# Index arbitrage with Options when not all underlyings have options listed?

One arbitrage strategy involves looking at the price of the Index Futures price compared with the prices of the options contracts for the underlyings.

My question is, can this arbitrage strategy still be performed when not all the underlyings have listed options contracts (like on the FTSE100)? Is there anything which can be done to account for the underlyings with no listed option contracts?

(Would also be interested in the answer when not all underlyings having Futures contracts listed for an Index)

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Why do you keep tagging your questions as "backtesting" when they clearly have nothing to do with backtesting? –  chrisaycock Sep 10 '13 at 20:41
Why are you leaving comments when they clearly have nothing to do with the answer? –  user997112 Sep 10 '13 at 20:50
Because I'm the moderator. –  chrisaycock Sep 10 '13 at 20:52
What are you attempting to arbitrage here? Your setup would only make sense if you look to extract alpha through trading implied volatility. Can you be more specific in order to understand what you are attempting to do? "Arbitrage" is nowadays such a misnomer describing anything and everything while in actuality arbitrage has a very closely defined meaning. –  Matt Wolf Sep 11 '13 at 4:33
What is this arbitrage 'strategy' are you trying to construct? An Index futures prices is most easily arbitraged (and replicated) via a basket of the underlying cash equities plus relevant financing - classic forward cash & carry. Why bring stock options into the equation? If you are trying replicate the index future via a basket of synthetic underlying forwards (call minus put - hence your focus on options?) you will go through far too much effort to get a worse result than that available from cash and carry. Please elaborate on the proposed strategy. –  tfb Sep 12 '13 at 7:04

it looks like you're talking about arbitraging options on multiple stocks vs options on the futures. if you can't buy options on some of the component stocks, you'd need to substitute for related stocks with high correlation (so that you are overall vega flat) and hope for the best

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Is there anything which can be done to account for the underlyings with no listed option contracts?

Classical options pricing theory relies on the idea that any option contract can be simulated with the appropriate dynamic hedging strategy. Options pricing practice indicates that this is sort-of true. So one thing you can do is synthesize the given options by dynamically trading the given equities.

Another common approach is to trade options in closely related companies, with extra hedges to account for the difference. But, you have a more serious problem here....

You are considering this an "arbitrage" strategy without (apparently) taking into account the key difference between a FTSE option and the component options, namely that the former is an option on a portfolio with correlated elements.

The FTSE option value will increase with increasing correlation, even if individual component volatilities remain unchanged. The two-element version with log returns $A_{1,2}$ and correlation $\rho$ shows why:

$$\text{Var}\left(\alpha_1 A_1 + \alpha_2 A_2\right) = \text{Var}\left(\alpha_1 A_1\right) + \text{Var}\left(\alpha_2 A_2\right) + \rho \sqrt{\text{Var}\left(\alpha_1 A_1\right) \text{Var}\left(\alpha_2 A_2\right)}$$

The value of an option position increases with increasing variance of its underlying.

Thus, a position in FTSE options hedged with individual equity options is considered a long (or short) correlation play, and certainly not an arbitrage strategy.

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very good points. +1. Clearly "arbitrage" might as well be voted one of the most misused terms in today's finance. –  Matt Wolf Sep 11 '13 at 4:36