Serving the Quantitative Finance Community

arr164
Topic Author
Posts: 8
Joined: February 7th, 2018, 9:49 pm

### perpetual callable bonds with stochastic yield

I would like to know how the FDM models gets adapted when bonds have very long maturity, e.g. 150Y. There are a number of fixed-to-float perpetuals on the market that I would like to be able to value.
One valuation approach is to consider the price to yield formula and compute the lowest yield. That will produce a workout date for which I can compute all the risks. Very often it is the first call date and essentially shortest duration possible.
Another approach is to assume some stochastic model for the yield. But that allows every exercise date to have a possibility of a call and as a result, even call dates that are 150Y away do have a chance of a call as on the FDM grid I have a range of values and for some nodes I am in the money. As a result, the workout date doesn't have to coincide with the workout date of the price to yield and can make a big impact on the risk. Even though the first date is the mode for the distribution for call dates, it is not one single call date.
I wonder if there is a special way of handling those fixed-to-float perpetuals when pricing with finite differences so the risk is more consistent? I see big risk numbers differences and I think this is due to the very different nature of the models where the latter assumes the distribution for callability and price to yield returns one fixed value.

Cuchulainn
Posts: 65000
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: perpetual callable bonds with stochastic yield

FDM approximate PDE, yes?
what's the PDE? there are several.
FDM is the solution, PDE is the problem.

//
the old perp was in 1648 in Holland for dike repair etc. Last week the dikes caved in Limburg, Belgium and Germany

https://en.wikipedia.org/wiki/Water_boa ... ter_boards

The Yale connection
https://rechtsgeschiedenis.wordpress.co ... -years-on/

I pay several hundred Euro per year to the water board to keep dry feet, much more than coupon to Yale.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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

arr164
Topic Author
Posts: 8
Joined: February 7th, 2018, 9:49 pm

### Re: perpetual callable bonds with stochastic yield

yes, very simple model where I end up with a backward heat equation(yield-vol model where yield is a lognormal process). I use Crank Nicolson to solve it and it is working well but conceptually they are different approaches, the models(their risks to be precise) diverge a lot on long maturities(first model is yield to worst and another one where the yield is a stochastic process). Again, stochastic yield(when vol is present) always allows for possibility to hold a bond to maturity(where risk is large) while yield to worst will pick a single call date as maturity for the callable bond and will calculate all the risks with respect to that date.

bearish
Posts: 6640
Joined: February 3rd, 2011, 2:19 pm

### Re: perpetual callable bonds with stochastic yield

Yield is simply the wrong concept here. Once you’re past the fixed rate period, this is just a callable credit risky bond with a fixed spread to the reference rate (historically Libor, but not for long). So while this can be reasonably modeled as a 1-D problem, the relevant dimension is credit spread (or risk adjusted default intensity). The fact that the reference interest rate is stochastic is pretty much irrelevant at this stage. Before that period, though, you ought to worry about the joint dynamics of the credit spread and the reference rate. If the bond is not callable during this period, you can probably find a way to finesse it. As for your key rate duration (or dv01) profile, you should include the effect of the “extension risk” that arises from the possibility that the bond is not called at the first possible opportunity. That being said, these bonds are usually sold with a wink and a nod and soft promise to call. Which means something. Until it doesn’t. At this stage I’d worry more about the underlying model than the specific numerical solution.

arr164
Topic Author
Posts: 8
Joined: February 7th, 2018, 9:49 pm

### Re: perpetual callable bonds with stochastic yield

@bearish: I must have specified, I use yield model for the fixed bonds indeed and I use a similar model to model the stochastic spread for the floaters. But in either case I end up with backward heat PDE. My model performs well on short maturities and is consistent with price to yield, even for floaters, but as I increase maturity over 30Y the risk numbers diverge. I think this is due to the fact that I create a distribution for call dates, it is no longer a single date and as a result the risk is skewed. Is there a different type of model used for long dated fixed to float or whatever model I come up with for pricing floaters should work for all maturities? My goal is to get something that remains consistent with Price to Yield risk numbers. Are there some references you can point to? What type of model could I use for that? "Extension risk" is something that gets build into a model? I assume that affects also price to yield model? I would like to get as much info as possible on the topic.

Cuchulainn
Posts: 65000
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: perpetual callable bonds with stochastic yield

At this stage I’d worry more about the underlying model than the specific numerical solution.
That's what I was intimating as well; what is the (PDE) model which was not specified? For large $T$ I suspect the solution should converge to a steady-state if the problem is well-posed.

The evergreen Crank-Nicolson: what is the reason for using it, It would only one of a short list? I have not tried it for large T, it is only A-stable, but fully implicit and Richardson are L-stable (aka 'good'). Another worthwhile try-out is adaptive ODE solves as in NDSOLVE or Boost C++ odeint, See an example (in my case I used ADE).
https://onlinelibrary.wiley.com/doi/epd ... wilm.10366

The PDE and/or FDM model can be wrong. Nothing useful can be said until resolution of this question.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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

Cuchulainn
Posts: 65000
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: perpetual callable bonds with stochastic yield

Some random remarks

. If it is a real perpetual bond on $(0,\infty)$ then a transformation $\tau = t/(1 + t)$ reduces PDE/ODE to one on $(0,1)$. It does work for othe classes of problems dU/dt = AU, where A is dissipative.
. Asymptotically, it is an elliptic PDE, so use ellptic solver.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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

arr164
Topic Author
Posts: 8
Joined: February 7th, 2018, 9:49 pm

### Re: perpetual callable bonds with stochastic yield

It is not about the PDE question or method question really. Before I even solve it or state it, I would like to explore the intuition behind 2 different approaches. That could guide me about results I expect to obtain from my numerical problem. In the fixed "Price to best/lowest Yield" there is one single workout date and it could be first call date which defines the duration. Whenever I am creating any random process for the bond though, I would allow paths not to go "the best" route, and essentially every possible path has a chance and that chance is driven by the model I assume. As a result, the duration of the bond in the random model gets pushed forward and as a result all the risks would increase. When maturity is short the results are fairly consistent because the domain is small, but when I enlarge it to 150Y and as a result give a chance to a particular random path bypass all call dates and reach the maturity that pushes the duration much further. I am looking for the intuition of replacing price to yield model with some random process driven one. It will yield a PDE and I will find a way to solve it but what am I expecting to change from switching from one model to another?

Alan
Posts: 10717
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: perpetual callable bonds with stochastic yield

"but as I increase maturity over 30Y the risk numbers diverge".

Not my area, but this sounds a little suspicious to me. Intuitively, for scenarios where rates at the 30Y mark are relatively high, present values for whatever happens beyond 30Y are small. And for scenarios where the rates then are relatively small, the bond gets called -- then or earlier. What am I missing?

Cuchulainn
Posts: 65000
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: perpetual callable bonds with stochastic yield

@arr164
As Alan said, it is not the area of us here. So, paraphrasing the model in terms of undefined terms is not going going to help on its own.
The challenge in this case is to explain it unambiguously to otherwise clever cookies who are not in this area. Hard maths might be a start.
Or is the whole point to discover the model?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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

arr164
Topic Author
Posts: 8
Joined: February 7th, 2018, 9:49 pm

### Re: perpetual callable bonds with stochastic yield

@Alan: we create a random process, so when you say "if the rates are relatively high" at 30Y then what's after doesn't matter. But when we create a stochastic process we allow for all the possibilities. Essentially the whole point is that on the FD grid I have points where I am in the money and where I am not. Most of them can be one way or the other but if vol is high most likely you have "a path" such that even if on average rates are high at 30Y it is better to hold it until 50Y and call then. When we have a stochastic process the range of possibilities is endless and they all have a chance to take place. Unlike deterministic model where there is only one solution and one implied path. Therefore, when pricing bonds with stochastic model you always get a distribution of call dates, you don't get a single answer. As a result that distribution can be skewed one way or the other. Perhaps I am looking for some references from a fundamental modeling point of view. If there are any papers I would be happy to look into those. I am not concerned with the model or implementation to be honest, I am trying to fundamentally understand the financial/mathematical interpretation of different approaches.

Alan
Posts: 10717
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: perpetual callable bonds with stochastic yield

Just out of curiosity, if the bond were non-callable with maturity T, does your stochastic interest rate process lead to a finite value for the bond as $T \rightarrow \infty$?

bearish
Posts: 6640
Joined: February 3rd, 2011, 2:19 pm

### Re: perpetual callable bonds with stochastic yield

He is using a lognormal yield process, so the answer should be yes. Even if the yield goes to zero almost surely (my best guess) the bond price will be bounded from above by the face value.

Cuchulainn
Posts: 65000
Joined: July 16th, 2004, 7:38 am
Location: Drosophila melanogaster
Contact:

### Re: perpetual callable bonds with stochastic yield

He is using a lognormal yield process, so the answer should be yes. Even if the yield goes to zero almost surely (my best guess) the bond price will be bounded from above by the face value.
If you have a PDE for this problem you can prove this hypothesis by the maximum principle if the problem is well-posed.
and I reckon as $T \rightarrow \infty$ you get the solution of a steady-state elliptic PDE, mutatis mutandis.
Caveat: Just a guess, don't know what your PDEs are..
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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