Option Greek Delta
Delta (Δ) is a measure of the sensitivity of an option’s price changes relative to the changes in the underlying asset’s price. In other words, if the price of the underlying asset increases by $1, the price of the option will change by Δ amount. Mathematically, the delta is found by:
Where:
- ∂ – the first derivative
- V – the option’s price (theoretical value)
- S – the underlying asset’s price
Delta is also known as a hedge ratio. If a trader knows the delta of the option, he can hedge his position by buying or shorting the number of underlying assets multiplied by delta.
In simple terms, Delta measures the change in the value of premium with respect to change in the value of underlying. For a call option, the value of Delta varies between 0 and 1 and for a Put option, the value of Delta varies between -1 and 0.
The above Option chain is for Nifty at 09:57 am. Nifty spot is trading at 9320.
The above Option chain is for Nifty at 10:07 am. Nifty spot is trading at 9316.
Now, form the above two tables, it is clear that with a small change in the value of Nifty, the premium for the option changes. The premium for 9100 CE in the first option chain is 291.65 and in the second option chain is 289.40.
Now, say if I were bullish on the market, so how would I find the premium for all the strike price if I were to expect the Nifty spot to be trading at 9400 by End of Day. So, this is where
Delta comes into the picture.
For a call option, assume the delta for a strike price is 0.40. So for every 1 point change in the value of underlying, the value of premium will change by .40 points. Say, if I had bought 9350 CE at a premium of 142.70. The Nifty spot price is 9316 and the Delta for this option is .40. And if by the End of the day, the spot price of Nifty jumps to 9350.
So the change in the Premium will be = (9350-9316)*0.40 = 14.4 points. So the new Premium will be = 157.1. Similarly, if the spot price were to come down to 9250, then the change in the Premium will be = (9250-9316)*0.40 = 26.4 points. So the new premium in this case will be = 142.7-26.4 = 116.3.
Delta value dependency on the Moneyness of an Option
The value of the Delta is derived using the Black & Scholes model. Delta is one of the output form this model. The Moneyness of the contract helps in deciding the value of Delta:
| Moneyness | Delta Value (Call Option) | Delta Value (Put Option) |
|---|
| In the Money | 0.6 to 1 | -0.6 to -1 |
| At the Money | 0.45 to 0.55 | -0.45 to -0.55 |
| Out of Money | 0 to 0.45 | 0 to -0.45 |
Delta of a Put Option: The delta of a Put option is always negative. The value ranges between -1 to 0. Let us understand it with the help of a situation. Say the spot price of Nifty 9450. And the strike price in consideration is 9500 PE (Put option). The Delta for this option is (-) 0.6 and the premium is 110.
Now, in Scenario 1, if the spot price of Nifty goes up by 80 points, then
New Spot price = 9530
Change in Premium = 80*(-.6) = -48 points
So the New Premium = 110-48 = 62. In case of Put options if the spot price of the underlying asset goes up, then the premium is reduced (the premium and spot price of a Put option are negatively co-related)
In Scenario 2, if the Spot price goes down by 90 points, then
New Spot price = 9360
Change in Premium = 90*(-.6) = 54 points
The New premium = 110+54 = 164 points
Risk profiling for choosing Delta
The risk-taking ability of a trader has an impact on choosing the right strike price. It is always advisable to avoid trading in Deep out of Money Options as the chances of those options expiring In the money is like their Delta (5% to 10%). For a Risk Taker trader, a slight out of Money or At the Money contracts are the best strategy. A Risk-Averse trader should always avoid trading Out of Money contracts. They should always trade At the Money or In the Money contracts as the chances of trade expiring in their favour is significantly higher than Out of Money contracts.
Gamma
Gamma (Γ) is a measure of the delta’s change relative to the changes in the price of the underlying asset. If the price of the underlying asset increases by $1, the option’s delta will change by the gamma amount. The main application of gamma is the assessment of the option’s delta.
Long options have a positive gamma. An option has a maximum gamma when it is at-the-money (option strike price equals the price of the underlying asset). However, gamma decreases when an option is deep-in-the-money or out-the-money.
As we have seen, the Delta of an option measures the change in the value of premium with respect to change in the value of underlying. The value of delta also changes with the change in the value of underlying.
But how does one measure the change in the value of delta?
We introduce you to ‘GAMMA’.
Gamma measures the change in the value of Delta with respect to change in the value of underlying. Gamma calculates the Delta gained or lost for a one-point change in the value of underlying. One important thing to remember here is that Gamma for both Call and Put option is positive. Let’s understand:
Spot price of Nifty: 10000
Strike price: 10100 CE
Call Premium: 25
Delta of option: .30
Gamma of option: .0025.
Now if Nifty goes up by 100 points, then
New Premium = 25 + 100(.3) = 55
Change in Delta will be = Change in Spot price * Gamma = 100*.0025 = .25
New Delta will be = .30+.25 = .55 (Option is now an At the Money contract)
Similarly if Nifty goes down by 70 points, then
New premium = 25 – 70(0.3) = 4
Change in Delta will be = Change in Spot Price * Gamma = 70*.0025 = 0.175
New Delta Will be = .30-.175 = 0.125 (Option is now a Deep Out of Money contract)
Gamma Movement
The movement of the gamma changes and varies with the change in the Moneyness of a contract. Just like Delta, the movement in Gamma is the highest for At the Money contracts and it is least for Out of Money contracts. So, one should ideally avoid selling/writing At the Money contracts. Out of money contracts are the best ones to write as they have a very good chance of expiring worthless for option buyer and the seller can pocket the premium.
Option Greek Vega
Vega (ν) is an option Greek that measures the sensitivity of an option price relative to the volatility of the underlying asset. If the volatility of the underlying asses increases by 1%, the option price will change by the vega amount.
Where:
- ∂ – the first derivative
- V – the option’s price (theoretical value)
- σ – the volatility of the underlying asset
The vega is expressed as a monetary amount rather than as a decimal number. An increase in vega generally corresponds to an increase in the option value (both calls and puts).
Vega as a Greek is sensitive to the current volatility. It is one of the most important factors in determining the option pricing. Volatility is simple terms is the rate of change. Vega simply signifies the change in the value of an option for a 1% change in the price of an underlying asset. Higher the volatility of an underlying asset, the more expensive it is to buy the option and vice versa for lower volatility.
Say the spot price of XYZ Company is Rs. 250 on 5th May and the 270 call option is trading at a premium of 8.
Let’s assume that the Vega of the option is 0.15. And the volatility of the XYZ Company is 20%.
If the volatility increases from 20 % to 21%, then the price of the option will be 8+0.15 = 8.15
And similarly, if the volatility goes down to 18%,
then the price of the option will drop to 8 – 2(0.15) = 7.7
Theta
Theta (θ) is a measure of the sensitivity of the option price relative to the option’s time to maturity. If the option’s time to maturity decreases by one day, the option’s price will change by the theta amount. The Theta option Greek is also referred to as time decay.
Where:
- ∂ – the first derivative
- V – the option’s price (theoretical value)
- τ – the option’s time to maturity
In most cases, theta is negative for options. However, it may be positive for some European options. Theta shows the most negative amount when the option is at-the-money.
Theta is an important factor in deciding option pricing. They use time as an ingredient in deciding the premium for a particular strike price. Time decay eats into the option Premium as it nears expiry. Theta is the time decay factor i.e., the rate at which option premium loses value with the passage of time as we near expiry. If we could recall, Premium is simply the summation of Time Premium and Intrinsic value.
Premium = Time premium + Intrinsic value.
Say, The Nifty spot is trading at 9450 and the strike taken into consideration is 9500 CE (call option). So the option is currently out of Money. There are 15 days to expiry and the premium charged for this option is 110. Now, the Intrinsic Value (IV) of this option = 9450-9500 = -50 = 0 (Since IV cannot be negative)
Now, Premium = Time value + IV
=> 110 = Time value + 0, hence the time value for this Out of the Money option is 110 i.e., the buyer is willing to pay a premium for an Out of the Money option. So, the analogy “
TIME IS MONEY” holds true in case of options pricing.
Let’s take another example:
- Say, Time to expiry = 15 days, Spot price of share of XYZ company = Rs. 95, Strike price = 100 CE, Premium = 5.5
- Now, if the spot price of XYZ = 96.5, time to expiry = 7 days, then for the same strike the Premium reduces to 3
- Again if the share price increases to 98.5, for the same strike price and with just 2 days to expiry, the premium reduces to 1.75
- Therefore, from the above example it is clear that even though the spot price is moving towards the strike price, the premium is reduced as the time remaining to make a substantial move above strike price is reduced. The option has less chances of expiring In the Money. The Greek Theta is a friend to Option writers. It is advisable for option writers to write/sell the option at the starting of contract as they will be able raise the premium erosion with the passage of time.
So from the above example, it is clear that the value of
Premium is Depreciating with the passage of time.
Rho
Rho (ρ) measures the sensitivity of the option price relative to interest rates. If a benchmark interest rate increases by 1%, the option price will change by the rho amount. The rho is considered the least significant among other option Greeks because option prices are generally less sensitive to interest rate changes than to changes in other parameters.
Where:
- ∂ – the first derivative
- V – the option’s price (theoretical value)
- r – interest rate
Generally, call options have a positive rho, while the rho for put options is negative.
Key Takeaways
If options is a team, then it has various players are Option Greeks like Delta, Gamma, Theta, Vega, volatility, etc. Each and every Greek has its own pivotal role in finding the exact pricing of the option. They play a pivotal role in deciding the Moneyness of the option.
A simple and clear understanding of all the Greeks goes a long way in deciding the right strike price and right option strategy. Risk Management both for option writers can be handled with a better understanding of the Greeks. Option buyers should ideally avoid trading Out of Money options and Option sellers should ideally write/sell Out of Money Options.
THANK YOU !!
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