# Physical or Real-world Probability Measure

For counterparty credit risk, in particular, for potential future exposure computation, people use the real-world probability measure to evolve the underlying risk factors. My question is that whether there is only a single real-world probability measure for the whole market, or an individual one for each individual market, for example, one for the US market and another one for the European market.

For risk-neutral probability measure, we certainly have one for each individual market, and thus we can say domestic risk-neutral measure and foreign risk-neutral measure and so on. Is there any such thing as domestic physical measure or foreign physical measure? That is, for an equity basket with underliers from various markets, do we need the so-called quanto adjustment in domestic physical measure? Is there any references for discussions?

Acknowledgements: Thanks to every one for your participation. Your insights, ideas, or debates are very helpful for myself and many people here.

• I believe one would simply multiply everything by the exchange rate. – emcor Jun 11 '15 at 22:00

There is only one real world! You would use the measure that best describes all the markets together. Bear in mind that for credit you are really interested in portfolio effects. What is the potential credit risk we could have to a particular name? This depends on all the contracts we have them regardless of currency and they need to be modelled simultaneously.

• Can you elaborate on your statement because in its current form I would tend to disagree. There are many real world probability measures, in fact there are countless real world probability measures even for a single asset hence the interest of market participants to price in risk neutral terms. Real world probability measures are a function of utility. – Matthias Wolf Jun 12 '15 at 1:15
• I disagree -- the actual market you observe has some measure. Our job when doing PFE is to estimate what that measure is rather than to calibrate to a utility function. – Mark Joshi Jun 12 '15 at 2:26
• so then let me ask you, how do you discount your terminal payoff when you price a derivative under the real-world probability measure ? – Matthias Wolf Jun 12 '15 at 5:21
• every single random variable already has two expectations, one under the "real world" probability measure, one under a "new" probability measure. Radon Nikodym comes to mind and the same Radon Nikodym derivative can be applied to a whole derivative process. So maybe I misunderstood and we are talking on one hand about many different expectations vs a single probability measure. Certainly, it might be possible to not just derive a single additional "risk-free" probability measure but several such, but yes, only one "real world" probability measure exists... – Matthias Wolf Jun 12 '15 at 5:37
• i wouldn't price a derivative using the real world measure. For PFE we aren't pricing however. – Mark Joshi Jun 12 '15 at 10:36

The quanto adjustment is required to achieve the martingale property for the discounted payoff after currency transformation. Since you do not require discounted asset values to be martingales for risk measurement you do not need a quanto adjustment. But of course you need to include the distribution of future FX-rates in your modelling (which might be what emcor was alluding to). I see a danger of getting confused by loose language here. One needs to distinguish different events and random variables and using different measures.

The question of one or more Real-World measures is a very practical one. While most would agree that there is just one Real World measure describing a fair six-sided die, things are more complicated with more complex random variables. Different Real World measures are regularly a result of different calibrations. Obviously fitting a GARCH-model over 1 year or 10 years of past data will produce different distributions hence different measures for your target variables. From a practical risk management perspective I would encourage (and the regulator might require) testing different assumptions (i.e. different measures) of the real world.

As a side remark, if phrased carefully, there are no different Martingale measures if you price in different markets. The measure without quanto adjustment is simply not a Martingale measure for a cash stream with currency transformation. The reason why this distinction is more than hairsplitting, is that having two different Martingale measures for the same asset would mean two prices for the same asset. Which some people feel is undesirable, since it is ruled out in liquid complete markets.

• Fully concur that terminology is key here. It is easy to get confused about probability space, probability measures, driving brownian motions/ random variables. And I agree that for the purposes OP described all 3 assets (domestic money market account, stock, and foreign money market account) do not have to be martingales. – Matthias Wolf Jun 12 '15 at 11:11

I think the terminology often leads to confusion. Risk neutral pricing is essentially based on the idea of state prices or Arrow securities. One imagines or attempts to replicate securities that represent a state of the market. The price that is the consensus price of the market participants is the price of the Arrow security and this defines the state price density and this the risk neutral measure.

The reason different markets often require different risk neutral measures is that the markets are not interconnected enough so that the beliefs of one market's participants determine the state prices in the other market.

In the case of the real world, as mark Joshi said, there is only one real world. It doesn't make sense to have more than one real world measure unless you are alluding to Bayesian interpretations of probability.

• what about the example of a 6-sided vs 10-sided dice? Both reside in the real world, both in the same probability space, yet they both represent 2 very different probability measures, meaning, both represent their own real valued functions, defined on events in a shared probability space, and both satisfy measure properties. – Matthias Wolf Jun 15 '15 at 0:42
• @MattWolf I find it strange that you refer to two random variables on the same probability space as representing probability measures. You can use THE probability measure that defines the probability space to take probabilities of events involving the random variables. That doesn't mean random variables are probability measures per se. – Chan-Ho Suh Jun 15 '15 at 1:11
• that is not what I said. But to clear up the terminology confusion I seemingly have, you are basically saying both dice experiments are underlying one and the same identical real-valued probability function (because that is how your cited Wiki article defines a probability measure. – Matthias Wolf Jun 15 '15 at 2:01
• ...but admittedly the dice example is not a good one, I apologize for that. But it does not change my stance that in the same probability space you can have random variables that have two expectations, "one under the original probability measure $P$..., and the other under the new probability measure $\tilde P$.,... (Shreve, Stochastic Calculus for Finance II, 2004 Edition, P.210). Both probability measures live in the same probability space. – Matthias Wolf Jun 15 '15 at 2:31
• Shreve is not asserting you can have TWO probability measures as part of ONE probability space. If your point is that you can have multiple measures on the same measurable space, that's true and what Shreve is saying. – Chan-Ho Suh Jun 15 '15 at 3:46