Quantitative Analysis.
Trading Platform.
Python for Excel.
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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

Self financing strategies. Arbitrage. Fundamental theorem of finance.

efinition

(Financial model on lattice). The financial model is a collection MATH of adapted processes. The $A_{0}$ is tradditionally denotes the money-market account. The "tilde" notation MATH is reserved for discounted processes.

Definition

(Trading strategies). The trading strategy is a collection of adapted processes MATH. The value process of the strategy MATH is the process MATH:MATH

The self financing strategy is the strategy that draws no money. For continuous space process the self-financing requirement readsMATH meaning "the change in the value of the portfolio $\Pi$ is driven by the price changes of the components". This is different from the formal differentiation ruleMATH Hence, the self financing requirement may be stated as MATH meaning "when rearranging the portfolio allocations $\zeta$ the total portfolio value $\Pi$ has to remain unchange\d".

On a lattice the requirement $Ad\zeta=0$ has spacial and time components as stated in the following definition.

Definition

(Self financing strategy on the lattice).The trading strategy MATH is self financing if the following two conditions hold:

1. MATH,$~\forall t$,

2. MATH,MATH.

Definition

(Arbitrage on lattice). A self-financing strategy is called "arbitrage" iff the corresponding $\tilde{\Pi}$ is increasing at some time t.

Proposition

(Fundamental theorem of finance on lattice). If there is a change of measure that turns all discounted price processes into martingale then

1. The discounted price of any self financing strategy is a martingale under such measure.

2. There is no arbitrage.

Conversely, if there is no arbitrage then there exists a change of measure that turns all dicounted price processes into martingales.

Proof

Let us fix some $x_{0}$ and $t_{0}$ and simplify notation:MATH We restate the nontrivial part of the theorem in the low level terms as follows.

Suppose that for any trading strategy MATH s.t.

MATH(Self financing 1)
the process MATH either fails to have both of the following properties or fails to have at least one of them in the strict form:
MATH(No arbitrage 1)
 Then one should be able to construct MATH such that
MATH(Change of measure 1)
MATH(Martingale 1)

We start the construction of such $G$ by examining the differenceMATH for $z\in D_{0}$, MATH, math and $t_{1}-t_{0}$ being small. We haveMATHMATHMATH Hence,MATH Consequently, the no arbitrage condition ( No arbitrage 1) transforms to the assertion that MATH fails to have both

MATH(No arbitrage 2)
for all $z_{t_{1}},t_{1}$. Note that this is only a property of $D_{0}$. Other properties of the probability distribution do not enter the non-arbitrage condition.

Our goal is the relationships ( Change of measure 1) and ( Martingale 1). We substitute ( Change of measure 1) into ( Martingale 1) :MATH Since $x_{0}\notin D_{0}$ we need to find MATH with all positive components such thatMATH

Equivalently,MATH Such property has to hold for all n. Geometrically, we need to prove that the vectors MATH do not have components pointing in all coordinate directions. However, we cannot simply say "there are not enough $\zeta_{n}$ to point in every direction, hence, we are done" because we don't just want any direction $G$ orthogonal to every $\zeta_{n}$ but rather we want the direction $G$ with all positive components. Hence, existence of such $G$ merits a proof.

By the formulas ( No arbitrage 2) and ( Markov generator properties) the MATH has to have both positive and negative components. Hence, the hyperplane MATH certainly has a direction with all positive components. We take a projection of remaining MATH on $H^{1}$ and note, that the projection of $\zeta_{2}$ on $H^{1}$ is a linear combination of $\zeta_{1}$ and $\zeta_{2}$ and, by no-arbtrage ( No arbitrage 2), it has positive an negative components. Hence, the hyperplane MATH has a direction with all positive components. We continue for $n=3,...,N$ to obtain MATH with the desired properties.





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