The Log Forward Contract – A Beautiful Product
Jul 06, 2005
We would like to suggest to sell side structurers and salespersons to revisit one of the unique products that was synthesized in the mid-nineties, the log forward contract.
The log forward – the natural logarithm of any forward contract – is quite a unique product though it never gained any popularity on its own. That is, as a unique and separate product there never has been and is a market for it. The reason for this eludes us because we think it is an amazing product. However, the log contract in some form, such as the natural logarithm of a spot or strike price, is widely used by quants to price instruments such a volatility and variance swaps, two very popular products.
A log forward is a simple contract that has a payoff like:
A log forward can be considered an exotic option because it is a nonlinear function of the underlying asset (the spot or the forward) and hence can be regarded as an option. The uniqueness of a log forward contract, or in general any log contract, is that though it is not a pure option it never loses its option like character, something that could of great value to volatility traders and hedgers. An option retains its option like character only when it is at-the-money (ATM). As an option moves away from the strike and becomes in the money it starts to behave like the underlying and a deep in the money option is nothing but the underlying asset. On the other hand as the option move away towards out of the money zone it again loses its option like character and deep out of the money options become worthless assets (though of course sometimes, these deep out of the money options could display strong volatility skews). Therefore, as the volatility moves the spot away from the strike an option starts to lose its option like characteristics.
On the other hand a log forward contract displays option like characteristics at all times, regardless of the movement of the spot.
The partial differential equation that holds true for a log contract is:
This valuation shows that the log forward contract is a function of the volatility of the asset. The delta of the log forward is a simply the reciprocal of the forward price and is independent of the forecast level of volatility. This is in sharp contrast to an option delta which is a fairly complex function of the asset price, time to maturity and the asset volatility.
The same is true for gamma and vega of a log forward contract as can be seen from the valuation equation above. Also, the gamma and the vega of a log forward contract are highly stable and predictable as opposed to options. This makes gamma hedging and volatility trading a much more tractable task for the investors. These features of the greeks make log forward contract ideal for volatility trading (pure bets on volatility) as well as an instrument to hedge a portfolio of options because the log forward contract offers a straightforward way of managing vega.
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