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Since a market index is nothing more than a basket of stocks, you can create your own index by putting together stocks of your choice. The only difference is that you can trade options on major indexes unlike your custom index.

So, I'm wondering if it's possible to use stocks and their options to replicate the option of a custom index, i.e. to closely match the profit/loss graph of a hypothetical option on that index. There are synthetic calls and puts on individual stocks, so I figured that it might be feasible to create synthetic calls and puts on a basket of stocks.

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I disagree with Tal. It can be done, but, unless you assume perfect liquidity and zero commissions, the incidental costs would be too high. –  barrycarter Oct 20 '11 at 16:06
@barrycarter don't forget lack of jumps. What you say is not really disagreeing with me, just illustrating why I'm right, since in the real world, liquidity is not perfect and commissions are strictly positive. –  Tal Fishman Oct 24 '11 at 13:45
We're quants. There are no commissions. There is no spread. Our biggest problem is getting hit by those damn frictionless physicists! –  barrycarter Oct 25 '11 at 14:58
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2 Answers

up vote 8 down vote accepted

Interesting question. Unfortunately for you, the answer is no, it cannot be done. The principal difference between a basket of options and an option on the basket (or index) is correlation risk. In fact, there is a systematic difference between the implied volatility of the basket and the (properly weighted) sum of implied volatilities on the components. Trading this difference is known as dispersion trading (see Deng (2008)). You may also be interested in the answers to this question.

As for the part of your question regarding synthetic replication of the option using the underlying (by which I assume you mean dynamic replication), this is only possible in theory. In practice, the transaction costs are too large and the jump risk / discontinuity of trading prevents one from achieving perfect replication. In fact, if it were so easy to replicate an option, there would be no reason to trade them. It is all the more difficult to try to dynamically replicate an option on an entire basket of stocks.

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Is it possible to replicate the option of a custom index? Yes and you can find OTC market-makers who will make a price. They use portfolio replication to mimic the payoff of the option with a position in the underlying (Black-Scholes, '73). Even though the underlying custom index is not traded it can be perfectly constructed via its traded constituents. So once you sell the option to your customer, you reconstruct the payoff of a long option by dynamically adjusting a position in the underlying. The underlying position is kept delta matched to the option. Delta matched positions are PL neutral over small changes, eg if the custom index changes price by 1USD, an at-the-money call option with a face value of 1,000,000USD will see about equivalent changes in value as a custom index holding of around 500,000USD (excuse the hand-waving).

Here the delta is based on your multivariate price model of the underlying basket. You know the volatilities of the single stocks from the options market. You have to input a correlation matrix, which is the hard part. The upper limit of this correlation matrix is where all the stocks move together, which of course will result in a high price for the option. However the price history of the stocks will generally show some negative correlation, which is what you see under normal conditions.

The problem with baskets it that this negative-correlation-under-normal-conditions makes them seem far less volatile than single assets, leading to people holding massive positions in baskets and their derivatives. However when extreme events occur, correlation can approach 1 and at the same time constituent volatility spikes. Extreme events though are rare (!), so people usually do not have the historic data to be prepared for extreme movements in baskets. In fact the underestimation of basket price volatility is the source of some spectacular financial failures. Nick Leeson bankrupted Barings Bank by selling massive options on Japanese stock indices, the positions blew up when an earthquake toppled Asian markets. The Global Financial Crisis was principally caused by the belief that baskets of credit risks were less risky than they were.

So make the positions small, and know that you are unlikely to have enough price history to confidently predict future outcomes, so account for that in your price.

BTW not sure where the idea that transaction costs prohibit dynamic replication strategies. The options markets of the 80's, 90's boomed because of dynamic hedging, and today automated trading has only reduced transaction costs further.

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How do you hedge the correlation risk? –  user5155 Apr 18 '13 at 23:05
Do you mean you expect the strategy to usually have no correlation risk? Well the point of trading an option on a basket is to create a position in their covariance matrix. Just as the point of trading an option on a single asset is to create a position in the asset's variance (albeit in terms of implied volatility). A covariance matrix is to a basket as scalar variance is to a single asset. That said I imagine you could hedge it to some extent with an option spread. –  Riaz Rizvi Apr 19 '13 at 16:51
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