A question about non-typical Black-Scholes equation

[kernel]

Hi everyone, this is my first question so it may be not particularly well-posed. And I can't find anything similar on the forums. Anyways, there's a Black-Scholes like PDE:

\(\large -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0\)

With boundary conditions:

\(V\large(0,S\right) = max\large(E-S,0\right)\)
\(V\large(t,S^*\right) = E-S^*\large(t\right)\)
\(V\large(t,\infty\right) = 0\)

Where
\(\left. \frac{\partial V}{\partial S} \right|_{S=S^*} = -1\)

and for \(S \le S^*\), \(\frac{\partial V}{\partial t} = 0\)

Sigma, r and E are given. The goal is to find numerical solution for V(a, b), where a and b is some values.

Seems like it's a free boundary problem and numerical solution is the only way. So I tried to transform this PDE to heat equation in order to get rid of free boundary later (this described, for example, in "The Mathematics of Financial derivatives" by Wilmott p.166).
But, it seems, that resulting heat-like equation \(\frac{\partial v}{\partial t} =
-\frac{\partial^2 v}{\partial x^2}\) is typical ill-posed PDE. That minus is from original equation. I wonder if I skipped some transformation which would allow me to come to normal PDE or original equation is ill-posed from the beginning.

Since i have really minimal knowledge of options and Black-Scholes atm, maybe someone could clarify, is this equation makes any sense from financial point of view? Or someone would give any advice how to succeed in finding correct solution.
Thanks in advance.

 

[dmv]

if u want to see relationship between heat equation and BS pde - read Steele`s Stochastic calculus. i did not understand what is E?

 

[kernel]

i did not understand what is E?

E - exercise price of an option.

 

[Timmi]

Hi kernel, I can tell you that the problem is not ill-posed and it is exactly the same as the free boundary problem to be solved for the arbitrage-free price of an American Put Option. In financial literature, the free boundary problem to be solved for the arbitrage-free price of an American Put Option is posed as

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0​

V(T,S)=max(E−S,0)​

; T is the time the option expires

V(t,S∗)=E−S∗(t)​

V(t,∞)=0​

Where

∂V∂S∣∣S=S∗=−1​

and for

S≤S∗​

,

∂V∂t=0​

If we apply a simple change of variable as u = T - t, this problem becomes

−∂V∂u+12σ2S2∂2V∂S2+rS∂V∂S−rV=0​

V(0,S)=max(E−S,0)​

;

V(u,S∗)=E−S∗(u)​

V(u,∞)=0​

Where

∂V∂S∣∣S=S∗=−1​

and for

S≤S∗​

,

∂V∂u=0​

Which is exactly the boundary value problem you are trying to solve.

​

 

[Timmi]

Sorry the symbols look awful

 

[Daniel Duffy]

I don't see how you get a backward heat equation from a forward BS PDE after a transformation. And the free boundary is irrevelant IMO to the issue.

Are you confusing t and T - t?

BTW, what makes your PDE 'non-typical'?

 

[kernel]

Hi kernel, I can tell you that the problem is not ill-posed and it is exactly the same as the free boundary problem to be solved for the arbitrage-free price of an American Put Option.

Thanks, Timmi. I already found that my problem is exactly american put with free boundary, which well described in various papers. Anyway, thanks for your reply.

 

[kernel]

I don't see how you get a backward heat equation from a forward BS PDE after a transformation. And the free boundary is irrevelant IMO to the issue.

Are you confusing t and T - t?

BTW, what makes your PDE 'non-typical'?

You right about the fact that there's nothing special about this PDE. I just got confused.
However, I think that problem is directly connected with free boundary problems. For instance in
this paper almost identical problem being solved.

 

[Daniel Duffy]

You right about the fact that there's nothing special about this PDE. I just got confused.
However, I think that problem is directly connected with free boundary problems. For instance in
this paper almost identical problem being solved.

Good. This is just the usual early exercise PDE. No backwards PDE in sight!

Another way is ADE.

http://www.datasimfinancial.com/forum/viewtopic.php?t=327

 

[Timmi]

I think if the time-step size is chosen carefully, a simple explicit finite difference scheme will suffice - or you could go for a Crank-Nicolson finite scheme with a PSOR solver.

 

[Daniel Duffy]

I think if the time-step size is chosen carefully, a simple explicit finite difference scheme will suffice - or you could go for a Crank-Nicolson finite scheme with a PSOR solver.

Nope, on all these counts. Each count for a different reason why not.

 

[Timmi]

Hi Daniel,

If we express the PDE as Vt = LV(t, S) with L denoting the operator, take the forward difference approximation of Vt between time levels n and n + 1, and discretise LV(t, S) at time level n, with the time step sufficiently small, I believe one should be able to march the solution from the initial values (payoff)
V(0,S) to the solution V(t,S), by explicitly computing the values at time level n + 1, from the values
at time level n, and applying the free boundary constraints if necessary. It would be identical to the explicit
finite difference scheme used to solve the Black-Scholes inequality for an American Put Option, whereby
one would march backwards in time from the terminal values V(T,S) to the solution V(t,S), by explicitly
computing the values at time level n, from the values at time level n + 1, and applying the free boundary
constraints.

I think this analogy should also enable a Crank-Nicolson-type finite difference scheme with PSOR.

 

[Daniel Duffy]

Timmi,
This is doable of course _but_

1. Euler is conditionally stable dt = O(h^2)
2. Applying constraint reduces accuracy to O(dt)
3. CN 'wiggles' at the strike E (and delta is real bad)
4. PSOR is too slow in general.

These are well-discussed on forums.

http://www.wilmott.com/messageview.cfm?catid=34&threadid=69722&FTVAR_MSGDBTABLE=

http://www.wilmott.com/messageview.cfm?catid=34&threadid=60415&FTVAR_MSGDBTABLE=
regards

Daniel

 

[Timmi]

Many thanks Daniel,

Very interesting results, and great food for thought!

Best Regards
Timmi

 

[Bastian Gross]

I think this analogy should also enable a Crank-Nicolson-type finite difference scheme with PSOR.

3. CN 'wiggles' at the strike E (and delta is real bad)

Rannacher smoothing combined with Crank Nicolson may be a good way to work against wiggling at strike.