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Say I hold an equity and I want to calculate the Value-at-Risk over some period. Would one calculate the Value-at-Risk of the equity under a risk-neutral (as in martingale) measure or under the initial measure (arbitragefree and complete market assumed)? If for instance the asset price is assumed to be a Brownian motion, Risk neutrality (possibly) changes the drift and thus has significant consequences on the Value-at-Risk. So what is appropriate? And why?

My guess is that one should indeed use the risk-neutral measure because the risk under the martingale measure is the ginuine risk that cannot be hedged. But I need a more throrough and formal explanation.

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I am not a believer in Value at Risk (VaR) and when I think about true exposure to firm-specific risk then you are in effect at the mercy of the specific equity return and hence risk. The classic theoretical model of assuming a BM driven process with specific drift is anyway a flawed concept no matter how you turn it. For equity linked or direct equity exposure other models, such as those that incorporate jumps, are way superior vs the traditional Ornstein–Uhlenbeck process, for example. My point is that you deliberately chose to expose yourself to the equity drift -> use it. –  Matt Wolf Nov 30 '13 at 15:04
    
Thanks for the comment, I am aware of these facts. I am not a fan of VaR either and I guess jump diffusions are great but one needs to be carefull and compare the extra cost of modelling with the benefit gained from employing such models. Anyways, I used these simple concepts to illustrate my problem which should be independent of the risk measure or process at hand. –  user1332526 Nov 30 '13 at 21:20
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The reason why one gets away pricing derivatives in the risk-neutral probability space (given certain conditions are met) is because the drift is already accounted for through the underlying and because of the hedge argument. This, however, is not the case when you take a straight exposure to cash equity. Of course you can always setup an equity pricing model where you end up with a stochastic differential in which the drift term "vaporized" but I question its usefulness. For pricing its perfectly fine as long as everyone agrees on the same way (such as B-S) but risk? –  Matt Wolf Dec 1 '13 at 4:12
    
...just adding one additional thought: Does it matter what your professor or even Ito thinks how risk should be defined if you adhere to theoretical models no matter how remote they are from reality if your trading desks lose tens of millions just because traders were not forced earlier to reduce risk. And all that because your "models" did not yet flag limit violations because they are so abstracted from reality. –  Matt Wolf Dec 1 '13 at 4:19
    
As for my part, I am not advocating conventional risk models! I am interested in robust and coherent methods in risk management. However research in this field is not advanced enought to make these models attractive for funds and banks. So by the end of the day I have to produce a risk estimate that will make people believe that they have a grasp of their exposure, when it's written on the wall that they don't! –  user1332526 Dec 1 '13 at 14:35
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I strongly recommend not assesing risk using the risk neutral measure. Doesn't this already sound like a contradiction (risk and risk-neutral)?

The risk neutral measure is there to derive prices (for derivatives e.g.) that fit to the prices of related contracts and traded assets. With "fit" I mean not allowing for arbitrage. For example if I calculate the forward price of a stock then I implicitely use the risk neutral probability measure and avoid arbitragy of trading the underlying spot.

The drift/expected value is related to interest rates and dividends due to arbitrage considerations - neither due to any risk considerations nor due to any idea where the spot price of the underlying could really be in the future.

If you call the other measure the physical measure then this is the one that should be used to measure risk. The problem is that it is by far not unique. For example if you measure risk by volatility (just as a starting point) then it is known that many volatility estimators exist. You can look at different observation periods, you can use weighted methods or e.g. GARCH.

Very often the expected value/drift of the asset whose risk you want to measure is assumed to be zero. For short periods of time this is very reasonable. Coming up with an expected value different from zero you risk to mix up your "trading idea" with your risk measure - which is in my mind not a good thing to do.

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Interested readers could have a look at the recent preprint: What is the best risk measure in practice? A comparison of standard measures by Susanne Emmer, Marie Kratz, Dirk Tasche. –  Richard Dec 6 '13 at 10:10
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