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oislah
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Posts: 13
Joined: March 4th, 2005, 8:40 am

EAD and SA-CCR

February 5th, 2019, 10:19 pm

Do you think capital rules are adequate?
Think.no more. With my colleagues Chris and Mourad, we have cracked an inverse problem. We offer both a very simple platform for the control of capital adequacy and a simple extendible framework for a transparent discussion of the calibration of parameters imposed by regulators...watch this space
https://papers.ssrn.com/sol3/papers.cfm ... id=3329538
 
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amike
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Joined: October 21st, 2005, 12:57 am

Re: EAD and SA-CCR

February 6th, 2019, 1:34 pm

Why would anyone think that?  I personally would have preferred a simpler SA-CCR -- admittedly probably with bigger flaws -- but with a somewhat flexible  approach to its use as a floor.  But that's me.

A question about your `3-factor' model: equation (3.11) combined with (3.13).  I assume that your [$]dW^i_t[$] are uncorrelated.

Then in [$]t<1[$] there is a singe process active, and:
[$]
dr_t=(\phi(t)-a r_t)dt+\sigma dW_t^1
[$]
In [$]1\le t < 5[$] there are two:
[$]
dr_t=(\phi(t)-a r_t)dt+\sigma \left( \rho_1 dW_t^1 +\sqrt{1-\rho_1^2}dW^2_t \right)
[$]
But of course, the sum of two Brownian motions is another brownian motion, and since that combination is standardized I can introduce the equivalent process:
[$]
dZ_1=\begin{cases}
dW_t^1 & t<1 \\
\rho_1 dW_t^1 +\sqrt{1-\rho_1^2}dW^2_y  & 1\le t < 5 \\
\end{cases}
[$]
and recognize that up to [$]t=5[$] the process is just a one factor model with constant parameters in disguise:
[$]
dr_t=(\phi(t)-a r_t)dt+\sigma dZ_t
[$]
(I assume that I could continue this into the next interval, but there is at least one typo in your equation (3.13), so I don't want to get distracted by that.)

I don't see how this model (which appears to be equivalent to a 1-factor model) can lead to the correlation structure you describe.  Probably I don't understand something about your construction...
 
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oislah
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Joined: March 4th, 2005, 8:40 am

Re: EAD and SA-CCR

February 6th, 2019, 4:44 pm

Hi Mike
On the first point simpler approaches have shown their limits and an excess of conservatism in parameters attempts to mask their flaws. We advocate transparency and comparability with IMM but still remaining in a simple approach just slightly more complex than current SA-CCR.
I like your question because I am sure it is going to get asked a lot by Quants ! The short answer is: this is how a multi-factor Affine short rate model works.  I will elaborate though.
To convince yourself that the correlation structure for zero coupon bond is recovered at time 0 is to start from the zero coupon bond dynamics given by equation (3.12) and compute correlations at time 0 between zero coupon bonds belonging to different maturity buckets.  Once this is checked. You derive from the bonds processes the processes for the instantaneous forward  f(t,T) processes for all times t and maturities T. Setting T=t gives you back the short rate process as given by (3.11)
Reciprocally one can start from (3.11) and derive the processes (3.12) for the zero coupon bonds. This is a bit more involved but can be done by writing what is the analytical solution of the bond pricing PDE given that we are in a multi-factor affine
These are classical Gaussian HJM arguments I believe.
 
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amike
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Joined: October 21st, 2005, 12:57 am

Re: EAD and SA-CCR

February 6th, 2019, 7:56 pm

I think maybe you missed my point. 

Starting from just the short rate process you provided (3.11) and the alpha's (3.13) it looks to me like the process (up to t=5 at least) does not depend on the correlation [$]\rho_1[$] at all.  So it's hard to see how you can arrive at the correlation structure you claim.

If I find some time I'll look into this some more...
 
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oislah
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Joined: March 4th, 2005, 8:40 am

Re: EAD and SA-CCR

February 6th, 2019, 10:08 pm

You need to ignore equations 3.11 as they are not used in the paper. Only zero coupon bond equations 3.12 are used and that is a 3 factor gaussian HJM with the correct correlation structure.
There is a typo in 3.11 and they should have been 3 processes r_{i} that equation is there just for completeness but we will fix it and write correct equations..thanks you very much for pointing this out
 
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amike
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Joined: October 21st, 2005, 12:57 am

Re: EAD and SA-CCR

February 6th, 2019, 10:29 pm

ah, ok, thanks.
 
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oislah
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Joined: March 4th, 2005, 8:40 am

Re: EAD and SA-CCR

February 8th, 2019, 5:03 pm

We have updated the model section .  The naming of the modelling framework was kind of wrong and the market model framework is a better set-up than the  continuous curve model (HJM types/Multi-factor short rate)which is a bit akward given the correlation structure and likely to lead to Non-Markovian processes(but doable). This is just for completeness and has no impact on the rest of the paper and is there just to give a consistent justification of the modelling framework.
@amike : thanks for the comment. This was very useful