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Cuchulainn
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Re: PWOQF, expected hitting time

May 24th, 2021, 10:37 am

Alan/Cuch:
Honestly, with greatest respect to the PDE gurus here, these days I generally do anything in the world to avoid solving PDEs, and usually try martingale approaches. Just my 2 cents, ha ha.
BTW Alan your link is really great.
I reached that point approximately 30 years ago.
OK, amateur armchair cultural anthropology time:-)

Why do quants like closed-form solutions?
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Orbit
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Re: PWOQF, expected hitting time

May 24th, 2021, 4:20 pm

Cuch, For me I think of pencil-and-paper solutions of PDEs as being really, really easy to screw up. But that's just me. Think of the original solution to Black-Scholes. There are a thousand ways to goof up on the way toward transforming into the heat equation.

BTW I would prefer your Plan C, ha ha!!!
 
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Paul
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Re: PWOQF, expected hitting time

May 24th, 2021, 4:59 pm

Cuch, For me I think of pencil-and-paper solutions of PDEs as being really, really easy to screw up. But that's just me. Think of the original solution to Black-Scholes. There are a thousand ways to goof up on the way toward transforming into the heat equation.
That's interesting because I would say that there are far more ways to goof up other methods!

Anyway, unless it's an exam question you might not have to transform anything. 

If your problem is novel then you should use whatever tools are applicable. Without prejudice! 

I find the following with martingale methods:

1. If the problem is trivial then you can use them. 

2. If the problem is non trivial then you usually end up with "...and then you solve by Monte Carlo simulation." So why bother with all that heavy machinery if the destination is always the same?

And the second case is usually only when you have the possibility of hedging. Which is pretty much all that martingalists ever consider!
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 24th, 2021, 6:53 pm

Cuch, For me I think of pencil-and-paper solutions of PDEs as being really, really easy to screw up. But that's just me. Think of the original solution to Black-Scholes. There are a thousand ways to goof up on the way toward transforming into the heat equation.

BTW I would prefer your Plan C, ha ha!!!
1. "Screw up"... how do you get to Carnegie Hall? work out each step, 1 per line (See mundane example below, at least it works).
2. What's the compelling reason to want a heat equation??
3. Why do quants love closed-form solutions?

It is better to solve one problem five different ways, than to solve five problems one way.
George Pólya.
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bearish
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Re: PWOQF, expected hitting time

May 24th, 2021, 10:45 pm

One doesn’t have to work too hard to find cases in the forum where an exact solution is presented, only to be met with a wide-eyed, slack-jawed response of “but, but it’s not a PDE!”

I find exactly zero intuition in the PDE approach, although absolutely appreciate the engineering benefits of leveraging hundreds of years of research and development in terms of providing numerical solutions. The probabilistic (aka martingale) approach is a lot more intuitive, especially when it comes to locating convexity effects. And, frankly, once Monte Carlo is called for, it is usually because of some path dependency that tends to complicate the PDEs too. And I don’t actually think the possibility of hedging plays a first order role in the choice of approach. Once you have a well defined valuation rule, a natural choice of numeraire will usually present itself.
 
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Paul
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Re: PWOQF, expected hitting time

May 25th, 2021, 1:32 am

bearish:

I find exactly zero intuition in the PDE approach, although absolutely appreciate the engineering benefits of leveraging hundreds of years of research and development in terms of providing numerical solutions. The probabilistic (aka martingale) approach is a lot more intuitive, especially when it comes to locating convexity effects. And, frankly, once Monte Carlo is called for, it is usually because of some path dependency that tends to complicate the PDEs too. And I don’t actually think the possibility of hedging plays a first order role in the choice of approach. Once you have a well defined valuation rule, a natural choice of numeraire will usually present itself.
Classically the "well defined valuation rule" follows from a clever, indeed Nobel Prize winning, hedging argument. Without that...

By "change of numeraire" you mean first-year undergrad change of (dependent) variables? You're not fooling anyone with this foreign stuff!
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 25th, 2021, 7:54 am

One doesn’t have to work too hard to find cases in the forum where an exact solution is presented, only to be met with a wide-eyed, slack-jawed response of “but, but it’s not a PDE!”

I find exactly zero intuition in the PDE approach, although absolutely appreciate the engineering benefits of leveraging hundreds of years of research and development in terms of providing numerical solutions. The probabilistic (aka martingale) approach is a lot more intuitive, especially when it comes to locating convexity effects. And, frankly, once Monte Carlo is called for, it is usually because of some path dependency that tends to complicate the PDEs too. And I don’t actually think the possibility of hedging plays a first order role in the choice of approach. Once you have a well defined valuation rule, a natural choice of numeraire will usually present itself.
Knock me over with a feather. No offence, I have found econometry and PDE live in different worlds.
If you study Paul's oeuvre you will find lots of financial motivation for PDEs; I particularly like the UVM chapters.
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 25th, 2021, 8:09 am

[2. If the problem is non trivial then you usually end up with "...and then you solve by Monte Carlo simulation." So why bother with all that heavy machinery if the destination is always the same?

And to add insult to injury, the treatment of MC us relegated to a 1/2 page and a cute plot. We cannot fact check it.
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 25th, 2021, 8:28 am

I find exactly zero intuition in the PDE approach

read

Friedman, A. (1976) Stochastic Differential Equations and Applications. Dover New York. (SDE<-> PDE are like kinda in 1:1 correspondence)
Friedman, A. (1982) Variational Principles and Free-Boundary Problems. Dover New York.
Friedman, A. (1992) Partial Differential Equations of Parabolic Type. Dover New York.
Lewis, A. L. (2000) Option Valuation under Stochastic Volatility. Finance Press Newport Beach California.
Lewis, A. L. (2016) Option Valuation under Stochastic Volatility II. Finance Press Newport Beach California.
Wilmott, P. (2006) Paul Wilmott on Quantitative Finance Wiley.
Wilmott, P., Lewis, A., and Duffy, D. (2014) Modelling Volatility and Valuing Derivatives under Anchoring, Wilmott Magazine 73: pp. 48-57.

paraphrasing
..close connection between statistics, stochastic processes and Monte Carlo simulation on the one hand and PDE theory on the other hand. In a sense there a one-to-one correspondence between the solution of a pde and its stochastic representation. Furthermore, each representation can be approximated by numerical methods, for example by Monte Carlo simulation of stochastic differential equations and of course by the finite difference method.
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 26th, 2021, 11:29 am

A related heat equation is transforming time-dependent boundaries (many have forgotten chain rule). It causes much agony.

How many students know the chain rule, let's say transforming from (x,t) to (z, tau)

z = z(x,t) = x/s(t)
tau = t

Now transform heat equation/Stefan problem

u_t = u_xx for 0 < x < s(t)

to a PDE in z and tau for 0 < z < 1.

// 95% get it wrong first time.

Hint; use the chain rule and z is a function of TWO variables.
 
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Orbit
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Re: PWOQF, expected hitting time

May 26th, 2021, 8:16 pm

Ok I feel as though I've started some kind of shitstorm.
I just wanted help with an integral.
Sorry.
 
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Paul
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Re: PWOQF, expected hitting time

May 26th, 2021, 9:06 pm

The main thing is have you understood where the integral comes from and how that integration by parts was achieved?
 
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Cuchulainn
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Re: PWOQF, expected hitting time

May 27th, 2021, 8:14 am

Ok I feel as though I've started some kind of shitstorm.
I just wanted help with an integral.
Sorry.
It's a minor incident. On the positive side, the root cause is now clear.
As William Feller said, better to be wrong than dithering.
 
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Orbit
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Re: PWOQF, expected hitting time

May 28th, 2021, 12:20 am

The main thing is have you understood where the integral comes from and how that integration by parts was achieved?
Yes. Yes, I do.
 
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bearish
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Re: PWOQF, expected hitting time

May 28th, 2021, 12:47 am

The main thing is have you understood where the integral comes from and how that integration by parts was achieved?
Yes. Yes, I do.
I think the proper line is “Thank you sir! May I have another?”