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bearish
Posts: 3956
Joined: February 3rd, 2011, 2:19 pm

### Re: Shifted log normal short rate model vs hull white short rate model

The first order problem with models that lack a relatively simple mapping from state variables to discount factors arises when you have a short dated option on a long dated bond or swap.

What kinds of problem? model, numerical, time scales?(?)
To take an example that I will admit bordering on the silly - let's say you want to value a three month option on a futures contract where the underlying is a set of 25-30 year coupon bonds, and where you want to consistently value the underlying bonds, the futures contract (via a cheapest-to-deliver embedded option) and the option contract. In a classic lognormal short rate model you need to build a 30-year lattice to value the bonds, but you would also like a relatively fine spatial resolution at the three month point to get reasonable precision in the option pricing part. I am not saying you can't do it, just that it is a lot more demanding than building a three month lattice and determining the terminal bond values (and thus futures price and option payoff) analytically, as e.g. in a Gaussian model.

berndL
Posts: 165
Joined: August 22nd, 2007, 3:46 pm

### Re: Shifted log normal short rate model vs hull white short rate model

At my shop, we are running Hull White short rate model for mortgage risk management purpose.  So no LMM.   I ran some tests by switching to shifted lognormal short rate model.   The risk analytic results look pretty reasonable.  Just wondering why shifted log normal short rate model never gains industry popularity.  Thanks for your insight.
Just a thought: short rate is not a tradable asset. So thinking of lets say zero bond prices as the natural asset in rates it appears natural to have some normal model for the  short rate.
Or another thought. The displaced diffusion seems to be first mentioned (only my confined knowledge of course) in the paper of rubinstein about displaced diffusion (you can find it easily on the net). Here it servers to model the rather special issue of viewing the value of an investment in  a firm as an investement in a risky asset and a riskless asset. Thus modeling the capital structure of the firm. Now how would this translate to a short rate. I would say: not at all.

berndL
Posts: 165
Joined: August 22nd, 2007, 3:46 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Sorry, yes! It's Monday.
So, you can use the closed to check the numerical solution; what about vice versa?
e.g. is there a closed solution for a  Bermudan HW2 (Hull-White 2 factor) bond?

Problem with NR is that the guess must be 'close' to the real solution.

Is it so that some quants eschew PDE/FDM for fixed-income models? There's no inherent reason for doing so? I'm just wondering how efficient the bespoke KL solution is? e.g. computing SDE paths for MC entails truncating a Fourier series + computing $sin(t)$ multiple times.
Aside from degenerate cases there is obviously no closed form solution for “Bermudan Bonds” (I am assuming this means a callable bond) in any non-trivial model. I am perfectly happy to use PDE methods to solve interest rate valuation problems but much prefer models where the boundary conditions can be computed from the state variables without too much pain.

The first order problem with models that lack a relatively simple mapping from state variables to discount factors arises when you have a short dated option on a long dated bond or swap.
Are these the same terms as in the pdf?
One of my quant students solved the Bermudan PDE using about 4 different FD methods in about 2 months before he heads off for UCB (PDE savvy mandatory to get in). I did a few FDM as well for comparison. The schemes are super fast. Next is to compute sensitivities.
This PDE is benign (compared to Heston etc.) because PDE coefficients are constant.

boundary conditions
We took Dirichlet BC on truncated domain. ?

Other cases give strange BCs,
Hi Daniel,

seems you truncated the domain at 0. If a normal model why did you do that? Did you estimate the error compared to moving the truncation point into the negative short rate domain?

bearish
Posts: 3956
Joined: February 3rd, 2011, 2:19 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Aside from degenerate cases there is obviously no closed form solution for “Bermudan Bonds” (I am assuming this means a callable bond) in any non-trivial model. I am perfectly happy to use PDE methods to solve interest rate valuation problems but much prefer models where the boundary conditions can be computed from the state variables without too much pain.

The first order problem with models that lack a relatively simple mapping from state variables to discount factors arises when you have a short dated option on a long dated bond or swap.
Are these the same terms as in the pdf?
One of my quant students solved the Bermudan PDE using about 4 different FD methods in about 2 months before he heads off for UCB (PDE savvy mandatory to get in). I did a few FDM as well for comparison. The schemes are super fast. Next is to compute sensitivities.
This PDE is benign (compared to Heston etc.) because PDE coefficients are constant.

boundary conditions
We took Dirichlet BC on truncated domain. ?

Other cases give strange BCs,
Hi Daniel,

seems you truncated the domain at 0. If a normal model why did you do that? Did you estimate the error compared to moving the truncation point into the negative short rate domain?
As an exercise in comparing numerical PDE methods it may not matter much, but truncating both r and u at zero will certainly not produce results consistent with the 2-factor HW model, which is just a particular (and not terribly useful) way to write down a slightly restricted version of the generic Gaussian 2-factor model with 2 state variables. One can attempt a version of the model where rates are floored at zero (a la Black 1995), but this loses all analytical tractability and does not correspond to P1=1. A simpler and cleaner way to get a 2-factor model with positive rates is Cairns (2004) or a square root affine type process.

Cuchulainn
Posts: 57319
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Shifted log normal short rate model vs hull white short rate model

As an exercise in comparing numerical PDE methods it may not matter much, but truncating both r and u at zero will certainly not produce results consistent with the 2-factor HW model, which is just a particular (and not terribly useful) way to write down a slightly restricted version of the generic Gaussian 2-factor model with 2 state variables. One can attempt a version of the model where rates are floored at zero (a la Black 1995), but this loses all analytical tractability and does not correspond to P1=1. A simpler and cleaner way to get a 2-factor model with positive rates is Cairns (2004) or a square root affine type process.

edit: I see now that the above was a response to berndl,

When you say PDE, do you mean  PDE/FDM or lattices which are 'kinda' FDM schemes?

We see no need to truncate at zero (how did this question arise?) but far-field truncation is needed in PDE model (plan B is to transform to [0,1]^2)

but this loses all analytical tractability and does not correspond to P1=1.
Are you saying BC P1 = 1 is correct or incorrect?
Last edited by Cuchulainn on March 19th, 2018, 9:34 am, edited 1 time in total.

Cuchulainn
Posts: 57319
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Shifted log normal short rate model vs hull white short rate model

Aside from degenerate cases there is obviously no closed form solution for “Bermudan Bonds” (I am assuming this means a callable bond) in any non-trivial model. I am perfectly happy to use PDE methods to solve interest rate valuation problems but much prefer models where the boundary conditions can be computed from the state variables without too much pain.

The first order problem with models that lack a relatively simple mapping from state variables to discount factors arises when you have a short dated option on a long dated bond or swap.
Are these the same terms as in the pdf?
One of my quant students solved the Bermudan PDE using about 4 different FD methods in about 2 months before he heads off for UCB (PDE savvy mandatory to get in). I did a few FDM as well for comparison. The schemes are super fast. Next is to compute sensitivities.
This PDE is benign (compared to Heston etc.) because PDE coefficients are constant.

boundary conditions
We took Dirichlet BC on truncated domain. ?

Other cases give strange BCs,
Hi Daniel,

seems you truncated the domain at 0. If a normal model why did you do that? Did you estimate the error compared to moving the truncation point into the negative short rate domain?
HI Bernd,
No truncation at 0! Please see my post to @bearish. See the pdf again for the way we pose the PDE.
We use rmax = 0.2 and umax = 0.1 for the far fields only which seems to be OK (in truth, we are using 'heuristic/pragmatic' thinking for the truncation part:))

The tentative conclusion is that our FDM models (exact HW2, ADE*, MOL and Soviet Splitting) all give the same results. Now just the Bermudan part. Let's see.
Attachments
untitled.pdf
Last edited by Cuchulainn on March 19th, 2018, 9:28 am, edited 1 time in total.

berndL
Posts: 165
Joined: August 22nd, 2007, 3:46 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Are these the same terms as in the pdf?
One of my quant students solved the Bermudan PDE using about 4 different FD methods in about 2 months before he heads off for UCB (PDE savvy mandatory to get in). I did a few FDM as well for comparison. The schemes are super fast. Next is to compute sensitivities.
This PDE is benign (compared to Heston etc.) because PDE coefficients are constant.

boundary conditions
We took Dirichlet BC on truncated domain. ?

Other cases give strange BCs,
Hi Daniel,

seems you truncated the domain at 0. If a normal model why did you do that? Did you estimate the error compared to moving the truncation point into the negative short rate domain?
HI Bernd,
No truncation at 0! Please see my post to @bearish. See the pdf again for the way we pose the PDE.
We use rmax = 0.2 and umax = 0.1 for the far fields only which seems to be OK (in truth, we are using 'heuristic/pragmatic' thinking:))
Hi,
i took a look again at your attached pdf. You are saying: "Numerically the relevant domain is (r,u,tau) = {[0,r_max],[0,u_max],[0,T]} = Omega". So you consider [0,r_max] as relevant? This was the point where i thought you are only considering non negative  short rates. Am i missing something? Cant see this atm.

Cuchulainn
Posts: 57319
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Shifted log normal short rate model vs hull white short rate model

BTW, we have not done domains for negative r, u. Is that useful/necessary?

berndL
Posts: 165
Joined: August 22nd, 2007, 3:46 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Are these the same terms as in the pdf?
One of my quant students solved the Bermudan PDE using about 4 different FD methods in about 2 months before he heads off for UCB (PDE savvy mandatory to get in). I did a few FDM as well for comparison. The schemes are super fast. Next is to compute sensitivities.
This PDE is benign (compared to Heston etc.) because PDE coefficients are constant.

boundary conditions
We took Dirichlet BC on truncated domain. ?

Other cases give strange BCs,
Hi Daniel,

seems you truncated the domain at 0. If a normal model why did you do that? Did you estimate the error compared to moving the truncation point into the negative short rate domain?
HI Bernd,
No truncation at 0! Please see my post to @bearish. See the pdf again for the way we pose the PDE.
We use rmax = 0.2 and umax = 0.1 for the far fields only which seems to be OK (in truth, we are using 'heuristic/pragmatic' thinking for the truncation part:))

The tentative conclusion is that our FDM models (exact HW2, ADE*, MOL and Soviet Splitting) all give the same results. Now just the Bermudan part. Let's see.
Hi, apart from my question on you pdf. Your arguments given to bearish are true. If you allow for example the short rate to become infinitly negativ then there is no easy boundary condition for - inf.
I tried out the PDE given in PWQF (Vol2) for a defaultable Zero Bond under the assumption that default will not occur before payment. Found it interesting. In PWQF no boundary conditions where given and the model war apperantly a normal one (looking at the sde for the default spread p there). I solved the equation (btw with mol) but truncated at 0 to. Well in fact i did a domain transformation from x -> r mapping [0,1] to [0,+inf].
so i did the same thing and i didnt analyze what happens if i truncate at negative short rates (or in this case negative default spreads).
To my (or maybe out both) excuses i like to say the choice of the truncation point in PDE Methods for normal ir models is mostly treated heuritically in what i found in papers. A recent paper (dont have the link atm) just did a domain transformation r->ln(ar+b) without commenting this much. This results of course also in domain truncation. But this time somewhere in the negative r domain.
And i remember a thread here on the site where one of the conclusion was afaik "Just choose an empirical truncation point slightly negative".
So i kindof wondered if you got access to some new arguments....

bearish
Posts: 3956
Joined: February 3rd, 2011, 2:19 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Lots of noise here. My comments was directed at the first attachment where you claim to truncate the domains for r and u at 0, and setting the corresponding bond prices P1 and P2 equal to 1. That is entirely inconsistent with a Gaussian interest rate model.

Cuchulainn
Posts: 57319
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Shifted log normal short rate model vs hull white short rate model

Lots of noise here. My comments was directed at the first attachment where you claim to truncate the domains for r and u at 0, and setting the corresponding bond prices P1 and P2 equal to 1. That is entirely inconsistent with a Gaussian interest rate model.
I see. So you are saying the PDE should be defined on $(-\infty, \infty)^2$? I see examples where $r_{min} = -0.5$. Is that it, for example?

A popular BC is  $\frac{\partial^2 V}{\partial r^2} =0$ but something simpler would be nicer.

// this reminds me a bit of BCs for CIR depending on whether Feller is satisfied or not. But the elliptic part is much different here.

berndL
Posts: 165
Joined: August 22nd, 2007, 3:46 pm

### Re: Shifted log normal short rate model vs hull white short rate model

Lots of noise here. My comments was directed at the first attachment where you claim to truncate the domains for r and u at 0, and setting the corresponding bond prices P1 and P2 equal to 1. That is entirely inconsistent with a Gaussian interest rate model.
I see. So you are saying the PDE should be defined on $(-\infty, \infty)^2$? I see examples where $r_{min} = -0.5$. Is that it, for example?

A popular BC is  $\frac{\partial^2 V}{\partial r^2} =0$ but something simpler would be nicer.
Hi Daniel,
in theory yes. (-inf,+inf).
I did [0,+inf) + domain transformation but had a bad conscience in doing so.
Given todays rates EUR-EONIA for example are negative its hard to argue why rates should stop at 0.
But i have no solution. So far i just truncated at 0 i admit

bearish
Posts: 3956
Joined: February 3rd, 2011, 2:19 pm

### Re: Shifted log normal short rate model vs hull white short rate model

I would pick a domain wide enough to cover +/-4 standard deviations around the means for r and u. I would also emphatically not use the lattice to solve for zero coupon (or other bullet) bond prices, unless as a test or exercise, since you have these in closed form as a function of u, r, and t, and you can use them as needed in the callable bond valuation. The idea of setting zero coupon bond prices equal to 1 when r hits zero implies that zero is an absorbing barrier.

Cuchulainn
Posts: 57319
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Shifted log normal short rate model vs hull white short rate model

I would pick a domain wide enough to cover +/-4 standard deviations around the means for r and u. I would also emphatically not use the lattice to solve for zero coupon (or other bullet) bond prices, unless as a test or exercise, since you have these in closed form as a function of u, r, and t, and you can use them as needed in the callable bond valuation. The idea of setting zero coupon bond prices equal to 1 when r hits zero implies that zero is an absorbing barrier.
It was a slight oversight on our part to truncate at 0 but the PDE approach is flexible enough for r < 0 IMO. Let's see. Even in the 1 factor case they use this trick.
https://cs.uwaterloo.ca/~paforsyt/numcall.pdf
Let's see how this pans out.
Another successful approach for equity PDE (see the article with Alan and Paul) is to transform the infinite axis to $(-1,1)$ using the transformation $y = tanh(ax)$, or  $y = coth(ax)$ where $a$ is a scale factor. tbd
This approach works for CIR.

// Lattices are indeed off-limits here. It's mathematically and numerically too intractible IMHO.

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: Shifted log normal short rate model vs hull white short rate model

Berndl,
A few "thinking out loud" questions:

I realised that I had a PDE model for HW1 by transforming the real line (positive, negative short rate) to (-1,1). The PDE is fine and  I hide the pesky reaction term by using $P = Pnew \exp(-rt)$. AFAIR the values were OK but will check. The transformed PDE is "Fichera-degenerate" at {-1,1} so BCs are simple (BTW I take Pnew(-1) = 1 and Pnew(1) = 0, is that also financially OK?).

What do you think of this approach? The transform is $y = tanh(x)$ as always.
//
I suppose the two-factor PDE(r,u) will be transformed in a similar way?

I must say truncation is adhoc and scary; how much do we need to truncate?