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The Greeks Part I

Introduction

Options "Greeks" are sensitivities of the option to various exposures of risk including time decay and volatility.  The names are taken from the actual Greek names.

Greek Sensitivity to
Delta Δ Change in option price relative to change in underlying asset price (ie Speed)
Gamma Γ Change in option Delta relative to change in underlying asset price (ie Acceleration)
Theta Θ Change in option price relative to change in time left to expiration (ie Time Decay)
Vega Κ Change in option price relative to the change in the asset's volatility (ie Historical Volatility)
Rho Ρ Change in option price relative to changes in the Risk Free Interest Rate (ie Interest Rates)
Zeta Ζ Percent change in option price per 1% change in implied volatility (ie Implied Volatility)

Delta - Δ

The Basics

  • The option delta is the rate of change of the option price compared with the rate of change of the underlying asset price.

    In other words delta measures the speed of the option position compared with that of the underlying asset.

    Delta = rate of change in Option price
    rate of change in underlying asset price
  • When the asset price is At the Money (ATM) the delta value will be around 0.5 (as a general rule).  This means that for every $1 the stock moves, the option will move at a speed of around half of that.  Obviously as the asset price deviates away from the ATM point, then the delta will change too, away from 0.5.

    ATM = +/- 50 deltas, ie moves at half the speed of the underlying asset

    Remember that 1 share has a delta of +/- 1, and 1 option contract represents 100 shares, therefore 1 ATM option will have a delta of +/- 50.

    You can think of delta as being the probability of the option expiring In the Money.  So a delta of +/- 50 is saying the option has a 50/50 chance of expiring In the Money.

Example

If you buy 100 shares of AMZN (+100 deltas), you would need to buy 2 ATM Puts (-50 deltas each) for a Delta Neutral Position.

Delta Neutral Trading

Remember that Delta means speed.  The greater the leverage of your position, the greater potential exposure to speed.  For example, if you buy a call option, the underlying stock may increase by 10% whilst your call option may increase by 100%.  This leverage is great when it's in your favour, but not so good when it's against you.  Taking the same example, if you buy a call and the underlying stock decreases by 10%, your call options may decrease in value by near 100%.  This risk needs to be hedged.  The term "hedge" is associated with the process of reducing risk. 

Delta Neutral Trading is a vast topic in itself, which will be covered in Special Article sessions within this site.  It is a method of trading whereby your position delta on the totality of your spread trade is one where the sum of the deltas equals zero.  The idea is that this conveys a "hedged" position, whereby the risk is reduced.

Delta Neutral Traders do this on the basis that they can continually make profitable adjustments to their trade as the asset price fluctuates.  The adjustments (usually selling part of the profitable side) bring the spread trade back to a delta neutral position (ie where the sum of the deltas for that position equals zero), whilst also capitalising on profitable side of the trade.

A popular technique is to make the profitable adjustments back to delta neutral when the underlying asset has moved by 20% in either direction.

Remember that Delta Neutral does NOT mean risk free!  Deltas are NOT linear.

Other points to remember

  • Delta Neutral still requires you to manage the Time Decay.
  • Longer term options will generally have lower deltas to shorter term options.
  • Your position Delta on your trade is also known as your "Hedge Ratio".
  • Delta is principally affected by Time left to expiration and Price of the underlying asset.
  • Some futures Delta neutral trades can require no margin sometimes (and with certain brokers)
  • With calls, Delta increases as the underlying asset price increases.  Call deltas are always positive.  Note that when you sell a call (naked) your position is delta negative.
  • With puts, Delta decreases as the underlying asset price decreases.  Put deltas are always negative.  Note that when you sell a put (naked) your position is delta positive.

Delta Trading

TRADE (US stock options) DELTA (+ or -) COMMENT
Buy 1 share +1 Buying a share has a delta of +1
Sell 1 share -1 selling a share has a delta of -1
Buy ATM call +50 Why +50?  Because 1 stock option contract represents 100 shares.  So 100 * 0.5 = +50
Buy Call + A long call always has a positive delta
Sell ATM call -50 Why -50?  Because 1 stock option contract represents 100 shares.  So 100 * -0.5 = -50
Sell Call - A short call always has a negative delta
Buy ATM put -50 Why -50?  Because 1 stock option contract represents 100 shares.  So 100 * -0.5 = -50
Sell ATM put +50 Why +50?  Because 1 stock option contract represents 100 shares.  So 100 * 0.5 = +50
Buy Put - A long put always has a negative delta
Sell Put + A short put always has a positive delta
ATM Straddle 0 The long call delta and long put delta effectively cancel each other out
ATM Strangle 0 The long call delta and long put delta effectively cancel each other out
Bull spreads + With a Bull Call Spread, the long call has a higher delta than the short call.  With a Bull Put Spread, the short put has a higher negative delta than the long put, but remember that because you're selling it, 2 minuses make a plus.
Bear spreads - With a Bear Call Spread, the short call has a higher delta than the long call, but the delta is negative because you're selling it.  With a Bear Put Spread, the long put has a higher negative delta than the short put.

Example: (Bull Call spread)

XYZ = $100

Buy 10 Jan 100c Delta = +500
Sell 10 Jan 105c Delta = -470 (say)
Hedge Ratio = +30

Why Does Speed Matter?

Example

  • You buy 100 shares of a stock.  Each $1.00 your stock rises, you make $100 * $1.00 = $100.  Each $1.00 your stock falls, you lose $100.
  • Alternatively, by buying call options you could make $300 when your stock rises by $1.00?  However, you can also lose $300 for every dollar the stock falls? 

This is the concept of leverage.

  • You buy a stock at $50.00.  Buying 100 shares costs you $5,000.
  • Let's compare this to buying the equivalent in call options: 1 contract at $7.00 will cost you $700.  (remember that 1 contract represents 100 shares for US stocks)
  • For illustration purposes only , let's say that your Delta is 1, ie for every one point the stock moves, the call option you've bought also moves by 1 point.
  • If the stock rises from $50 to $55:

  • Your shares will increase by $5.00 per share and you'll make $500 in extra profit, a profit of 10%.
  • Your calls will increase by $5.00 and you'll make $500 in profit, a profit of over 170%.

If the stock falls from $50 to $45:

  • Your shares will decrease by $5.00 per share and you'll lose $500, a loss of 10%.  Out of the $5,000 you started with, you now have $4,500.
  • Your options will decrease by $5.00 and you'll lose $500, a loss of over 70%.  Out of the $700 you started with, you now only have $200.

Can you now see why we might want to do something about the speed of the options price movements and why we might want to offset (or hedge) Delta?

When we buy an option, we always want enough time to be right.  We also want to make sure that modest swings in the stock price aren't causing uncomfortably fast and wild movements in our options position.  This is why we want to hedge Delta, or in other words, slow down the speed of the percentage movement of our options position compared with that of the underlying asset.

Delta Picture Summary

Figures assume trading 1 contract

Gamma - Γ

  • Gamma measures Delta's sensitivity to changes in the stock price, in other words "the speed of speed" or acceleration of the options position.

    Gamma = rate of change in Delta
    rate of change in underlying asset price
  • By knowing the Gamma of an option, we know how quickly the Delta will change and how quickly we should adjust our position in advance of this.
  • Gamma is significant because it helps the trader measure risk, particularly for Delta Neutral Traders.  Gamma effectively shows us how quickly the odds change of the option expiring in the money. 
  • Gamma tends to be large when the option is Near the Money (NTM).  This means that the Delta is highly sensitive (when the option is NTM) to changes in the stock price.  In other words the odds of the option changing from being OTM to ITM or vice versa are high.  Therefore, it is logical that ATM options have higher Gammas.
  • When options are Deep In the Money (DITM), the Delta is close to 1 and is not too sensitive itself to changes in the underlying asset price.  Therefore, the Gamma of DITM options is low.
  • Similarly, Gamma is low for Deep Out of the Money (DOTM) options.
  • The Gamma for puts and calls is always identical and can be positive or negative.
Underlying Asset Price Delta Gamma
ATM Around 0.5 High
NTM Around 0.5 High
Deep ITM Around 1 (high) Low
Deep OTM Low Low

Gamma Picture Summary