This document defines and explains the key Greeks (derivatives risk measures) - Delta, Gamma, Theta, Vega and Rho. It also discusses delta hedging to maintain a portfolio's delta neutral position, and how portfolio insurance strategies use options to protect against market declines while allowing gains when the market rises. However, these strategies can also potentially increase stock market volatility as portfolio managers trade in response to market movements.

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