Delta
- Optionâ€™s delta is defined as the rate of change in an option's price relative to a one-unit change in the price of the underlying asset
- The delta for European Call option is given by N(d1) and for the Put option it is given by N(d1)-1
- Holding delta share and selling one call lead to a risk neutral outcome, other things remaining same
- This is an example of delta hedging
- Such a package of option and share is called a delta neutral portfolio
- Delta hedging strategy is an improvement over the simple stop-loss strategy
- However, the optionâ€™s delta changes with changes in stock price
- Requires traders to frequently readjust their positions in order to remain delta neutral. This process is called rebalancing
- Delta hedging therefore is a dynamic hedging strategy
- Properties of Optionâ€™s delta
- Call has a positive delta and Put has a negative delta
- Option delta will lie between -1 and +1
- The delta increases in absolute terms as the option goes further in-the-money and decreases as the option goes out-of-the-money
- At-the-money call and put options would have a delta around 0.50 and -0.50 respectively
- The delta of a Put option is negative reflecting an inverse relationship with the price of the underlying
- Deep in-the-money call options have a delta that approaches +1.00 as they are most likely to be exercised. Similarly, deep in-the-money put options would have a delta tending towards -1.00
- Deep out-of-the-money calls and puts have deltas that approach zero as the probability that they will be exercised is nearly zero
- The delta of the underlying asset is always 1.00
- The delta of a portfolio of derivatives (such as options) with the same underlying asset, can be found out if the deltas of each of these derivatives are known
- Example:
- Lets find the delta of the following portfolio constructed from the options of the same underlying asset and also estimate the trade required in order to turn the portfolio delta neutral
- The delta of the portfolio is given by:
- âˆ†_{portfolio }= 50,000 * 0.53 â€“ 100,000 * 0.47 â€“ 25,000 * (-0.51) = -7750
- It follows from the calculations above that 7,750 stocks need to be bought in order to get a delta neutral portfolio
Option (Strike Price, Expiration in Months) |
Quantity |
Position |
Delta of Option |
Call (55, 3 months) |
50,000 |
Long |
0.53 |
Call (56, 5 months) |
100,000 |
Short |
0.47 |
Put (56, 2 months) |
25,000 |
Short |
-0.51 |
- Cost involved in Delta Hedging
- Although effective as a hedging strategy, delta hedging carries a cost
- Every time a portfolio is rebalanced, the difference between the price paid for a stock and the price received for it creates this cost
- This is so because this hedging strategy calls for buying when stock prices rise and selling when they fall