Preparation for Stochastics Calculus Interview?

[Once_Set]

I often get asked stochastic calculus questions, and I read the book by Baxter and Rennie, and it's not helpful in that context.

In the interviews, they just want you to manipulate correlated processes, various stochastic integrals, different SDEs...

Does anyone know any good books/materials to practise applying basic Stochastic Calculus techniques?

thanks!

 

[Ole Bueker]

Shreve's Stochastic Calculus for Finance II: Continuous Time ( http://www.springer.com/mathematics/quantitative finance/book/978-0-387-40101-0 ) is the most frequently used, but can be a bit tough (the exercises aren't exactly easy) but since you are a math PHD you shouldn't have too many problems.

If you're have too many problems with Shreve's book, I would recommend Arbitrage Theory in Continuous Time by Bjoerk (http://www.amazon.com/Arbitrage-Theory-Continuous-Oxford-Finance/dp/019957474X ). Also contains many exercises, which are easier to solve.

If you find these books to easy, I would recommend some pure mathematical books focused on Stochastic Calculus. Depending on your mathematical maturity, here are some more books (sorted from easy to more difficult based on my opinion):

1. Introduction to Stochastic Calculus by Klebaner
2. An Introduction to Continuous-Time Stochastic Processes by Capasso and Bakstein
   
3. Stochastic Differential Equations by Øksendal
4. Introduction to Stochastic Integration by Chung and Williams
5. Adventures in Stochastic Processes by Rednick
6. Stochastic Calculus A practical Introduction by Durrett

From here on, these books are more dry and I wouldn't recommend them if you are only interested in interviews. These would be for math students who are looking to pursue a graduate degreee in the field of Probability / Stochastics.

1. Stochastic Calculus by Prokhorov and Shiryaev
   
2. Stochastic Processes by Doob
   
3. Brownian Motion and Stochastic Calculus by Karatzas and Shreve
   
4. Continuous Martingales and Brownian Motion by Revuz and Yor

Many of the authors should be familiar to you if you're a mathematics student.
Best of luck in your interviews (and studies)!

 

[Once_Set]

Thank you! that's a great list, I have heard of Shreve's and Bjoerk's books but wasn't sure whether they were applied enough.

I assume Baxter and Rennie would be easier than any of the others? I am thinking of buying Bjoerk then, but does he have lots of very applied problems, with solutions?

Thanks!

 

[Ole Bueker]

Yes, the book by Baxter and Rennie is easier than Shreve/Bjoerk. I would say it's around the same level as Klebaner, maybe even easier (although the book by Baxter and Rennie is more about general introduction to finance, and Klebaner is solely for Stochastic Calculus).

For Bjoerk's book, every core chapter (which isn't marked with a * in the table of contents) has around 5-20 exercises, and I think there's a solution manuals somewhere on the internet.

BTW, I've updated my post above because I forgot one very good book (Stochastic Differential Equations by Øksendal).

 

[yetanotherquant]

I have heard of Shreve's and Bjoerk's books but wasn't sure whether they were applied enough.

These are rather theoretical (though in no way "ivory tower") books. But you have asked to recommend you a book for a Stochastic Calculus Interview, not for practical application, haven't you?
And these are the books that I, myself, used for self-study while doing my Master.
I also read Baxter and Rennie but I do not like it, imho they tried to make it as simple as possible and finally oversimplified.

I would also recommend you my tutorial, which also covers some important aspects: change of numeraire, multidimensional Girsanov theorem, T-Forward measures. There are also a couple of exercises with solutions:
http://www.yetanotherquant.de/libor/tutorial.pdf

 

[bigbadwolf]

I would say it's around the same level as Klebaner, maybe even easier (although the book by Baxter and Rennie is more about general introduction to finance, and Klebaner is solely for Stochastic Calculus).

There's no comparison. Other than Klebaner I also recommend Wiersema's Brownian Motion Calculus.

 

[Daniel Duffy]

Why no mention of how to compute solutions of SDEs? Like Kloeden and Platen?

http://sdetoolbox.sourceforge.net/

 

[Daniel Duffy]

There's no comparison. Other than Klebaner I also recommend Wiersema's Brownian Motion Calculus.

Ubbo Wiersema's book is quite practical IMO. Kind of 1/2 way between measure theory (which is almost a dead-end) and Euler's scheme.

http://www.wilmott.com/messageview.cfm?catid=8&threadid=95199&FTVAR_MSGDBTABLE=

 

[Ole Bueker]

I didn't mention them since he specifically asked about Stochastic Calculus, and didn't say anything about actual computational approaches. If you mention Kloeden and Platen, we can of course also mention (my favourite) Glassermann.
Wiersema's book looks quite practical, also contains many exercises (which seem to be at a similar level of Oksendal's book).

 

[Daniel Duffy]

I didn't mention them since he specifically asked about Stochastic Calculus, and didn't say anything about actual computational approaches. If you mention Kloeden and Platen, we can of course also mention (my favourite) Glassermann.
Wiersema's book looks quite practical, also contains many exercises (which seem to be at a similar level of Oksendal's book).

Glasserman's book indeed is essential IMO.

 

[yetanotherquant]

measure theory (which is almost a dead-end)

I would not be so pessimistic about measure theory.
The problem with it is that there are very few lecturers that can teach it properly with financial math in mind.
If you are taught by an analysis Prof, he will probably tell you the Lebesque story with a semiring on \mathbb{R}, which is very intuitive. But he will tell you nothing on filtrations.
And if you are taught by a probabilist, he will definitely start with Caratheodory construction: elegant, concise ... and at the first glance totally counterintuitive...
In my notes on measure theory I try to marry the best ideas from analysis and probability:
http://www.yetanotherquant.de/#MeasureTheoryBook

I am not sure whether one definitely needs measure theory to be a quant. But if you what to understand the risk-neutral pricing deeply then IMO the measure theory is a must.

 

[Daniel Duffy]

I would not be so pessimistic about measure theory.
The problem with it is that there are very few lecturers that can teach it properly with financial math in mind.
If you are taught by an analysis Prof, he will probably tell you the Lebesque story with a semiring on \mathbb{R}, which is very intuitive. But he will tell you nothing on filtrations.
And if you are taught by a probabilist, he will definitely start with Caratheodory construction: elegant, concise ... and at the first glance totally counterintuitive...
In my notes on measure theory I try to marry the best ideas from analysis and probability:
http://www.yetanotherquant.de/#MeasureTheoryBook

I am not sure whether one definitely needs measure theory to be a quant. But if you what to understand the risk-neutral pricing deeply then IMO the measure theory is a must.

I got my measure theory courses from a prof who had been a PhD student of William Feller at Princeton. We used it for Lebesgue integral and probability in Banach spaces.

I only once saw an article on computational measure theory.

 

[yetanotherquant]

We used it for Lebesgue integral and probability in Banach spaces.

And they use it for the interest rate modeling
http://www.amazon.com/Interest-Rate-Models-Dimensional-Perspective/dp/3540270655/

I only once saw an article on computational measure theory.

I saw a book on computability / theoretical informatics, which was heavily measure-theoretic and topological.

What I would like to say is that in practice a quant can do in more 90% without measure theory.
But those who want to deeply understand such things as risk-neutral pricing or T-Forward measures in LIBOR model need it.

 

[Daniel Duffy]

And they use it for the interest rate modeling
http://www.amazon.com/Interest-Rate-Models-Dimensional-Perspective/dp/3540270655/


I saw a book on computability / theoretical informatics, which was heavily measure-theoretic and topological.

What I would like to say is that in practice a quant can do in more 90% without measure theory.
But those who want to deeply understand such things as risk-neutral pricing or T-Forward measures in LIBOR model need it.

I always got the impression that parsimonious (few factor) model like HW, CIR and BDT were more popular with practitioners.

 

[yetanotherquant]

I always got the impression that parsimonious (few factor) model like HW, CIR and BDT were more popular with practitioners.

Likely they really are.
But the LIBOR model is also often used in practice. (At least I did use it )

 

[Daniel Duffy]

Likely they really are.
But the LIBOR model is also often used in practice. (At least I did use it )

Of course
But IR in infinite-dimensional Banach spaces is sometime in the future?

 

[Once_Set]

What level would be expected from an entry level quant? If one had to put a range on it...

For example, is it enough to have read rennie and baxter and then solved some SDEs, Stochastic integrals, applied some of that stuff to risk neutral pricing, checked whether something is a martingale?

Would I really need the more advanced books towards the end of the list?

 

[yetanotherquant]

For example, is it enough to have read rennie and baxter and then solved some SDEs, Stochastic integrals, applied some of that stuff to risk neutral pricing, checked whether something is a martingale?

IMO it should be enough for an entry level. But it really depends. In either case do not expect to cope with these tasks after reading only Baxter&Rennie. Shreve (if you read it with pencil in hand) should do.
And they also like testing, so to say, your Passion for the markets. A not uncommon question is where the DJ30 currently is. Or what is the most complicated derivative you know. Or...

 

[Once_Set]

I had an interview recently where the guy said "Shreve is definitely not enough, I need you to study book X, Y and Z before you come back to me next time"

Who has time for that? There are so many other things to study as well, apart from stochastic calculus! If you developed this kind of knowledge in all fields that are required, it would take a year.

 

[bigbadwolf]

I had an interview recently where the guy said "Shreve is definitely not enough, I need you to study book X, Y and Z before you come back to me next time"

Who has time for that? There are so many other things to study as well, apart from stochastic calculus! If you developed this kind of knowledge in all fields that are required, it would take a year.

Many years. The interviewer was either an imbecile and/or should have exclusively been interviewing people with PhDs in stochastic calculus. The MFE is not designed to engender this kind of in-depth stochastic expertise (it's also an open question to me about how applicable this stochastic calculus is in the world of finance).