Abstract.

Notes on Quantitative Analysis in Finance.
Konstantin G. Aslanidi
Version of August 17, 2003 (since June 01, 2003).
    I wrote these notes for the same reason that regularly compels people to write notes. I do not claim that the Notes are in any way better then the text books written by known grandmasters. However, I hope that this text is fit to be an introduction to basic concepts. It may save you hours of time if you are a regular practitioner  (like I am) interested in fast getting in. I welcome constructive critical e-messages sent to the [email protected]. 
    I do not claim originality. Some of the proofs I moved from literature with slight changes and some I reinvented as I was struggling to understand nature behind clever proofs in various text books. Natural degree of rigor and a clearly motivated "brutal force" approach to every problem is maintained.
    The notes are aimed to cover ideas and techniques rather then statements and results. For this reason some important theorems are proved more then once and other standard statements are cited without proof or omitted.
    Please, mind the fact that these notes are a developing project. Some of the topics are omitted because I did not get to them. The date of the present version is kept on the front page and updated on every upload.
    Master level of mathematical sophistication is presumed.
    One thing that I can state with total confidence is that I can be wrong. There may be misprints and there may be mistakes. Use these notes on your own risk.
    Excuse my English.

Contents.

1. Basic Math

   1. Conditional probability.

      1. Definition.
      2. Example: a bomb on a plane.
      3. Example: the famous three door game.
      4. Example of Bayesian reasoning.
      5. Example of a recursive Bayesian computation.
      6. Example of backward induction.
      7. Conditional expectation. Filtration. Flow of information. Stopping time.
   
   

   2. Normal distribution.

      1. Definition.
      2. Calculus of normal variables.
      3. Central limit theorem (CLT).
      4. Linear transformation.
      5. Multivariate normal distribution. Choleski decomposition.
   
   

   3. Brownian motion.

      1. Definition.
      2. Example: Brownian motion passing through gates.
      3. Reflection principle.
      4. Example: Brownian motion hitting a barrier.
   
   

   4. Topics in stochastic analysis.

      1. Ito integral.
      2. Ito calculus.
      3. Change of measure.
      4. Girsanov's theorem.
      5. Forward Kolmogorov's equation.
      6. Backward Kolmogorov's equation.
      • Direct proof for one dimension.
      • Several dimensions.
      • Martingale proof.
      • Localized backward Kolmogorov's equation.
   
   

   5. Poisson process.

      1. Definitions.
      2. Distribution.
      3. Poisson stopping time.
   
   

   6. Integration of some SDEs.

      1. Black-Scholes formula.
      2. Mean reverting equation.
      3. Brownian bridge technique.



2. Pricing and Hedging.

   7. Basics of derivative pricing I.

      1. Why Ito process?
      2. Existence of the risk neutral measure via Girsanov's theorem.
      3. Existence of the risk neutral measure via backward Kolmogorov's equation. Delta hedging.
   
   

   8. Change of numeraire.

      1. Existence.
      2. The set up.
      3. A useful computation.
      4. Transformation of SDE via moment matching.
      5. Transformation of SDE via Girsanov's theorem.
      6. Transformation of SDE by direct term matching.
      7. Example. Change of numeraire in the Black-Scholes economy.
      8. Transformation of SDE. Useful formula.
   
   

   9. Basics of derivative pricing II.

      1. Option pricing formula for an economy with stochastic riskless rate.
      2. T-forward measure.
      3. Delta hedging in economy with stochastic riskless rate.
      4. HJM.
   
   

   10. Market models.

       1. Forward LIBOR.
       2. LIBOR market model.
       3. Swap rate.
       4. Swap measure.
   
   

   11. Topics in Currency Exchange.

       1. Exchange Rate.
          • Definitions.
          • Drift.
          • Change to foreign probability measure.
          • Forward exchange rate.
       2. Forward contract to purchase a foreign stock for domestic currency.
       3. Quanto forward contract.
       4. Quanto caplet.
       5. Fixed-for-floating quanto swap.
   
   

   12. Intensity based approach to credit risk.

       1. Stopping time.
          • Definition.
          • Handling conditional expectation.
          • Stochastic intensity.
          • Martingale properties of the stopping time.
       2. Single stopping time case.
          • Cash flow.
          • Expressing risk neutral expectation through the intensity of the default time.
          • Credit default swap.
          • Defaultable bond with certain recovery rate.
          • Reasonable parameters of the credit default swap.
          • Drift of defaultable contract.
       3. Basket credit derivatives.
       • Canonical construction of conditionally independent default times.
       • Definition of i-th-to-default basket contract.
       • Pricing of i-th-to-default contract under assumptions of independence.
       • Pricing of i-th-to-default contract under assumptions of conditional independence.
       • Pricing of first-to-default contract.
       4. Credit migrations. Discrete time Markov chain approach.



3. Data Analysis.

   13. Topics in Time Series.

       1. Forecasting.
       2. Updating a linear forecast.
       3. Kalman filter.
       4. Simultaneous equations.
   
   

   14. Basic concepts of classical statistics.

       1. Sample.
       2. Chi squared distribution.
       3. Student's t-distribution.
       4. Estimation.
          • Sufficient statistics.
          • Example. Sufficient statistic for the normal population.
          • Maximal likelihood estimation (MLE).
          • Asymptotic consistency of the MLE. Fisher's information number.
          • Asymptotic efficiency of the MLE. Cramer-Rao low bound.
       5. Pattern recognition.
          • Decision rule based on a loss function.
          • Hypothesis testing problem.
          • Neyman-Pearson Lemma.
   
   

   15. Topics in Bayesian statistics.

       1. Basic idea.
       2. Estimating the mean of a normal distribution with known variance.
       3. Estimating unknown parameters of a normal distribution.
       4. Hierarchical analysis of a normal model with known variance.



4. Implementation tools.

   16. Finite differences.

       1. Basics.
       • Definitions and the main convergence theorem.
       • Approximations of basic operators.
       • Stability of the general evolution equation.
       • Spectral analysis of the finite difference Laplacian.
       2. One dimensional heat equation.
       • The finite difference schemes.
       • Stability.
       • An effective way to solve the scheme: the factorization procedure.
       3. Two dimensional heat equation.
       • The Peaceman-Rackford (alternating directions) scheme.
       • Stability.
       4. One dimensional Black equation.
       • Transformation to the heat equation.
       • Localization.
       5. Two dimensional Black equation.
   
   

   17. Gauss-Hermite Integration.

   
   

   18. Asymptotic expansions.

   
   

   19. Generation of random samples.

       1. Uniform [0,1] random variable.
       2. Inverting cumulative distribution.
       3. The accept/reject procedure.
       4. Normal distribution. Box-Miller procedure.
       5. Gibbs sampler.

Bibliography.

Fixed Income and Foreign Exchange.

1. Damiano Brigo, Fabio Mercurio. Interest Rate Models. Theory and Practice.
2. Riccardo Rebonato. Interest Rate Models.
3. Lars Tyge Nielsen. Pricing and Hedging of Derivative Securities.
4. Marco Avellaneda, Peter Laurence. Quantitative Modeling of Derivative Securities.
5. Damien Lamberton, B. Lapeyre. Introduction to Stochastic Calculus Applied to Finance.
6. Steven Shreve's Lectures on Stochastic Calculus and Finance.

Credit.

1. Janet Tavakoli. Credit Derivatives & Synthetic Structures.
2. Tomasz R. Bielecki, Marek Rutkowski, Credit Risk: Modeling, Valuation and Hedging.

Programming.

1. Bjarne Stroustrup. C++ programming language.
2. Nicolai M. Josuttis. The C++ Standard Library.
3. Erich Gamma, Richard Helm, Ralph Johnson, John Vlissides. Design Patterns.
4. Scott Meyers. Effective C++: 50 Specific Ways to Improve Your Programs and Design.
5. Scott Meyers. More Effective C++: 35 New Ways to Improve Your Programs and Designs.
6. Scott Meyers. Effective STL: 50 Specific Ways to Improve Your Use of the Standard Template Library.
7. Cay S. Horstmann. Practical Object-Oriented Development in C++ and Java.
8. Ken Arnold, James Gosling, David Holmes. The Java Programming Language.
9. Kate Gregory. Using Visual C++ 6.
10. David Kruglinski, Scot Wingo, George Shepherd. Programming Visual C++ 6.0.

Relevant general theory books.

1. N.N.Gikhman, A.V. Skorokhod. Introduction to theory of stochastic processes.
2. James Hamilton. Time Series Analysis.
3. George Casella. Roger L. Berger. Statistical Inference.
4. Adrew Gelman, John B. Gaarlin, Hal S. Stern, Donald B. Rubin. Bayesian Data Analysis.
5. G.I. Marchuk. Methods of computational mathematics.
6. A.A.Samarski. Theory of finite differences.
7. John Hull. Options, Futures & Other Derivatives.
8. Paul Wilmott, Sam Howison, Jeff Dewynne. The Mathematics of Financial Derivatives: A Student Introduction.

Other free sources on the web of the kind.

1. http://quantlib.org/
2. Steven Shreve's Lectures on Stochastic Calculus and Finance.
   http://www-2.cs.cmu.edu/~chal/shreve.html
3. Philipp J. Schönbucher's Homepage. Contains articles of all levels on the subject of credit derivatives.
   http://www.finasto.uni-bonn.de/~schonbuc/
4. Marco Avellaneda’s web page. Contain Mathematical Finance I, II course notes.
   http://www.math.nyu.edu/faculty/avellane/index.html
5. Jonathan Goodman’s teaching page. Contains notes on computational methods of mathematical finance.
   http://www.math.nyu.edu/faculty/goodman/teaching/teaching.html
6. Robert V. Kohn’s teaching page. Contains notes from various math. finance classes.
   http://www.math.nyu.edu/faculty/kohn/index.html