Physical Probability Measures for Cross-Market Credit Risk

# Physical or Real-world Probability Measure


# 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.

## Answer by Mark Joshi (score 6, accepted)

https://quant.stackexchange.com/a/18312

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.

## Answer by g g (score 4)

https://quant.stackexchange.com/a/18317

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.

## Answer by Chan-Ho Suh (score 0)

https://quant.stackexchange.com/a/18343

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.