In the option pricing examples discussed thus far, the option holder's payoff at the time of exercise was computed by comparing the value of the stock price (or spot index value or spot currency rate) at the time of exercise with the value of the option strike. Unlike these options, options on futures contracts are settled upon exercise by comparing the value of a futures price (or rate) at the time of exercise to the option strike price.^{[1]}

Futures contracts and options on futures contracts are both extensively traded on the Chicago Mercantile Exchange (CME).^{[2]} Like the options discussed earlier, options on futures also tend to be American-style in nature and usually physically settled (with the underlying futures contract) upon exercise. In the event that the holder of the futures-option contract has no desire to take physical delivery of the underlying futures contract, the holder can either easily unwind the entire futures options position prior to option expiry or transact in the OTC markets (which tend to be cash settled). Although the OTC version of the futures options can have early-exercise features, it is just as easy for one to transact in a European-style option that is cash settled based on the level of the futures contract at the time of the option expiry. As a consequence, I will discuss the valuation of a European-style futures option in this section.

Given the nature of a futures contract, the reader should note that the model that is used to mimic stock-price movements does not lend itself naturally to the modeling of movements in a futures price. The reader is referred to Black (1976) for further details relating to the reasons. The consequence of this is the diffusion process:

(3.9)

where

F is the price of the futures.

σ is the annualized volatility of futures price returns.

dz is the random variable drawn from a standard normal probability density function.

dF is the small change in the futures price over a small time interval dt.

As before, one first can observe that the diffusion equation given by equation (3.9) is equivalent to assuming that In FT is normally distributed with a mean of and a variance of where

Ft is the price of the futures contract at time t.

at,T is the spot-volatility rate corresponding to a maturity (T) using spot volatility curve at time t.

t is the time today.

T is the time when option matures.

It readily follows that the formula for pricing the call option on a futures contract paying stock is similar to equation (3.6) and takes the form

(3.10)

Using the distributional assumption for FT, it can be seen that equation (3.10) can be simplified to obtain

(3.11a)

(3.11b)

Equations (3.11a) and (3.11b) have been implemented on an Excel spreadsheet as shown in Table 3.6.

Table 3.6 shows the valuation of European-style futures options. The setup provided in the table can be customized to allow for the pricing of the European-style options contract on any type futures contract (e.g., Eurodollar futures, oil futures, grain futures, and so on) by incorporating contract sizes, lots per contract size, and other settlement conventions. The reader is referred to the CME website for examples of such applications.

[1] It is implicitly assumed that the maturity of the futures contract usually exceeds that of the option contract unlike that for an option on a forward contract (where the maturity of the forward contract is assumed to usually match that of the option contract).

[2] See Chicago Mercentile Exchange (CME) website (cmegroup.com) for contract specifications.

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