Is it wrong to use 'real world' probabilities for option valuation, even when the market is not liquid enough to delta hedge the option?
My instinct is that it is wrong, because the time value of the option is determined by the cost of delta hedging. But if I am selling the option without hedging, then I do not have that cost. Indeed, we can imagine my company is split into two separate desks: Desk A which completely delta hedges with desk B, at an agreed reference rate, and Desk B, which trades the delta hedge only, with A. When the option expires, the profit/loss of desk A should reflect the tracking error of the hedging, caused by the difference between realised and implied volatility. But since A is directly trading with B, and since the positions are equal and opposite, whatever A loses, B will make, and conversely. So volatility is irrelevant.
Indeed there seems to be a bit of a paradox here. Suppose desk A significantly underestimates volatility, e.g. suppose it prices the option using 1% vol instead of 20%. Then A will lose massively in tracking error. But desk B will make it all back! So, oddly I as the owner of both desks am indifferent to the premium that A charges for the option. And so it makes no sense to value my unhedged option position. I can charge the counterparty a fee, and that can be whatever they are willing to pay. But from my point of view it makes no sense to use option pricing methodology. My concern about real-world probabilities are relevant only to risk management, i.e. reserving an appropriate amount of capital in the event that the market collapses and I am left with a large trading loss.