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krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Martingale Correction

Am I allowed to do the following to exactly match the term structure of rates?

P( t, T)_coorected  = P( t, T)_simulated - P( t, T)_simulated_mean + P( t, T)_market

The expectation of the above expression  will be equal to the market value of P(t, T)

If this is the case, then this approach would be model (and product), independent.

thanks,

bearish
Posts: 6396
Joined: February 3rd, 2011, 2:19 pm

### Re: Martingale Correction

Your notation leaves a little bit to be desired, but if I make a reasonable guess as to what you are suggesting doing, I think it’s OK. Not great, but probably OK. There are thick yellow books and innumerable papers dedicated to the topic of efficient Monte Carlo simulation. Your approach may be mentioned in the intro to the first chapter on “variance reduction”, also known as “cheating”, but it’s almost certainly better than doing nothing. A really good idea would be to test it out in your particular setting. Pick some non-trivial instruments that you want to analyze and see how your relatively simple control variate scheme performs relative to, say, increasing your sample size by a few orders of magnitude or, if available, a closed form solution or (just to keep somebody around here happy) the output of a PDE solver. If you haven’t tried using low discrepancy sequences (Sobol, Halton, etc.), you probably should.

hs16022021
Posts: 2
Joined: February 18th, 2021, 4:52 pm

### Re: Martingale Correction

My two cents - its a practical solution which I've seen implemented before - though not sure its arbitrage free. However, it would be worth invetigating whether there is a systematic bias in your simulation which is causing the problem, and if so, potentially correct that bias "at source", rather than the output. I'm not sure what model you are using, but, for example, in a LMM model there are various ways (predictor-corrector, picard iteration, BB, choice of numeraire etc) that can help in correcting such issues. If the error you are seeing is unbiased and due to numerical noise, for example, errors in the random variates you are drawing, then I agree with bearish that low-discrepancy sequences should be considered if you are not already doing so. And for Sobol, having a BB is a must.

krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Re: Martingale Correction

Thanks, Bearish and HS.

I am working on G2++ and the curve that I am using in calibration (to calculate swap rates to price swaptions) is different from the curve that I have to simulate. Insurance analytics problem; we have to simulate the EIOPA curve.

So, it is basically a moment matching problem (first moment). I have tried QMC, it improves the result. but not good enough to price liabilities at T = 30 years. there is an error like 3% in bond prices at T= 60Y

thanks,
K

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Martingale Correction

My two cents too: this can work only for martingale processes. I don't know G2++, but it seems that it is not a martingale process ? Could this explain the 3% diff ?
Considering Sobol or Halton, I think we can provide sequences that outperforms them.

Cuchulainn
Posts: 64338
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: Martingale Correction

Brigo and Mercurio 2006
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

bearish
Posts: 6396
Joined: February 3rd, 2011, 2:19 pm

### Re: Martingale Correction

My two cents too: this can work only for martingale processes. I don't know G2++, but it seems that it is not a martingale process ? Could this explain the 3% diff ?
Considering Sobol or Halton, I think we can provide sequences that outperforms them.
There ought to be a martingale there somewhere. E.g., pick the zero coupon bond with the latest maturity date that you care about and look at the ratio of other prices to that. That will be a martingale in the relevant forward measure, independent of any details of the model dynamics, and this measure is normally fairly convenient to work with.
I was just singling out Halton and Sobol because they’re probably the best known. And when used with some care they’re not usually that bad, although I agree that you can undoubtedly do better.

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Martingale Correction

There ought to be a martingale there somewhere. E.g., pick the zero coupon bond with the latest maturity date that you care about and look at the ratio of other prices to that. That will be a martingale in the relevant forward measure, independent of any details of the model dynamics, and this measure is normally fairly convenient to work with.
I agree to say that in every stochastic process there is a martingale part and a drift (hyperbolic) part.
Last edited by JohnLeM on March 2nd, 2021, 8:09 am, edited 1 time in total.

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Martingale Correction

My two cents too: this can work only for martingale processes. I don't know G2++, but it seems that it is not a martingale process ? Could this explain the 3% diff ?
Considering Sobol or Halton, I think we can provide sequences that outperforms them.
There ought to be a martingale there somewhere. E.g., pick the zero coupon bond with the latest maturity date that you care about and look at the ratio of other prices to that. That will be a martingale in the relevant forward measure, independent of any details of the model dynamics, and this measure is normally fairly convenient to work with.
I was just singling out Halton and Sobol because they’re probably the best known. And when used with some care they’re not usually that bad, although I agree that you can undoubtedly do better.
I agree to say that in every stochastic process there is a martingale part and a drift (hyperbolic) part. However, and I think that it is the question of the author of this post, it is not clear to me how to separate both parts just from price observations. Indeed, the author provides a moment matching method,  that should be able to match any price of bonds, being martingale in the relevant forward measure, exactly what you said. His method should work for that purpose, but might fail to match swaption prices.
Passing by, I found this alternative reference expliciting the calibration with G2++ from bonds or swaptions prices (Brigo is almost surely a good one too).
Concerning Halton and Sobol I agree, they are known and provide already good results (for low dimensions, take care). I was simply advertizing that we can propose better alternatives if more accurate results are needed. In the same spirit, we can also provide a general method to calibrate from any set of prices of derivatives, be they bonds or swaptions or whatever. The calibrated process is computed as "as close as possible" to the initial process (here a G2++) while constrained to reproduce this set of prices. Both methods works, but they require a more sophisticated approach.

Cuchulainn
Posts: 64338
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: Martingale Correction

The question is now: will you do it?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

JohnLeM
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: Martingale Correction

The question is now: will you do it?
@Cuchulainn I suppose that this question is adressed to me (since the author is probably unfamiliar with the method I quoted) ? Implementation will be a quite big amount of work : it means testing a new process, the G2++, testing sequences for this process, testing the calibration procedure with swaptions and bonds under this process, documentation etc etc..I think this is a full month work for me. Thus the answer is no, I won't do it, unless there is a real gain to be expected for insurances with such a sophisticated approach. What are the authors thoughts here ?

bearish
Posts: 6396
Joined: February 3rd, 2011, 2:19 pm

### Re: Martingale Correction

Well, when doing a project that is ultimately motivated by regulatory considerations, it would be interesting to know what the regulators demand, as well as what they will not accept. That’s outside of my scope.

krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Re: Martingale Correction

thanks, Bearish and JohnLem!

In my opinion, I must work always under the risk-adjusted measure for this particular problem. Please consider that the goal is to match the entire yield curve and not only a specific maturity T.

Let me put it again in more detail: I am calibrating the G2++ model to ATM swaption prices, I get ATM vol matrix from some data provider, I also have a discounting curve, the ATM vols in the matrix are linked to this discounting curve (correct? there has to be a unique curve for that particular swaption vol matrix to calculate implied Normal vols).

I can do the calibration using some approximation on swaption prices. In the simulation part, the curve simulated for insurance analytics (liability pricing) is the EIOPA curve and it is different from the curve used in the calibration procedure in the first place. The deterministic part of the G2++ model depends on the five parameters of the calibrated model. So, the deterministic part of the calibrated model will match the initial discounting curve only.

To match the EIOPA curve exactly, I was thinking to introduce a moment-matching approach, and instead of making adjustments in the two factors (x and y), I was thinking to play around with the deterministic part of the bond price P(t, T), but in this case, I would lose some information on the swaption prices, as mentioned by JohnLem.

I don't see any other way to match the bond prices. I have to simulate the term-structure of rates for the next 50 years at monthly frequency.

thanks,
K

krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Re: Martingale Correction

Using quasi-MC techniques won't make any big difference, I have already tried the Halton sequence. that would fix some variance issues with the factors x and y, but the real problem lies in the deterministic part of the short-rate, which is actually driving the short-rate process.

krs
Topic Author
Posts: 20
Joined: August 20th, 2019, 12:40 pm

### Re: Martingale Correction

solved it...

I work on both the curves for model calibration:

part1: squared error in swaption prices.

part2: matching P(t, t+1) from the analytical and the model formula

Of course, I lose some information on the swaption vol part, but this seems to be a reasonable approach to deal with the two curves.