Revisiting our post on calibration, there’s a fairly stupid example, that at least illustrates the opportunity cost of letting other people analyse your data for you (and callibrating to their prices rather than making your own).

Suppose there is a share trading with a return of 10% and a volatility of 0. A common approach to calculating volatility ignores the underlyings mean return (assumes a mean of 0). If the volatility of the underlying is calculated, ignoring the return, it equates to a vol of 31%.

Suppose the above method is the method used by the market. One can trade a call at 31%, assuming an underlying spot of ZAR 10, a call struck at 10 will cost ZAR 1.256 (assuming risk free rate of 0). The return will be realised, and so one could trade this option, hedge it out as an option and make profit on whatever premium one is able to charge.

Alternatively, we know the return will be 10%, so one could trade the call as a forward knowing that the spot price will be ZAR 11 at the end of the first year. Again assuming a risk free rate of 0, this contract would be worth ZAR 1. Again you could trade and hedge the contract as a forward, and are guaranteed to make any profit you build into the contract premium.

But,  what happens if the entire market is essentially treating the contract as a forward, but someone is unaware of this and see only a contract price of ZAR 1? This equates to an implied vol of 25% (what they’d actually see, but never mind). So, they “calibrate” their options price/trading strategy to the market, sell at 25% and try to hedge out. They will lose money in the process.

These two prices are determined by the same underlying distribution, but differing approaches to hedging the contract allow for different prices. It may be a silly example, but it demonstrates that prices can be context specific, and it is important to also relate calibrated (assuming one must calibrate) prices back to a supporting hedge strategy. It also shows that through the use of clever approaches to hedging and  data analysis it may be possible to outprice the market.

Incidentally, because the risk free rate is 0%, one could also value the contract at 0, but maybe best not to open up that can of worms.