## Relative Value And Statistical Arbitrage

In the previous section we saw that appropriately constructed combinations of prices can be largely immunised against market-wide sources of risk. Such combinations of assets are potentially amenable to statistical arbitrage because they represent opportunities to exploit predictable components in asset-specific price dynamics in a manner which is (statistically) independent of changes in the level of the market as a whole, or other market-wide sources of risk. Furthermore, as the asset-specific component of the dynamics is not directly observable by market participants it is plausible that regularities in the dynamics may exist from this perspective which have not yet been "arbitraged away" by market participants.

To motivate the use of statistical arbitrage strategies, we briefly relate the opportunities they offer to those of more traditional "riskless" arbitrage strategies. The basic concept of riskless arbitrage is that where the future cash-flows of an asset can be replicated by a combination of other assets, the price of forming the replicating portfolio should be approximately the same as the price of the original asset. Thus the no-arbitrage condition can be represented in a general form as:

where Xt is an arbitrary asset (or combination of assets), SA(Xt) is a "synthetic asset" which is constructed to replicate the payoff of Xt and "transaction cost" represents the net costs involved in constructing (buying) the synthetic asset and selling the "underlying" Xt (or vice versa). This general relationship forms the basis of the "no-arbitrage" pricing approach used in the pricing of financial "derivatives" such as options, forwards and futures.1 From this perspective, the price difference Xt — SA(Xt) can be thought of as the mispricing between the two (sets of) assets.

A specific example of riskless arbitrage is index arbitrage in the UK equities market. Index arbitrage (see for example Hull (1993)) occurs between the equities constituting a particular market index, and the associated futures contract on the index itself. Typically the futures contract Ft will be defined so as to pay a value equal to the level of the index

1 See Hull (1993) for a good introduction to derivative securities and no-arbitrage relationships.

at some future "expiration date" T. Denoting the current (spot) stock prices as Slt, the no-arbitrage relationship, specialising the general case in equation (2.7), is given by:

where wt is the weight of stock i in determining the market index, r is the risk-free interest rate, and qt is the dividend rate for stock i. In the context of equation (2.7) the weighted combination of constituent equities can be considered as the synthetic asset which replicates the index futures contract.

When the "basis" Ft —J21 wiS'te(r~qi)(T-t) exceeds the transaction costs of a particular trader, the arbitrageur can "lock in" a riskless profit by selling the (overpriced) futures contract Ft and buying the (underpriced) combination of constituent equities. When the magnitude of the mispricing between the spot and future grows, there are frequently large corrections in the basis which are caused by index arbitrage activity, as illustrated in Figure 2.5 for the UK FTSE 100 index.

Many complex arbitrage relationships exist and "riskless" arbitrage is an important subject in its own right. However such strategies are inherently self-limiting - as competition amongst arbitrageurs grows, the magnitude and duration of mispricings decreases. Furthermore, in practice, even arbitrage which is technically "riskless" will still involve a certain level of risk due to uncertain future dividend rates qt, trading risks, and so on. From this perspective the true attraction of index arbitrage strategies lies less in the theoretical price relationship than in a favourable property of the mispricing dynamics - namely a tendency for the basis risk to "mean revert" or fluctuate around a stable level.

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