On Feb 25, 2011, at 1:52 PM, JORGE ESPINOSA wrote:
>> i've already solved many examples and they look very well.
I'm glad to hear it.
> I have to solve a nonlinear black scholes equation (the nonlinearity is in
> the
> convection term)
>
> My equation in tex is:
>
> \frac{\partial V}{\partial t}+rs\frac{\partial V}{\partial s}
> +\frac{1}{2}\sigma^2a^2\frac{\partial^2 V}{\partial s^2}[1+exp{r(T-
> t)}a^2s^2\frac{\partial^2 V}{\partial s^2}]-rV=0
>
> With other notation:
>
> V_t+rsV_{s}+\frac{1}{2}\sigma^2a^2V_{ss}[1+exp{r(T-t)}a^2s^2V_{ss}]-rV=0
>
> a, \sigma are constants.
>
> with the same final condition that the linear Black Scholes equation.
>
> ¿Do you think that this equation is solvable with Fipy?
Yes, although the current release may not do a good job of it, particularly
respecting boundary conditions.
You can *always* write everything out explicitly
TransientTerm()
+ r * s * V.getGrad()
+ ((sigma**2 * a**2 / 2.) * V.getFaceGrad().getDivergence() * (1. + exp(r * (T
- t)) * a**2 * s**2 + V.getFaceGrad().getDivergence())
- r * V == 0
however this will have poor numerical performance and FiPy will not do the
right thing with your boundary conditions.
I don't have time to work through it in detail, but remember that as much as
possible you want to arrange terms into implicit forms like
\frac{\partial}{\partial s} [D \frac{\partial V}{\partial s}]
which is a DiffustionTerm
\frac{\partial}{\partial s} [u V]
which is a ConvectionTerm
and
sV
which is a ImplicitSourceTerm
You can get a term like rsV_{s} by doing the chain rule on
\frac{\partial}{\partial s}[r s V]
and subtracting off the parts you don't want, i.e.,
rs \frac{\partial V}{\partial s} = \frac{\partial}{\partial s}[r s V] - rV
or
ConvectionTerm(coeff=rs) - ImplicitSourceTerm(coeff=r)
\frac{1}{2}\sigma^2a^2\frac{\partial^2 V}{\partial s^2}
is easy because \sigma and a are constants, so you can rewrite this as
\frac{\partial}{\partial s}[\frac{1}{2}\sigma^2a^2 \frac{\partial V}{\partial
s}]
which is a
DiffusionTerm(coeff=sigma**2 * a**2 / 2.)
\frac{1}{2}\sigma^2a^2\frac{\partial^2 V}{\partial
s^2}exp{r(T-t)}a^2s^2\frac{\partial^2 V}{\partial s^2}
is trickier because FiPy doesn't have anything like (\frac{\partial^2
V}{\partial s^2})^2 in it, but I'd be inclined to run the chain rule on
\frac{\partial}{\partial
s}[\frac{1}{2}\sigma^2a^4s^2\exp{r(T-t)}\frac{\partial^2 V}{\partial s^2}
\frac{\partial V}{\partial s}]
and see what you had left over.
That term can be written as
DiffusionTerm(coeff=sigma**2 * a**4 * s**2 * exp(r*(T - t)) *
V.getFaceGrad().getDivergence())
the remainder are hopefully going to be some combination of ConvectionTerm and
ImplicitSourceTerm, perhaps with explicit forms of higher-order derivatives.
That's not ideal, but probably the best you can do.
The problem with V.getFaceGrad().getDivergence() is that it won't see any of
your boundary conditions.
In the development trunk/ of FiPy, which we will hopefully be releasing soon,
I'd be inclined to cast this as two equations
\frac{\partial V}{\partial t}+rs\frac{\partial V}{\partial
s}+\frac{1}{2}\sigma^2a^2\Lambda[1+exp{r(T-t)}a^2s^2\Lambda]-rV=0
and
\Lambda = \frac{\partial^2 V}{\partial s^2}
and to solve them in coupled form. This would then be
eq1 = (TransientTerm(var=V)
+ ConvectionTerm(coeff=r*s, var=V)
+ ImplicitSourceTerm(coeff=sigma**2 * a**2 * (1 - exp(r * (T - t)) * a**2 *
s**2 * Lambda), var=Lambda)
- ImplicitSourceTerm(coeff=2*r, var=V) == 0)
eq2 = (ImplicitSourceTerm(var=Lambda) == DiffusionTerm(var=V))
coupled_eq = eq1 & eq2
coupled_eq.solve(...)
We also have a new model for boundary conditions that will be seen by
V.getFaceGrad().getDivergence(), but I still think the coupled form is going to
be clearer and have fewer "weird" terms.
Daniel Wheeler and I are off to two different conferences next week, so I don't
know how much further we'll be able to offer until after that.
