The only major difference between these two tables is the time to expiration. Figure 1 shows contracts 45 days from expiration, while Figure 2 shows contracts 10 days from expiration. What is interesting is that while the delta of the ATM strike is roughly the same for both expirations, the 145 strike delta is almost five times as much for the 45-day expiration as it is for the 10-day expiration. Also, it seems like the market is convinced that underlying prices equal to or less than 130 are not expected to be realized regardless of time frame.
In the above example, we see how delta represents the sensitivity of changes in an options contract’s price to changes in the price of the underlying. Three other options Greeks—vega, theta and rho—represent options price sensitivity to a 1 percent change in volatility, change in time to expiration and a 1 percent change in interest rates, respectively. The fifth and final Greek is gamma, which represents the rate of change of delta as underlying prices move. Out-of-the-money (OTM) contracts tend to have low delta and high gamma, while ATM and in-the-money (ITM) contracts have increasingly higher delta and lower gamma. Delta moving 0.05 from 0.05 to 0.10 equates to gamma of 1 ("100 gamma"), whereas delta moving 0.05 from 0.5 to 0.55 equates to gamma of 0.1 ("10 gamma").
How can the Greeks assist in explaining the observations made earlier in the comparison of Figures 1 and 2? Without devolving into yet another treatise on the Black-Scholes options pricing model, suffice it to say that while the price of the underlying, volatility of price of the underlying, time-to-expiration and prevailing interest rate all contribute to options contract price discovery, the volatility of underlying price changes drives this equation. Current observed, or historical, 10- and 45-day volatilities of SPY are each approximately 15 percent, which means that over any given 10- or 45-day period in the past year, investors could reasonably expect (68 percent of the time given a normal distribution) a 15 percent movement around any given price of SPY. In considering just the left tail of this distribution, this translates to an expected 7.70 percent move, or in this case, about $10 given the current $140 SPY price. This explains the 100 delta for contracts at or below the 130 strike regardless of time-to-expiration. Given that both observed volatilities in prices of SPY are held constant when considering any of the contracts in question, as are interest rates, the only variable left is time-to-expiration. That there is more time-to-expiration explains why the delta of the August 145 call is 0.05 while the delta of the September 145 call is 0.24: There is simply more time for the unknown to become known.
Consider a 100-share position of SPY at $140 per share. This position has a notional value of $14,000. What if an investor didn’t have or didn’t want to tie up $14,000 but still wanted exposure to the price movement of 100 shares of SPY? To participate in the upside of this trade for approximately the next month would require getting exposure to the ATM SPY call expiring as close to one month from now as possible. The call in question would be the SPY 140 September call. This contract is priced at $2.93 per share and, with option contracts almost always representing notional exposure of 100 shares per contract, the contract would cost $293. As the 140 strike call has a delta of roughly 0.5, investors would expect to observe roughly half of the price movement of the underlying reflected in the price of the option. To fully replicate exposure to 100 shares of SPY, an investor would have to purchase two calls in order to bring the overall position delta up to 1.
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