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Cuchulainn
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One-liner questions of a numerical kind

April 22nd, 2019, 4:59 pm

Subject:

gamma (second derivative of option  price wrt S). Does it have a value at t = 0?
Can we model gamma as a PDE?

This is for t == 0 -> t = 1.0e-3
When t = T for initial conditions gives a better approximation..

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bearish
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 12:11 am

I’m pretty sure Peter Carr derives a PDE for gamma in his “Deriving derivatives of derivative securities” paper. Since delta at time zero is given by the Heaviside function, gamma at time zero is given by the Dirac delta function.
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 12:34 pm

Subject:

gamma (second derivative of option  price wrt S). Does it have a value at t = 0?
Can we model gamma as a PDE?

This is for t == 0 -> t = 1.0e-3
When t = T for initial conditions gives a better approximation..

Image
The reason for this red shift is that I forgot a convection term when differentiating the PDE for gamma. Now it's fine.
 
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bearish
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 2:31 pm

Subject:

gamma (second derivative of option  price wrt S). Does it have a value at t = 0?
Can we model gamma as a PDE?

This is for t == 0 -> t = 1.0e-3
When t = T for initial conditions gives a better approximation..

Image
The reason for this red shift is that I forgot a convection term when differentiating the PDE for gamma. Now it's fine.
Your gamma Dopplered to the left?
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 2:49 pm

I’m pretty sure Peter Carr derives a PDE for gamma in his “Deriving derivatives of derivative securities” paper. Since delta at time zero is given by the Heaviside function, gamma at time zero is given by the Dirac delta function.
Found the article by Peter C. Will read it. Actually, I attended a talk by Peter around 2001 in Broad Street (51?) when he actually introduced the topic! 

Now, I am fairly confident the maths is solid and the numerics are more robust than other stuff. I tried both Gaussian and box deltas and they are fine. Moving to multiple stat variables with this approach should be easy as well. 

An open question is how to compute [$] \frac{\partial C}{\partial K} [$]; the strike [$]K[$] is 'hidden' in the payoff and is nowhere to be seen in the PDE.
Last edited by Cuchulainn on April 23rd, 2019, 4:18 pm, edited 2 times in total.
 
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bearish
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 3:17 pm

Given that standard options are essentially symmetric in S and K, you could try a change of numeraire. Then the “strike delta” becomes the regular delta. This is usually easier to think about in an FX context.
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 4:11 pm

Given that standard options are essentially symmetric in S and K, you could try a change of numeraire. Then the “strike delta” becomes the regular delta. This is usually easier to think about in an FX context.
Don't suppose you have a useful link? (BTW PC also excludes this case from his article).
 
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Paul
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 4:47 pm

I had taken it for granted that everyone here knows how to differentiate. Looks like I was wrong.
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 5:55 pm

I had taken it for granted that everyone here knows how to differentiate. Looks like I was wrong.
Mathematica and Maple have a lot to answer for. Just like no one can do log tables not long division any more.
Last edited by Cuchulainn on April 23rd, 2019, 8:23 pm, edited 1 time in total.
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 7:15 pm

http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf

This is the article. I use the approach as in eq (1) but then for all greeks in my case, but not in the article.. 
Theorem 2 shows a formula for the derivatives wrt S written as series of Hermite polynomials. These are computationally difficult in a number of ways. The alternative here is that I only need to embed the  greeks in a PDE (qualitative solution) and them embed them in a parabolic PDE system, much easier.

Eqs. (28)-(30); look like PDEs.
 
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FaridMoussaoui
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Re: One-liner questions of a numerical kind

April 23rd, 2019, 10:22 pm

You have to look into Dupire work.
 
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Alan
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Re: One-liner questions of a numerical kind

April 24th, 2019, 12:26 am

An open question is how to compute C/K; the strike is 'hidden' in the payoff and is nowhere to be seen in the PDE.

That makes it easy. You can still differentiate the PDE => [$]u = C_K[$] follows the same PDE with the differentiated payoff. 
 
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Cuchulainn
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Re: One-liner questions of a numerical kind

April 25th, 2019, 9:15 am

An open question is how to compute C/K; the strike is 'hidden' in the payoff and is nowhere to be seen in the PDE.

That makes it easy. You can still differentiate the PDE => [$]u = C_K[$] follows the same PDE with the differentiated payoff. 
Excellent! I was over-engineering it! When I came back from the dojo it came in a flash.

Here  is   [$] \frac{\partial P}{\partial K} [$] as a function of the underlying [$]S[$], Is it correct(*)? It is the same as the BS PDE and the same BC/IC as detta.It looks a bit like a call delta.
(*) compared against [$]e^{-rT}N(-d2)[$]

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Cuchulainn
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Re: One-liner questions of a numerical kind

April 25th, 2019, 10:44 am

Strike gamma (output as expected I suppose)

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Cuchulainn
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Re: One-liner questions of a numerical kind

April 30th, 2019, 10:42 am

This counterexample was salvaged from the ashes and saved for posterity.
Some related question to follow..

Good remarks by @fogsnow.

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