Serving the Quantitative Finance Community

 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American Barrier options - reference prices

September 18th, 2017, 8:26 pm

V0=0 is uninteresting for real world applications, beter focus on relevant parameter ranges?
No. This is incorrect logic. The issue cannot be brushed under the carpet.
Look at the spike! And the PDE model in Heston 1993.
Articles put v0 = 0.4 because they are scared of hyperbolic PDEs. Sheppard solved it.
The spike is due to the S-direction discontinuity (at the strike) and has essentially nothing to do with v0, or the Heston model for that matter.  You'll get similar error spikes near the strike even if you use simple Black Scholes.
Still, v0 = 0 is more elegant, yes?
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 18th, 2017, 8:35 pm

that's exactly the same as saying S0=0 is more elegant.
?!
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 18th, 2017, 8:55 pm

This thread is very strange.. Maybe you think v0 is the grid lower bound? It's not, it's the initial (latent) vol level at t=0.
 
User avatar
Billy7
Posts: 262
Joined: March 30th, 2016, 2:12 pm

Re: American Barrier options - reference prices

September 18th, 2017, 9:36 pm

Yes I agree, if you think v0=0 is more elegant, then you are referring to the grid truncation boundary. I now understand why you asked "why are we truncating the grid". As outrun points out, v0 here is the initial variance level, apologies if that wasn't crystal clear. So v0 the initial level, vL the long term level where it mean reverts to, etc... Again, the plots I posted before depict a cross section of the error surface at v=v0, which is where we want the pricing for.
Last edited by Billy7 on September 18th, 2017, 10:29 pm, edited 1 time in total.
 
User avatar
Billy7
Posts: 262
Joined: March 30th, 2016, 2:12 pm

Re: American Barrier options - reference prices

September 18th, 2017, 9:52 pm

One question we could ask is what cause RE spikes, i.e. why?

Some people try to smooth the payoff to resolve spikes ..
The spike isn't RE (Richardson Extrapolation) related, it's just how numerical methods work, i.e. less good where/when there's a discontinuity/singularity etc. So it's confusing when you're saying RE-spikes. If anything I showed how this error spike can be significantly reduced using RE. Yes some people (including me!) try to smooth the payoff and in general do anything possible to make the solution as accurate as possible.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American Barrier options - reference prices

September 19th, 2017, 7:30 am

This thread is very strange.. Maybe you think v0 is the grid lower bound? It's not, it's the initial (latent) vol level at t=0.
You seem to be contradicting yourself.

This is what Billy wrote
Last edited by Cuchulainn on September 19th, 2017, 7:32 am, edited 1 time in total.
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 7:32 am

This thread is very strange.. Maybe you think v0 is the grid lower bound? It's not, it's the initial (latent) vol level at t=0.
You seem to be contradicting yourself.
What are you taking about. Why is this so confusing for you?
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 7:34 am

v0 is the current spot vol at t=0.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American Barrier options - reference prices

September 19th, 2017, 7:34 am

This is what Billy wrote

Here's a Heston example I just made up, parameters are v0=0.04, vL=0.08, 

I am sure he can explain. Billy, please define and put this to bed.
Last edited by Cuchulainn on September 19th, 2017, 7:37 am, edited 1 time in total.
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 7:36 am

Yes.. so what do you think that means?
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 7:40 am

It all clear to me and others. It's the convention used in Heston everywhere.

It has been explained three times now? Just read back, Billy's posts or my posts.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American Barrier options - reference prices

September 19th, 2017, 7:41 am

It's clear now, Billy has defined it. Notation was not defined.
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 7:42 am

If you think these parameters mean something different then that's obviously the reason why this dialog is so strange., Right?
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American Barrier options - reference prices

September 19th, 2017, 8:04 am

Like  I said, my impression that the v boundary was truncated. A remedy is to write the system of equations and then you see what v0 etc. But that is NDA!

Something like this and then it is clear.

http://ac.els-cdn.com/S0898122111002811 ... bc7e3d7e8c
 
User avatar
outrun
Posts: 4573
Joined: January 1st, 1970, 12:00 am

Re: American Barrier options - reference prices

September 19th, 2017, 8:50 am

 Give it a try, it would be interesting to see how that hold up against Billy's results? Can you?

About the paramters:  when you say "these are the Heston parameters", then those will refer to the Heston model (i.e. the SDE parmeters) and not specific numerical methods (like grid parameters, time steps, MC samples, quadrature points,..whatever numerical method ..) . That's why it wasn't confusing.
However when you have a hammer, you see nails everywhere!