Finding Mispriced Options    • 151
$20-strike volatility. If you were interested in buying an ITM call option, 
you would pay less time value for the $20-strike than for the $21-strike op-
tions—essentially the same investment. I will talk more about the volatility 
smile in the next section when discussing delta.
In a similar way, sometimes the implied volatility for puts is different 
from the implied volatility for calls struck at the same price. Again, this is 
one of the market frictions that arises in option markets. This effect also 
has investing implications that I will discuss in the chapters detailing dif-
ferent option investing strategies.
The last column in this price display is delta , a measure that is so 
important that it deserves its own section—to which we turn now.
Delta: The Most Useful of the Greeks
Someone attempting to find out something about options will almost 
certainly hear about how the Greeks are so important. In fact, I think that 
they are so unimportant that I will barely discuss them in this book. If you 
understand how options are priced—and after reading Part I, you do—the 
Greeks are mostly common sense. 
Delta, though, is important enough for intelligent option investors 
to understand with a bit more detail. Delta is the one number that gives 
the probability of a stock being above (for calls) or below (for puts) a given 
strike price at a specific point in time.
Deltas for calls always carry a positive sign, whereas deltas for puts are 
always negative, so, for instance, a call option on a given stock whose delta is 
exactly 0.50 will have a put delta of −0.50. The call delta of 0.50 means that there 
is a 50 percent chance that the stock will expire above that strike, and the put 
delta of −0.50 means that there is a 50 percent chance that the stock will expire 
below that strike. In fact, this strike demonstrates the technical definition of 
ATM—it is the most likely future price of the stock according to the BSM.
The reason that delta is so important is that it allows you one way 
of creating the BSM probability cones that you will need to find option 
investment opportunities. Recall that the straight dotted line in our BSM 
cone diagrams meant the statistically most likely future price for the stock. 
The statistically most likely future price for a stock—assuming that stocks