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Index Options Hedging and the Correlation Risk Behind Index Arbitrage

Article Quant Q&A · Author: user997112
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Summary

The document asks whether index arbitrage using options can work when some index constituents have no listed options. It describes two possible substitutes: dynamically hedge with the underlying shares to replicate option exposure, or use options on related companies and add hedges for differences. These approaches can help approximate missing component options, but they do not make the trade riskless.

The key issue is that an index option depends on the joint behavior of its constituents. The variance of a two-stock portfolio includes a term for correlation, so index option value can change when correlation changes even if each stock’s volatility stays constant. Hedging an index option with individual-stock options therefore creates exposure to correlation. The discussion is conceptual and gives no empirical test or implementation details; substitutions and dynamic hedges leave risks that must be assessed rather than assumed away.

Key ideas

  • Missing constituent options can sometimes be approximated by dynamically hedging with the underlying shares.
  • Options on related companies can serve as substitutes, but their differences require additional hedges.
  • Index option value depends on correlations among constituents as well as their individual volatilities.
  • A hedge using index options and constituent options carries correlation exposure and is not necessarily arbitrage.

Tags

OptionsEquitiesArbitrageVolatilityRisk management
Full text
# Index arbitrage with Options when not all underlyings have options listed?


# 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)

## Answer by Brian B (score 4)

https://quant.stackexchange.com/a/8910

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.

## Answer by test (score 0)

https://quant.stackexchange.com/a/8939

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.