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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
1. Black-Scholes formula.
2. Change of variables for Kolmogorov equation.
3. Mean reverting equation.
4. Affine SDE.
A. Ricatti equation.
B. Evaluation of option price.
C. Laplace transform.
D. Example: CDFX model.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
9. Hamilton-Jacobi Equations.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Affine SDE.

he reference for this section is [Duffie1999] .

We consider $N$ -dimensional process $X_{t}$ given by the equations

MATH (Affine equation ab)
where the $a_{t}$ and $b_{t}$ are column and matrix valued functions of $X_{t}$ . We aim to calculate the characteristic function MATH Observe that MATH is a $t$ -martingale. We look for a representation MATH where $\alpha,\beta$ are deterministic functions of time, $\beta$ is a column, MATH is a scalar product.


If the quantity $Z_{t}$ is a martingale then there exists a choice of final conditions for $\alpha,\beta$ such that MATH


Note that MATH MATH Hence it suffices to choose MATH to have MATH

We explore conditions for $Z_{t}$ to be a martingale. MATH MATH MATH We want the dt term to be zero under some analytically tractable conditions. Hence, the presence of the term MATH leads to the requirements of affinity:

MATH (Affinity pq)
with $u,w,p,q$ being some deterministic functions of time. The $b$ are matrixes. The notation $q\cdot X$ means that we multiply every component of $X$ with a matrix: MATH for some matrixes $q_{i}$ . We require the dt term to vanish and arrive to MATH Here we used the fact that MATH Hence, we require MATH We separate the terms by powers of $X$ :

MATH The first equality defines $\alpha$ , the second equality is a system of Ricatti equations for $\beta$ . The functions $u,p,w,q$ are given.


Suppose an $N-$ dimensional process $X_{t}$ is given by SDE MATH where $a_{t}$ and $b_{t}$ are given affine functions of $X_{t}$ : MATH MATH and $u,w,p,q_{i}$ are matrix valued deterministic functions of time $t$ .

Then the characteristic function

MATH (Affine characteristic function 1)
of the $R^{N}$ -valued argument $z$ is given by the expression
MATH (Affine characteristic function 2)
where the $R$ -valued function MATH and $R^{N}$ -valued function MATH are determined by the ODEs
MATH (Affine alpha component)
MATH (Affine beta component)
with the boundary conditions
MATH (Affine boundary conditions)

A. Ricatti equation.
B. Evaluation of option price.
C. Laplace transform.
D. Example: CDFX model.

Notation. Index. Contents.

Copyright 2007