IPO Insight

Option Greek Delta

Option Greek Delta

Table of Contents

What is Delta ?

  • Delta is the most important of the option Greeks.
  • It measures sensitivity of an option’s price (premium) to changes in underlying price.
  • For example, if an option has delta of 0.45, it means that when the underlying stock’s price increases by $1,
    the option’s price increases by 45 cents.
  • Delta can reach values from 0 to 1 for call options and from -1 to 0 for put options.

Call Option Delta

  • Call options are generally more valuable when the underlying security is more valuable.
  • A call option’s value increases when the underlying price goes up.
  • Therefore, it makes sense that call delta is always a non-negative number.
  • At the same time, a call option’s value can’t grow faster than underlying price.
  • As a result, call delta can never be greater than 1.

Call delta value range is from zero to positive one.

Example - Call Option Delta

  • Consider a $55 strike call option on a stock.
  • The stock is currently trading at $57 (underlying price) and the option at $2.60 (option premium).
  • The option’s delta is 0.75.
  • The delta tells us how the option premium will approximately change if the underlying price increases by $1.
  • If the stock grows by $1 to $58, we can expect the call option premium to grow by approximately $0.75 to 2.60 + 0.75 = $3.35.
  • Delta is the ratio of option price change and underlying price change.

Delta Inaccuracy for Big Moves

  • Notice the word approximately.
  • In fact, delta is only accurate for very small price changes.
  • As we will discuss later, delta itself also changes with underlying price (this is measured by gamma).
  • As a result, for bigger changes in underlying price, the actual option price change may be a little different than predicted by delta.
  • In our example above (the option has 20 days to expiration and implied volatility of 25%), the new option premium with underlying price at $58 would more likely be closer to $3.39, four cents higher than predicted by the delta.

Put Option Delta

  • Put options become more valuable when underlying price goes down, and lose value when underlying goes up.
  • Therefore, put delta is generally negative.
  • It can’t be smaller than -1, because the speed of option price changes can’t be faster than the underlying price change (other factors being equal).
  • Same logic as calls, just opposite direction.

Put delta value range is from zero to negative one.

Example - Put Option Delta

  • Consider a $55 strike put option on the same stock as in our call example.
  • With the stock trading at $57, the put option’s premium is $0.52 and its delta is -0.25.
  • If the stock’s price grows by $1 to $58, the put option’s premium goes down by approximately $0.25 to 0.52 – 0.25 = $0.27.
  • The delta is negative, which indicates the put option’s premium moving in opposite direction from the underlying price.

Delta and Option Moneyness

  • Delta is closely related to option moneyness.
  • You may already know that in the money options are generally more sensitive to underlying price changes than out of the money options.
  • In general, the deeper in the money, the greater sensitivity to underlying price.
  • Conversely, the further out of the money, the smaller sensitivity.
  • At the money options are something in between.

Delta and Call Option Moneyness

  • At the money calls have delta close to 0.50 (moderate sensitivity to underlying price).
  • In the money calls have delta from 0.50 to 1.00 (high sensitivity).
  • The deeper in the money (lower strike), the higher delta.
  • The underlying security could be considered an extremely deep in the money call option on itself with strike price of zero. Its delta is 1
  • Out of the money calls have delta from 0 to 0.50 (low sensitivity). The higher strike, the smaller delta.
  • In other words, call option delta is inversely related to strike price.

Delta and Put Option Moneyness

  • ATM puts have delta close to -0.50 (moderate sensitivity, opposite direction).
  • ITM puts have delta from -1.00 to -0.50 (high sensitivity, opposite direction).
  • The deeper ITM (higher strike), the lower (more negative) delta.
  • A short position in the underlying security could be considered an extremely deep in the money put option on itself (with infinitely high strike price) and has delta equal to -1
  • Finally, OTM puts have delta from -0.50 to 0 (low sensitivity, opposite direction).
  • The further OTM (lower strike), the closer delta is to zero.

Put option delta is also inversely related to strike price: close to zero for low
strikes, close to -1 for high strikes.

Delta as Probability of Expiring in the Money

  • One interpretation of delta is that its absolute value indicates the approximate probability of the option expiring in the money.
  • For instance, a deep in the money call option ($30 strike with underlying price at $40) is under normal circumstances (I am using 30% volatility and 50 days to expiration) almost certain to expire in the money, as its delta of 0.996 also suggests.
  • A $35 strike call on the same underlying is still very likely to expire in the money, although slightly less likely that the $30 strike call. It has delta of 0.90, indicating a 90% probability.
  • On the contrary, a $50 strike call is far out of the money, has delta of 0.025, and is unlikely to expire in the money (the underlying would have to increase by more than 25% from its current level).
  • It works the same with puts – you just need to ignore the minus sign. A deep in the money put with delta of -0.95 has approximately 95% probability of expiring in the money.

Factors Affecting Delta

Factors Affecting Delta

Using Delta in Practice

  • The role of all option Greeks is to measure risk – quantify how an option position’s value would change under various possible market developments.
  • Quantifying these effects enables us to do several useful things:
  • Choose the most suitable option strategy, expiration(s) or strike(s) for a potential trading idea
  • Decide the size (number of contracts) of a potential trade.
  • Know what would happen under different scenarios and plan in advance how we would react.
  • Make adjustments to an existing position – keep it exposed to the risks we want to take (and profit from) and at the same time reduce or eliminate (hedge) its exposure to the risks we are not willing to take.

Delta and Underlying Price

  • The fact that delta changes with underlying price should also be obvious from the relationship between delta and moneyness.
  • In the money call options have higher delta than out of the money call options.
  • As an option changes from out-of-the-money to in-the-money its delta must increase.
  • Similarly for puts, with increasing underlying price puts generally move out of the money and their deltas also increase .
  • The speed of delta changes (which is measured by gamma) also changes with underlying price:
  • It is generally fastest (highest gamma) at the money and approaches zero both deep in the money and far out of the money.

Delta and Time to Expiration

  • Delta also changes with passing time, even when nothing else happens in the markets.
  • In the money options become more sensitive to underlying price changes as they approach expiration and their delta gets closer to +1 for calls or -1 for puts.
  • Out of the money options become less sensitive to the underlying price moves and their delta gets closer to zero near expiration.
  • Longer time to expiration, the closer an option’s delta is to the middle of its range (0.50 for calls, -0.50 for puts), because with plenty of time remaining, it is still unclear whether the option eventually ends up in the money or out of the money.
  • Shorter Time to Expiration, the delta is pushed to one of the extremes: towards zero for out of the money options, towards +1 for in the money calls, and towards -1 for in the money puts.

Delta and Volatility

  • The effect of volatility on delta is very similar to the effect of time to expiration.
  • Increasing volatility pushes call delta closer to 0.50 and put delta closer to -0.50
  • Decreasing volatility pushes delta to the extremes – out of the money call and put delta closer to zero, in the money call delta closer to +1, and in the money put delta closer to -1.
  • More volatility and longer time to expiration generally make options more valuable, and they also affect the delta in similar ways.

Conclusion

  • Delta measures how option price will change if underlying price increases by $1.
  • Call option delta is from 0 to +1. Put option delta is from 0 to -1.
  • Out of the money options have delta near zero. In the money options near +1 (calls) or -1 (puts).
  • Delta itself changes with underlying price (this is measured by gamma). Therefore, delta is only accurate for small underlying price changes.
  • Like all other Greeks, delta is additive. Total delta of a position with multiple options is the sum of all options’ deltas.
  • Delta hedging makes delta zero – makes a position immune to small underlying price changes. It requires ongoing monitoring and rebalancing.
  • Delta also changes with volatility and passing time. Lower volatility or lower time to expiration push delta closer to the extremes (0 or +1 or -1).

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