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A Complete Guide to the Futures mArket
and selling “overpriced” options would be justified only if empirical evidence supported the conten-
tion that, on balance, the model’s volatility assumptions proved to be better than implied volatility in 
predicting actual volatility levels.
If a model’s volatility estimates were demonstrated to be superior to implied volatility estimates, 
it would suggest, from a strict probability standpoint, a bullish trader would be better off selling puts 
than buying calls if options were overpriced (based on the fair value figures indicated by the model), 
and buying calls rather than selling puts if options were underpriced. Similarly, a bearish trader would 
be better off selling calls than buying puts if options were overpriced, and buying puts rather than 
selling calls if options were underpriced. The best strategy for any individual trader, however, would 
depend on the specific profile of his price expectations (i.e., the probabilities the trader assigns to 
various price outcomes).
 ■ Delta (the Neutral Hedge Ratio)
Delta, also called the neutral hedge ratio, is the expected change in the option price given a one-unit 
change in the price of the underlying futures contract. For example, if the delta of an August gold 
call option is 0.25, it means that a $1 change in the price of August futures can be expected to result 
in a $0.25 change in the option premium. Thus, the delta value for a given option can be used to 
determine the number of options that would be equivalent in risk to a single futures contract for small 
changes in price. It should be stressed that delta will change rapidly as prices change. Thus, the delta 
value cannot be used to compare the relative risk of options versus futures for large price changes.
Table 34.3 illustrates the estimated delta values for out-of-the-money, at-the-money, and in-the-
money call options for a range of times to expiration. Where did these values come from? They are 
derived from the same mathematical models used to determine a theoretical value for an option pre-
mium given the relationship between the strike price and the current price of futures, time remaining 
table 34.3  Change in the premium of an e-Mini S&p 500 Call Option for 20.00 ($1000) Move in the 
Underlying Futures Contracta
Increase in the 2000 call option premium if the futures price rises:
From 1900 to 1920 From 2000 to 2020 From 2100 to 2120
Time to expiration $ Delta $ Delta $ Delta
1 week $10 0.01 $500 0.5 $1,000 1
1 month $120 0.12 $510 0.51 $870 0.87
3 months $260 0.26 $510 0.51 $750 0.75
6 months $330 0.33 $520 0.52 $690 0.69
12 months $390 0.39 $520 0.52 $650 0.65
aAssumed volatility: 15 percent; assumed interest rate: 2 percent per year.
Source: CMe Group (www .cmegroup.com).