The dramatic growth of equity index futures has been accompanied by the substantial growth of equity index options. These options come in three forms:

1. Options on the spot index itself such as the contract traded in the London International Financial Futures and Options Exchange — like the futures, these options are settled in cash, rather than by delivery of the underlying securities. Sometimes both European and American options are traded on the same index.
2. Options on the index futures are American options that call for delivery of an equity index futures contract at expiry.
3. Over-the-counter equity index warrants, issued by a number of securities houses, are effectively longer-term options on the spot index, with a different credit risk than the market traded options. Similar valuation considerations apply to these instruments as to other equity index options. However, the problems of forecasting volatility become greater as maturity increases.

Options on spot equity indices

Options on the spot value of equity indices are available on a number of the major indices. In some instances, both European and American options are traded simultaneously. Typical contracts are the American and the European options traded on the FTSE 100 spot index and listed on the London Traded Options Market. Both contracts represent an option on a notional index fund equal to £10 multiplied by the level of the index. The American option contract has an expiry cycle of four nearby consecutive months, plus two additional months; whereas the European option has, a March, June, September and December cycle, plus the two nearest months. These contracts are cash settled.

The Black—Scholes model adjusted for dividends can be used for valuing European options on indices that pay discrete dividends; the binomial model can be used; for valuing all American options; or the Barone-Adesi and Whaley model can be used for American options where the index is characterized as paying continuous dividends. In all these cases, the index level has to be adjusted for the amount and type of the dividend flow. Moreover, the use of these models does assume that the diffusion process of the index is identical to that of the individual securities.

Strictly speaking, if the assumption of a log-normal distribution of asset price relatives holds for individual equities, it will not hold for a weighted average of those equities — i.e. the index. However, the log-normal assumption is considered to be reasonably accurate, and the models discussed make that assumption.

The assumptions about flow of dividends, discussed above in the context of futures, also have important implications for the valuation of both European and American options. Brenner, Courtadon and Subrahmanyam found that for a broad-based index, such as the New York Stock Exchange Composite Index, the assumption of continuous dividends was not unreasonable but for the narrower MMI such an assumption could not be supported. They reported considerable mispricing of options on the MMI using a model that assumes a continuous dividend flow. Thus the correct assumption of the dividend flow is highly important in index option valuation, as indeed it is for index futures valuation.

Option valuation theory using the risk-neutral assumptions does assume that a riskless hedge can costlessly be created by combining the underlying asset with fairly priced options. One problem with index options is that the underlying asset itself is not traded, only the individual constituents. Thus it is more difficult (costly) to establish the risk-free portfolio. When there is a liquid futures market in the same underlying index, market participants may prefer to hedge the options with transactions in the futures market rather than in the cash market. Yet we know from our earlier discussions of equity index futures pricing that arbitrage in the futures market is also difficult and value basis can be considerable. Thus an option on the spot index may be priced not off the cash index, but off the index future, which itself may be mispriced. Moreover, the volatility of that future will be influenced by the volatility of the index, the volatility of short-term interest rates and of dividends (i.e. net cost of carry) and the volatility of the value basis. Thus it may be different to the volatility of the index itself. These factors may account for some of the mispricing that has been observed in empirical studies of equity index options

As with index futures, the procedure of cash settlement makes arbitrage less than riskless. Although the writer of the option may try to hedge the written options in the spot equity market, if exercised, settlement is in cash. Thus the option writer suffers the execution risk of adjusting the equity position. This problem is greatest with American options because of the multiple potential for early exercise and the fact that exercise notices are issued after the spot equity market has closed and is not communicated until the next business day.

Although it is to be expected that (because an index is a diversified portfolio of individual assets) the volatility of the index will be lower than that of the individual equity constituents, the volatility of the index will not be constant if the volatility of the constituents is not constant.

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