SERVING THE QUANTITATIVE FINANCE COMMUNITY

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Paul
Posts: 8961
Joined: July 20th, 2001, 3:28 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Let me look at the book tomorrow, and I’ll refer to the relevant equations.

inozemtsev
Posts: 8
Joined: January 17th, 2019, 6:47 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Thank you very much, Mr. Wilmott!

inozemtsev
Posts: 8
Joined: January 17th, 2019, 6:47 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Cuchulainn
Thank you for trying to help me
This PDE has  many characteristics as I understood, they all are described in Uncertain Interest Rate Modelling (David Epstein).
That derivative can change sign, as well it is described in that paper too.

Paul
Posts: 8961
Joined: July 20th, 2001, 3:28 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Mr. Wilmott, as for me, I can not give an assessment :
I will try to show what I understood

1)  Model with not diffusive equation means that there is no parabolic PDE (parabolic PDE like heat equation, 2nd order). So, we have only 1st order PDE. As well, in this model we don't have any stochastic processes.
2) I guess, I have a misunderstanding here,
Value of portfolio = Value(contract + n*hedging instrument) - n*cost of h.i.
3) The fact of nonlinearity gives us that the value of sum of the contracts does not equal to the sum of values of the contracts.
4) Briefly, we maximize portfolio value equation with some constraints on hedging instrument.
But how to derive it (Where is nonlinearity? For me it looks like trying to optimize linear function...)

On the other hand we have a solution of PDE
V(r,t)=f(r,c,t)∙e^(-c/2 (T-t)^2-r(T-t)) (sorry for this texting)
So, the value is the solution of that PDE.
What is f()? What is c? Where do you get that solution from?

Cuchulainn
Posts: 58122
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

I think this formula comes from David Epstein PhD (D Phil?) thesis..

inozemtsev
Posts: 8
Joined: January 17th, 2019, 6:47 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

c is speed of interest rate change (defined as dr/dt), can change its sign in best and worst case.
f() is an arbitrary function, that we can find if we have Cauchy problem.

I found this general solution of PDE in David Epstein's paper. Cuchulainn gave a link to this file.
But I guess it was unnecessary to mention PDE solution.

Paul
Posts: 8961
Joined: July 20th, 2001, 3:28 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Ok. Still trying to home in on which bit you are struggling with, and also what your knowledge is.

Are you comfortable with the concept of "characteristics" in these sort of PDEs? I'm asking this to see how much maths you know. Most people in finance won't know about them. And I'm not suggesting they are particularly helpful here for anything but simple problems.

The key things to understand is that if you have a non-linear problem and you have two separate solutions A and B then generally A+B is not also a solution. Are you ok with that?

inozemtsev
Posts: 8
Joined: January 17th, 2019, 6:47 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

As for me, I do not enjoy these characteristics in PDE, but I can figure out with it.
Although, in that paper were quite difficult cases...

When I was studying for bachelor's degree, I solved some sort of pde "semi-infinite string", "wave equation", "heat equation" and so on.

Regarding non-linear problem, I understand that fact, I am ok with that.

Cuchulainn
Posts: 58122
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Courant and Hilbert discuss characteristics for 2nd/1st order hyperbolic, elliptic and parabolic PDE in great detail.

They are of fundamental importance all over the place. For example, you can use them to simplify Black-Scholes PDE to canonical form (no mixed derivatives).  But I can imagine 1st order PDEs can be a bit of a culture shock. Those who did fluids mechanics in degree program are exposed to it at an early age.

When I was studying for bachelor's degree, I solved some sort of pde "semi-infinite string", "wave equation", "heat equation" and so on.
How? Separation of variables? {Laplace, Fourier} transform?

I suspect many of the challenges in this project will be seen here but I may be overstepping the mark.

inozemtsev
Posts: 8
Joined: January 17th, 2019, 6:47 pm

### Re: non probabilistic model for interest rate derivative by Epstein&Wilmott

Thank you for helping!
But I didn't figure out with that static hedging