,000+ (0.18)2,000
+ (0)5,000 + (0.34)(-10,000)
= $4,030.
= - $2,380.
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= 4,030 a “small” change of the oil price, say one cent per barrel, will change the value of the above portfolio by $40.30 in the same direction.
= - 2,380 a “small” change in the oil price, say one cent per barrel, will change the delta by $23.80 in the opposite direction.
Also, Gamma is negative when the price per barrel moves away from $48.57, the portfolio value will decrease.
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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.
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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
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EXAMPLE:
The underlying asset is the a stock. The options on this stock are European.
S = $300; K = $300; T = 1yr; = 18%; r = 8%; q = 3%.
c = $28.25.
= .6245
= .0067
= .0109
= .0159
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DELTA-NEUTRAL
Short 100 calls. n0 = - 10,000; Long nS = 6,245 shares
Case A1: S increases from $300 to $301.
Portfolio Initial Value New value Change
-100Calls - $282,500 - $288,800- $6,300
6,245S $1,873,500 $1,879,745 $6,245
Error: - $55
Case A2: S decreases from $300 to $299.
Portfolio Initial value New value Change
-100Calls - $282,500 - $276,200+ $6,300
6,245S $1,873,500 $1,867,255 - $6,245
Error: + $55
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Case B1: S increases from $300 to $310.
Portfolio Initial Value New value Change
-100Calls - $282,500 - $348,100- $65,600
6,245S $1,873,500 $1,935,950 $62,450
Error: -$3,150
The point here is that Delta has changed significantly and .6245 does not apply any more.
S = $300 $301 $310
= .6245 .6311 .6879.
We conclude that the delta-neutral portfolio must be adjusted for “large” changes of the underlying asset price.
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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
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A DELTA-GAMMA-NEUTRAL PORTFOILO
(portfolio) = 0: nS + n0(.6245) + n1(.4952) = 0
Γ(portfolio)= 0: n0(.0067) + n1(.0148) = 0
Solution:
n0 = -10,000
n1 = - (-10,000)(.0067)/.0148 = 4,527
nS = - (-10,000)(.6245) – (4,527)(.4952) = 4,003
Short the initial call : n0 = -10,000
Long 45.27 of call #1 n1 = 4,527
Long 4,003 shares nS = 4,003
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THE DELTA-GAMMA-NEUTRAL PORTFOLIO
Case A1: S increases from $300 to $301.
Portfolio Initial value New value Change
0) -10,000 - $282,500 - $288,800 -$6,300
1) 4,527 $45,360 $47,657 $2,297
S) 4,003 $1,200,900 $1,204,903 $4,003
Error: 0
Case B1: S increases from $300 to $310.
Portfolio Initial value New value Change
0) -10,000- $282,500 - $348,100 - $65,600
1) 4,527 $45,360 $70,930 $25,570
S) 4,003 $1,200,900 $1,240,930 $40,030
Error: 0
80
- 144 - 82 0 0 Risk
0 0 0 4,003 4,003S
15 27 67 2,242 4,527
- 159 - 109 - 67- 6,245-10,000
RhoVegaGammaDeltaPortfolio
The above numbers reveal that the Delta-Gamma-neutral portfolio is exposed to risk
associated with
the volatility and the risk-free rate
If we examine the exposure level to all parameters, however, we observe that:
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Case C1:
S increases from $300 to $310
and simultaneously,
r increases from 8% to 9%.
Portfolio Initial value New value Change
-10,000 - $282,500 - $330,500 - $48,000
4,527 $45,360 $73,166 $27,806
4,003S $1,200,900 $1,240,930 $40,030
Error: - $10,756
82
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
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Delta-Gamma-Vega-Rho-neutral portfolio
CALL
0 .6245 .0067 .0109 .0159
1 .4954 .0148 .0059 .0034
2 .6398 .0138 .0055 .0044
3 .5931 .0100 .0080 .0079
S 1.0 0.0 0.0 0.0
84
The DELTA-GAMMA-VEGA-RHO-NEUTRAL-PORTFOLIO
In order to neutralize the portfolio to all risk exposures, following the sale of the initial call, we now determine the portfolio’s holdings such that all
the portfolio’s sensitivity parameters
are zero simultaneously.
= 0 and = 0 and = 0 and = 0simultaneously!
85
= 0nS+n0(.6245)+n1(.4954)+n2(.6398)+n3(.5931)
=0
= 0
n0(.0067)+n1(.0148)+n2(.0138)+n3(.0100)=0
= 0n0(.0109)+n1(.0059)+n2(.0055)+n3(.0080)
=0
= 0n0(.0159)+n1(.0034)+n2(.0044)+n3(.0079)
=0
86
The solution is:
Exact nShort 100 CBOE calls #0; -10,000
Short 339 calls #1; -33,927
Long 265 calls #2; 26,534
Long 204 calls #3; 20,420
Short 6,234 shares. -6,234
87
Case D: S increases from $300 to $310 r increases from 8% to 9% increases from 18% to 24%
Portfolio Initial Value New value
0) - 10,000 - $282,468 - $428,071
S)- 6,234 - $1,870,200 - $1,932,540
1) - 33,927 - $340,023 - $664,552
2)26,534 $405,668 $694,062
3)20,420 $379,677 $622,240
TOTAL - $1,707,356 $1,708,861
Error:$1,505 or .088%.
88
DYNAMIC DELTA - HEDGING
The market stock price keeps changing all
the time. Thus, a static DELTA- neutral
hedge is not sufficient.
A continuous delta adjustment is not
practical.
An adjusted Delta-neutral Position:
1.Every day, week, etc.
2.Following a given % price change.
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DYNAMIC DELTA HEDGING
Market makers provide traders with the options they wish to trade. For example, if a trader wishes to long (short) a call, a market maker will
write (long) the call. The difference between the buy and sell prices is
the market maker’s
bid-ask spread.bid-ask spread.
The main problem for a market maker who shorts calls is that the premium received, not only may be lost, but
the loss is potentially unlimited.
90
DYNAMIC DELTA - HEDGING
Recall: The profit profile of an uncovered call is:
At expirationP/L
ST
c
K
91
DYNAMIC DELTA - HEDGING
Recall: The profit profile of a covered call is:
Strategy IFC At expirationST < K ST > K
Short call
c 0 -(ST – K)
Long stock
-St ST ST
Total - St + c ST K
P/L ST - St + c
K - St + c
92
DYNAMIC DELTA - HEDGING
Recall: The profit profile of a covered call is:
At expirationP/L
ST
K –St+ c
K
-St + c
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DYNAMIC DELTA - HEDGING
The Dynamic Delta hedge is based on the
impact of the time decay on the call Delta.
Recall that:
Δ(c))N(d
andtTσ
t]][T.5σ[r]KS
ln[d
1
2t
1
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DYNAMIC DELTA - HEDGING
Observe what happens to d1
when T-t 0.
1. For: St > K d1 and
N(d1) = (c) 1
2. For: St < K d1 - and
N(d1) = (c) 0
95
DYNAMIC DELTA - HEDGING
The Dynamic hedge:
1.Write a call and simultaneously, hedge the call by a long Delta shares of the underlying asset. As time goes by, adjust the number of shares periodically.
Result: As the expiration date nears, delta:
goes to 0 in which case you wind up without any shares.
goes to 1 in which case you call is fully covered.
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DYNAMIC DELTA - HEDGING
St : St < K St > K
(c): 0 1
Call: uncovered fully covered
n(S): 0 n(c;S) 1
97
Table 15.2 Simulation of Dynamic delta - hedging.(p.364) Cost of
Stock Shares shares Cummulative Interest Week przce Delta purchased purchased cost
cost ($000) ($000)
($000) 0 49.00 0.522 52,200 2,557.8 2,557.8
2.5 1 48.12 0.458 (6,400) (308.0) 2,252.3
2.2 2 47.37 0.400 (5,800) (274.7) 1,979.8
1.9 3 50.25 0.596 19,600 984.9 2,966.6
2.9 4 51.75 0.693 9,700 502.0 3,471.5
3.3 5 53.12 0.774 8,100 430.3 3,905.1
3.8 6 53.00 0.771 (300) (15.9) 3,893.0
3.7 7 51.87 0.706 (6,500) (337.2) 3,559.5
3.4 8 51.38 0.674 (3,200) (164.4) 3,398.5
3.3 9 53.00 0.787 11,300 598.9 4,000.7
3.810 49.88 0.550 (23,700) (1,182.2) 2,822.3
2.7 11 48.50 0.413 (13,700) (664.4) 2,160.6
2.1 12 49.88 0.542 12,900 643.5 2,806.2
2.7 13 50.37 0.591 4,900 246.8 3,055.7
2.9 14 52.13 0.768 17,700 922.7 3,981.3
3.8 15 51.88 0.759 (900) (46.7) 3,938.4
3.8 16 52.87 0.865 10,600 560.4 4,502.6
4.3 17 54.87 0.978 11,300 620.0 5,126.9
4.9 18 54.62 0.990 1,200 65.5 5,197.3
5.0 19 55.87 1.000 1,000 55.9 5,258.2
5.1 20 57.25 1.000 0 0.0 5,263.3
98
Table 15.3 Simulation of dynamic Delta - hedging. (p. 365) Cost of
Stock Shares shares Cumulative I nterestWeek price Delta purchased purchased cost cost
($000) ($000) ($000) 0 49,00 0.522 52,200 2,557.8 557.8
2.5 1 49.75 0.568 4,600 228.9 2,789.2 2.7 2 52.00 0.705
13,700 712.4 3,504.3 3.4 3 50.000.579 (12,600) (630.0) 2,877.7 2.8
4 48.38 0.459 (12,000) (580.6) 2,299.9 2.2 5 48.25 0.443 (1,600) (77.2)
2,224.9 2.1 6 48.75 0.475 3,200 156.0 2,383.0 2.3 7 49.63 0.540 6,500
322.6 2,707.9 2.6 8 48.25 0.420 (12,000) (579.0) 2,131.5 2.1 9 48.25 0.410 (1,000) (48.2) 2,085.4 2.0 10 51.12 0.658 24,800 1,267.8 3,355.2 3.2 11 51.50 0.692 3,400 175.1 3,533.5 3.4 12 49.88 0.542(15,000) (748.2) 2,788.7 2.7 13 49.88 0.538 (400) (20.0) 2,771.4 2.7 14 48.75 0.400 (13,800) (672.7)
2,101.4 2.0 15 47.50 0.236 (16,400) (779.0) 1,324.4 1.3 16 48.00 0.261 2,500
120.0 1,445.7 1.4 17 46.25 0.062 (19,900) (920.4) 526.7 0.5 18 48.13 0.183 12,100 582.4 1,109.6 1.1 19 46.63 0.007 (17,600) (820.7) 290.0
0.3 20 48.12 0.000 (700) (33.7) 256.6