Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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I.Basic math.
II.Pricing and Hedging.
III.Explicit techniques.
IV.Data Analysis.
V.Implementation tools.
VI.Basic Math II.
VII.Implementation tools II.
1.Calculational Linear Algebra.
2.Wavelet Analysis.
3.Finite element method.
4.Construction of approximation spaces.
5.Time discretization.
6.Variational inequalities.
7.Lattice approach to derivative pricing.
A.Basic definitions of the lattice approach.
B.Markov Generator for smooth process. Dyson decomposition. Kolmogorov equations.
C.Path-Integral representation of Markov generator.
D.Markov generator for piecewise smooth process.
a.Change of measure on lattice between jumps.
b.Change of measure at jump points.
c.Connection to generic change of numeraire.
d.Monotonic processes and martingales.
e.Self financing strategies. Arbitrage. Fundamental theorem of finance.
E.Fast exponentiation.
Bibliography.
Forum Notation Index Contents

Change of measure on lattice between jumps.

et us introduce a positive function MATH and use it as a way to change a propagator MATH. We define the following transformation

MATH(Change of measure on lattice)
and explore $\tilde{P}$ as a candidate for a new propagator. Here $\Delta$ is a normalisation function chosen to satisfy the definition ( Markov propagator)-2. We wish to obtain the most general form for such transformation that produces a valid propagator.

According to the definition ( Markov propagator) we choose $\Delta_{m}$ to satisfy the following:

MATH(Markov Generator normalization)
Let us introduce the notationsMATH We proceed to calculate the MATH consequences of the formula ( Markov Generator normalization). We pass to the limit and obtain
MATH(Lattice normalization 1)
 In addition, we differentiate with respect to T and then pass to the limit:MATH thusMATH Consequently,
MATH(Lattice normalization 2)

Let us now apply the operation MATH to the relationship ( Change of measure on lattice). We calculateMATH ThusMATH We substitute the first two terms from the formula ( Lattice normalization 2):MATH We do some argument renaming and present the result as follows:

MATH(Change of measure on lattice 1)
We summarize our calculations with the following definition.

Definition

Let MATH be a family of non negative functions that are non zero on MATH and zero otherwise. The measure change defined by MATH is the transformation of the Markov generator $L_{m}$ into an equivalent generator $\tilde{L}_{m}$ according to the formula ( Change of measure on lattice 1).





Forum Notation Index Contents


















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