Chapter 19 Solutions
1. The delta (∆) of a call options is represented the sensitivity of the call option price to
the current price of its underlying asset. ∆=0.7 implies that the purchase of 0.7 times
the number of underlying shares in the contract of the call option to hedge the loss
from value of call option when the underlying asset’s price changes.
2. Theta (Θ) is represented the sensitivity of the option price to the time to maturity. The
negative Theta value implies that the time value of the option decreases when time
passes.
Gamma (Γ) is represented the sensitivity of the change of delta (∆) to the rate of
change of the underlying asset price. The positive Gamma implies that the value of
option increases when the changes in the underlying stock price increase.
3. The vega (ν) is represented the sensitivity of the option price to the volatility of the
underlying asset. The positive vega implies that the option price is positively related
to the volatility of stock price. Increase 1% in volatility of underlying stock price will
contribute to the increase of vega % in option price.
4. When we utilize a Taylor series expansion to evaluate the option price on the base of
the current price of the underlying asset (S0) , then the option price (V) can be written
as
V ( S0 )
1  2V ( S0 )
1  nV ( S0 )
2
( S  S0 ) 
(
S

S
)


( S  S0 ) n
0
2
n
S
2! S
2! S
2
V ( S0 )
1  V ( S0 )
 V ( S0 ) 
( S  S0 ) 
( S  S 0 ) 2  o( S )
2
S
2! S
V ( S )  V ( S0 ) 
If we ignore the higher terms of error, then the change of option price can be arranged
as
V ( S0 )
1  2V ( S0 )
1
( S  S0 ) 
( S  S 0 ) 2   ( S  S 0 )  ( S  S 0 ) 2
2
S
2 S
2
On the right side of equation above, the first term is Delta multiplied by the change of
V ( S )  V ( S0 ) 
underlying asset price and the second term is the Gamma times the square of the
change of underlying asset price.
5. For an European call option on a non-dividend stock, the formula to calculate Delta is
Equation (20.5) as follows:
2
S    
ln  t    r  s 
2 
X 
  N(d1 ) , d1 
 
s
When r = 5% per year,  s = 15%, and the maturity  = 2, the Delta of an at-themoney call option is
(0.15) 2 
X 
ln     0.05 
2
2 
X 
d1 
 0.5775
(0.15) 2
  N(0.5775)  0.7182
6. For an European call option on a non-dividend stock,
  N(d1 )

1
N  d1 
St s 
  St   N  d1 

St s
2 
 N(d1 )  rX  e  r N(d 2 )
rho  X  e  r N(d 2 )
2
S    
ln  t    r  s 
2 
X 
d1 
s 
2 
S  
ln  t    r  q  s  
2 
X 
d2 
 d1  s 


s
,
Given the information as follows:
S= $110, X = $100, r = 0.05,  s = 15%, and  = 7 months (  = 7/12 year)
(0.15) 2  7
 110  
ln 

0.05


( )

2  12
 100  
d1 
 1.1438,
7
(0.15) ( )
12
7
d 2  d1   s   1.1438  (0.15) ( )  1.0292
12
  N(1.1438)  0.8736

1
7
(110)(0.15) ( )
12
N 1.1438   0.0165
7
  (110) ( )  N 1.1438   17.4249
12

7
0.05( )
(110)(0.15)
12
 N(1.1438)  (0.05)(100)  e
N(1.0292)  6.36
7
2 ( )
12
7
0.05( )
7
12
rho  (100)( )  e
N(1.0292)  48.0627
12
7. If the stock pays dividend with 1% dividend yield, then the Greek letters of an
European call option on the stock are
  e  q N(d1 )

e  q
N  d1 
St  s 
  St e q   N  d1 
  q  St eq N(d1 ) 
St e qs
 N(d1 )  rX  e r N(d 2 )
2 
rho  X  e  r N(d 2 )
2 
S  
ln  t    r  q  s  
2 
X 
d1 
s 
2 
S  
ln  t    r  q  s  
2 
X 
d2 
 d1  s 
s 
Therefore,
(0.15) 2  7
 110  
ln 
   0.05  0.01 
( )
2  12
 100  
d1 
 1.0929
7
(0.15) ( )
12
7
d 2  d1  s   1.0929  (0.15) ( )  0.9783
12
e

0.01(
7
)
12
e
N(1.0929)  0.8578
0.01(
7
)
12
7
(110)(0.15) ( )
12
  (110)e
0.01(
7
)
12
N 1.0929   0.0173
7
( )  N 1.0929   18.3386
12
  (0.01)  (110)e
0.01(
7
)
12
0.01(
N(1.0929) 
(110)e
7
)
12
(0.15)
7
2 ( )
12
 N(1.0929)  (0.05)(100)  e
0.05(
7
)
12
 5.4743
7
7 0.05( 12 )
rho  (100)( )e
N(0.9783)  47.3673
12
8. The put-call parity for European options on non-dividend stock is shown as
P = C + Xe-rτ – S
Then the Delta of a put option can be represented as the formula below in terms of
the Delta of a call option:
P (Ct  Xe r  St ) Ct Xe r St Ct
P  t 




 0 1  C  1
St
St
St
St
St St
Therefore, the delta of put option is equal to the delta of call option minus one.
For European options on stock with dividend yield q%, the put-call parity as follows:
P = C + Xe-rτ - Se-qτ
Then the Delta of a put option on stock with dividend can be represented as the
formula below in terms of the Delta of a call option with dividend:
P  (Ct  Xe r  St e q ) Ct Xe r St e q Ct
P  t 




 0  e q   C  e q
St
St
St
St
St
St
9. Use the relationship between the delta of put option and delta of call option from
previous question (#8), we can get
N(0.9783)
P 
Pt C t

1  C 1
St St
P 
Pt C t
 e  q   C  e  q
St St
The Gamma is defined as  
non-dividend payment case
with dividend payment case

, then the gamma of put option can be derived as follows:
S
 P  ( C  1)  C


 C
S
S
S
 ( C  e q )  C

P  P 

 C
S
S
S
P 
non-dividend payment case
with dividend payment case
Therefore, for European options, the Gamma of put option is the same as the Gamma of call
option whether the underlying asset pays dividends or not.
10. The traded option with Delta=0.8, Gamma = 2.0, Vega=0.8, we want to compute
a. Delta neutral portfolio,
If we set x as the number of put options with delta, -0.5, to make a delta
neutral portfolio, then
(-0.5)x+0.8=0, x=0.8/0.5=1.6
Therefore, we can buy 1.6 put options with delta (-0.5) and the traded option
with delta (0.8) to make a delta neutral portfolio.
b. Gamma neutral portfolio,
If we choose x call options with gamma 0.4 to make a gamma neutral
portfolio, then
(0.4)x+2.0=0, x=-5
Therefore, we can sell 5 call options with gamma (0.4) and buy the traded
option with gamma (2.0) to make a gamma neutral portfolio.
c. Both Delta and Gamma neutral portfolio,
If we buy 3 call options with delta (0.3) and gamma (0.4), sell 1.5 call options
with delta (0.9) and gamma (1.3), sell 0.5 call potions with delta (0.7) and
gamma (2.5), and buy the traded option with delta (0.8) and gamma (2.0), then
(0.3)(3)-(1.5)(0.9)-(0.5)(0.7)+0.8=0 for delta of the portfolio
(3)(0.4)-(1.5)(1.3)-(0.5)(2.5)+2.0=0 for gamma of the portfolio
Therefore, we can obtain a delta and gamma neutral portfolio by the
combination of buying 3 call options with delta (0.3) and gamma (0.4) and
buying a traded option with the sale of 1.5 call options with delta (0.9) and
gamma (1.3) and 0.5 call potions with delta (0.7) and gamma (2.5).