I am looking for good books to learn quantitative finance. As I have strong background in physics, I would appreciate introductions that do not hesitate to show the equations, but in the same time cover the finance rather comprehensively. Most of what I have seen up till now errs either a) in the direction of explaining elementary probabilistic concepts, or b) towards formal math/statistics, or c) giving just a gist of it.
Physicists typically know PDEs but not stochastic calculus
I have a masters in physics, so have a reasonable idea of the usual skillsets a physicist will know (at least at undergraduate level), and also then a masters in mathematical finance, so learnt the hard way the bits of maths physicists typically don't know but will need to know for quantitative finance.
Typically physicists are very strong with linear algebra, and PDEs, but the world we work in is largely deterministic (I'll overlook QM for now) and we rarely do much with distributions. If you are happy enough to have a 5-minute overview of Ito calculus and focus on the PDEs that appear in finance and options pricing, then it is possible to take a very PDE centered approach. A great book in this regard is the book The Mathematics of Financial Derivatives (1995) by Wilmott, Howison, and Dewynne.
If you want to know stochastics
If you take the PDE approach then much of quantitative finance will be inaccessible to you, as you can only go a small way before learning Ito calculus is required. A great resource I found for this was Introduction to stochastic calculus with applications by Klebaner. This will give you pretty much all the stochastic calculus skills you will need.
Some more advanced stochastics and control theory
At this point you will be able to go into much of quantitative finance (or at least have the core skills to). However there are some branches where I think you will need a fair chunk more of mathematics, and the biggest is likely control theory (and the HJB equations), for which there are only really graduate books, and the best I can think of is Stochastic Controls: Hamiltonian Systems and HJB Equations by Yong and Zhou.
So far all of this is largely focussed on financial modelling, but from a theory based perspective rather than from an empirical or statistical perspective. Of course a huge number of hedge funds (and investment banks) model financial behaviour through statistical trends, or even just through blackbox machine learning. A great book for time series and statistics is Introduction to Time Series and Forecasting by Brockwell and Davis, and the standard book (amongst several) for statistics and machine learning is The Elements of Statistical Learning by Friedman, Tibshirani, and Hastie.
At this point you can now cover the main items including: options pricing, fixed income, statistical arbitrage, time series modelling, numerical methods, optimal control, etc.
Since you didn't study measure theoretic probability, that would be the first thing I recommend. In my opinion that's the main gap that many physicists on math side, because stochastic calculus is not in mainstream physics curriculum.
Whether you first study measure theory in calculus then take on probability, or jump right into measure theoretic probability is up to you. I took the first approach:
- I studied Kolmogorov's classical text in Russian. It's very clearly written, and surprisingly accessible to non mathematicians. I had one of my math professors help me digest the content.
- Then I took PhD course with Billingsley's text "probability and measure," which covers both subjects at once. I think it's possible in principle to learn both following this book, but I had a feeling that everyone in the class room knew measure theory, sets etc.
- I also took a PhD seminar on continuous stochastic calculus and we used Shreve's text's second volume. Again, it is not impossible to start with this book, but it's written for mathematicians, unless you're a theoretical or math physicist it will not be a comfortable read.
If you want to follow this path then I recommend enrolling/auditing PhD classes on this subjects in a local university.
A completely different approach would be to start from the end, e.g. read Wilmott's three volume book, Hull's "options..." text or Neftci's stochastic calculus text. I've seen people going this route too. It depends on your background and how much time you allocate for this project.
Then you need to study finance itself. That's a whole different ball game. If you have funds and time, then maybe getting MBA or CFA Level 1 exam is the most comprehensive approach.
As a long practicing plasma physicist who moved into quantdom (now retired), I suggest focusing on stochastic calculus and modeling. How deep you go down the rabbit hole of measure theory will depend on what you do. Simulation will be your friend and help you in many situations. To the excellent suggestions above, I add Paul Glasserman's Monte Carlo Methods in Financial Engineering. Build up a repertoire of solved derivative models as soon as you can. Have Fun, I did. ntgladd
As a former physicist you will certainly enjoy Jean-Philippe Bouchaud’s approach. Pragmatic and empirical with simple models that are sophisticated enough to be useful.
Check out “Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management” and “Trades, Quotes and Prices: Financial Markets Under the Microscope” in that order.