Option Valuation in Illiquid and Incomplete Markets

# Is it wrong to use 'real world' probabilities for option valuation?


# Is it wrong to use 'real world' probabilities for option valuation?












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.

Any ideas?

## Answer by g g (score 4)

https://quant.stackexchange.com/a/10245

I think you need to go even one step further than vonjd went in his reply. If liquid trading of the underlying is not possible, not only the arbitrage argument underlying risk neutral pricing breaks down. In that case there is simply no reason why the prices of those two assets (the option and its underlying) should be related in any way at all. So in my opinion the real question is not risk neutral versus real world probabilities but whether the option has a uniquely defined price at all. These markets are called "incomplete" and one workaround (at least in theory) is to work with multiple risk-neutral measures all compatible with the available market data.

In practice all markets are incomplete and the solution is either to constantly recalibrate your model (i.e. to change the risk neutral measure all the time) and otherwise ignore the issue or to come up with non-probabilistic ways to arrive at prices. This process is actually the standard approach to pricing. It is called "negotiating" and you seem to allude to this in your second paragraph as well. Real world probabilities might enter negotiations among other things.

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/10234

I think the main point of your question lies in the assumption that you cannot (delta) hedge your option. When you cannot hedge the argument for risk-neutrality breaks down and you have to use real world probabilities.