The following is an excerpt from Managing Expectations by Tony Saliba

# Option Delta

The risks related to options are many and include path dependency, implied volatility, and the passage of time. These risks can be calculated with figures produced by simple mathematical formulas known as greeks, as most use greek letters as designations. Each greek estimates the risk for one particular variable.

Delta is the sensitivity of an option’s price with regards to the movement of the related underlying future or security. It is expressed both as a percentage and a total. A call option with an estimated 25 delta suggests that the call option is one-quarter as sensitive as compared to the corresponding underlying. It implies that you would need 4 25 delta options to replicate the performance of a one-point move in the underlying.

• Professional traders think of an option’s delta as a hedge ratio; to what extent the option offsets or emulates the underlying. Professional traders learn very quickly that an option’s delta is only useful for a fractional move within that precise snapshot of time which it is calculated. An option’s delta can and does lose its relevance when there are changes in time, movement, and implied volatility.

• From a pedestrian viewpoint, it appears logical to envision an option’s delta using a simple equiprobable decision tree (i.e. a 50% chance of either an up or down move in the underlying) to price a call option. Yet this mind thought is dangerously flawed due to the conceptual problem of linking the resultant delta value with a probability.

Probabilities are beneficial when assessing risk with defined and limited outcomes. Applying probability or overemphasizing them in a financial world chock-full of infinite combinations can be dangerous indeed. Delta is a best-guess estimate at a given point of time and place – it’s nothing more and nothing less.

## Delta Details – Positive & Negative

To reiterate, an option’s delta is a mathematical expression that estimates how much the theoretical value of an option will change with a 1-point move in the corresponding underlying. It is the amount whereby an options trader would consider himself “dollar-neutral” compared to the underlying.

The delta of a call option spans from 0.00 to 1.00; the delta of a put option spans from 0 to (-1.00). Positive delta means that the option is estimated to rise in value if the asset price rises and is estimated to drop in value if the asset price falls. Negative delta means that the option position will theoretically rise in value if the asset price falls and theoretically drop in value if the asset price rises.

• Long (purchased) calls always have a positive delta; short (sold) calls always have negative delta.

• Long (purchased) puts always have a negative delta; short (sold) puts always have a positive delta.

• Long (purchased) underlying always have a positive delta; short (sold) underlying always have a negative delta.

• The nearer an option’s delta is to 1.00 or (-1.00), the more the price of the option responds like the actual long or short underlying when the underlying price moves.

### Delta Details – Changes in Volatility & Time

Delta is a best-guess estimate – susceptible to changes in volatility and time to expiration. The delta of at the money options (i.e. .50 delta call or put) is relatively invulnerable to changes in time and volatility. This means that at the money options with six months remaining to expiration compared to at the money options with one-month to expiration both have deltas very similar to .50.

However, the further divergence away from the money an option is, the more susceptible its delta will be to alterations in volatility or time to expiration. Fewer days to expiration or a decrease in volatility push the deltas of in the money calls closer to 1.00 (-1.00 for puts) and the deltas of out of the money options closer to 0.00.

Consequently an in the money option with 10 days to expiration and a delta of .80 could see its delta grow to .90 (or more) with only a couple days to expiration without any movement in the underlying.

Similarly, an out-of-the-money option with 10 days to expiration and a delta of .15 could see its delta drop to a .10 delta without any movement in the underlying. Lastly, an at the money option with 10 days to expiration and a delta of .50 will see its delta remain at .50 up through and including expiration day.

#### Delta & Synthetic Relationships

Synthetic long underlying is constructed with a long call and short a put at the same strike price in the same month. Therefore, the delta of a long call plus the delta of a short put (at the same strike in the same month) must equal the delta of long underlying. Conversely, synthetic short underlying is short a call and long a put at the same strike in the same month.

It must be recognized that options delta can be calculated with various input formulas. Using the Black-Scholes model for European style options, the total of the absolute values of the call and put is equal to

1.0.

Using varied input models for American style options and other exclusive circumstances, the sum of the absolute values of the call and put (at the same strike in the same month) can be marginally more or slightly less than 1.00.

#### Options Portfolio & Position Delta

Realistically speaking an option’s delta becomes more complex – less reliable – with the complexity of a position. A successful trader will view their delta holistically – balancing it with the risks of time and volatility.

That said, a trader can add, subtract, and multiply deltas to determine the “net delta” of a position and underlying. The position delta is a way to estimate the risk/reward character of your position in terms of sensitivity to the underlying. The calculation is very straightforward:

Position Delta = Option’s Theoretical Delta x Amount of Options Contracts

A trader owns five of the ABC June 60 calls, each with an estimated delta of +.40, and short (sold) one hundred shares of ABC stock. The traders position delta would be +100 or (short 100 shares of ABC or

-100 deltas, long 5 +.40 delta = +.40 delta x 5 – 100 = +100).

What does +100 mean? The mathematics estimates that if ABC stock should increase by $1.00, the trader will earn $100. On the other hand, if ABC drops $1.00, the trader will lose $100. Once again, it is imperative to realize that these numbers are mere approximations. Remember that delta is relevant for insignificant moves and for brief time periods. Beyond that it gets fuzzy fairly quickly.

#### The Relationship between Volatility and Delta

As mentioned earlier in this chapter, delta is an estimate and that estimate is partially produced on the trader’s assumption about implied volatility levels.

At its core, options implied volatility embodies the degree of uncertainty in the market and the extent to which the prices of the underlying asset are expected to change over time. When there is relatively more uncertainty, people will pay more for options – thus raising the level of implied volatility.

In August 2015, for example, as the markets reflected on China and its currency devaluation, participants became fearful and bid up the prices of options or the implied volatility. But when people feel more secure, they tend to collect option premium through the sale of options. This would cause implied volatility levels to drop.

#### The Change in Delta with Changes in Implied Volatility

All other factors (movement, time to expiry) being constant, an increase in implied volatility causes all option deltas to converge towards .50. In fact, during the unprecedented volatility spike of “Black Monday” (1987) option models did produce .50 deltas for every strike available for trading!

During a rising implied volatility environment, in-the-money call option deltas will decrease towards .50 while out-of-the-money call options will increase towards .50. Reiterating our Chapter 2 discussion on synthetics would imply the opposite would hold true for put deltas. Other words, in a rising volatility environment, in the money put option deltas will decrease towards -.50 while out of the money put options delta would increase towards -.50.

This should begin to make sense, for when ambiguity increases (the reason for higher implied volatility levels) it becomes less clear where the underlying will wind up at expiration. Thus, the absolute value of an in the money option delta will decrease, the absolute value of an out of the money option delta will increase, while an at the money option delta will always remain near a .50 delta.

A somewhat drastic yet helpful approach to understanding this is to look at expiration. At expiry, volatility is 0; all deltas are either 0 or 1, finishing either out of the money or in the money. Any increase in volatility – like an increase in time – causes probabilities to move away from 0 and 1, reflecting a higher level of uncertainty.