Stochastic Calculus for Finance Ii

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1. pvitzJan 2021
   Someone with a strong math background should cut Wilmott and go directly to Shreve: Stochastic Calculus for Finance II (or Björk: Arbitrage Theory in Continuous Time).
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2. veyronOct 2011
   You need to be a little more specific regarding your interests and background. If you are interested in theory, there are plenty of books discussing specific models. However, most practical or buzzword (e.g. HFT) books are rubbish (think about it: if you have a profitable operation, why would you write a book? Unless, like me, you have a screw loose :P )

   If you want some practical discussions, "Trading and Exchanges" by Larry Harris is a bit dated, but nothing comes close. Most of the descriptions are still valid today (even if some of the mechanics have changed)

   If you want something more along the lines of what a financial engineer would know:

   - "Options, Futures, and other Derivatives" by John Hull (goes over basic models without delving too deeply into theoretical math aspects)

   - "Stochastic Calculus for Finance II" by Steve Shreve (goes over the basic stuff but has enough stats to keep a grad student happy)

   - "Monte Carlo Methods in Financial Engineering" by Paul Glasserman (much more practical, and goes over subtleties of monte carlo simulation and other stuff like low discrepancy sampling)

   - "Modelling Fixed Income Securities and Interest Rate Options" by Robert Jarrow (walks through how to perform certain simulations, covers lots of little details most theoretical books skip)

   If you want something more theoretical:

   - "Introduction to Stochastic Calculus Applied to Finance" by Lamberton and Lapeyre (Nice little intro)

   - "Arbitrage Theory in Continuous Time" by Tomas Bjork (Slower discussion, larger breadth)

   - "Brownian Motion and Stochastic Calculus" and "Methods of Mathematical Finance" by Karatzas and Shreve (Solid theoretical foundation, for the more mathematically inclined)

   There was a good non-measure theoretic discussion of financial models but the name escapes me ATM.

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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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