# The Urban Legend and the Options Delta

Within the knitting of the options industry are numbers used to estimate the risks associated with the movement of the underlying, implied volatility, and the passage of time.  They are numbers produced by fuzzy mathematical formulas and together known as “greeks” because most use Greek letters as names.  Delta is the first and foremost greek and it simply measures the theoretical change in an options price with respect to a change in the underlying price.

As a young boy, I was captivated by Robert Daley’s 1959 book, “The World beneath the City,” and its retelling of sewer workers reporting large colonies of alligators thriving in the New York City sewer system.  The tale made perfect sense – wealthy families returning from Florida vacations toting these baby reptiles as pets. When they grew too large for comfort, the family would proceed to flush them down the toilet.  And, firmly believing this logic to be so profoundly reordered my motivation on “if” and “when” I ventured to a public restroom – particularly when the gossip grew to where these creatures were determinedly crawling up the pipes at Gimbels department store.

An options delta is also victim of a curious urban legend of sorts whereas countless options participants, for unexplainable reasons, equate the delta of an option to the probability of the option expiring in-the-money.  For the record, and contrary to half-baked storytelling, delta is NOT measuring the odds that the option will expire in-the-money!  Sadly that idea too often sticks until experience—such as a movement in the underlying coupled with time and/or volatility will abruptly prove otherwise.

Delta is the measure of the relationship between the price of an option and the price of its underlying. Delta is also referred to as a hedge ratio: The amount by which an options trader needs to “hedge” to be dollar/delta neutral at that point in time. Positive delta indicates the option will rise in value if the asset price rises, and 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.

The delta of a call can range from 0.00 to 1.00; the delta of a put can range from 0.00 to -1.00. Long calls have positive delta; short calls have negative delta. Long puts have negative delta; short puts have positive delta. Long underlying has positive delta; short underlying has negative delta. The closer 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.

Calls and puts with the same strike and month will have an absolute value of 1.00.  For example the SPY November 180 call currently has a positive delta of .80 – this would imply the November 180 put is assigned a negative delta of - .20.  Because of this, one can synthetically replicate the purchase or sale of underlying with options.  In the previous example, if one would purchase one SPY November 180 calls and simultaneously sell one November 180 puts; this would be the synthetic equivalent of purchasing 100 shares SPY.  Conversely, sell the call and buy the put would synthetically provide you with the exact same profit-and-loss characteristics as selling shares of SPY.

Delta and Movement in Volatility, Time

Delta can be sensitive to changes in volatility and time to expiration. The delta of at-the-money options is relatively impervious to changes in time and volatility. This means at-the-money options with 100 days to expiration and at-the-money options with 20 days to expiration both have deltas very similar to 50. But the more in-the-money or out-of-the money an option is, the more sensitive its delta will be to changes 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. Hence, as an example, an in-the-money call 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 as compared to the same strike option with 100 days till expiry.

Assume the XYZ December 50 call has a value of 1.00 and a delta of .40, and the price of XYZ is at \$47. If XYZ rises to \$48, the value of the XYZ December 50 call will theoretically rise to \$1.40. If XYZ falls to\$46, the value of the XYZ December 50 call will theoretically drop to \$.60.

If the December 50 put has a value of \$3.25 and a delta of -.60 with the price of XYZ at \$47, if XYZ rises to \$48, the value of the XYZ December 50 put will theoretically drop to \$2.65. If XYZ falls to \$46, the value of the XYZ December 50 put will theoretically rise to \$3.85.

These numbers assume that nothing else changes, such as a rise or fall in volatility or the passage of time. Alterations in any one of these parameters can change delta, even if the price of the underlying asset doesn’t move.