Option

1. Option Greeks
Sneha Chawla M-15-7
Rahul Joshi M-15-26
Vishal Katkam M-15-31

2. DELTA
 Delta (D) is the rate of change of the option
price with respect to the underlying asset.
 c is the price of the call option.
 S is the stock price
S
c



D

3. DELTA HEDGING

4. DELTA HEDGING
 For a European call option on a non-dividend-
paying stock
 For a European put option on a non-dividend-
paying stock
)
(
)
( 1
d
N
call 
D
1
( ) ( ) 1
put N d
D  

5. DELTA HEDGING

6. DELTA HEDGING

7. DELTA HEDGING
 Π is the value of the portfolio
 The delta of a portfolio of options or other
derivatives dependent on a single asset whose
price is S.
 A portfolio consists of a quantity wi of option i
(1≦i≦n)
 Δi is the delta of the ith option.
S





D

D
n
i
i
i
w
1

8. THETA
 The theta (Q) of a portfolio of options is the
rate of change of the value of the portfolio
with respect to the passage of time with all
else remaining the same.
 Theta is sometimes referred to as the time
decay of the portfolio.

9. THETA
 Because N(-d2)=1-N(d2), the theta of a put
exceeds the theta of the corresponding call by
rKe-rT
)
(
2
)
(
)
( 2
1
'
0
d
N
rKe
T
d
N
S
call rT




Q

2
/
' 2
2
1
)
( x
e
x
N 


)
(
2
)
(
)
( 2
1
'
0
d
N
rKe
T
d
N
S
put rT




Q 


10. THETA

11. THETA

12. GAMMA
 The gamma (G) is the rate of change of the
portfolio’s delta (D) with respect to the price
of the underlying asset.
2
2
S




G

13. GAMMA

14. GAMMA

15. GAMMA

16. RELATIONSHIP BETWEEN DELTA,
THETA, AND GAMMA













r
S
S
S
rS
t 2
2
2
2
2
1

t




Q
S




D 2
2
S




G


G

D

Q r
S
rS 2
2
2
1
 

G

Q r
S 2
2
2
1

rf
S
f
S
S
f
rS
t
f









2
2
2
2
2
1


17. VEGA
 The vega (n) is the rate of change of the
value of a derivatives portfolio with respect
to volatility of the volatility of the underlying
asset.

n





18. VEGA
 For a European call or put option on a non-
dividend-paying stock )
( 1
'
0 d
N
T
S

n

19. RHO
 The rho(ρ) of a portfolio of options is the rate of change
of the value of the portfolio with respect to the interest
rate.
 It measures the sensitivity of the value of a portfolio to a
change in the interest rate when all else remains the
same.
r





)
(
)
( 2
d
N
KTe
call rT



)
(
)
( 2
d
N
KTe
put rT


 


20. PORTFOLIO INSURANCE
 A portfolio manager is often interested in acquiring a put
option on his or her portfolio.
 The provides protection against market declines while
preserving the potential for a gain if the market does
well.
 Options markets don’t always have the liquidity to
absorb the trades required by managers of large funds.
 Fund managers often require strike prices and exercise
dates that are different from those available in
exchange-traded options markets.

21. STOCK MARKET VOLATILITY
 Portfolio insurance strategies such as those just
described have the potential to increase volatility.
 When the market declines, they cause portfolio
managers either to sell stock or to sell index
futures contracts.
 When the market rises, the portfolio insurance
strategies cause portfolio managers either to buy
stock or to buy futures contracts.

22. THANK YOU