The Concepts and Practice of Mathematical Finance

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1. fatdogSep 2016
   Mark Joshi's "The Concepts and Practice of Mathematical Finance" came recommended to me by some people in the field as a foundation. I found it quite readable.
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2. shogunmikeOct 2011
   If by 'quantitative finance' you mean option pricing then check out the books by Mark Joshi:

   - 'The Concepts and Practice of Mathematical Finance' - 'C++ Design Patterns and Derivatives Pricing'

   Also of note is Baxter & Rennie:

   - 'Financial Calculus: An Introduction to Derivative Pricing'

   Once you've studied those and have a good grasp of Measure Theory, you'll want to tackle Shreve, Vol II.

   And a brief plug of my (slightly out of date!) quant finance website, Quantstart.com.

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3. backlash875Nov 2019
   Are you looking for a proof of Girsanov's theorem or an explanation of how it is used to price? A good reference is Oksendal [0], but it's still quite tough going. As for how it's used, a good reference is Shreve's second volume [1], which also contains a proof of Girsanov's theorem. Joshi's book [2] is a little bit lighter on the mathematical rigour.

   A quick explanation of how it's used:

   Taking the stochastic differential equation for geometric Brownian motion, apply Girsanov's theorem to change measure via a drift change such that we now have a discounted stock price that is a martingale. The discounted stock price is the stock price divided by a short term bond or cash account asset. In this new measure the discounted short term bond/cash account asset is also trivially a martingale since it's being divided by itself. So we have that our two key assets (discounted) are martingales. We then define the time zero price of the option (divided by the time zero price of the bond/cash asset) to be the discounted expected value of its value at maturity in this newly constructed measure. By construction this discounted option price is a martingale and we now have three assets that are all martingales which implies there is no arbitrage possible. With this option price, called the "risk neutral" price, no arbitrage is possible under our newly constructed measure, but, because the original measure is an equivalent measure no arbitrage is possible here in the "real world" either and so this is our actual price.

   I appreciate there are a few steps here that seem like a bit of a leap. It took me a while to appreciate them. The key things to appreciate are:

   How everything being a martingale implies a lack of arbitrage. Girsanov allows you to make your (discounted) underlying a martingale.

   How you can then just make the option price a martingale by construction. And then how lack of arbitrage under one measure means lack of arbitrage in any equivalent measure.

   The discounting can also be a little confusing, but it's really just incorporating the time value of money into the calculations.

   [0] Oksendal B . Stochastic Differential Equations. [1] Shreve S E. Stochastic Calculus for Finance II. [2] Joshi M S. The Concepts and Practice of Mathematical Finance

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4. nadamMar 2012
   Thanks!

   Currently I am reading these books:

   Mark Joshi: The Concepts and Practice of Mathematical Finance

   Daniel Duffy: Introduction to C++ for Financial Engineers (I already know C++, but I read this because of the finance related stuff.)

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