- 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**

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...

- 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**

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...

- October 3rd, 2020, 12:25 pm
- Forum: Numerical Methods Forum
- Topic: What are the boundary conditions for the Forward contract PDE?
- Replies:
**45** - Views:
**2829**

oopsyNo, that's another thread!

- 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**

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...

- October 3rd, 2020, 8:51 am
- Forum: Numerical Methods Forum
- Topic: What are the boundary conditions for the Forward contract PDE?
- Replies:
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**2829**

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 ...

- 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**

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

- 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**

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) = ...

- 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**

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) = ...

- 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**

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.Yes, that was also one of my suggestions and is standard.

- 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**

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...

- 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**

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...

- 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**

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...

- 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**

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 ?

Maybe you are confusing payoff and fair-value ?

- 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**

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 !Can you write down the precise 'ehancements'? e.g. for heat equation and then BS?

- 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**

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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