ADVANCED OPTION GREEKS

Black Scholes Model ReCALL Option

( ) ( )2rT1 dNKedSNC =where N(di) = the cumulative standard normal distribution function, evaluated at di, and:

T/2)T(rln(S/K)d

2

1++

=

Tdd 12 =

N(-di) = 1-N(di)

Black Scholes Model - R

There are many packages available for option pricing in R. We will be using fOptions

install.packages(fOptions)

library(fOptions)

Black Scholes Model - R

Trading Option Greeks

Option can be used to take a bullish or bearish position, given a directional forecast

We can trade sideways nontrending stocks or volatile stocks as well

Profit or loss can be independent of whether stock rises or falls

Option trading is non-linear. Hence, more opportunities can be exploited by trading options than trading stock

Option Trader

Option can be used to take a bullish or bearish position, given a directional forecast

We can trade sideways nontrending stocks or volatile stocks as well

Profit or loss can be independent of whether stock rises or falls

Option trading is non-linear. Hence, more opportunities can be exploited by trading options than trading stock

GREEKS

Greeks

The greeks are a product of option pricing model and each greek letter represents a specific sensitivity to influences on the options value

The five greeks commonly used by traders are,

Delta

Gamma

Theta

Vega

Rho

Delta () Formula

Formula for delta of call option is given by,

Delta of put option can be obtained using put-call parity

Delta ()

Delta is the rate of change of an option value relative to a change in the underlying stock price

Delta Call Option

Delta for a call option has positive correlation with the stock price. Therefore delta for call option is positive.

If a bullish trader buys a call instead of stock, he expects the call option price to go up, but by how much? To answer that question consider the delta of the option

Delta is stated as a percentage.

Delta gives us an approximate change in price of option w.r.t. underlying in case of small changes. As the change in underlying increases, error in approximation will increase.

Delta Call Option

S=60, X=60, r=0.04, t=26/365, sigma=0.3

Call Option Price = 1.91

Delta of Option = 0.514

S=61, X=60, r=0.04, t=26/365, sigma=0.3

Call Option Price = 2.47 ~ 1.91+0.514

S=59, X=60, r=0.04, t=26/365, sigma=0.3

Call Option Price = 1.44 ~ 1.91 - 0.514

Delta Put Option

Price of put option will increase with decrease in stock price. So put option value has negative correlation with stock price.

Delta of a put option will be negative. It will vary between range of 0 and -1.

Delta Definition 2

Delta can be thought of as owning that many number of underlying shares

So options gives us a way to leverage

E.g. S=60, X=60, r=0.04, t=26/365, sigma=0.3

Call Option Price = 1.91

Delta of Option = 0.514

Here you own 0.514 shares for just 1.91 instead of 60*0.514~30.84

Delta Definition 3

Delta is also known as hedge ratio

As we had seen in case of binomial pricing model. We can construct a self financing portfolio with delta number of shares. Market makers use delta as hedge ratio to hedge themselves.

Delta Definition 4

Traders would say that delta is a statistical approximation of the likelihood of option expiring in the money.

This definition is mathematically imprecise but is used nonetheless as general rule of thumb by option traders.

Deltas of Call and Put of same strike

Strike Call Delta Put Delta

50 0.99 -0.01

55 0.87 -0.13

60 0.51 -0.48

65 0.17 -0.83

70 0.03 -0.97

Because of put-call parity we get sum of absolute value of same strike call and put as 1.

Dynamic Inputs

Option Deltas are NOT constant.

They are calculated from the dynamic inputs of the pricing model stock price, time to expiration, volatility and so on.

When these variables change, the changes affect the delta.

Instead of learning underlying math, we will consider the effects from a more qualitative perspective.

Delta and Moneyness

Delta and Moneyness

Just by looking at delta you can tell the moneyness of the option

Far out of the money options have delta close to zero

Deep in the money call options have delta close to +1

Deep in the money put options have delta close to

-1

At the money options have delta about 0.5 (or -0.5 for puts)

Effect of Time on Delta

The more time left untill option expiry, the less certain it is whether the option will be ITM or OTM at expiration.

The deltas of both the ITM and OTM option reflect that uncertainty.

The more time left in the life of option, the closer the deltas gravitate around 0.5.

Effect of Time on Delta (Call)

Effect of Time on Delta (Put)

Call Delta vs. Time to Expiration

Put Delta vs. Time to Expiration

Effect of Volatility on Delta

For In the Money options higher the volatility then higher the chances that it will end less ITM or OTM (keeping all other parameters constant).

Similarly, for OTM higher the volatility, more will be the chance for the option to expire in the money. Hence for OTM options, higher volatility option will have higher deltas compared to lower volatility and same time to expiry.

Effect of Volatility on Delta (Call)

Effect of Volatility on Delta (Put)

Call Delta vs. Volatility (Call)

Call Delta vs. Volatility (Put)

Gamma ()

The strike price is the only constant in the pricing model.

When the stock price moves relative to the constant, the option in question becomes more in-the-money or out-of-the-money. This means delta changes.

This isolated change is measured by options Gamma, sometime called as curvature.

Gamma () is the rate of change of an options delta given a change in the price of underlying security.

Gamma is conventionally stated in terms of deltas per one point change in underlying.

Gamma () Formula

Gamma ()

Gamma () Example (Call)

Example 1

Example 2

Gamma () Example (Put)

Example 1

Example 2

Gamma () Explained

When traders buy options, they acquire positive gamma.

Since gamma causes options to gain value at a faster rate and lose value at a slower rate, gamma helps the option buyer.

When traders sell options, gamma works against them. The delta value moves toward zero at slower rate. When the underlying moves adversely, gamma speeds up losses.

The effect of gamma is less significant for small moves in the underlying than it is for bigger moves.

Gamma () Definition 2

Gamma is the second derivative of the graph of the option price relative to stock price.

Put another way, gamma is the first derivative of the graph of delta relative to stock price(s shaped curve)

So not only delta changes, it changes at a changing rate.

Gamma is not constant. Moneyness, time to expiration and volatility each have an effect on the gamma of the option.

Gamma and Moneyness

Gamma and Moneyness

When options are far in-the-money or out-of-the-money, they are either at 1 delta or 0 delta, respectively.

At the extremes, small changes in the stock price will not cause the delta to change much.

When option is at the money, its delta changes very quickly.

Summary, ITM and OTM options have low gamma

ATM options have relatively high gamma

Gamma vs. Time to Expiry

A decrease in the time to expiration solidifies the likelihood of ITMs or OTMs remaining as such. But an ATM options moneyness at expiration remains to the very end uncertain.

As expiration draws nearer, the gamma decreases for ITMs and OTMs, and increases for the ATM strike.

Gamma vs. Time to Expiry

Gamma vs. Time to Expiry

Gamma vs. Volatility

An increase in the volatility will increase chances of ITM to expire out of money and OTM to expire in the money. Hence gamma for ITMs and OTMs will increase with increase in volatility and vice versa

As volatility increases gamma for ATMs decreases.

Gamma vs. Volatility

Gamma vs. Volatility

Gamma vs. Volatility

Gamma Summary

Short-term ATM options with low volatility have the highest gamma.

Lower gamma is found in ATMs when volatility is higher and it is lower for ITMs and OTMs and in longer-dated options.

Theta ()

Option price can be broken down into two components,

Intrinsic Value (ITM part of the premium)

Time Value (leftover)

All else held constant, the more the time left in the life of the option, the more valuable it is (there is more time for the stock to move).

And as the useful life of an option decreases, so does its time value.

Theta ()

The decline in the value of an option because of passage of time is called time decay, or erosion.

Theta is the rate of change in an options price given a unit change in the time to expiration.

What is unit change in time? One days worth of time decay

Seven Days or Business Days week

Seven days worth of time decay

We will use one days worth of time decay based on seven days week

Theta ()

Commonly after afternoon, option price can often be observed getting cheaper relative to the stock price. This is because traders will normally take a day out of their model once prices are stabilized.

On Fridays and sometimes late Thursdays, traders will take all or part of the weekend out. Commonly by Friday afternoon traders will be using Mondays days to value their options.

Is Theta () good or bad?

Theta can be good or bad depending on the position

Theta hurts long positions; whereas it helps short option positions.

E.g. The 32 day 80 call has a theta of 0.03. If trader is long this call, the traders position would lose 0.03, as the time until expiration changes from 32 to 31.

A higher theta for the call than for the put of the same strike price is common when interest rate is greater than zero is used in the pricing model (We will see this later).

Theta vs. Moneyness

Theta vs. Moneyness

The more the ITM an option gets, the higher its theoretical value. But when studying an options time decay, one needs to be concerned only with the options time value, because intrinsic value is not subject to time decay.

The ATM option have more time value than ITMs or OTMs option. Hence they have more time premium to lose in the same period.

Effect of time on Theta

As the number of days to expiration decreases, the rate at which an option decays may change, depending on the relationship of the stock price to the strike price.

ATM options tend to decay at non-linear rate, whereas the time values of ITM and OTM option decay at a steady rate.

Effect of time on Theta

Effect of volatility on Theta

The volatility input to the pricing model has a direct relationship to the option values.

The higher the volatility, the higher the value of the option.

Higher valued option decay at faster rate.

All else held constant, the higher the volatility assumption, the higher the theta.

Vega

The volatility component of options value is called implied volatility. The higher the volatility input, the higher the theoretical option value, holding all other variables constant.

Vega is the rate of change of options value relative to a change in value of implied volatility.

Specifically if the implied volatility rises or declines by one percentage point, the value of the option rises or declines by amount of the options vega.

Vega

Buying a call or put establishes a long vega position

An increase in IV and subsequent increase in options value helps the P&L of long option positions and hurts short option positions

A put with same expiration month and same strike on same underlying will have the same vega value as its corresponding call.

Vega is also referred as Kappa sometimes.

Vega vs. Moneyness

Implied Volatility affects only time value portion of an option.

Because ATM options have the greatest amount of time value, they will have higher vegas.

ITM and OTM options have lower vega values than those of ATM options

Volatility vs. Call Price

X = 60Sigma = 0.3

Vega vs. Moneyness

X = 60Sigma = 0.3

Effect of Implied Volatility on Vega

Effect of Implied Volatility on Vega

Effect of Implied Volatility (IV) on Vega

At all IV ATM strike option maintains a similar vegavalue. This makes it easy to estimate the theoretical value of an at-the-money option under a wide variety of volatility scenarios.

Lower IV inputs tend to cause ITM and OTM vegas to decline. Higher IV inputs tend to cause vegas to increase for ITM and OTM options.

Effect of Time to Expiry on Vega

Effect of Time to Expiry on Vega

Vega of all options declines as expiration approaches.

Therefore, a long-term option will always be more sensitive to a change in volatility than a short term option with otherwise identical contract specification.

Interconnection of Time and Volatility

More time to expiration means more time for volatility to take effect, while less time to expiration may mean that any change in volatility will have only a minor effect on an options value.

Moreover, changes in the amount of time remaining to expiration and changes in volatility often have similar effect on an options value.

Decreasing volatility is similar to decreasing time to expiration.

Rho

Sensitivity of an options theoretical value to a change in interest rate is given by its rho.

Unlike the other sensitivities, one can not generalize about the rho since its characteristics depend on the type of underlying instrument and the settlement procedure.

The interest rate is usually the least important of the inputs into a pricing model. For this reason, rho usually takes a backseat to other sensitivities.

Rho Summary

In all cases the option which have the highest rho are those which are deeply-in-the-money, because they require greatest cash outlay.

And the greater the amount of time to expiration, the greater the rho.

Greeks Sign Summary

Greeks Effect of changing market conditions

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