483
AN INTrOduCTION TO OPTIONS ON FuTureS
investing the cash received and so is willing to accept less; the value of the options fall. 
However, in futures markets, part of the value of distant contracts in a carry market 
reflects the interest costs associated with owning the commodity. An increase in the 
interest rate might cause the futures price to increase, leading to the value of existing 
calls going up. The net effect on calls is ambiguous, but puts should decline in value 
with increasing interest rates, as the effects are reinforcing.
 ■ Theoretical versus Actual Option Premiums
There is a variety of mathematical models available that will indicate the theoretical “fair value” for an 
option, given specific information regarding the four factors detailed in the previous section. Theoret-
ical values will approximate, but by no means coincide with, actual premiums. 
does the existence of 
such a discrepancy necessarily imply that the option is mispriced? definitely not. The model-implied 
premium will differ from the actual premium for two reasons:
 1. The model’s assumption regarding the mathematical relationship between option prices (premi-
ums) and the factors that affect option prices may not accurately describe market behavior. This 
is always true because, to some extent, even the best option-pricing models are only theoretical 
approximations of true market behavior.
 2. The volatility figure used by an option-pricing model will normally differ somewhat from the 
market’s expectation of future volatility. This is a critical point that requires further elaboration.
recall that although volatility is a crucial input in any option pricing formula, its value can 
only be estimated. The theoretical “fair value” of an option will depend on the specific choice of a 
volatility figure. Some of the factors that will influence the value of the volatility estimate are the 
length of the prior period used to estimate volatility, the time interval in which volatility is mea-
sured, the weighting scheme (if any) used on the historical volatility data, and adjustments (if any) 
to reflect relevant influences (e.g., the recent trend in volatility). It should be clear that any specific 
volatility estimate will implicitly reflect a number of unavoidably arbitrary decisions. 
different 
assumptions regarding the best procedure for estimating future volatility from past volatility will 
yield different theoretical premium values. Thus, there is no such thing as a single, well-defined fair 
value for an option.
All that any option pricing model can tell you is what the value of the option should be given 
the specific assumptions regarding expected volatility and the form of the mathematical relationship 
between option prices and the key factors affecting them. If a given mathematical model provides a 
close approximation of market behavior, a discrepancy between the theoretical value and the actual 
premium means the market expectation for volatility, called the implied volatility, differs from the 
historically based volatility estimate used in the model. The question of whether the volatility assump-
tions of a specific pricing model provide more accurate estimates of actual volatility than the implied 
volatility figures (i.e., the future volatility suggested by actual premiums) can only be answered 
empirically. A bias toward buying “underpriced” options (relative to the theoretical model fair value)