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ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Silly questions

The formula (22) in the paper looks doable but equation (27) not so because we have to sole for both the price and the free boundary S*.

https://ijpam.eu/contents/2014-97-3/3/3.pdf

Even if we had an extra condition in (27) it would be a big numerical challenge to compute P and S*?
(22) should just be the Green's function expression. I never remember these things (easier to look them up) but I think there's a typo and the exponential inside the integral should be "squared" as it were, like (26)
Certainly for European options, either Laplace in time or Mellin in stock price should just lead you to the Green's function.

(27) is a fairly well-known expression, again coming from the Green's function. From memory the name Kolodner is associated with it for heat conduction. I've got several handbooks by a couple of russians (Polyanin and another one). The books are in English. One is a handbook of solvable ODEs, an updated version of Kamke. Another is a handbook of linear PDEs, and I think they give Kolodner's formula in there
Handbook of Linear Partial Differential Equations for Engineers and Scientists 2nd Edition
(I have the older first edition)
Black-Scholes transforms into the heat conduction equation fairly easily, so you can use Kolodner's result
It's one of those problems people seem to keep doing without realizing it's been done before. People like Kim and Jacka did it in  the 1990s, seemingly unaware of Kolodner, and one of the Kellers (Joe I think) did it in the 2000s seemingly unaware of either Kim/Jacka/... or Kolodner

Cuchulainn
Topic Author
Posts: 62928
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Silly questions

I have Polyanin/Zaitsev handbook .. it's full of PDE solutions.
Met Herbie Keller a few times around 1976 at international conferences organised at TCD. He was mainly a numerical analyst. A very humorous man AFAIR. Sad to read how he met his demise.
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Silly questions

Mellin transform seems appropriate since the elliptic part of BSPDE is an Euler equation. Most methods' solution end up as an infinite series or integral when they run their course. At some stage they have to solved numerically (Maple, Mathematica?).

The formula (22) in the paper looks doable but equation (27) not so because we have to sole for both the price and the free boundary S*.

https://ijpam.eu/contents/2014-97-3/3/3.pdf

Even if we had an extra condition in (27) it would be a big numerical challenge to compute P and S*?
If you have a few hundred hours to spare, there's a nice PhD thesis,
Robert Frontczak: On the Application of Mellin Transforms in the Theory of Option Pricing
or a shorter paper
[url=http://8.
www.fbv.kit.edu/symposium/11th/Paper/19 ... hoebel.pdf]http://www.fbv.kit.edu/symposium/11th/P ... hoebel.pdf[/url]

which seems to have been presented at the 11th Symposium on Finance, Banking, and Insurance December 17 - 19, 2008,
Karlsruhe Institute of Technology (KIT), Germany

Sadly, that conference series seems to have died: there's a 12th (2011) on the site http://www.fbv.kit.edu/symposium/ but I can't see a 13th or subsequent ones

Cuchulainn
Topic Author
Posts: 62928
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Silly questions

Frontczak's thesis looks very good .. he must have locked himself in a room for a couple years.

I went through the thesis focusing on 'free boundary' What's nice is the benign integral equation 5.180 (a fixed point algorithm works here AFAIR) and the explicit formulae on pages 104-106.

A possibility (?) is to use S*(t) in front-fixing/front-tracking FDM/FEM methods when now there will only be one unknown, namely P, and not coupled P/S*(t) nasty system. For front-fixing instead of a DAE we get an ODE.

https://en.wikipedia.org/wiki/Different ... _equations
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

neauveq
Posts: 10
Joined: May 28th, 2018, 10:25 pm

### Re: Silly questions

Would calculating the variance of the variance of a data set ever be useful? I can't figure out if this statistic would convey anything new?

bearish
Posts: 5802
Joined: February 3rd, 2011, 2:19 pm

### Re: Silly questions

That sounds like the kurtosis, up to scaling. It will tell you something about the nature of the tails of the distribution, if nothing else. If you are dealing with a normal distribution, nothing new would be conveyed.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Silly questions

neauveq
Posts: 10
Joined: May 28th, 2018, 10:25 pm

### Re: Silly questions

Thank you, both!

JackiBones
Posts: 1
Joined: December 13th, 2018, 6:31 am

### Re: Silly questions

With asset prices falling across all markets, proving them unstable, and yield curves of short to long run bonds inverting . What will the impact of  what Federal reserve does to interest rates in the coming quarter? If they continue to engage in quantitative tightening policy will we see the emergence of a bear market? If they go back on their progress in reducing the monetary base from 2009, will it save assets or only drive inflation? What are they most likely to choose, neither?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: Silly questions

the Fed would happily crash the economy in order to get rid of the president. It wouldn't be the first recession they've caused and it probably wouldn't be the last
The onion in the ointment for the Fed is that increasingly Wall Street has realized what is afoot and doesn't want a recession.

bearish
Posts: 5802
Joined: February 3rd, 2011, 2:19 pm

### Re: Silly questions

If it only were that easy to get rid of him! The Fed ought to continue along the long path toward normalization of interest rate levels and their own balance sheet. With the dynamic trio of Trump, Ryan and McConnell having blown out both the domestic deficit and the US trade balance, they are pretty much the only rational economic force operating at the federal level. And that, somewhat shockingly, has remained the case under Trump appointee Powell.

Cuchulainn
Topic Author
Posts: 62928
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Silly questions

Sometim deterministic + random functions is like mixing oil and water.

Let's take the nonlinear RODE

$dx/dt = - grad f(x) = -\nabla f(x) + \sqrt T dW$

Question; Can I do a 'splitting"

leg 1 al level n
$dz/dt = -\nabla f(z)$

leg 2 at level n+1 ($dz/dt$ known)
$dx/dt = dz/dt + \sqrt T dW$

Roughly, leg 2 looks like simulated annealimg with a non-homogeneous term $dz/dt$.

Is this mathematical heresy?
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

Cuchulainn
Topic Author
Posts: 62928
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Silly questions

Can I differentiate the following PDE with respect to $\sigma$ to get an initial boundary value problem for $\frac{\partial V}{\partial {\sigma}}$?
$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0,$

And if so, what at the BCs? Is vega defined at $t = 0$?
Last edited by Cuchulainn on April 9th, 2019, 3:34 pm, 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

Paul
Posts: 10796
Joined: July 20th, 2001, 3:28 pm

### Re: Silly questions

Yes. But you'll need to know gamma, it's a source term. (Typo in your PDE.)

Cuchulainn
Topic Author
Posts: 62928
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Silly questions

Yes, gamma enters the fray when I differentiate. Let's take $r = 0$ and $v$ be vega Then it satisfied the PDE (assuming mixing my derivatives are allowed..)

$\frac{\partial v}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 v}{\partial S^2} + \sigma S^2 \frac{\partial^2 V}{\partial S^2}=0$

So, we indeed need to know gamma in some way. What about about solving simultaneously for $V,v$ as a parabolic PDE system, like with chooser option PDEs?

I have never tried it nor seen it approached this way (on paper) before but with the binomial method (ouch) it can be done by bumping $\sigma$. And a tree is almost like a PDE, kind of.

At least, the parabolic system's matrix for $(V,v)$ is positive definite, so it's kind of hopeful.

Corollary: for $\rho$ we need to know $\Delta$ and $V$?
Last edited by Cuchulainn on April 19th, 2019, 1:27 pm, 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

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