Risk Latte - Using Logarithm to create an Asset:The Log Contract

Using Logarithm to create an Asset:The Log Contract

Team Latte
September 6, 2011

Of all the assets that we trade and invest in today, none is more esoteric and mysterious - and if we made add, complex - than "volatility". How did volatility become an asset from being a mathematical parameter in an option pricing formula?

In 1994, Anthony Neuberger, then at the Institute of Finance and Accounting at the London Business School, introduced a simple, yet a very peculiar product, the Log contract. The publication of his paper, “The Log Contract” in the Journal of Portfolio Management has since then become a watershed event in the history of quantitative finance. This contract would forever change how we look at volatility and lay the foundation for the introduction of variance swaps in the late 1990s and the new volatility index, VIX by CBOE in 2003. Though Neuberger may not have realized it then but what he had done was initiated the process by which volatility would no longer remain a mathematical and an abstract concept but become a tangible – well, almost tangible – asset. Perhaps, the last frontier of quantitative finance was about to be conquered.

Something was bothering Neuberger and it may have come as flash of imagination. In a recent interview he said to me, "One of the things that always bugged me about vanilla options was their kinked pay-off; the Black-Scholes approach to option pricing works with any non-linear pay-off, and it would surely be easier to work with smooth curved pay-offs which did not have infinite derivatives. From a mathematical perspective, the formulae would be simpler, and that is a great advantage because it focuses attention on what is fundamental. From a practical perspective, it would get round the problem of buying a risk management instrument that will almost surely become either innocuous (if it ends up deep in or out of the money) or excessively risky (if it ends up close to the money). With a smooth curved pay-off, your option would remain close to the money whatever happens."

Was such a simple product possible, which behaved like an option, retained all its derivatives like characteristics and yet was far simpler to manage in terms of its risk? The log contract was indeed such a product. In hindsight, it is truly amazing that none of the great minds in the field of Quantitative Finance never ever thought of this product. Even though it would be another five years before Emanuel Derman and his colleagues at Goldman Sachs would formalize the concept and theory of variance swaps by incorporating the notion of Log contract in their model, Neuberger had ushered in a silent revolution. Without the Log contract, there would be no variance or volatility swaps and no VIX, the volatility index.

What Neuberger had done was to create a contract whose payoff was simply the natural logarithm of the spot price of the asset. If the underlying was the forward, instead of the spot, then the contract became a log forward, and it is the log forward that forms the backbone of a variance swap. The Log contract did not look like an option at all and yet retained all characteristics of an exotic option. Its payoff was a non-linear function of the underlying asset. As Neuberger explained to me in the same interview, "The obvious claim to look at is a power contract (that pays the n’th power of the terminal price). Of all power contracts, the one that looks most interesting is the limit case as the power goes to zero - which turns out to be the Log Contract."

The delta or the hedge ratio of a log forward is simply the reciprocal of the forward price of the asset. Such is the simplicity of this derivative contract. If the forward price of an asset is $1.00 then the investor trying to replicate the log forward would have to buy one forward contract. In the next period if the forward price moves to $1.20 then she should reduce the hedge to 1/1.20 or 0.8333 forward contracts. Apart from the simplicity of the hedge ratio what is striking about the log contract is that the delta of this contract, unlike a vanilla call or a put option, does not depend on the volatility of the asset. Any wrong estimation of the volatility – or the future volatility – will cause hedging errors in a call option. Not so in a log contract. The delta is no longer a complex function of time to maturity, future volatility and asset price. The delta is simply the inverse of the forward. As Neuberger further explained to me, "I had also been doing some work on hedging errors with Black-Scholes, where it is easy to show that the hedge error is roughly proportional to the integral of the gamma times the difference between realized and forecast variance. So the Log Contract with its constant gamma had obvious potential as a risk management instrument."

Further, the gamma and the vega of a log forward contract are also highly stable and predictable unlike options. This makes gamma hedging and volatility trading a much more tractable task for the investors. The Log contract is indeed a wonderful product, not just because it is so simple to understand in terms of its payoff and risks, but because underneath that veneer of simplicity lies a very complex and rich tapestry of mathematics that has made creation of volatility swaps, the VIX index and the transformation of volatility into an asset possible.

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