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by JohnLeM
October 4th, 2020, 7:36 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

Very much like a market quoted swap rate for a given maturity and other (usually standard) terms and conditions. For an underlying asset that can be costlessly stored and doesn’t generate any intermediate value (like paying a dividend or a coupon), the forward price is simply the spot price divided...
by JohnLeM
October 4th, 2020, 7:29 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

I'll try to be less cryptic. Why price a simple forward contract in such a complicated fashion when no-arbitrage considerations relative to the underlying prices make delta 1 product valuation nearly trivial.   For code quality, it is desirable to test trivial input to show that any algorithm (here...
by JohnLeM
October 3rd, 2020, 12:04 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

Has OP's question been resolved yet? You mean constructing monotone conservative dissipative schemes ? For the simple FD heat equation yes, but it is really a straightforward, very simple idea.  The difficulty is to generalize the proof to the schemes I really use, that is more challenging. It seem...
by JohnLeM
October 3rd, 2020, 8:51 am
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

I was trying to answer to your point above : " At the end of the finite difference scheme, to obtain the value of the option at time 0 we compute  [ltr] V ( S 0 , 0 ) = ( F ( S 0 , 0 ) − K ) e − r T V(S0,0)=(F(S0,0)−K)e−rT   which in general is not 0, in contrast with what I said above about  F F  ...
by JohnLeM
October 2nd, 2020, 5:33 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

Following Mars objection, I also checked my answer to jherekhealy. I tested on a very simple example: whatever tau is, I obtain a non monotone scheme (the matrix (I-\tau A)^{-1}(I+\tau A) has some negative entries). Excel sheet here for those who want to toy with it
by JohnLeM
October 2nd, 2020, 11:20 am
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

I do not agree, if  [$] \lambda_i[$] are thre eigenvalues of [$]A[$] then [$]\frac{1 - \lambda_i}{1 + \lambda_i}[$] are the eigenvalues of  [$](I_d + \tau A)^{-1}(I_d - \tau A)[$].   Let us suppose that you missed a [$]\tau[$] here. So there exists eigenvectors [$] v^i [$] such that [$]v^i(\tau) = ...
by JohnLeM
October 2nd, 2020, 7:17 am
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

I do not agree, if  [$] \lambda_i[$] are thre eigenvalues of [$]A[$] then [$]\frac{1 - \lambda_i}{1 + \lambda_i}[$] are the eigenvalues of  [$](I_d + \tau A)^{-1}(I_d - \tau A)[$].   Let us suppose that you missed a [$]\tau[$] here. So there exists eigenvectors [$] v^i [$] such that [$]v^i(\tau) = ...
by JohnLeM
September 30th, 2020, 8:32 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

Yes, that was also one of my suggestions and is standard.
The only bad point is that the matrix A is no longer symmetric with these BCs, and I don't know what happens to monotonicity and conservation properties of the FD scheme. But the numerics seemed ok AFAIR.
by JohnLeM
September 30th, 2020, 7:56 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

This is the plot with spot prices at time T (vector  ST  in the code) on x-axis and option prices at time 0 ( (F-K)*exp(-r*T)  in the code) on y-axis. As you can see the there is a problem when the option price approaches the biggest value of the spot price, since the curve goes from linear to expo...
by JohnLeM
September 30th, 2020, 5:49 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

I am not getting your point: suppose that we enter today a contract where I will pay to you 1 euro in one year. This contract worth the zero-coupon today, hence is not null, even if we do not exchange money today.. Maybe you are confusing payoff and fair-value ? I don't fully understand your argume...
by JohnLeM
September 30th, 2020, 4:23 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

When I look in the Lawson and Morris article you provide, just below eq 1.7 there is a discussion where the oscillation is linked to the negative (close to -1) eigenvalues when timestep is too large. The problem studied is heat equation, without first order derivatives the coefficient below and abo...
by JohnLeM
September 30th, 2020, 4:04 pm
Forum: Numerical Methods Forum
Topic: What are the boundary conditions for the Forward contract PDE?
Replies: 45
Views: 2829

Re: What are the boundary conditions for the Forward contract PDE?

I am not getting your point: suppose that we enter today a contract where I will pay to you 1 euro in one year. This contract worth the zero-coupon today, hence is not null, even if we do not exchange money today..

Maybe you are confusing payoff and fair-value ?
by JohnLeM
September 29th, 2020, 12:36 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

Can you write down the precise 'ehancements'? e.g. for heat equation and then BS?
I started to work on it. I need a half day more to finish a first readable draft. I hope you will receive it at end of this week !
by JohnLeM
September 29th, 2020, 12:19 pm
Forum: Numerical Methods Forum
Topic: Monotone Schemes: what are they and why are they good?
Replies: 78
Views: 4875

Re: Monotone Schemes: what are they and why are they good?

The Crank-Nicolson scheme becomes monotone for small enough time-steps (in relation to the space-steps ^2). Either the limit corresponds to the explicit scheme stability limit or to twice the latter, I don't remember exactly. Hélas non :/ Crank Nicolson approach always produces non monotone schemes...
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