Pricing and risk management of options and exotics is always performed under some martingale measure associated to some numeraire, typically the risk-neutral measure associated with the bank account or the forward-neutral measure associated with some zero-coupon.
In addition, the parameters of pricing models are generally calibrated to implied market data, as opposed to being historically estimated.
This approach is universal in front office risk management, and generally adopted for CVA and other xVA too.
However, for other regulations like CCR, a conventional approach is simulate under a "historical" probability, where drifts and volatilities are "estimated", as opposed to be implied from tradeable market prices.
Opinions differ on whether the "correct" approach is implied, historical or a mix of both. In a context where regulations affect the capital and bottom line of investment banks in major manner, so that most bank quants spend most time on the implementation of complex regulatory computations, a serious discussion on this fundamental topic is overdue. We propose to collect and discuss arguments from both sides of the fence in this forum.
What about simulating according to the R measure, that is, the predictive measure ? And use as much info we have to get a sense of what R looks like ? The only reason I would see to use Q rather than P is if we think that Q is closer to R than P (which might be the case indeed...)
Dear Pr. Raphael Douady , thank you for joining the discussion.
Contrarily to the usual, mathematically correct definition of P and Q, in this context, "measures" refer to different ways of setting drift and volatility parameters on the simulation model.
If we call P a simple historical estimate (drift = average, vol = standard dev), Q the parameters implied from market prices of futures and options and R a "best" estimate of future realized drift and vol:
1) From a theoretical viewpoint, there must be strong restrictions on the model parameters. For consistency and non-arbitrage, drifts of asset prices must incorporate the same risk premia, one per risk factor.
2) These restrictions are naturally satisfied under Q, which, under the assumption of efficient markets, also provides the best estimate of future dynamics (meaning R = Q).
Does somebody know of reliable statistical research on how well implied parameters predict dynamics?
2) From a practical point of view, to let different institutions i apply their "best" prediction Ri for model parameters opens the door to arbitrary, inconsistent methods between different institutions.
If the regulator was to dictate a prediction method for the determination of R, what should it be?
Researchers, quants and risk managers, please leave your opinion, comments and suggestions.
Well, the different methods used in regulation and what usually used by institutions in practice is not a surprising fact, since their goals are different. Pricing models aim to accurately measure/estimate the price, while the regulation's main goal is usually either to ensure that the market does not collapse or to ensure fairness. So loosely speaking, pricing models try to be accurate in average cases, while regulations try to be accurate in the worst case. Choosing for a "correct" parameter without knowing the background is an impossible task. Define what you mean by "correct" but never assume that the other method is "incorrect".
Thank you Christopher Paulus Imanto , to define and discuss what "correct" simulations mean in the context of regulatory calculations is precisely the ambition of this discussion, and we are looking forward to reading your take on the question.
Query: which regulation are you refering to? In this context, I suppose you meant CCR as in Counterparty Credit Risk (which is not a regulation). Or did you mean the EU-regulation CRR as in Capital Requirement Regulation?
CCR (as in Counterparty Credit Risk) is an excellent example. It is a computation investment banks perform for regulatory as well as economic reasons. How CCR numbers are produced is a tightly regulated and monitored process. Banks models may be approved or disproved by the regulator.
The debate is, then, what are the "correct" parameters for the simulation model for CCR (and, yes, what does "correct" means in this context to start with) by scientific and economic standards?
With regard to simulation wrt a measure it is always necessary to validate/argue for your decision. The latter seems to be more easy if you use P.
If you use Q, for instance people might argue why you have chosen the domestic money market account and not T-Fwd for instance (See the papers by H. Stein).
Have you any ideas how to argue here?
However, also if you work under P the actual quantities depend on your estimation method... But not so much questions or objectives coming in here if you stick to some common method...
What are your opinions here?
Joerg Kienitz Nice to see you here.
" If you use Q, for instance people might argue why you have chosen the domestic money market account and not T-Fwd for instance (See the papers by H. Stein)." is very interesting.
Maybe you could invite H. Stein to leave a comment here too?
Leif Andersen also made a point (although I can't find the exact reference) that he could set the expected exposure wherever he wants changing the numeraire/measure of the simulation.
The reality is that (discounted) values are numeraire/measure independent but (un-discounted) expectations are not. I think that your comment is absolutely correct, but this does not mean that P is more correct than Q, it hints that we should probably compute exposures in today's dollar units, as discounted expectations (and perhaps divide by today's discount factor to the exposure date in the end) so as to make results numeraire independent.
Yes, you are right. I think it was first mentioned by L. Andersen and then H. Stein gave a recipe for constructing RN measures for getting any exposure you like...
Regulation, from Antoine Savine question. I refer, the problem in the financial markets as the non-common rule and fundamental. The key of regulation will be to clear on the "historical" the prospect of the trade or trading with the counterpart and the fit of the required position in the market. The historic CCR or CRR measures are necessary. Sufficiently, for the price of risk, the risk-neutral measure assert a stable money performance. However, the R measure for any market performance scenario is keen to ballance the predicted outcome.
I think that from the qualitative consideration of the non-unicity of pricing measures (a money market measure for each currency, multiple forward measures,...) suggests that any particular pricing measure can't be the universal 'correct' measure. For those interested we wrote up some thoughts and practicable approaches for dual measure modelling a couple of years ago:
https://ssrn.com/abstract=2831726.
I am not a 100% convinced with using the real world measure and advocating it as the measure of choice.
First, the real world measure also is a bit arbitrary since it depends on your estimation method and/or your choice of the market price of risk process.
Second, if one uses the standard arguments with the realized against the projected forward rates I wonder if this is a reason for choosing the real world measure or some flaw since this is not always the case. I remember there have been some trading around this issue and people argued that they make profits ... However, most of the time (FRB Bonds, etc.)
Third, is there a kind of quality check for your real world measure apart from backtesting. For risk neutral we have the calibration quality and the backtesting.
Looking fwd to your comments and I also would like to hear Antoine's thought about the topic...
I think rough volatility is an ideal framework to consolidate the two worlds. From a set of requirements (stylised facts) on the relationship between the P and Q measures we could derive a consistent model satisfying these requirements. What would the the three most important stylised facts be in your opinion?
The simulation for CCR type of applications has to occur under P a.ka. real world measure.
1- The validation is done through backtesting and backtesting makes sense only under P!
2- Realized data is deviating a lot from market implied one (USD.LIBOR.3M realised vs. projected; spot vs. fwd etc)
3. Of course, the challenge is the estimation / calibration of the drift or market price of risk and the answer (subjective ?) is in the data -- data driven models are far more powerful (over long term horizons (60Y in CCR etc)) that simple risk neutral dynamics (GBM for FX USDJPY or BRLUSD with a drift drift of r_d -r_f over 60Y is non sense! How to control the tails (by introducing mean reversion))?
4- People tended to mask the poor backtesting performance of Q-dynamics with higher volatilities (over-conservative and very penalizing to the business).
5- How would one capture the the distorted term structure of vol observed in the market over the last 10Y (using any of the conventional models ; HJM ; Cheyette etc...)
The answer is P and learning from realized data.
Thx,
Fahed
We often throw around notions like Q measure without justification. Q meausure is just a construct, a by-product of hedging away all the risks and arguing that the resulting portfolio must return the risk-free rate (quibble: which risk-free? Uncollateralized like Fed funds or EONIA or collateralised like SOFR?)
I think it is best to simulate under P and to simulate all the hedging activity under P on top of this. If the firm or position is fully hedged (ie risk-free) then this should be equivalent to simulating under Q.
It is probably never going to be fully hedged, in which case siulation under Q was never justified.
However care must be taken since if you try to optimize your position you may find yourself putting on many risk premia (eg should you hedge with a 1y swap or a 10y swap? In Q it possibly shouldnt matter but in P you might be putting on a carry trade).
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