SERVING THE QUANTITATIVE FINANCE COMMUNITY

• 1
• 2

bearish
Posts: 2849
Joined: February 3rd, 2011, 2:19 pm

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

Cuchulainn wrote:
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: 157
Joined: August 22nd, 2007, 3:46 pm

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

atlwjs wrote:
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: 157
Joined: August 22nd, 2007, 3:46 pm

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

Cuchulainn wrote:
bearish wrote:
Cuchulainn wrote:
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.

[size=100] boundary conditions
We took Dirichlet BC on truncated domain. ?[/size]
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: 2849
Joined: February 3rd, 2011, 2:19 pm

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

berndL wrote:
Cuchulainn wrote:
bearish wrote:

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.

[size=100] boundary conditions
We took Dirichlet BC on truncated domain. ?[/size]
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.