Chapter 20
Comparative Static
Analysis of the Option
Pricing Models
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Outline
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2
20.1 DELTA (∆ )
20.1.1 Derivation of Delta for Different Kinds of Stock Options
20.1.2 Application of Delta ( ∆)
20.2 THETA (Θ )
20.2.1 Derivation of Theta for Different Kinds of Stock Options
20.2.2 Application of Theta (Θ )
20.3 GAMMA (Γ )
20.3.1 Derivation of Gamma for Different Kinds of Stock Options
20.3.2 Application of Gamma ( Γ)
20.4 VEGA (𝜈)
20.4.1 Derivation of Vega for Different Kinds of Stock Options
20.4.2 Application of Vega (𝜈 )
20.5 RHO (𝜌 )
20.5.1 Derivation of Rho for Different Kinds of Stock Options
20.5.2 Application of Rho (𝜌)
20.6 DERIVATION OF SENSITIVITY FOR STOCK OPTIONS WITH
RESPECTIVE TO EXERCISE PRICE
20.7 RELATIONSHIP BETWEEN DELTA, THETA, AND GAMMA
20.1 Delta (∆)
•
The delta of an option, ∆, is defined as the rate of change of the option price
respected to the rate of change of underlying asset price:


S
•
3
Where ∏ is the option price and S is underlying asset price. We next show
the derivation of delta for various kinds of stock option.
20.1.1 Derivation of Delta for Different
Kinds of Stock Options
•
From Black–Scholes option pricing model, we know the price of call option
on a nondividend stock can be written as
Ct  S t N d1   Xe  r N d 2 
•
(20.1)
And the price of put option on a nondividend stock can be written as
Pt  Xer N  d2   St N  d1 
•
(20.2)
Where
2
 St    s 

ln     r 
X
2
  

d1 
2
 St    s 
ln     r   
2 
X 
d2 
 d1  s 
s 
s 
N  is the cumulative density function of normal distribution
N d1  
4

d1

f u du 

d1

1
e
2

u2
2
du
 T t
•
First we calculate N d1  
N d1 

d1
N d 2  
1
e
2

d12
2
(20.3)
N d 2 
d 2
d2
1  22

e
2
d1  s  2

1
2

e
2
d2
1  21 d1 s

e e
2


•
5
d12
1 2
e e
2
1
e
2

d12
2


e

 s2
2
2
S   
ln  t    r  s
2
 X  




e

 s2
(20.4)
2
S t r
e
X
Equations (20.2) and (20.3) will be used repetitively in determining following
Greek letters when the underlying asset is a non-dividend paying stock.
•
For a European call option on a non-dividend stock, delta can be shown as
(20.5)
𝛥 = 𝑁(𝑑1
The derivation of Equation (20.5) is in the following:

N  d1 
N  d 2 
C t
 N  d1   St
 Xe  r
St
St
St
 N  d1   St
N  d1  d1
N  d 2  d 2
 Xe  r
d1 St
d 2 St
2
 N  d1   St
 N  d1   St
 N  d1 
6
2
1  d21
1
1  d21 St r
1
 r
e 
 Xe
e  e 
X
2
St  s 
2
St  s 
1
St s 2
e

d12
2
 St
1
St s 2
e

d12
2
•
For a European put option on a non-dividend stock, delta can be shown as
  N(d1 )  1
•
(20.6)
The derivation of Equation (20.6) is

N  d 2 
N  d1 
Pt
 Xe  r
 N  d1   St
St
St
St
 Xe r
 (1  N  d 2 ) d 2
 (1  N  d1 ) d1
 (1  N  d1 )  St
d 2
St
d1
St
2
 Xe  r
 St
1
St s 2
 N  d1   1
7
2
1  d21 St r
1
1  d21
1
e  e 
 (1  N  d1 )  St
e 
X
2
St  s 
2
St  s 
e

d12
2
 N  d1   1  St
1
St s 2
e

d12
2
•
If the underlying asset is a dividend-paying stock providing a dividend yield
at rate q, Black–Scholes formulas for the prices of a European call option on a
dividend-paying stock and a European put option on a dividend-paying stock
are
Ct  St eq N  d1   Xe r N  d 2 
(20.7)
and
Pt  Xe r N  d2   St eq N  d1 
•
where
s2 
 St  
ln     r  q   
2 
X 
d1 
s 
s2 
 St  
ln     r  q   
2 
X 
d2 
 d1  s 
s 
8
(20.8)
•
To make the following derivations more easily, we calculate Equations (20.9)
and (20.10) in advance.
N  d 2 
N(d 2 ) 
d 2
(20.9)
N(d1 ) 
N  d1 
d1

1

e
2
d12

2




9
1
e
2
1
e
2
1
e
2
1
e
2
1
e
2

d 22
2
(20.10)
 d1 s  

2
2

d12
2
d12

2
d12

2
 ed1s
e


e

s2 
2
2 
S  
ln  t    r  q  s  
2 
 X  
St (r q) 
e
X
e
s2 

2
•
For a European call option on a dividend-paying stock, delta can be shown as
  e  q N(d1 )
•
(20.11)
The derivation of (20.11) is
C t
 q
 q N  d1 
 r N  d 2 

 e N  d1   St e
 Xe
St
St
St
 e  q N  d1   St e  q
N  d1  d1
N  d 2  d 2
 Xe  r
d1 St
d 2 St
2
 e  q N  d1   St e  q
 e  q N  d1   St e  q
 e  q N  d1 
10
2
1  d21
1
1  d21 St (r q) 
1
 r
e 
 Xe
e  e

X
2
St  s 
2
St  s 
1
St s 2
e

d12
2
 St e  q
1
St s 2
e

d12
2
•
For a European call option on a dividend-paying stock, delta can be shown as
  eq  N(d1 )  1
•
(20.12)
The derivation of (20.12) is
Pt
 r N  d 2 
 q
 q N  d1 

 Xe
 e N  d1   St e
St
St
St
 Xe  r
 (1  N  d 2 ) d 2
 (1  N  d1 ) d1
 e  q (1  N  d1 )  St e  q
d 2
St
d1
St
2
  Xe r
 St e  q
1
St s 2
 e  q (N  d1   1)
11
2
1  d21 St (r q) 
1
1  d21
1
 q
 q
e  e

 e (1  N  d1 )  St e
e 
X
2
St  s 
2
St  s 
e

d12
2
 e  q (N  d1   1)  St e  q
1
St s 2
e

d12
2
20.1.2 Application of Delta
•
Figure 20-1 shows the relationship between the price of a call option and the
price of its underlying asset.
•
The delta of this call option is the slope of the line at the point of A
corresponding to current price of the underlying asset.
Figure 20-1 The Relationship between the Price of a Call Option
and the Price of Its Underlying Asset
12
13
•
By calculating delta ratio, a financial institution that sells option to a client
can make a delta neutral position to hedge the risk of changes of the
underlying asset price.
•
Suppose that the current stock price is $100, the call option price on stock is
$10, and the current delta of the call option is 0.4.
•
A financial institution sold 10 call option to its client, so that the client has
right to buy 1,000 shares at time to maturity.
•
To construct a delta hedge position, the financial institution should buy 0.4 ×
1,000 = 400 shares of stock.
•
If the stock price goes up to $1, the option price will go up by $0.40. In this
situation, the financial institution has a $400 ($1 × 400 shares) gain in its
stock position and a $400 ($0.40 × 1,000 shares) loss in its option position.
•
The total payoff of the financial institution is zero. On the other hand, if the
stock price goes down by $1, the option price will go down by $0.40.
•
The total payoff of the financial institution is also zero.
14
•
However, the relationship between option price and stock price is not linear,
so delta changes over different stock price.
•
If an investor wants to remain his portfolio in delta neutral, he should adjust
his hedged ratio periodically.
•
The more frequent adjustments he does, the better delta-hedging he gets.
•
Figure 20-2 exhibits the change in delta affecting the delta-hedges.
•
If the underlying stock has a price equal to $20, then the investor who uses
only delta as risk measure will consider that his or her portfolio has no risk.
•
However, as the underlying stock prices changes, either up or down, the delta
changes as well and thus he or she will have to use different delta hedging.
•
Delta measure can be combined with other risk measures to yield better risk
measurement. We will discuss it further in the following sections.
Figure 20-2 Changes of Delta Hedge
15
20.2 Theta(Θ)
•
The theta of an option, Θ , is defined as the rate of change of the option price
with respect to the passage of time:


t
•
where ∏ is the option price and t is the passage of time.
•
If τ = T − t, theta (Θ) can also be defined as minus one timing the rate of
change of the option price respected to time to maturity. The derivation of such
transformation is easy and straight forward:
  



 (1)
t
 t

•
16
where τ = T − t is time to maturity. For the derivation of theta for various kinds
of stock option, we use the definition of negative differential on time to
maturity.
20.2.1 Derivation of Theta for Different
Kinds of Stock Options
•
For a European call option on a non-dividend stock, theta can be written as

17
St s
2 
 N(d1 )  rX  e  r N(d 2 )
(20.13)
•
The derivation of (20.13) is

C t
N(d1 )
N(d 2 )
 St
   r   X  e  r N(d 2 )  Xe  r



N(d1 ) d1
N(d 2 ) d 2
 St
 rX  e  r N(d 2 )  Xe  r
d1

d 2


 st 
s2
s2 
ln
r

r





1
X
2 
2   rX  e  r N(d )
 St
e

2
3
2
2s  
 s 
2s  2





 st 
s2 
2
ln
r
d



 1
 
 1
St
r
X

 r
r
2
2


 Xe  
e

e 


3
 2
  
2
X
2



2s 
s

  s




 st 
s2
s2 
2
ln
r

r

d




 1
1
X
2 
2   rX  e  r N(d )
 St
e 2 
2
3
2
2s  
 s 
2s  2





s 
s2 
ln  t 
r

d12



1
r
X

2
2


 St
e



3
2
2


2


 s

2s 
s




 s2 
d12



S
1
 St
e 2   2   rX  e  r N(d 2 )
  t s  N(d1 )  rX  e  r N(d 2 )
2
2 
 s  




d2
 1
2
18
•
For a European put option on a non-dividend stock, theta can be shown as

19
St s
2 
 N(d1 )  rX  e r N(d 2 )
(20.14)
•
The deviation of (20.14) is:

Pt
N( d 2 )
N( d1 )
    r   X  e  r N(d 2 )  Xe  r
 St



 (1  N(d 2 )) d 2
 (1  N(d1 )) d1
 ( r)X  e  r (1  N(d 2 ))  Xe  r
 St
d 2

d1


 st 
s2 
ln
  r

 1

St r   r
X


 r
 r
2
2

 ( r)X  e (1  N(d 2 ))  Xe  
e
 e 


 2
    2  3 2
X
2



s

  s
s




s2 ln  s t 
s2 
2
d
  r
r

 1
1
X

2
2
2


 St
e



3
2
2
2


 s 

2s 
s




2 

s 
ln  t  r  s 
d12 

1
r
X
2 
 rX  e  r (1  N(d 2 ))  St
e 2 
  3 
2
2s  
 s  2s  2





 s2
s2 ln  s t 
s2 
2
2
d
d
  r
r


 1
 1
1
1
X

 r
2
2
2
2


 St
e




rX

e
(1

N(d
))

S
e

 2
2
t
3
2
2s  
2
 s 
 s 
2s  2






S
S
 rX  e  r (1  N(d 2 ))  t s  N(d1 )  rX  e  r N( d 2 )  t s  N (d1 )
2 
2 
d12
20






•
For a European call option on a dividend-paying stock, theta can be shown as
 q
S
e
s
  q  St eq N(d1 )  t
 N(d1 )  rX  e r N(d 2 )
2 
(20.15)
21
•
The derivation of (20.15) is

C t
N(d1 )
N(d 2 )
 q  St e  q N(d1 )  St e  q
   r   X  e  r N(d 2 )  Xe  r



N(d1 ) d1
N(d 2 ) d 2
 q  St e  q N(d1 )  St e  q
 rX  e  r N(d 2 )  Xe  r
d1

d 2

 q  St e  q N(d1 )  St e  q

1
e
2
d12
2

 st 
s2
s2 
ln

 r q
 rq

X
2 
2   rX  e  r N(d )

2
3
2s  
 s 
2s  2





 st 
s2 
ln
r

q





 1

St
r q
X

 r
( r q ) 
2


 Xe

e

e



3
 2
  
2
X
2



2s 
s

  s



2
2

s 
s
s 
ln  t 
r

r

d12



1
X
2 
2   rX  e  r N(d )
 q  St e  q N(d1 )  St e  q
e 2 
2
3
2
2s  
 s 
2s  2




d2
 1
2

s 
2 
ln  t 
r s 

1
r
X
2 
 St e  q
e


3
2
2s  
 s 
2 s  2




2
 s 
d12



1
 q  St e  q N(d1 )  St e  q
e 2   2   rX  e  r N(d 2 )
2
 s  




 q
S e s
 q  St e  q N(d1 )  t
 N(d1 )  rX  e  r N(d 2 )
2 
d2
 1
2
22
•
For a European call option on a dividend-paying stock, theta can be shown as
St e qs
  rX  e N(d 2 )  qSt e N(d1 ) 
 N(d1 )
2 
 r
23
 q
(20.16)
The derivation of (20.16) is
•
𝜕𝑃𝑡
𝜕𝑁(−𝑑2
𝜕𝑁(−𝑑1
= − −𝑟 ⋅ 𝑋 ⋅ 𝑒 −𝑟𝜏 𝑁(−𝑑2 − 𝑋𝑒 −𝑟𝜏
+ (−𝑞 𝑆𝑡 𝑒 −𝑞𝜏 𝑁(−𝑑1 + 𝑆𝑡 𝑒 −𝑞𝜏
𝜕𝜏
𝜕𝜏
𝜕𝜏
𝜕(1 − 𝑁(𝑑2 𝜕𝑑2
𝜕(1 − 𝑁(𝑑1 𝜕𝑑1
= 𝑟𝑋 ⋅ 𝑒 −𝑟𝜏 (1 − 𝑁(𝑑2 − 𝑋𝑒 −𝑟𝜏
− 𝑞𝑆𝑡 𝑒 −𝑞𝜏 𝑁(−𝑑1 + 𝑆𝑡 𝑒 −𝑞𝜏
𝜕𝑑2
𝜕𝜏
𝜕𝑑1
𝜕𝜏
𝑠𝑡
𝜎𝑠2
2
ln
𝑟
−
𝑞
+
𝑑
1 − 1 𝑆𝑡 (𝑟−𝑞 𝜏
𝑟−𝑞
𝑋 −
2
= 𝑟𝑋 ⋅ 𝑒 −𝑟𝜏 (1 − 𝑁(𝑑2 + 𝑋𝑒 −𝑟𝜏 ⋅
𝑒 2 ⋅ ⋅𝑒
⋅
−
3
𝑋
𝜎
𝜏
2𝜎
𝜏
2𝜋
𝑠
𝑠
2𝜎𝑠 𝜏 2
𝑠𝑡
𝜎𝑠2
𝜎𝑠2
2
ln
𝑟
−
𝑞
+
𝑟
−
𝑞
+
𝑑
1 − 1
𝑋 −
2 −
2
−𝑞𝑆𝑡 𝑒 −𝑞𝜏 𝑁(−𝑑1 − 𝑆𝑡 𝑒 −𝑞𝜏
𝑒 2 ⋅
3
𝜎
𝜏
2𝜎
𝜏
2𝜋
𝑠
𝑠
2𝜎𝑠 𝜏 2
𝑠𝑡
𝜎𝑠2
2
ln
𝑟
−
𝑞
+
𝑑
1 − 1
𝑟−𝑞
𝑋 −
2
= 𝑟𝑋 ⋅ 𝑒 −𝑟𝜏 (1 − 𝑁(𝑑2 + 𝑆𝑡 𝑒 −𝑞𝜏
𝑒 2 ⋅
−
3
𝜎𝑠 𝜏 2𝜎𝑠 𝜏 2
2𝜎𝑠 𝜏
2𝜋
𝛩=−
−𝑞𝑆𝑡 𝑒 −𝑞𝜏 𝑁(−𝑑1 − 𝑆𝑡 𝑒 −𝑞𝜏
= 𝑟𝑋 ⋅ 𝑒 −𝑟𝜏 (1 − 𝑁(𝑑2
= 𝑟𝑋 ⋅ 𝑒
24
(1 − 𝑁(𝑑2
− 𝑞𝑆𝑡 𝑒
−𝑞𝜏
𝑁(−𝑑1
2𝜋
𝑠
𝜎2
𝜎2
ln 𝑋𝑡
𝑟 − 𝑞 + 2𝑠
𝑟 − 𝑞 + 2𝑠
⋅
−
3 −
𝜎𝑠 𝜏
2𝜎𝑠 𝜏
2𝜎𝑠 𝜏 2
− 𝑞𝑆𝑡 𝑒 −𝑞𝜏 𝑁(−𝑑1 − 𝑆𝑡 𝑒 −𝑞𝜏
−𝑞𝜏
−𝑟𝜏
1
𝑑2
− 1
𝑒 2
1
2𝜋
𝑑2
− 1
𝑒 2
𝜎𝑠2
2
⋅
𝜎𝑠 𝜏
𝑆𝑡 𝑒 𝜎𝑠
𝑆𝑡 𝑒 −𝑞𝜏 𝜎𝑠
′
−𝑟𝜏
−𝑞𝜏
−
⋅ 𝑁 (𝑑1 = 𝑟𝑋 ⋅ 𝑒 𝑁(−𝑑2 − 𝑞𝑆𝑡 𝑒 𝑁(−𝑑1 −
⋅ 𝑁 ′ (𝑑1
2 𝜏
2 𝜏
20.2.2 Application of Theta (Θ)
25
•
The value of option is the combination of time value and stock value. When
time passes, the time value of the option decreases.
•
Thus, the rate of change of the option price with respect to the passage of
time, theta, is usually negative.
•
Because the passage of time on an option is not uncertain, we do not need to
make a theta hedge portfolio against the effect of the passage of time.
•
However, we still regard theta as a useful parameter, because it is a proxy of
gamma in the delta neutral portfolio. For the specific detail, we will discuss in
the following sections.
20.3 Gamma (Γ
•
The gamma of an option, Γ , is defined as the rate of change of delta
respective to the rate of change of underlying asset price:
  2

 2
S S
26
•
where ∏ is the option price and S is the underlying asset price.
•
Because the option is not linearly dependent on its underlying asset, deltaneutral hedge strategy is useful only when the movement of underlying asset
price is small. Once the underlying asset price moves wider, gamma-neutral
hedge is necessary. We next show the derivation of gamma for various kinds
of stock option.
20.3.1 Derivation of Gamma for
Different Kinds of Stock Options
•
For a European call option on a non-dividend stock, gamma can be shown as
1

N  d1 
(20.17)
St s 
•
The derivation of (20.17) is
 C t 




S
 2Ct

  t
2
St
St

N  d1  d1

d1
St
 N  d1  

27
1
St
s 
1
N  d1 
St  s 
•
For a European put option on a non-dividend stock, gamma can be shown as

•
1
N  d1 
St s 
The derivation of (20.18) is
 Pt 



2

S
 Pt
 t


St 2
St
 (N  d1   1) d1


d1
St
 N  d1  

28
1
St
s 
1
N  d1 
St  s 
(20.18)
•
For a European call option on a dividend-paying stock, gamma can be shown
e  q
as

•
St  s 
N  d1 
(20.19)
The derivation of (20.19) is
 C t 


2

S
 Ct
 t


St 2
St

  e  q N(d1 ) 
 e  q 
St
N  d1  d1

d1
St
 e  q  N  d1  
1
St
s 
e  q

N  d1 
St  s 
29
•
For a European call option on a dividend-paying stock, gamma can be shown
as
 q
e

N  d1 
St  s 
•
The derivation of (20.20) is
(20.20)
 Pt 




S
 2 Pt

  t
2
St
St

  e  q (N  d1   1) 
St
 (N  d1   1)  d1
 e  q  

d1
St
 e  q  N  d1  
1
St
s 
e  q 

N  d1 
St  s 
30
20.3.2 Application of Gamma (Γ)
•
One can use delta and gamma together to calculate the changes of the option
due to changes in the underlying stock price. This change can be
approximated by the following relations.
1
change in option value    change in stock price     (change in stock price) 2
2
•
31
From the above relation, one can observe that the gamma makes the
correction for the fact that the option value is not a linear function of
underlying stock price.
•
This approximation comes from the Taylor series expansion near the initial
stock price. If we let V be option value, S be stock price, and S0 be initial
stock price, then the Taylor series expansion around S0 yields the following.
V ( S0 )
1  2V ( S0 )
1  nV ( S0 )
2
n
V ( S )  V ( S0 ) 
( S  S0 ) 
(
S

S
)


(
S

S
)
0
0
S
2! S 2
2! S n
V ( S0 )
1  2V ( S0 )
2
 V ( S0 ) 
( S  S0 ) 
(
S

S
)
 o( S )
0
2
S
2! S
•
If we only consider the first three terms, the approximation is then
V ( S0 )
1  2V ( S0 )
2
V ( S )  V ( S0 ) 
( S  S0 ) 
(
S

S
)
0
S
2! S 2
1
  ( S  S 0 )  ( S  S 0 ) 2
2
32
•
For example, if a portfolio of options has a delta equal to $10,000 and a
gamma equal to $5,000, the change in the portfolio value if the stock price
drop to $34 from $35 is approximately
1
change in portfolio value  ($10000)  ($34  $35)   ($5000)  ($34  $35) 2
2
 $7500
•
33
The above analysis can also be applied to measure the price sensitivity of
interest rate related assets or portfolio to interest rate changes.
•
Here we introduce Modified Duration and Convexity as risk measure
corresponding to the above delta and gamma.
•
Modified duration measures the percentage change in asset or portfolio value
resulting from a percentage change in interest rate.


Change in price
Modified Duration  
Price

 Change in interest rate 
  / P
Using the modified duration,
Change in Portfolio Value    Change in interest rate
 (  Duration  P)  Change in interest rate
we can calculate the value changes of the portfolio. The above relation
corresponds to the previous discussion of delta measure.
34
•
We want to know how the price of the portfolio changes given a change in
interest rate.
•
Similar to delta, modified duration only shows the first-order approximation
of the changes in value.
•
In order to account for the nonlinear relation between the interest rate and
portfolio value, we need a second-order approximation similar to the gamma
measure before, this is then the convexity measure.
•
Convexity is the interest rate gamma divided by price,
Convexity   / P
and this measure captures the nonlinear part of the price changes due to interest
rate changes.
35
•
Using the modified duration and convexity together allows us to develop first as
well as second-order approximation of the price changes similar to previous
discussion.
Change in Portfolio Value   Duration  P  (change in rate)
1
  Convexity  P  (change in rate) 2
2
•
As a result, (−Duration × P) and (Convexity × P) act like the delta and gamma
measure, respectively, in the previous discussion.
•
This shows that these Greeks can also be applied in measuring risk in interest-rate
related assets or portfolio.
36
•
Next we discuss how to make a portfolio gamma neutral. Suppose the gamma
of a delta-neutral portfolio is Γ, the gamma of the option in this portfolio is Γo ,
and ωo is the number of options added to the delta-neutral portfolio. Then,
the gamma of this new portfolio is
o o  
37
•
To make a gamma-neutral portfolio, we should trade ωo ∗= − Γ Γo options.
•
Because the position of option changes, the new portfolio is not in the deltaneutral.
•
We should change the position of the underlying asset to maintain deltaneutral.
•
For example, the delta and gamma of a particular call option are 0.7 and 1.2.
A delta-neutral portfolio has a gamma of −2,400. To make a delta-neutral and
gamma-neutral portfolio, we should add a long position of 2,400/1.2 = 2,000
shares and a short position of 2,000 × 0.7 = 1,400 shares in the original
portfolio.
20.4 Vega (𝜈)
•
The vega of an option, 𝜈 , is defined as the rate of change of the option price
respective to the volatility of the underlying asset:



•
38
Where ∏ is the option price and σ is volatility of the stock price. We next
show the derivation of vega for various kinds of stock option.
20.4.1 Derivation of Vega for Different
Kinds of Stock Options
•
For a European call option on a non-dividend stock, vega can be shown as
  St   N  d1 
39
(20.21)
•
For a European put option on a non-dividend stock, vega can be shown as
  St   N  d1  (20.22)
•
The derivation of (20.22) is
𝜕𝑃𝑡
𝜕𝑁(−𝑑2
𝜕𝑁(−𝑑1
= 𝑋𝑒 −𝑟𝜏
− 𝑆𝑡
𝜕𝜎𝑠
𝜕𝜎𝑠
𝜕𝜎𝑠
𝜕(1−𝑁(𝑑2 𝜕𝑑2
𝜕(1−𝑁(𝑑1 𝜕𝑑1
𝑋𝑒 −𝑟𝜏
− 𝑆𝑡
𝜕𝑑2
𝜕𝜎𝑠
𝜕𝑑1
𝜕𝜎𝑠
𝜈=
=
=
−𝑋𝑒 −𝑟𝜏
•
+𝑆𝑡
= −𝑆𝑡
1
𝑒
2𝜋
𝑑2
− 21
1
𝑒
2𝜋
1
𝑒
2𝜋
𝑑2
− 1
2
𝑑2
− 21
2
⋅
𝑆𝑡
𝑋
⋅
2
𝜎𝑠2 𝜏
2
3
⋅
𝑋
1
2
1
+ 𝑆𝑡
𝑑
1
− 21
𝑆𝑡
𝑒
2𝜋
1
𝑒
2𝜋
𝑑2
− 21
3
⋅
𝜎𝑠2 𝜏 2
𝜎𝑠2 𝜏
= 𝑆𝑡 𝜏 ⋅ 𝑁 ′ 𝑑1
40
𝜏
1
⋅𝜏2
𝜎𝑠2 𝜏
2
=
⋅
−
𝜎2
𝑟+ 𝑠
2
2
𝜎𝑠 𝜏
𝑆
𝜎
𝜎𝑠2 𝜏 2 − ln 𝑡 + 𝑟+ 𝑠 𝜏 ⋅𝜏2
𝑆
𝜎
− ln 𝑡 + 𝑟+ 𝑠 𝜏 ⋅𝜏2
𝑋
⋅
𝑒 𝑟𝜏
𝑆
ln 𝑡 +
𝑋
2
3
⋅
1
𝑆
𝜎
𝜎𝑠2 𝜏 2 − ln 𝑡 + 𝑟+ 𝑠 𝜏 ⋅𝜏2
𝑋
𝜎𝑠2 𝜏
2
•
For a European call option on a dividend-paying stock, vega can be shown as
  St e q   N  d1 
•
(20.23)
The derivation of (20.23) is

C t
N(d1 )
N(d 2 )
 St e  q
 Xe  r
s
s
s
 S t e  q
N(d1 ) d1
N(d 2 ) d 2
 Xe  r
d1 s
d 2 s
 2 3  St 
2   1 
 s  2  ln   r  q  s      2 
2  
1 2 

 X 
 S t e  q
e 
2

s 
2






  St 
s2   12

ln

r

q

 

   
d2


X
2
 1 S
1

 
 Xe  r 
e 2  t  e(r q)     
2
 2

X
s 

 


 2 3  St 
s2   12 
2
 s   ln   r  q       
2
2  
1  d21 
  S e  q
 X 
 q
 St e
e 
2
 t
s 
2






3
2
1  d21  s2  2 
 St
e  2 
 s  
2


d12
 St e  q   N  d1 
41







d12
1 2
e
2
  St 
s2   12
  ln   r  q      
X 
2  
  
s2 










•
For a European call option on a dividend-paying stock, vega can be shown as
  St e q   N  d1 
The derivation of (20.24) is

Pt
N(d 2 )
N(d1 )
 Xe  r
 St e  q
s
s
s
 Xe  r
 (1  N(d 2 )) d 2
 (1  N(d1 )) d1
 St e  q
d 2
s
d1
s
  St 
s2   12 

ln

r

q


2
    
 1  d1 St (r q)     X 
2
 

 r
 Xe 
e 2  e

2

 2

X
s 

 





 2 3  St 
s2   12 
2
 s   ln   r  q       
2
2  
1  d21 

 X 
 q
 St e
e 
2

s 
2






  St 
s2   12 

ln

r

q


2


    
d2
X
2
 1 
1
1  d21

 


 q
 q
2
 St e
e 
e
  St e
s2 
2
2






3
2
1  d21  s2  2 
 q
 St e
e  2 
 s  
2


 St e  q   N  d1 
42
(20.24)
 2 3  St 
2   1
 s  2  ln   r  q  s      2
2  
 X 
 
s2 










20.4.2 Application of Vega (ν)
43
•
Suppose a delta-neutral and gamma-neutral portfolio has a vega equal to
ν and the vega of a particular option is νo .
•
Similar to gamma, we can add a position of − ν νo in option to make a veganeutral portfolio.
•
To maintain delta-neutral, we should change the underlying asset position.
•
However, when we change the option position, the new portfolio is not
gamma-neutral.
•
Generally, a portfolio with one option cannot maintain its gamma-neutral and
vega-neutral at the same time.
•
If we want a portfolio to be both gamma-neutral and vega-neutral, we should
include at least two kinds of option on the same underlying asset in our
portfolio.
•
For example, a delta-neutral and gamma-neutral portfolio contains option A,
option B, and underlying asset.
•
The gamma and vega of this portfolio are −3,200 and −2,500, respectively.
•
Option A has a delta of 0.3, gamma of 1.2, and vega of 1.5.
•
Option B has a delta of 0.4, gamma of 1.6, and vega of 0.8.
•
The new portfolio will be both gamma-neutral and vega-neutral when adding
ωA of option A and ωB of option B into the original portfolio.
Gamma Neutral:  3200  1.2A  1.6B  0
Vega Neutral:  2500  1.5A  0.8B  0
•
From two equations shown above, we can get the solution that ωA =1000 and
ωB = 1250.
•
The delta of new portfolio is 1000 × .3 + 1250 × 0.4 = 800.
•
44
To maintain delta-neutral, we need to short 800 shares of the underlying asset.
20.5 Rho (𝜌)
•
The rho of an option is defined as the rate of change of the option price
respected to the interest rate:
rho 
45

r
•
Where ∏ is the option price and r is interest rate.
•
The rho for an ordinary stock call option should be positive because higher
interest rate reduces the present value of the strike price which in turn
increases the value of the call option.
•
Similarly, the rho of an ordinary put option should be negative by the same
reasoning.
•
We next show the derivation of rho for various kinds of stock option.
20.5.1 Derivation of Rho for Different
Kinds of Stock Options
•
For a European call option on a non-dividend stock, rho can be shown as
rho  X  e  r N(d 2 )
•
(20.25)
The derivation of (20.25) is
rho 
C t
N(d1 )
N(d 2 )
 St
     X  e  r N(d 2 )  Xe  r
r
r
r
N(d1 ) d1
N(d 2 ) d 2
 St
 X  e  r N(d 2 )  Xe  r
d1 r
d 2 r
2
 St
1  d21
e
2
 1  d1 St r    
 
 r
 r
 
e 2   e   
  X  e N(d 2 )  Xe  
 s 

X
2

s




 
2
 
1  d21
 r
 St
 
e
  X  e N(d 2 )  St

2

 s 
 X  e  r N(d 2 )
2
1  d21
e
2
46
2
 
 


 s 
•
For a European put option on a non-dividend stock, rho can be shown as
rho   X  e  r N(d 2 )
•
(20.26)
The derivation of (20.26) is
Pt
N(d1 )
 r
 r N( d 2 )
rho 
     X  e N(d 2 )  Xe
 St
r
r
r
 (1  N(d 2 )) d 2
(1  N(d1 )) d1
 X  e  r (1  N(d 2 ))  Xe  r
 St
d 2
r
d1
r
d
d




 1 S
 1
1

1
2
 X  e  r (1  N(d 2 ))  Xe  r  
e 2  t  e r   

S
e

t
 2
 s 
X
2


 
2
 r
 X  e (1  N(d 2 ))  St
 X  e  r N(d 2 )
47
1
e
2

d12
2
2
 
1  d21
 
e
  St
2
 s 
2
 
 


 s 
 
 


 s 
•
For a European call option on a dividend-paying stock, rho can be shown as
rho  X  e  r N(d 2 )
•
The derivation of (20.27) is
rho 
C t
N(d1 )
N(d 2 )
 St e  q
     X  e  r N(d 2 )  Xe  r
r
r
r
N(d1 ) d1
N(d 2 ) d 2
 St e q
 X  e  r N(d 2 )  Xe  r
d1 r
d 2 r
 St e q
 St e
 q
1
e
2

d12
2
 1  d1 St (r q)     
 
 r
 r
 
e 2  e
  
  X  e N(d 2 )  Xe  



X

2

 s 

  s 
1
e
2

d12
2
d
 
 1
1
 r
 q
 
e 2
  X  e N(d 2 )  St e
2
 s 
 X  e  r N(d 2 )
48
(20.27)
2
2
 
 


 s 
•
For a European put option on a dividend-paying stock, rho can be shown as
rho   X  e  r N(d 2 )
•
(20.28)
The derivation of (20.28) is
rho 
Pt
N(d 2 )
N(d1 )
     X  e  r N(d 2 )  Xe  r
 St e  q
r
r
r
 (1  N(d 2 )) d 2
(1  N(d1 )) d1
 X e r (1  N(d 2 ))  Xe  r
 St e  q
d 2
r
d1
r
d
d




 1 S
 1
1

1
 X e r (1  N(d 2 ))  Xe  r  
e 2  t  e(r q)       St e  q
e 2
 2
 s
X
2

  
2
 r
 X e (1  N(d 2 ))  St e
 X e r N(d 2 )
49
 q
1
e
2

d12
2
2
 
1  d21
 q
    St e
e

2
 s
2
 
  
 s 
 
  
 s 
20.5.2 Applications of Rho (𝜌)
•
Assume that an investor would like to see how interest rate changes affect the
value of a three-month European put option she holds with the following
information.
•
The current stock price is $65 and the strike price is $58. The interest rate and
the volatility of the stock is 5% and 30% per annum, respectively.
•
The rho of this European put can be calculated as follows.
Rho put
50
1
ln(65 58)  [0.05  (0.3) 2 ](0.25)
2
 Xe  r N(d 2 )  ($58)(0.25)e  (0.05)(0.25) N(
)  3.168
(0.3) 0.25
•
This calculation indicates that given 1% change increase in interest rate, say
from 5% to 6%, the value of this European call option will decrease 0.03168
(0.01 × 3.168).
•
This simple example can be further applied to stocks that pay dividends using
the derivation results shown previously.
20.6 Derivation of Sensitivity for Stock
Options with Respect to Exercise Price
•
For a European call option on a nondividend stock, the sensitivity can be
shown as
C t
 e  r N(d 2 )
X
•
(20.29)
The derivation of (20.29) is
Ct
N (d1 )
N (d 2 )
 St
 e r N (d 2 )  Xe r
X
X
X
N (d1 ) d1
N (d 2 ) d 2
 St
 e r N (d 2 )  Xe r
d1 X
d 2 X
d
 1  d1 S r  1
1  21
1
 1   r
 1
 r 
e 
     e N (d 2 )  Xe
e 2  t e 
 
 2
   X 
X
2
s   X 

 s
2
 St

1
e
 s 2

 e r N (d 2 )
51
d12
2
2
1
 S t  r
e
    e N (d 2 ) 
X
 s 2



d12
2
 St 
 
 X
•
For a European put option on a nondividend stock, the sensitivity can be shown
as
Pt
 e  r N(d 2 )
X
•
(20.30)
The derivation of (20.30) is
Pt
N(d 2 )
N(d1 )
 e  r N(d 2 )  Xe  r
 St
X
X
X
 (1  N(d 2 )) d 2
 (1  N(d1 )) d1
 e  r (1  N(d 2 ))  Xe  r
 St
d 2
X
d1
X
d
 1  d1 St r  1  1 
 1
1
1  1
 r
 r
 e (1  N(d 2 ))  Xe 
e 2  e 
     St
e 2 
 
 2

X
2
s   X 

 s   X 
2
2
d
 1
1
 r
 e (1  N(d 2 )) 
e 2
s 2
 e  r N(d 2 )
52
2
2
d
 1
1
 St 
e 2
 
 X  s 2
 St 
 
X
•
For a European call option on a dividend-paying stock, the sensitivity can be
shown as
C t
 e  r N(d 2 )
X
•
(20.31)
The derivation of (20.31) is
C t
N(d1 )  r
N(d 2 )
 St e  q
 e N(d 2 )  Xe  r
X
X
X
N(d1 ) d1  r
N(d 2 ) d 2
 St e  q
 e N(d 2 )  Xe  r
d1 X
d 2 X
 St e  q
1
e
2
1

e
s 2

 e  r N(d 2 )
53
d12
2

d12
2
d

 1  1
 1 S
1  1   r
1
 r
(r  q) 
t
2

     e N(d 2 )  Xe 
e  e
  



X
X
s  

 2
 s   X 
2
 St e   r
1


e
N(d
)

e


2
X
s 2


 q

d12
2
 St e  q 


X 

•
For a European put option on a dividend-paying stock, the sensitivity can be
shown as
Pt
 e  r N(d 2 )
(20.32)
X
•
The derivation of (20.32) is
Pt
N(d 2 )
N(d1 )
 e r N(d 2 )  Xe  r
 St e  q
X
X
X
 (1  N(d 2 )) d 2
(1  N(d1 )) d1
 e r (1  N(d 2 ))  Xe  r
 St e  q
d 2
X
d1
X
d
 1  d1 St (r q)   1  1 
 1
1
1  1
 r
 r

q

 e (1  N(d 2 ))  Xe 
e 2  e
     St e
e 2 
 

 2

X
2
s   X 

 s   X 
2
2
d
d
 q
 q
 1 S e
 1 S e


1
1
 r
t
t
2
2
 e (1  N(d 2 )) 
e 

e



s 2
 X  s 2
 X 
 e r N(d 2 )
2
54
2
20.7 Relationship Between Delta, Theta,
and Gamma
•
So far, the discussion has introduced the derivation and application of each
individual Greeks and how they can be applied in portfolio management.
•
In practice, the interaction or trade-off between these parameters is of concern
as well.
•
For example, recall the Black–Scholes–Merton differential equation with nondividend paying stock can be written as

 1 2 2  2
 rS
  S
 r
2
t
S 2
S
where is the value of the derivative security contingent on stock price, S is the
price of stock, r is the risk-free rate, and is the volatility of the stock price, and
t is the time to expiration of the derivative.
55
•
Given the earlier derivation, we can rewrite the Black–Scholes partial
differential equation (PDE) as
1 2 2
  rS    S   r
2
56
•
This relation gives us the trade-off between delta, gamma, and theta.
•
For example, suppose there are two delta neutral portfolios, one with positive
gamma and the other one with negative gamma and they both have value of
$1 .
•
The trade-off can be written as
1 2 2
  S   r
2
57
•
For the first portfolio, if gamma is positive and large, then theta is negative
and large.
•
When gamma is positive, changes in stock prices result in higher value of the
option.
•
This means that when there is no change in stock prices, the value of the
option declines as we approach the expiration date.
•
As a result, the theta is negative. On the other hand, when gamma is negative
and large, changes in stock prices results in lower option value.
•
This means that when there is no stock price change, the value of the option
increases as we approach the expiration and theta is positive.
•
This gives us a trade-off between gamma and theta and they can be used as
proxy for each other in a delta neutral portfolio.
Summary
•
In this chapter, we have shown the partial derivatives of stock option with respect to five
variables.
•
Delta (∆), the rate of change of option price to change in price of underlying asset, is first
derived.
•
After Delta is obtained, Gamma (Γ) can be derived as the rate of change of delta with respect to
underlying asset price.
•
Another two risk measures are Theta (Θ) and Rho (ρ); they measure the change in option value
with respect to passing time and interest rate, respectively.
•
Finally, one can also measure the change in option value with respect to the volatility of the
underlying asset and this gives us the Vega (ν).
•
•
The applications of these Greeks letter in the portfolio management have also been discussed.
In addition, we use the Black–Scholes PDE to show the relationship between these risk
measures.
•
In sum, risk management is one of the important topics in finance for both academics and
practitioners.
58