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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?

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"Is it wrong to use 'real world' probabilities for option valuation" ==> wouldn't say it's wrong, just that it's not "Neutral" anymore when you do this (but again, depending on what you mean by "Real World"?) –  Joanne Feb 13 at 15:32
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3 Answers

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.

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I agree: It is one of the dirty little secrets of the financial industry that in the end supply and demand again matter also for derivatives. +1 –  vonjd Feb 14 at 11:58
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This was my intuition (assuming I have understood correctly). –  quis est ille Feb 14 at 14:52
    
"Dirty little secret"? –  user4732 Feb 15 at 4:01
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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.

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@Downvoter: It is good practice here to give a reason for the downvote - Thank you! –  vonjd Feb 13 at 16:16
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Arnold and Crack (2000) is an extension of the binomial option pricing model that uses real world rather than risk neutral probabilities.

Our model, in both its one-period and multi-period forms, is a direct generalization of the Cox, Ross, and Rubinstein (CRR) binomial option pricing model (Cox et al., 1979). CRR do not give enough information to price options in the real world. Cox and Rubinstein (1985), however, do give enough information to deduce real-world option pricing (see discussion in our Appendix A.3), but the information is not used explicitly for that purpose ... We take their analysis one step further and generalize their model in the sense that options are priced under any discount rate.

Their proof seems sound. They do comment that real-world probabilities are not possible in continuous time, however, as the discrete nature of the model is essential for the inference of real world probabilities.

I scanned the 23 citations to this paper on Google Scholar and none appeared to contradict the basic premise that real-world probabilities can be used.

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The link does not work. –  vonjd Feb 13 at 14:22
    
Should be fixed –  Bob Jansen Feb 13 at 14:32
    
I think this ansatz does not clarify the issue. Have a look at p. 20 of the paper: "Using real-world probabilities, here does exist a single discount rate that discounts the FVs to give the correct present value, but it is a peculiar nonlinear weighted average of the path-dependent discount rates and is without direct and simple economic interpretation." So you'll get the same price either way (i.e. using risk-neutral probabilities or this strange mixture model) whereas when you cannot hedge and use real-world probabilities you will get a different price. –  vonjd Feb 13 at 14:41
    
@BobJansen: Thank you! –  vonjd Feb 13 at 14:44
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