98777 - COMPUTATIONAL FINANCE

Learning outcomes

In this course fundamentals of numerical methods are presented with focus on applications in mathematical finance.

Course contents

1 Generating Random Variables
  1.1 Introductions
  1.2 The Distribution Function
  1.3 The Normal Distribution
    1.3.1 The meaning of (0, 1)
  1.4 How to approximate an integral
  1.5 The weak law of large numbers
  1.6 The central limit theorem
  1.7 Acceptance Rejection Method

2 The Black and Scholes Model
 2.1 Introducing Interest Rates and Dividend Yields .
 2.2 Vanilla Options. The MC approach . . . . . . . .
  2.2.1 The European Put . . . . . . . . . . . . .
  2.2.2 Analytical Computation . . . . . . . . . .
  2.2.3 The European Call . . . . . . . . . . . .
  2.2.4 The Call-Put Parity . . . . . . . . . . . .
  2.2.5 Cap and Floor Option . . . . . . . . . .
  2.2.6 Asian Option . . . . . . . . . . . . . . . .
  2.2.7 Average Strike Option . . . . . . . . . .
 2.3 Vanilla Options. The Finite Difference Approach

3 Multi Dimensional Log Normal Processes
 3.1 An Example . . . . . . . . . . . . . . . . . . .
  3.2 The evolution . . . . . . . . . . . . . . . . . .
  3.3 Generation of a multivariate Gaussian

4 Numerical Utilities
   4.1 A Crash course on Interest rate
      4.1.1 Modeling the discount curve
  4.2 Building a discount curve
     4.2.1 Starting from the data
     4.2.2 Yearly compounded interest rates
   4.3 The Volatility Surface

5 The CIR model
   5.1 Introduction
   5.2 Numerical CIR for poets
  5.3 A helpful result . . . . . .

6 The Hull+White model
  6.1 The 1-factor Hull-White model
  6.2 The Gaussian HW Model
     6.2.1 Solution of the Gaussian HW Model
     6.2.2 The unbiased evolution algorithm
     6.2.3 The Distribution Parameters
     6.2.4 The Zero Coupon Bond

7 Jumps
  7.1 Basic facts
     7.1.1 The martingale process
  7.2 Building martingales
  7.3 Application
     7.3.1 Jumps
     7.3.2 Jump diffusion processes
     7.3.3 The Merton model
     7.3.4 The binomial model
     7.3.5 Bernoulli model
     7.3.6 Kou double exponential model

8 The Variance Gamma Model
8.1 The Variance Gamma Model
8.1.1 A time changed diffusion process
8.1.2 Vg as time changed Brownian motion
8.1.3 The characteristic function
8.1.4 The martingale process
8.1.5 The Characteristic function

9 The Heston Model
  9.1 The Heston Model
  9.2 The MC method
  9.2.1 Numerical simulation
  9.3 The Finite Difference method

10 Numeraire change: a recipe
  10.1 The standard story
  10.2 Generalized Vanilla Option Pricing Equation
  10.3 One dimensional formulation
  10.4 A simple change of measure
  10.5 The SDE for P (t, T )
  10.6 Evolution in the terminal measure
  10.7 Bond Options in the terminal measure

11 First Passage Time
  11.1 Ito calculus
    11.1.1 Basic definition
    11.1.2 The Ito’s theorem in one dimension
    11.1.3 Stochastic differential equations
  11.2 Feynman-Kac Formula
    11.2.1 Introduction
    11.2.2 Solution
  11.3 The Kolmogorov equations for the transition probabilities
    11.3.1 The backward Kolmogorov equation
    11.3.2 The Forward Kolmogorov equation
  11.4 The diffusion equation
    11.4.1 Solution of the equation
    11.4.2 Boundary conditions
    11.4.3 Finite distance barriers
  11.5 Passage Time
  11.6 Vanilla Options
    11.6.1 Notation
    11.6.2 High Barrier
    11.6.3 Low Barrier

12 American Options
  12.1 A Different View on Pricing
  12.2 The America Put Option
  12.3 Binary Trees
  12.4 Finite Difference methods
  12.5 Going Backward. The Longstaff Schwartz method
    12.5.1 Interpolation. A Quick Look
    12.5.2 The Least Square Problem
  12.6 The Forward MC

Readings/Bibliography

[1] Paul Glasserman, Monte carlo methods in financial engineering, vol. 53, Springer Science & Busi-
ness Media, 2013.
[2] Peter Jäckel, Monte carlo methods in finance, vol. 71, J. Wiley, 2002.
[3] Samuel Karlin and Howard M Taylor, A first course in stochastic processes, vol. 1, Academic Press,
1975.
[4] Donald Ervin Knuth, The art of computer programming, vol. 3, Pearson Education, 1997.

Assessment methods

At the end of the class, students will receive a couple of exercises. These should be addressed by small teams of two students working together towards the solution. Then each student will have to submit a short paper ( 7/10 pages max ) discussing the results of the work.

Office hours

See the website of Pietro Rossi