PDEs or Stochastic Process? - Question from an Econ&Fin Undergraduate

[Raymond HKU]

Dear all,

I am new to this forum so let me introduce myself first. I am currently an undergraduate, double-majoring economics and finance, which are probably not a choice in terms of preparation for a quant. The reason why I want to become a quant is that I find it interesting to put some structures on financial securities and dynamics using mathematics and statistics. I have finished all introductory courses in mathematics (multivariable calculus, linear algebra, introduction to mathematical analysis) . Although I did not do very well in these courses (B/B+ for all), I decide to pursue a minor in math.

My question is: since I have to finish two majors, I don't really have much credits left for taking math. electives. I am just wondering whether I should take PDEs (of course, before that, I would take Differential equations and Numerical Analysis first) or stochastic processes (certainly, before that, it comes Probability theory). As far as I know, PDEs and martingales are two main approaches to pricing of derivatives. Deeper understanding of which of the two subjects - PDEs or Stochastic Process - is more beneficial in terms of understanding quantitative finance? Or, which of the two approaches - PDEs or martingales - are now dominant in the real world practice of derivatives pricing? Thank you for your attention and correct me if I have any misunderstanding about "quant" or "math. used by quant"!

Cheers,
Raymond.

 

[mastertrader]

If you want to do a MFE then you should probably take PDE. A lot of schools want to see it on your resume. You'll have a better understanding of Black-Scholes since it's derived from the Heat Equation. However, you'll take Stochastic Calculus as part of a MFE program. That said I recommend double majoring in Finance and Math or Applied Math while doing a Minor in Economics if you can instead. Almost any job you can get with an Economics degree you can also get with a Finance degree. An Applied Math and Finance degree opens up more doors.

 

[Daniel Duffy]

PDE is very big area. Not all PDE university PDE theory is directly relevant.

It is more important IMO to learn Numerical Analysis and then apply it to Finite Difference for PDE.

You'll have a better understanding of Black-Scholes since it's derived from the Heat Equation.
Don't forget 1st order hyperbolic convection PDE.

 

[themarketprofessor]

Dear all,

I am new to this forum so let me introduce myself first. I am currently an undergraduate, double-majoring economics and finance, which are probably not a choice in terms of preparation for a quant. The reason why I want to become a quant is that I find it interesting to put some structures on financial securities and dynamics using mathematics and statistics. I have finished all introductory courses in mathematics (multivariable calculus, linear algebra, introduction to mathematical analysis) . Although I did not do very well in these courses (B/B+ for all), I decide to pursue a minor in math.

My question is: since I have to finish two majors, I don't really have much credits left for taking math. electives. I am just wondering whether I should take PDEs (of course, before that, I would take Differential equations and Numerical Analysis first) or stochastic processes (certainly, before that, it comes Probability theory). As far as I know, PDEs and martingales are two main approaches to pricing of derivatives. Deeper understanding of which of the two subjects - PDEs or Stochastic Process - is more beneficial in terms of understanding quantitative finance? Or, which of the two approaches - PDEs or martingales - are now dominant in the real world practice of derivatives pricing? Thank you for your attention and correct me if I have any misunderstanding about "quant" or "math. used by quant"!

Cheers,
Raymond.

Hi Raymond,

I am going be honest and sincere in helping you with your bid in making study choices which may potentially change your life. After reading your question, i understand you're a fresh student and of course students certainly have a very very narrow picture of the real world. Before you seriously take a MFE or whatever course to become a quant, let me give a practical advice: The use of martingales/stochastic calculus was popular some 30 years ago, and along thru those years mathematical models (Local Vol, Heston, Hull-White, BGM, HJM, Cheyette, etc...you name it) had been developed for pricing derivatives also some 30 years ago. They are still being used by one bank or another, and they became standard and well embedded in bank in-house systems. None of the new models (Stoch Local vol, Uncertain Vol model, etc..) are embraced in the more challenging markets. Plus, almost nobody cares about martingale approach anymore. As for PDE, it's just a numerical physical exercise to get numerical value of a derivative given a model. You know what most of C++ developers can just copy and paste free source codes they found somewhere on the web and implement it and wah lah...they get the price of a derivative by the end of the day. Can you compete that?!

More importantly, market professionals have been talking about an overhaul of financial and mathematical theories (including martingale approach, etc...) for pricing and hedging financial derivatives.

So before you go on choosing a career path (becoming a quant is one), you should spend some good amount of time to know the real world and see what it is becoming, ask what are the current big problems in the financial markets and why? and you ask yourself what kind of knowledge should obtain NOW in order to secure a meaningful career in the future.

Good luck!

 

[themarketprofessor]

PDE is very big area. Not all PDE university PDE theory is directly relevant.

It is more important IMO to learn Numerical Analysis and then apply it to Finite Difference for PDE.

You'll have a better understanding of Black-Scholes since it's derived from the Heat Equation.
Don't forget 1st order hyperbolic convection PDE.

Hi Daniel,

I have to correct you here: Black-Scholes formula is NOT derived from a heat equation, but is formally derived by a martingale approach which is endowed with non-arbitrage condition and risk-neutral measure. Black-Scholes equation is a heat equation, so it cannot be derived from itself. Black-Scholes formula can be said alternatively represented by a heat-equation, certainly not derived from it.

I have to argue and make that clear so that young minds like Raymond donot get confused in the already-confusing financial world they are about to enter.

 

[Daniel Duffy]

Hi MPSSOR,
You are right; I quoted Raymond in italics but it looks like a quote of mine.

Maybe the 'confusion' is that books transform BS PDE to heat equation.

"Black-Scholes equation is a heat equation,"

Actually, BS PDE is a convection-diffusion-reaction PDE.

 

[Berenger]

...
I am going be honest and sincere in helping you with your bid in making study choices which may potentially change your life. After reading your question, i understand you're a fresh student and of course students certainly have a very very narrow picture of the real world. Before you seriously take a MFE or whatever course to become a quant, let me give a practical advice: The use of martingales/stochastic calculus was popular some 30 years ago, and along thru those years mathematical models (Local Vol, Heston, Hull-White, BGM, HJM, Cheyette, etc...you name it) had been developed for pricing derivatives also some 30 years ago. They are still being used by one bank or another, and they became standard and well embedded in bank in-house systems. None of the new models (Stoch Local vol, Uncertain Vol model, etc..) are embraced in the more challenging markets. Plus, almost nobody cares about martingale approach anymore. As for PDE, it's just a numerical physical exercise to get numerical value of a derivative given a model. You know what most of C++ developers can just copy and paste free source codes they found somewhere on the web and implement it and WAH LAH...they get the price of a derivative by the end of the day. Can you compete that?

More importantly, market professionals have been talking about an overhaul of financial and mathematical theories (including martingale approach, etc...) for pricing and hedging financial derivatives.
...

Quite a few outlandish remarks made by the professor here (in bold).. Anyone cares to confirm?
BGM is from 1997, so scarcely more than half the shelf life of 30 years he declares in the post which is from 2014
wah lah is probably meant to be voila'

 

[CrossGamma]

I am an options quant at a proprietary trading firm. The bulk of our time is currently indeed not spent on pure pricing work but more on the strategy and signal generation stacks. However, a solid understanding of pricing theory is still extremely useful and definitely a plus. While the basics ideas of the models we employ have indeed been developed many years ago, we constantly refine them to include many real-world effects that are often ignored by academics. I cannot give specifics here unfortunately. For American options, the corresponding implementations will typically be PDE-based. An understanding of the interplay between risk-neutral distributions and implied volatility is also very valuable when e.g. working on volatility models. I need to refute two more claims of the "professor": (i) we don't copy paste any pricing code off the internet and (ii) we constantly look into recent academic publications also on the pricing side to see whether we find ideas we could monetize.

Working as a quant at a trading firm is quite different from derivatives quants at banks, who indeed typically focus mostly on pricing. As opposed to marker makers, their models are often a bit "rougher" as they are used to derive values for exotics off the vanillas and not to be able to make an on-screen market in the vanillas themselves. Much of the interesting ground work has been done here already though and projects are more driven by regulatory requirements.

 

[Daniel Duffy]

I would say it is better to learn both PDE and SDE methods. They are two sides of the same coin. I fact there is a 1:1 correspondence as Einstein showed in 1905, See also the book by Friedman.

So, both are very handy.

http://stat.math.uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf

 

[Daniel Duffy]

You might be interested in knowing that a PDE/FDM book is on the way.

 

[Daniel Duffy]

As for PDE, it's just a numerical physical exercise to get numerical value of a derivative given a model. You know what most of C++ developers can just copy and paste free source codes they found somewhere on the web and implement it and wah lah...they get the price of a derivative by the end of the day. Can you compete that?!

This is so wrong, on several levels. It shows a lack of understanding of what quants do.

// Back to basics: a PDE is not a numerical procedure; maybe you are thinking about FDM (FEM, FVM).

 

[Berenger]

I am an options quant at a proprietary trading firm. The bulk of our time is currently indeed not spent on pure pricing work but more on the strategy and signal generation stacks. However, a solid understanding of pricing theory is still extremely useful and definitely a plus. While the basics ideas of the models we employ have indeed been developed many years ago, we constantly refine them to include many real-world effects that are often ignored by academics. I cannot give specifics here unfortunately. For American options, the corresponding implementations will typically be PDE-based. An understanding of the interplay between risk-neutral distributions and implied volatility is also very valuable when e.g. working on volatility models. I need to refute two more claims of the "professor": (i) we don't copy paste any pricing code off the internet and (ii) we constantly look into recent academic publications also on the pricing side to see whether we find ideas we could monetize.

Working as a quant at a trading firm is quite different from derivatives quants at banks, who indeed typically focus mostly on pricing. As opposed to marker makers, their models are often a bit "rougher" as they are used to derive values for exotics off the vanillas and not to be able to make an on-screen market in the vanillas themselves. Much of the interesting ground work has been done here already though and projects are more driven by regulatory requirements.

Thanks for the input, is "strategy and signal generation", in 1 line of your post, more in line with the controversial technical analysis theory in investment, as opposed to fundamental analysis? Both opposed to classic option pricing which is more of a sell-side thing, as you also notice in your last paragraph.

Summing up the pde vs martingales diatribe, Pde and finite differences are to be avoided for multi-asset products as the multidimensionality of the problem hits hard on computational efficiency. On the flip side, Montecarlo is not a backward method and thus has problems when pricing early-optionality (although approaches like the alpha-parameter method exist); finite differences are better there because you can start from expiry and crawl back to present, at each node evaluating whether exercising may be a good thing or not.

 

[Daniel Duffy]

Pde and finite differences are to be avoided for multi-asset products as the multidimensionality of the problem hits hard on computational efficiency
PDE up to dimension 3 are doable.
They weren't built for higher dimensions. It's a no-brainer to use MC in those cases.

Some of the ML community can solve PDE up to dimension 100 but that's a Grimm fairy tale.

 

[Berenger]

You might be interested in knowing that a PDE/FDM book is on the way.

You already wrote that book in 2006, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach
From the book summary you posted above, the impression is you are going to add to the 2006 material the contents of the video you did with Thalesians daniel-duffy-some-perspectives-on-computational-finance-and-ml

I knew Paul Wilmott's books deal with the pde-based approach in option pricing; there is his mammoth Paul-Wilmott-Quantitative-Finance-Set, and also I have read recommendations to use this leaner but older treatment The Mathematics of Financial Derivatives: A Student Introduction
Not much newer around because, as opposed to what the "market professor" says, the risk neutral martingale approach is indeed quite alive, and has largely superseded thinking of problems in terms of solving a heat equation (or something reducible to it, let's not get picky on terms)

 

[Daniel Duffy]

Au contraire; it's a 2021 PDE/FDM book! '
Did you miss the two Word files I posted here.

Forget transforming to a heat equation, is not even wrong. It's a false sense of security. It's dumbing down.

Anyway, linear two-factor PDEs are of convection-diffusion-reaction type with mixed derivative terms.

 

[Berenger]

why is wrong, just transform Black-Scholes into a heat equation, transform the boundary conditions as well, then solve using the Fourier transform to find a solution, and revert back to the original Black-Scholes variables. If the domain shape is intractable, use finite differences. Do I need your book? The professor would approve of my line of reasoning, if he cared to reply.

 

[Daniel Duffy]

why is wrong, just transform Black-Scholes into a heat equation, transform the boundary conditions as well, then solve using the Fourier transform to find a solution, and revert back to the original Black-Scholes variables. If the domain shape is intractable, use finite differences. Do I need your book? The professor would approve of my line of reasoning, if he cared to reply.

BS PDE does not have/need boundary condition, but the FDM does need numerical BC. The above reasoning is outdated and crude, I must say.
Maybe ask your prof then. Or a PDE quant.
Send him/her my TOC and preface. I am available 24/7 for feedback.

Or ask a PDE quant to discover what the real challenges are.
My MSc students' theses at the University of Birmingham and UCB know how to do it all.

A snippet..


The chapters in this part fill a major gap in the application of PDE/FDM to finance. In general, most of the finance literature glosses over the niceties of analysing PDEs mathematically before approximating them using the finite difference method. The new approach resolves many of issues and heuristic approaches. Some new improvements for two-factor PDEs are:

. Transform a PDE with a mixed derivative term to one in which this term has been removed (the canonical pde).

. Why domain transformation is better than domain truncation in general.

. A rigorous set of mathematical techniques (Fichera theory, energy estimates) to discover the correct boundary conditions for finance problems.

. The deep relationship between PDEs and stochastic differential equations (SDEs). We discuss formulations and results that are important in calibration applications.

 

[Berenger]

BS PDE does not have/need boundary condition, but the FDM does need numerical BC. The above reasoning is outdated and crude, I must say.
Maybe ask your prof then. Or a PDE quant.
Send him/her my TOC and preface.

The option payoff acts as boundary (terminal) condition at expiry.

Perhaps the market professor will reply.

 

[CrossGamma]

Thanks for the input, is "strategy and signal generation", in 1 line of your post, more in line with the controversial technical analysis theory in investment, as opposed to fundamental analysis? Both opposed to classic option pricing which is more of a sell-side thing, as you also notice in your last paragraph.

A signal is something that is tries to predict a future change in a trade-able risk factor. Different companies will have slightly different terminologies though. If you e.g. consider implied volatility to be mean reverting then a current implied volatility above historical averages would induce a sell signal and vice versa. This is of course too simplistic to actually work in practice.

 

[Berenger]

A signal is something that is tries to predict a future change in a trade-able risk factor. Different companies will have slightly different terminologies though. If you e.g. consider implied volatility to be mean reverting then a current implied volatility above historical averages would induce a sell signal and vice versa. This is of course too simplistic to actually work in practice.

Interesting, if trades effected based on indicators such as mean-reverting implied volatility in your example above, were manageable by a human trader in terms of volumes and frequency, I reckon that trader would be following Technical Analysis