GREEK LETTERS

What are Greeks ??• Greeks are statistical values that show the sensitivity of

the price of an option to the factors that determine the value of an option

• They can be used as indicators to help monitor and analyze the risks associated with portfolios which include options

• Greeks include:· Delta · Gamma· Vega · Theta · Rho

• Delta - It is the rate of change in the price of the stock option to a change in the price of the underlying

• Gamma - It is the rate of change of delta with respect to the underlying asset price

• Theta - It is the rate of change of the value of the portfolio with respect to the passage of time

• Vega - It is the rate of change of the value of the portfolio with respect to the change in volatility

• Rho - It is the rate of change of the value of the portfolio with respect to the change in risk free interest rates

DELTA• Delta is the rate of change in the price of the stock option to a

change in the price of the underlying

• A delta of .5o means that a Re 1 increase in the price of the underlying asset will increase the price of the option by approximately Re .50

• Delta is also known as the ‘Hedge-Ratio’• Delta = change in the option price change in the stock price

Characteristics of delta:

• The delta of call options (long call and short put) has positive value• The delta of put options (short call and long put) has negative value• The delta of the underlying is always constant at 100 • All call and put options that are at the money have delta of around 050

and -050 respectively• Delta for long futures is +1• Delta for short futures is -1

CHARACTERISTICS

CHARACTERISTICS• The value of delta moves far from zero as the option moves into the

money• The value of delta approaches zero as the option moves out of the money

OptionDelta values

OTM ATM ITMLong Call / Short Put 0 +050 +100

Short Call / Long Put 0 -050 -100

Example: Strike price = Rs 30; time to expiry = 54 days, volatility = 25 % and rate of interest = 6%

CHARACTERISTICS

DELTA SENSITIVITY DELTA’S SENSITIVITY TO VOLATILITY AND TIME TO

MATURITY

• As time to maturity increases or volatility increases all options become more like ATM with their deltas approaching 05

• Conversely as time to maturity or volatility decrease, delta moves away from 05 levels

• For ITM calls delta moves towards 1 and for OTM calls delta moves towards 0 as time to maturity or volatility decreases

DELTA VS TIME TO EXP

CALL DELTA VS TIME TO EXP

0.000000

0.200000

0.400000

0.600000

0.800000

1.000000

1.200000

0.500

0

0.460

0

0.420

0

0.380

0

0.340

0

0.300

0

0.260

0

0.220

0

0.180

0

0.140

0

0.100

0

0.060

0

TIME TO EXP

DEL

TA

DELTA VS TIME TO EXP-ATMDELTA VS TIME TO EXPINMDELTA VS TIME TO EXP-OTM

DELTA HEDGING• DELTA HEDGING:• Delta is used for hedging of options• By taking an opposite position in the underlying instrument equal in

size to the option’s delta, we immunize the position against profit or loss variability due to small movements in the market This is referred to as delta hedging or creating a delta-neutral portfolio

• The delta neutral implies a strategy wherein the sum of all deltas in a portfolio is equal to zero

• Example: Assume that the stock price is Rs 200 and the option premium is Rs 20 and an investor has sold 10 call option contracts (1000 shares) Suppose delta in this case is 060

• Now to hedge, the investor would have to go long on the underlying by

• Delta x number of shares traded in option• Hence, he’ll buy 06 (delta) x 1000(no of shares short in option) =

600 shares

DELTA HEDGINGAssume that the price of the underlying goes up by Re1

Profit on the underling:Investor has purchased 600 shares @ Rs 200 = Rs 1,20,000Current price of share = Rs 201Hence his profit in the underlying will be (201 x 600) - 1,20,000 = Rs 600

Loss on the option:The option price will also go up by 06 x Re1 = 60 paise and now the option

price would be Rs 20.60The investor sold 10 call options @ Rs 20Hence his loss on the option will be (Rs 2060 - Rs 2000) x 1000 = Rs 600

Here, we can see that loss in options was offset by profit in underlying stock

DELTA HEDGINGNow, assume that the price of the underlying goes down by Re1 Loss on the underling:Investor has purchased 600 shares @ Rs 200 = Rs 1,20,000Current price of share = Rs 199Hence his loss in the underlying will be 1,20,000 - (199 x 600)= Rs 600

Profit on the option:The option price will also go down by 06 x Re1 = 60 paise and now the option

price would be Rs 1940 The investor sold 10 call options @ Rs 20Hence his profit on the option will be (Rs 200 - Rs 1940) x 1000 = Rs 600

An investor has to delta hedge constantly in order to make a profit or to maintain his risk less position because delta does not remains constant and changes during the life time of the option

ISSUES IN DELTA HEDGING

• The log- normal assumption may not be valid

• The volatility estimate may not be correct

• The hedge may not be done frequently enough to prevent losses due to or gamma risks

DELTA HEDGING

GAMMAGAMMA:

• Gamma is the rate of change of delta with respect to the underlying asset price

For example: Spot Price Delta Gamma

XYZ Aug 50 call Rs 480 +045 007 Rs 490 +052

Rs 470 +035• If gamma is small, delta changes slowly whereas if gamma is large delta is

very sensitive to the price of the underlying

gamma = change in the value of the portfolio * change in time change in stock price

GAMMACharacteristics of gamma:• Gamma of call and put is similar• Gamma of long positions (ie long calls and long puts)

is positive• Gamma of short positions (ie short calls and short puts) is

negative• Gamma of underlying stock is zero• Gamma is highest of at-the-money option• The further an option goes in or out-of-money, the smaller the

gamma becomes • Short-term options have higher gamma than long-term

options

GAMMASENSTIVITY OF GAMMA:

Before going into the details remember one important principle

• The magnitude of gamma is consistent with the uncertainty whether the option will expire “in” or “ out” of money

• Greatest for ATM options ( uncertainty is maximum)

• The gamma for ITM and OTM options increase with the increase in volatility and time to maturity ( uncertainty increases)

• Gamma for ATM option falls with the increase in volatility and increase in time to maturity ( is the uncertainty decreasing?)

GAMMA VS VOLATILITY

GAMMA VS VOL

0

0.1

0.2

0.3

0.4

0.5

0.6

0.020

0

0.050

0

0.080

0

0.110

0

0.140

0

0.170

0

0.200

0

0.230

0

0.260

0

0.290

0

0.320

0

VOLATILITY

GA

MM

A

GAMMA VS VOL INMGAMMA VS VOL OTMGAMMA VS. VOL ATM

GAMMA VS TIMETO EXPIRATION

GAMMA VS TIME TO EXP ATM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.03 0.01

TIME TO EXPIRATION

GA

MM

A

GAMMA VSEXP ATMGAMMA VSEXP INMGAMMA VSEXP OTM

GAMMA VS SPOT PRICEGAMMA VS SPOT PRICE

0

0.01

0.02

0.03

0.04

0.05

0.06

SPOT PRICE

GAM

MA

Series1

GAMMA Example: Strike price = Rs 115; volatility = 60 %, rate of interest = 10% and time to expiry = 3 months, 6 months and 1 year

GAMMA RISKMOST IMPORTANT THING TO UNDERSTAND:• The holder of an option who is short delta units of the underlying (long

call) will achieve a positive cash flow if subsequently the price movement is sufficiently large and a negative cash flow if price movement is sufficiently small

• Though theta and gamma are equal and opposite ,theta is more or less fixed in terms of loss per day where as gamma is directly proportional to change in the price of the underlying

• Now when there is a big move in the price of the underlying gamma change is more than theta and therefore for a delta hedged portfolio with long option and short underlying (positive gamma) position it results in a + cash flow

GAMMA HEDGINGGAMMA HEDGING : • Delta can protect against small changes in stock price, but for protection against large changes in stock price gamma position should be made neutral• Gamma neutral position can be achieved by selling options when gamma exposure is positive and buying options when gamma exposure is negative• For example: A stock’s spot price is Rs 50 and an investor wants create a spread and he also wants to keep his delta as well as gamma position neutral• In order to establish delta as well as gamma neutral position, first gamma position should be neutralized and then delta can be neutralized by taking an offsetting position in the underlying security or futures First the two gammas should be divided

• Here gamma neutral ratio = 005/0025 = 2:1

• So, if one wants to buy one June 50 call option, he should sell 2 gamma options to make gamma neutral portfolio

• Here we can see that after using the gamma neutral ratio, the spread is gamma neutralized but delta position is still long by 10 shares Hence we should take short position in stock by 10 shares to make delta neutral position

GAMMA HEDGING

VEGA

VEGA• Vega is also known as kappa, omega, tau

• Vega is the rate of change of the value of the portfolio with respect to the change in volatility

• A vega of 012 indicates that an option’s value will increase/decrease by Re 012 with every 1% increase/decrease in volatility

VOLATILITY• Volatility is one of the key inputs to an option pricing model

• Volatility is the degree to which the price of an underlying asset tends to fluctuate over time

• More generally, it is a measure of how uncertain we are about future

stock price movements

• If an underlying asset has a small volatility or price variability, then an option on that asset would not have much value to the holder

Types of Volatility• Future or projected volatility

– is based on the expected future distribution of prices for a particular underlying asset

• Implied volatility – is calculated based on the option price traded in the marketplace It is the

volatility which would have to be input into a theoretical pricing model in order to yield a theoretical value equal to the market value of the option

• Historical volatility – is calculated based on a range of historical prices At lease 20

observations are usually desirable to ensure statistical significance

• Seasonal volatility – comes into affect with certain commodities, for example as a

consequence of changes in weather conditions or demand

UNDERSTANDING VOLATILITY

• The volatility of a stock is measured by our uncertainity about the returns provided by a stock

• Typical volatilities of the stocks are in the range of 20% to 40% per annum• B&S model assumes a continuous framework and therefor while

computing returns we use: ln{[s(t+1)]/[st]}• We take these observations for 20-24 previous trading days typically to

estimate historical volatilty of the daily returns from the underlying• These returns are then annualised by multiplying by sqrt(252) iE multipied

by app Number of trading days in a year to get an annualise number• This annualisd number is then plugged in the b&s model to estimate

theoritical value of the options

IMPLIED VOLATILITY• Volatility that forces the value obtained from an option pricing model

(b&s) to equal the current market price of the option• The model is said to imply the level of volatility in the underlying asset and

thereby reveal to the market , what investors expect the volatility of the asset to be over the remaining life of the option

• It can also looked upon as the consensus volatility among the market participants

• It can be much more uselful in quoting optin prices, because regardless of the underlying price, time to expiration, or whether the option is put/call, iv is suppose to be of the underlying asset

• Sometimes option prices can reflect information about the fundamental value of the underlying that is not immediately reflected in the price of the stock itself, implying that the actual stock prices tend to move in the direction i”implied by the option prices”

Characteristics of Vega

• Vega of long positions (ie long calls and long puts) is positive• Vega of short positions (ie short calls and short puts)is negative• The vega of underlying security is zero• Vega of at-the-money options is generally the largest • Vega decreases as an option goes further in or out-of-the-money • Long-term options have higher vega than short-term options

Variation of Vega of a Call Option

• The figure below shows the variation of vega of a call option with the stock price

• For this assume – Strike price = Rs 115, – volatility = 60 %, – rate of interest = 10% and – time to maturity = 3 months

0.0700

0.0900

0.1100

0.1300

0.1500

0.1700

0.1900

0.2100

0.2300

75 86 97 108 119 130 141 152 163 174 185Stock Price

Veg

a

Variaton of vega of a call option with stock price

The figure below shows the variation of vega of options of different time to

expiry with the stock price

The figure below shows the variation of vega of options of different time to

expiry with the stock price

VEGA RISK• THE VEGA OF BOTH THE CALL AND PUT OPTIONS IS EQUAL AND POSITIVE REFLECTING THE

FACT THAT AN OPTION PRICE IS DIRECTLY RELATED TO VOLATILITY

• B&S MODEL IS ACTUALLY INCONSISTENT WITH CHANGING VOLATLITY THE MODEL ASSUMES THAT VOLATILITY WILL REMAIN CONSTANT FOR THE FULL LIFE OF THE OPTION

• VEGA INVREASES WITH THE INCREASE IN TIME TO MATURITY• ATM OPTIONS ARE MOST SENSITIVE TO TOTAL POINTS CHANGE IN VOLATILITY , WHERE AS

OTM IN PERCENTAGE TERMS

• VEGA OF AN ATM IS CONSTANT AT VARIOUS VOLATILITY LEVELS AND VEGA OF AN ITM OR OTM OPTION INCREASES WITH THE INCREASE IN VOLATILITY

Theta• The Theta of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to time with all else remaining the same. It is sometimes referred to as the time decay of the portfolio

theta = change in the value of the portfolio change in time

• Theta is always negative (positive) for the buyer (seller) of option, as the value of option loses value each day if the anticipated view is not realized

Example

• Theta of one month Reliance 420 call option is 1Spot =2000Call Premium = 150

• One day passes, the value for RIL 2000 call option will reduce by Re 1/-

THETA RISKSensitivity of Theta• A long option position will always have a negative theta and a short option position a positive theta

• The theta of an ATM option increases as expiration approaches. This implies that a short term option will decay more quickly than a long term option

• As we increase volatility the theta of all options increases Higher volatility means that there is a greater risk premium with the option and each day’s decay will also be greater when no movement occurs

THETA VS TIME TO MAT

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

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18.000

20.000

0.010 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500

Time to Maturity

thet

a

THETA VS EXP-ATM

THETA VS EXP INM

THETA VS EXP OTM

• Normally investors do not concentrate on theta of an individual option They are more concerned with delta and gamma However “position theta” can be calculated for an entire portfolio and an investor can get idea about the daily gain or loss in the portfolio due to time erosion• For example: Assume an investor has following positions in the month of June of a company named XYZ The spot price of XYZ is Rs 49

• This position is expected to make a gain of Rs 456 per day due to time decay • A negative position theta means the position has risk due to time, while a positive position theta means time is working in favour of the position

Rho

• The Rho of a portfolio of derivatives is the rate of change of the value of the portfolio to the interest rate. It is a measure of the sensitivity of the portfolio's value to interest rates

• rho = change in the value of the portfolio change in interest rates

• Interest rates are almost constant over the expiry hence are considered insignificant

LONG CALL

-150

-100

-50

0

50

100

150

200

1,000 1,100 1,200 1,250 1,350 1,450 1,550

SHORT CALL

-250

-200

-150

-100

-50

0

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100

150

200

1,000 1,100 1,200 1,250 1,350 1,450 1,550

LONG PUT

-150

-100

-50

0

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100

150

200

950 1050 1150 1250 1350 1450 1550

SHORT PUT

-250

-200

-150

-100

-50

0

50

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200

950 1050 1150 1250 1350 1450 1550

Option Greeks

Effect of Changing Market Conditions

Rise

Long Straddle• It involves selling a call and a put option with same exercise price and date of expiration

• Example:-

STRADDLE - Pay off

-300

-200

-100

0

100

200

300

400

500

SPOT

PAY-

OFF

Pay-Off

• It involves selling a call and a put option with same exercise price and date of expiration

• Example:-

Short Straddle

SHORT STRADDLE

-400

-300

-200

-100

0

100

200

300

400

SPOT

PAY-

OFF

Pay-Off

Synthetics

Synthetic pricing• Synthetic Call price

(Put Price + Stock price + CoC) – Strike price

• Synthetic Put price(Call Price + Strike price - CoC) – Stock price

• Synthetic Stock price(call Price - Put price) + Strike price

Note :- CoC = strike price * Interest Rate * days to Expire/365

Example: Actual vs. SyntheticAll factors relevant to this particular situation:

• Original stock purchase: 1000 shares of XYZ for 68• Original put purchase: Long 10 February 65 puts for 1.60• Current stock price: 69.70• Current put bid price: 1.05• Current Feb 65 call price: 6.10• Days to Exp: 30• The traders Long Rate: 2.5%

• Alternative A: Sell the put and the stock at market prices• Alternative B: Close the position by treating the married put as a call;

selling the actual call against it.

Choosing Alternative A

1.70 (stock profit) - .55 (put loss) = 1.15 x 1000 = $ 1150 profit

Choosing Alternative BThe position is equivalent to a long call purchased for 4.73;calculated as

follows:1. Cost of carry:• Interest Rate x Strike Price x Days to Expiration / 360• 2.5% x 65 x 30 / 365 = .13

2. The synthetic call price• (Put Price + Stock Price + Cost to Carry) - Strike Price• (1.60 + 68 + .13) - 65 = 4.73

3. Selling the actual to close out the synthetic• 6.10 (actual call)- 4.73 (synthetic call) = 1.37 x 1000 =$1370 profit

4. Conclusion:• Alternative B is more profitable by $220.

3. Selling the actual to close out the synthetic• 6.10 (actual call) . 4.73 (synthetic call) = 1.37 x 1000 =$1370 profit

4. Conclusion:• Alternative B• is more profitable by $220.

Delta Volatility

ITM strikedelta of higher volatility stock < delta of lower volatility stock delta

OTM strikedelta of higher volatility stock delta > delta of lower volatility stock

DeltaTime

ITM strikedelta for higher TtE stock < delta for lower TtE stock

OTM strikedelta for higher TtE stock > delta for lower TtE stock

GammaVolatility

ITM strikegamma for higher volatility stock < gamma for lower volatility stock

OTM strikegamma for higher volatility stock > gamma for lower volatility stock

GammaTime

ITM strikegamma for higher TtE stock < gamma for lower TtE stock

OTM strikegamma for higher TtE stock > gamma for lower TtE stock

THANK YOU

Black and Scholes Model

Assumptions of the Model

• Stock pays no dividend during options life• European exercise terms are used• Markets are efficient i.e no arbitrage opportunity • No commissions are charged• Interest rates are constant and known• Volatility is constant

Probability and Normal distributionPrice Range Profit range Probability of

occurrencesContribution to expectation of profit

1500-10 0-10 0.19 5*.19=0.951510-20 10-20 0.15 2.251520-30 20-30 0.09 2.251530-40 30-40 0.05 1.751540-50 40-50 0.01 0.451550-60 50-60 0.01 0.45

Total = 0.50 Total = 8.20

Black and Scholes Model for Option Valuation

The B–S model is as follows:

where C0 = the current value of call option S0 = the current market value of the share E = the exercise price e = 2.7183, the exponential constant rf = the risk-free rate of interest t = the time to expiration (in years) N(d1) = the cumulative normal probability density function

0 0 1 2( ) ( )fr tC S N d Ee N d

where ln = the natural logarithm; σ = the standard deviation; σ2 = variance of the continuously compounded annual return on the share.

2

1

2 1

ln ( / ) / 2fS E r td

td d t

• Stock price 12 months from expiration of contract is 100, the exercise price of option is 100. Risk free interest rate is 12% and volatility is 10% pa.

S = 100, E=100, r=0.12, σ=0.1, t=1 year

d1 = IN (100/100)+(0.12 *1) + 0.5 (.10)*sqrt(1) 0.1 * sqrt(1) = 1.25

d2 = d1 – σ * sqrt(t)= 1.25 – (0.1)( 1) =1.15

N (d1) = N(1.25) = 0.8944

N (d2) = N(1.15) = 0.8749

= 100(0.8944) – 100^(-0 .12*1) * 0.8749= 11.84

0 0 1 2( ) ( )fr tC S N d Ee N d