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1. Option Greeks
Dr. Trilochan Tripathy, XLRI

2. Intrinsic and extrinsic value of an option
value of an option =Time value of a call option+ intrinsic value
Time value of a call option= C- Max( 0, S-K)
Time value of a Put option= C- Max( 0, K-S)

3. Option Price Determinants and Option Greeks
Option Price
Intrinsic
value
Spot price
Strike price
Extrinsic
Value
Time to
maturity
Rate of
interest
Spot Price
I. Volatility
Option Delta
Option Gamma
Option Theta
Option Rho
Option Vega

4. Option Greeks
• 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, 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. Option Greeks
• Delta: It is the sensitivity of an options price to a change in the price of underlying variable. It is
calculated as percentage change in option price for a 1% change in the underlying asset price. It is
also known as hedge ratio.
• [Delta(call) = ∂c/∂s, Delta(put) = ∂p/∂s]
• Gamma: It is the change in the Delta with respect to change in the underlying asset price.
• [Gamma(call) = ∂2c/∂s2, Gamma(put) = ∂2p/∂s2]
• Theta: It is the sensitivity of the value of an option to changes in time, everything else remaining
constant (spot, volatility, strike and interest rate and forward)
• [Theta(call) = ∂c/∂t, Theta(put) = ∂p/∂t]
• Vega: It is the sensitivity of the option price with respect to changes in volatility. Vega is thus the
variation in percentage of the value of the option for a 1% change of implied volatility. Vega is
large if the option has a long time to expiry or close to ATM
• [Vega(call) = ∂c/∂σ, Vega(put) = ∂p/∂σ]
• Rho: It is the sensitivity of the option price with respect to changes in interest rate.
• [rho (call) = ∂c/∂r, rho (put) = ∂p/∂r]

6. Delta = 
• Delta is the rate of change of the call price with respect to the underlying
• Mathematically, 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., Rupee per share.
• For a Call: (c) = c/S
• For a Put: (p) = p/S
(p) = (c) – 1 ( approx.)
Proof: ??
• For the (S) = S/S = 1

7. Understand Call Delta (): From Traders’ perspective
• A trader who is bullish on a stock may chose to buy a call instead of buying
underlaying security.
• Suppose the price of a stock increases (decreases) by Rs. 1, the trader would expect
to profit (loss) on the call- but by how much – consider influence of Delta only?
• To answer the above question, the trader must consider the  of the call option.
Example: If stock price increases
Stock Price was Rs. 100 Rs. 101
Call value was Rs. 3 =0.50 Rs. 3.50
If stock price decreases
Stock Price was Rs. 100 Rs. 99
Call value was Rs. 3 =0.50 Rs. 2.50

8. Understand Put Delta () : From Traders’ perspective (contd.)
• Puts have negative correlation to the underlaying
• A trader who is bearish on a stock may chose to buy a put instead of buying underlaying security.
• Suppose the price of a stock decreases (increases) by Rs. 1, the trader would expect to profit
(loss) on the put- but by how much – consider influence of delta only?
• To answer the above question, the trader must consider the  of the put option.
Example: If stock price increases
Stock Price was Rs. 100 Rs. 101
Put value was Rs. 2.25 =-0.40 Rs. 1.85
Example: If stock price decreases
Stock Price was Rs. 100 Rs. 99
Put value was Rs. 2.25 =-0.40 Rs. 2.65

9. Relationship Between Call  and Put 
• Option s are not constant and rather they are estimated from the
dynamic inputs from the pricing models,
• When the input variable changes, option  also changes and thus,
they are dynamic in nature,
• Traders understanding on  behaviour is crucial for designing the
trading strategies
• See next slide for the relationship between call and put  of the nifty
50 option

10. Why ∑(IcI+ IpI) ≠ 1 in real market scenarios?
• Sometimes the difference is due to rounding
• Possibility of the early exercise of the American option
• When the puts have bigger chance of early exercise, they
begin to act more like a short stock and consequently have
bigger delta.
• Often dividend paying stocks will have higher  “ in the
money call” segment compared to the put in the pair.

11. Moneyness and Delta
• Moneyness describes the degree to which the option is in the or out
of the money.
• As a general rule an option with :
ITM >0.50 to 1.0 ATM ≈ 0.50 OTM <0.50 to 0
Higher the degree of OTM,
the closer the  is to ‘0’
ATM options are not
exactly ‘0.50’
Higher is the degree of ITM,
the closer the  is to ‘1’
The ATM  is larger for
call relative to put.

12. Effect of Time and 
Stock price  at Expiration  Before 1 Month  Before 3 month  Before 6 Month
42 0 0 0.11 0.23
44 0 0.05 0.20 0.31
46 0 0.15 0.31 0.40
48 0 0.32 0.43 0.49
50 0.50 0.53 0.56 0.58
52 1 0.73 0.67 0.66
54 1 0.87 0.77 0.73
56 1 0.95 0.85 0.79
58 1 0.98 0.90 0.85
Estimated  of 50 strike call

13. Effect of Volatility on 
Stock price  at 10% Volatility  at 20% Volatility  at 30% Volatility  At 40% Volatility
42 0 0.06 0.16 0.25
44 0.01 0.14 0.25 0.32
46 0.08 0.26 0.35 0.40
48 0.28 0.40 0.45 0.48
50 0.58 0.56 0.56 0.56
52 0.84 0.70 0.66 0.64
54 0.96 0.82 0.75 0.70
56 0.99 0.89 0.81 0.76
58 1 0.94 0.87 0.81
Estimated  of 50 strike call- Impact of volatility

14. 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 price change will not
have any impact on the call-shares position.
• Definition: A portfolio is Delta-neutral if
(portfolio) = 0
• DELTA neutral position in the simple case of call-stock portfolios.
• Vportfolio = n(S) + n(c;S)
• (portfolio) = n(S) +  n(c;S)=0
• 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.

16. GAMMA ()
• Gamma measures the rate of change in an option delta given a
change in the price of the underlying security.
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

17. Understand call Gamma (): From Traders’ Perspective
• A trader who is bullish on a stock may chose to buy a call instead of buying underlaying security.
• Suppose the price of a stock increases (decreases) by Rs. 1, the trader would expect to profit (loss) on the
call- but by how much-consider joint influence of Delta and Gamma?
• To answer the above question, the trader must consider the  and  of the call option.
Example: If stock price increases
Stock Price was Rs. 100 Rs. 101 Rs. 102
Call  was 0.50  =0.04  =0.54  =0.04  =0.58
Call value Rs. 3 Avg= 0.52 c=Rs.3.52 Avg= 0.56 c= Rs. 3.56
If stock price decreases
Stock Price was Rs. 100 Rs. 99 Rs. 98
Call  was 0.50  =0.04  =0.46  =0.04  =0.42
Call value Rs. 3 Avg = 0.48 c=Rs.2.52 Avg= 0.44 c=2.08

18. Understand Put Gamma (): From Traders’ Perspective
• A trader who is bearish on a stock may chose to buy a put instead of buying underlaying security.
• Suppose the price of a stock decrease (increase) by Rs. 1, the trader would expect to profit (loss) on the
call- but by how much- consider joint impact of Delta and Gamma?
• To answer the above question, the trader must consider the  and  of the put option.
Example: If stock price increases
Stock Price was Rs. 100 Rs. 101 Rs. 102
Put  was -0.40  =0.04  =-0.36  =0.04  =-0.32
Put value Rs. 2.25 Avg= -0.38 p=Rs.1.87 Avg= -0.34 p= Rs. 1.53
If stock price decreases
Stock Price was Rs. 100 Rs. 101 Rs. 102
Put  was -0.40  =0.04  =-0.44  =0.04  =-0.48
Put value Rs. 2.25 Avg= -0.42 p=Rs.2.67 Avg= -0.46 p= Rs.3.13

19. • When trader buy options, they acquire positive Gamma
• When trader sell options, Gamma negatively works against them
• When the underlaying moves adversely , Gamma speeds up losses
• ITM and OTM options have a low Gamma
• ATM options have a relatively high Gamma
Understand Gamma (): From Traders’ Perspective

20. THETA ()
• Decline in the value of an option because of the passage of time is termed as time decay
• Incremental measurement of time decay in an option are represented by .
• Thus,  is the rate of change in an option’s price given a unit change in time to expiration.
• Theta measures are given by:
(c)= c/(t)
(p)= p/(t)
• s are positive but the they are reported as negative values.
• The negative sign only indicates that as time passes, time to expiration t, diminishes and so does the
option’s value, ceteris paribus.
• This loss of value is labeled the option’s “time decay.”

21. Effect of Volatility and Time on Theta
• Higher the volatility higher the option value
• Higher valued options decayed at a higher rate than a lower valued option
• Ceteris paribus, higher is the volatility higher is Theta for an option
• Days to expiration have a direct relationship to option value
• Time value of options at expiration both for higher valued and lower valued
options are ‘0’.
• ATM options tend to decay at a nonlinear rate – they lose value at a faster rate as
expiration approaches
• OTM and ITM option decays value at a steadier rate

22. VEGA ()
• In a market in which options are traded actively, we can reasonably assume that the
market price of the option is an accurate reflection of its true value.
• Thus, by setting the Black-Scholes-Merton price equal to the market price, we can work
backwards to infer the volatility.
• This procedure enables us to determine the volatility that option traders are using to
price the option. This volatility is called the implied volatility.
• 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.

23. Implied Volatility estimation

24. 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.

25. Interpret the Greeks
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.
• Gamma measures the change in delta when the price of the underlying asset
changes.
• Vega measures the sensitivity of the option’s market price to “small” changes in
the volatility of the underlying asset’s return.
• Rho measures the sensitivity of the option’s price to “small changes in the rate of
interest.

27. Delta neutral portfolio
• Suppose price of a stock is Rs. 100 and an investor sold 20 call option
with each call size of 100 shares at c=10. if the delta of the call option
on the said stock is 0.6, how many stock the investor should buy to
make is delta neutral portfolio.
• Solu: The investor’s position can be hedged by buying : 0.6* 2000 =
1200 stocks.
• If stock increases by Rs. 1 : portfolio outcome 0
• If stock decreases by Rs. 1: portfolio outcome 0

28. Delta neutral portfolio
• Suppose a financial institution has a following three position in option
on a stock:
• A long position in 100, 000 call option with strike price of Rs. 55 and
expiration date of 3 M. The delta of each option is 0.533.
• A short position in 200, 000 call option with strike price of Rs. 56 and
expiration date of 5 M. The delta of each option is 0.468.
• A long position in 50, 000 put option with strike price of Rs. 56 and expiration
date of 3 M. The delta of each option is -0.508.
How many stock should buy to make the portfolio delta neutral?

29. Stochastic Volatility

30. Put Call Ratio (PCR)
No. of traded put/ No. Traded call option
• One of the indicators to understand the mood of the market
• PCR>1 ( PCR<1: in practice below 0.7) signals bearish (bullish) move
in the market ahead
• However, it is not wise to start evaluating the market sentiment just
based on PCR=1

31. Explore further in the next lecture