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Cuchulainn
Posts: 63177
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### Re: sigma root (T-t)

In that case everything I write must be highly relevant, because you pick on every single sentence. WHY, OH WHY?! Yawn...

Don't worry, these are my last few weeks here
Argument is an intellectual process. Contradiction is just the automatic gainsaying of anything the other person says.
Where you going?
Step over the gap, not into it. Watch the space between platform and train.
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Cuchulainn
Posts: 63177
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### Re: sigma root (T-t)

Some people just like to argue
No they don't.

P.S. I didn't know they made faucets at Caltech. I thought it was MIT's province.
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Cuchulainn
Posts: 63177
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### Re: sigma root (T-t)

Here's a way to 'recover' $\frac{1}{2}\sigma^2 T$ from first principles.

. Starting from the BS $Pde(S,t)$ define the $Pde(x, \tau)$ by  new variables $x = log(S), t = T - 2\tau/\sigma^2$.
. After some arithmetic we get a Cauchy initial value problem for a 1d heat equation defined by

$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$

with $-\infty \lt x \lt \infty$
$0 \lt \tau \le$  $\frac{1}{2}\sigma^2 T$

This IVP has an analytical solution and if you work it out you can retrieve the famous BS equation.

So, $\frac{1}{2}\sigma^2 T$ is a dimensionless time.
Rebump
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Alan
Posts: 10357
Joined: December 19th, 2001, 4:01 am
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### Re: sigma root (T-t)

Some people just like to argue
No they don't.

P.S. I didn't know they made faucets at Caltech. I thought it was MIT's province.
Haha!
Anyway, the Caltech mention is one of the reasons I remember that scene!

Cuchulainn
Posts: 63177
Joined: July 16th, 2004, 7:38 am
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### Re: sigma root (T-t)

Some people just like to argue
No they don't.

P.S. I didn't know they made faucets at Caltech. I thought it was MIT's province.
Haha!
Anyway, the Caltech mention is one of the reasons I remember that scene!
For me, this one does it. BTW the liberal arts types were really like that in the 70s.

There's a glimpse of the maths dept @1.34. In the summer we played tennis on the green. There were only 6 maths students left in junior and senior soph years. Now it's a factory.
Step over the gap, not into it. Watch the space between platform and train.
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complyorexplain
Topic Author
Posts: 120
Joined: November 9th, 2015, 8:59 am

### Re: sigma root (T-t)

To summarize this thread, as Bill Clinton famously almost said:
"It depends on what the meaning of the word ‘meaning’ is.”
No it doesn't. We agree on the meaning of the word 'meaning'.

The OP was to find out not what sig root t means, but what people here think it means. We have answered that question, more or less.

Collector
Posts: 4755
Joined: August 21st, 2001, 12:37 pm

### Re: sigma root (T-t)

Cuchulainn
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### Re: sigma root (T-t)

It's a sign! sig sqrt occurs in nature.
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riskneutralprob
Posts: 38
Joined: December 16th, 2005, 5:16 pm

### Re: sigma root (T-t)

I don't post a ton, but I believe this is a waste of most of these guys' time.  You require too much prerequisite information.  This should really be moved to the 'Student Forum' where it will be appropriately ignored as it has been covered by many texts already.    For stock options, start with a Bachelier (normal) model, do simple price diffs, and then find out why you don't want that.  Then move to logReturns where when you take the log(Px) then the diffs are actually log returns  with some nice features dLogPx = (log(Px_1) - log(Px_0)) = log(Px_1/Px_0) .   Doing this stuff over time, etc....  As was said earlier.  Kindergarten stuff.  Maybe I'd give it 8th grade, but I'm American. So everywhere else it may be like 4th grade

Cuchulainn
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### Re: sigma root (T-t)

See it as a bit of fun at the weekend. It's not easy being couped up with the wife and kids 24/7 when you'r trying to get ideas.

Seriously, I actually converted BS PDE to heat equation when I was waiting for my tea to arrive (my butler is not so fast) just to see what came out of the woodwork. It might have some mileage and be even better than FDM in this case (we get an integral involving payoff and transition density). We can also do FDM for heat equation.

And of course, Caltech, and not MIT is leader in faucet design.

I always knew that applied mathematicians like that sort of stuff but it is useful. Once you know why.
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bearish
Posts: 5883
Joined: February 3rd, 2011, 2:19 pm

### Re: sigma root (T-t)

See it as a bit of fun at the weekend. It's not easy being couped up with the wife and kids 24/7 when you'r trying to get ideas.

Seriously, I actually converted BS PDE to heat equation when I was waiting for my tea to arrive (my butler is not so fast) just to see what came out of the woodwork. It might have some mileage and be even better than FDM in this case (we get an integral involving payoff and transition density). We can also do FDM for heat equation.

And of course, Caltech, and not MIT is leader in faucet design.

I always knew that applied mathematicians like that sort of stuff but it is useful. Once you know why.

This quaint idea of yours to convert the Black-Scholes PDE to the heat equation may have some merit. It was independently done in a 1973 JPE paper by some bozos. Fischer and Myron, equation 10.

Cuchulainn
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### Re: sigma root (T-t)

I am not the inventor of this idea. In the past I saw PDE transformations as cheating. But it does have computational/numerically advantages such as robustness, performance etc. Pedagogically, it is  good way to show MFE/MSc students how it works (the Fourier transform and FPE can be dragged into the process).

Alan and Paul know this stuff forever (And no doubt bearish as well.) My 2 cents is moving the PDE to numerics. So, we can work it out A-Z.
Last edited by Cuchulainn on October 20th, 2020, 9:37 am, edited 1 time in total.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

complyorexplain
Topic Author
Posts: 120
Joined: November 9th, 2015, 8:59 am

### Re: sigma root (T-t)

I don't post a ton, but I believe this is a waste of most of these guys' time.  You require too much prerequisite information.  This should really be moved to the 'Student Forum' where it will be appropriately ignored as it has been covered by many texts already.    For stock options, start with a Bachelier (normal) model, do simple price diffs, and then find out why you don't want that.  Then move to logReturns where when you take the log(Px) then the diffs are actually log returns  with some nice features dLogPx = (log(Px_1) - log(Px_0)) = log(Px_1/Px_0) .   Doing this stuff over time, etc....  As was said earlier.  Kindergarten stuff.  Maybe I'd give it 8th grade, but I'm American. So everywhere else it may be like 4th grade
But this was not the question raised by my OP, which was about the semantics of sigma root t, i.e. its meaning. The fact my question has so consistently been misunderstood kind of answers my question.

If you look at the actual application of the heat equation, i.e. where there are terms referring to heat flux (flow) and so on, tells you that there are entirely different ways of *interpreting* the terms.

Cuchulainn
Posts: 63177
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### Re: sigma root (T-t)

Maybe you could have been less cryptic.
Put us out of our pain, and give us the answer. Life is all about communication.

If you look at the actual application of the heat equation, i.e. where there are terms referring to heat flux (flow) and so on, tells you that there are entirely different ways of *interpreting* the terms.

Most of us here already know that. I would remove the word 'entirely'.
Are you familiar with Avner Friedman's books on SDE/PDE?

BTW did I get the correct answer with my PDE solution?
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

complyorexplain
Topic Author
Posts: 120
Joined: November 9th, 2015, 8:59 am

### Re: sigma root (T-t)

BTW did I get the correct answer with my PDE solution?
Not sure - there were many intermediate steps missing, and the proposition to be proved was not clear.

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