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Greeks in Option pricing


Option pricing is based on some key parameters as discussed before in this chapter. Essentially the following factors, known as ‘Greeks’ should be grasped to understand the option pricing:

  1. Delta
  2. Gamma
  3. Theta
  4. Vega
  5. Rho
  6. Omega

These factors are discussed below in detail.

Delta

Delta is a measure that reflects the relationship between a change in the price of the underlying and the price of the option. Delta plays a vital role in determining market strategies for several analysts. Delta is a measure that determines how much the price of an option will change consequent to a change in the price of the underlying stock. Also this measure can be thought of as indicating the probability that the option will finish in-the-money.

A call option at the strike price that equals the market rate, the probability that the option will finish in-the-money is exactly a 50% chance. Hence the delta for a call option equals 0.5. The sign for a long call option is positive as the price of the call option would increase corresponding to an increase in the price of the underlying.

A put option at the strike price that equals the market rate, again the probability that the option will finish in-the-money is exactly a 50% chance. Hence the delta for a put option equals 0.5. However the sign for a long put option is negative since the price of a put option would decrease corresponding to an increase in the price of the underlying.

Delta is also referred to as the hedge ratio. Delta is a key factor that is usually relied upon to implement the delta-neutral strategies for several fund managers. Delta-neutral strategy creates a risk less position which means that a portfolio would be worth the same whether the stock price rose by a small amount or fell by a small amount. However, wide fluctuation in the stock price will not protect the position.

If a fund manager has a portfolio that contains short position of x number of option contracts in a stock then x multiplied by the delta gives the fund manager the number of shares that would be needed to create a risk-less position.

Type of holding

Delta

Long underlying stock

Positive

Short underlying stock

Negative

Long call

Positive

Short call

Negative

Long put

Negative

Short put

Positive

For a stock that trades at $100, the following table illustrates the likely delta value for both call and put options at various strike prices. As the strike price gets higher than the market price of the underlying, the call delta approaches zero value and as the strike price gets lower and lower than the market price of the underlying, the put delta approaches zero value. When the option is deep in the money the value of delta approaches 100, indicating that for every dollar increase in the price of the underlying, the option price would also increase by a dollar.

Market Price: $100

Option Strike Price

Call Delta

Put Delta

$25

+100

0

$50

+90

-10

$75

+75

-25

$100

+50

-50

$125

+25

-75

$150

+10

-90

$175

0

-100

  • $100 strike price option which is at-the-money option has a delta of +50 for calls and -50 for puts
  • For strike price $150 and above, the call option delta drops substantially and approaches zero for calls of strike price $175; correspondingly the put option delta approaches -100 for strike price $175 and above
  • For strike prices $50 and below, the put option delta drops substantially and approaches zero for calls of strike price $25; correspondingly the put option delta approaches -100 for strike price $25 and below

There is an interesting relationship between the delta of an option product and its time to expiry. As mentioned earlier, delta can be thought of as the probability that the option will finish in-the-money on expiry of the contract. Hence the delta for an in-the-money option will approach 100 as time to expiry decreases. The reason is that the probability of the option finishing in-the-money increases as the time to expiry decreases.

The delta of an at-the-money option will always tend to remain at 50, irrespective of the time to expiry as there is always a 50% chance of the option finishing in-the-money.

Similarly for an out-of-the-money option, the delta will approach zero as the time to expiry decreases, because the probability of the option finishing in-the-money decreases as the time to expiry decreases.

Gamma

While delta is the first derivative, gamma is the second derivative of option price. Gamma is the acceleration of the option’s price to its underlying stock. Gamma is the ratio of the change of an option’s delta to a small change in the price of the underlying stock. Like delta, gamma is also expressed as a number between zero and one. It is also expressed as a number between 100 and zero when the gamma is multiplied by 100 which is normally the number of stock represented by one contract.

Gamma can be thought of as a measure that determines how much an option’s delta changes for every $1 change in the price of the underlying stock. Like delta, gamma value can be either positive or negative. A positive gamma indicates that the option’s delta increases when the price of the underlying stock increases in value and decreases as the price of the underlying stock decreases. A negative gamma similarly indicates that the option’s delta decreases when the price of the underlying stock decreases in value and increases as the price of the underlying stock increases.

Unlike delta, both short call and short put have negative gamma and long call and long put have positive gamma.

Type of holding

Delta

Gamma

Long underlying stock

Positive

No Gamma

Short underlying stock

Negative

No Gamma

Long call

Positive

Positive

Short call

Negative

Negative

Long put

Negative

Positive

Short put

Positive

Negative

Delta of at-the-money option contracts which are close to 50 are susceptible to small changes in the price of the underlying stock and thus ATM calls have the greatest gamma. Deep in-the-money options or far out-of-the-money options have a gamma value close to zero.

Theta

Theta is the measure of time value of the option. At the time of expiry of the option contract, both the at-the-money and out-of-the-money options become worthless as there is no intrinsic value for these options. Only the intrinsic value would remain for in-the-money option contracts, leaving no time value.

For an option seller the value of the time premium that decays as time passes is a benefit while the same is a loss for someone who is long in an option.

Theta represents how much value of an option’s price is attributable to time value. Theta is expressed as the dollar value that is lost every day, including holidays, known as the time decay, while the other factors remain constant.

It is interesting note the relationship of gamma and theta. Options that have a positive theta have a negative gamma and the options contracts that have a negative theta have a positive gamma. As we have observed earlier, at-the-money options have the maximum time premium or maximum theta. For at-the-money option contracts, the theta accelerates towards expiry. For in-the-money and out-of-the-money option contracts, the decay is linear towards the expiry.

Rho

Rho is a measure of the sensitivity of option prices to changes in the interest rates. The rate of interest is another factor in the determination of the price of an option, but considered less significant when compared with other factors like delta, vega and theta. Higher the interest, higher will be the call option premium and lower the put option premium. Lower the interest, lower will be the call option premium and higher the put option premium. Rho is expressed as a positive number for calls and negative number for puts. Rho indicates the theoretical change in the option premium for every one percent change in the rate of interest.

Vega

Vega or volatility as discussed earlier, is a measure of how fast the underlying futures prices are moving. It is a measure of the speed and magnitude at which the underlying stock’s prices change. This is a key factor in the determination of the price of an option. This is also known as Lambda.

Omega

Omega is a measure of the change in an option’s value with respect to the percentage change in the underlying price. The omega gives option investors an idea of how the option price and the stock price that underlies it move together. Omega is the third derivative of the option price, and the derivative of gamma.

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