Quantitative Analysis.
Trading Platform.
Python for Excel.
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I.Basic math.
II.Pricing and Hedging.
1.Basics of derivative pricing I.
A.Single step binary tree argument. Risk neutral probability. Delta hedging.
B.Why Ito process?
C.Existence of the risk neutral measure via Girsanov's theorem.
D.Self-financing strategy.
E.Existence of the risk neutral measure via backward Kolmogorov's equation. Delta hedging.
F.Optimal utility function based interpretation of delta hedging.
2.Change of numeraire.
3.Basics of derivative pricing II.
4.Market model.
5.Currency Exchange.
6.Credit risk.
7.Incomplete markets.
III.Explicit techniques.
IV.Data Analysis.
V.Implementation tools.
VI.Basic Math II.
VII.Implementation tools II.
Bibliography.
Forum Notation Index Contents

Existence of the risk neutral measure via Girsanov's theorem.

he $\QTR{cal}{F}_{t}$ is a reference filtration and $dW_{t}$ is a vector of $\QTR{cal}{F}_{t}$-adapted independent standard Brownian motions. Consider the vector of prices of several traded assets $dS_{t}$:MATHMATH where the $a=a_{t}$ and $b=b_{t}$ are vector and matrix valued processes adapted to the filtration $\QTR{cal}{F}_{t}$. For simplicity we restrict our attention to assets without dividends. We assume a possibility to construct a portfolio MATH such that the value of the portfolio is deterministic during the next infinitesimally small time interval:MATHMATH The $\psi_{t}$ is another vector valued process adapted to $\QTR{cal}{F}_{t}$. It represents the trading strategy. By no-arbitrage assumption we conclude that such riskless portfolio must earn the riskless rate $r$:MATH Equivalently,MATH where the $r=r_{t}$ is the stochastic $\QTR{cal}{F}_{t}$-adapted riskless rate. We summarize the above argument with the following statementMATH We think of columns $b_{k}$, $k=1,...,M$ of the matrix $b$ as vectors in a linear space of some finite dimension. We introduce the linear span $H$ of the set MATH. Let $\Pr_{H}$ be an orthogonal projection to the linear span $H$. The statement (*) reads as MATH We would like to conclude that MATH. Indeed, suppose such conclusion is false and there is a nonzero vector MATHWe have $x\perp H$ by the form of the $x$. We arrive to a contradiction by taking $\psi=x$ in the statement (**). Therefore MATH. Equivalently, there are $\QTR{cal}{F}_{t}$-adapted processes MATH such that MATH or

MATH(Market price of risk)
This result is remarkable because the $\lambda$ is dependent on only one index. The formula ( Market_price_of_risk) states that the excess return of the asset over the riskless rate $r$ is proportional to the volatilities associated to the driving Brownian motions $dW_{k}$ and the coefficient of proportionality $\lambda_{k}$ does not depend on the type of the asset but is only dependent on the source of risk. Therefore, the coefficients $\lambda_{k}\,$ are called the market prices of risk. We haveMATH
MATH(Risk neutral Brownian motion)
 According to the Girsanov's theorem there exists a change of the probability measure that makes the process $W^{\ast}$ given by MATH a standard Brownian motion. Such probability measure is called the risk neutral measure. Note that MATH hence the discounted price of any traded asset $V_{T}$ is a martingale with respect to the risk neutral measure :
MATH(Risk neutral pricing)
 The above relation is the foundation of the classic derivative pricing.

Consider the following special situation:MATH where the $\mu$ and $\sigma$ are some deterministic functions of time. The ( Market price of risk) reads asMATH orMATH


Hence, in the setting (***), if $r_{t}$ is a deterministic function then the market prices of risk are deterministic functions as well.

A particular vector $\alpha-rS$ may be represented as a linear combination of $b_{s}$ in many different ways if the number of vectors $b_{s}$ is bigger then the dimensionality of their linear span. This is the first sign of trouble. If there more sources of uncertainty then traded assets then the risk neutral measure is not unique. This means there may be a variety of opinions about prices of contingent claims but still there is no arbitrage.





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