# The Log Contract

The article titled, “The Log Contract”, is a 20 year old article. It was first article that made a case for the need for a new instrument to hedge volatility. There is something nice about papers written in the old times. The authors give a healthy intuition about the stuff they are about to explain in the paper, use simple equations that do not require too much of “head banging” and at the end of it, the reader pretty much gets the gist of the paper. Such papers are rare nowadays. In today’s world, pick any finance/stats/quant paper, there are at least two dozen heavy references given in the appendix and a substantial preparation is needed to understand the key idea of the paper. Very rarely is a paper self-contained. May be that’s the way it is supposed to be.

Coming back to this paper…

The author, Anthony Neurberger, shows that a delta hedged contract is not completely risk free. A delta hedged portfolio might make money or lose money based on the difference between the volatility used for hedging and the actual volatility seen over the life of the option. Hence traders need an instrument to hedge pure volatility. An empirical observation made in the paper is that 80% of the hedging error that remains after delta-hedging an option is on account of incorrect forecast of the volatility over the life of the option. Hence an option writer will need some instrument to hedge the vol. What about a trader who just wants to go long volatility or short volatility ? Well, one go long or short a delta hedged portfolio. However this does not give a complete exposure to the volatility. Hence the need for an instrument that gives a direct exposure to the realized volatility.

This article talks about a specific kind of contract called the log contract. The contract is a futures-style contract that is tied to a conventional futures contract. If the conventional futures settlement price at expiration is F_T, then the settlement price of the log contract is log(F_T). What’s the advantage of having such log contract ? It can be seen that by forming a long short portfolio of future and log futures contract, a trader can get exposure to a contract whose payoff depends only one the realized volatility and not on the hedger’s forecast of volatility.

The greatest advantage of log contract is its simplicity in providing a pure play volatility trade. The contract keeps its sensitivity to volatility whatever be the asset price. This idea was later used in valuing a variance swap, which in turn lead to the CBOE VIX index. VIX is basically an interpolation between fair values of near month and mid month variance swaps.