# 93008 - Computational Finance

### Course Unit Page

• Teacher Pietro Rossi

• Credits 6

• SSD SECS-S/06

• Language English

• Campus of Bologna

• Degree Programme Second cycle degree programme (LM) in Quantitative Finance (cod. 8854)

• Course Timetable from Feb 14, 2022 to Mar 17, 2022

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

## Learning outcomes

The objective of the course is to address frontier topics in computational finance and muster the computational skills needed to tackle them. At the end of the course the student will have a working experience on selecting the most appropriate model and the capability to devise, and implement the numerical tools needed for problem at hand. The student will be able to judge the quality of the results from data analysis, and she will have enough knowledge to set up benchmarks to test for the accuracy of the numerical results. The topics addressed will refer to state-of-the-art issues in pricing, risk management and numerical techniques and programming languages. The student will be exposed, and gain proficiency, in rapid prototyping languages like Python and languages better suited to achieve high computational performance, like C/C++

## 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