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Analytical Model of Financial Market

Warning! Reading of the subsequent text assumes basic knowledge of probability theory.

The Analytical Model of Financial Market (or simply Analytical Model), utilized in SmartFolio, is based on Multidimensional Geometric Brownian Motion – the most common class of stochastic processes used in mathematical finance to model the dynamics of prices.

Consider risky assets available for the investments. Assume also that the investor has access to some
riskless asset , which yields continuously compounded rate of return that will be referred to as the riskless rate or risk-free rate. An investor expresses prices of assets in units of asset .
Discounted price of asset at time is denoted by . Upper asterisk in consequent text denotes transposition operation.

The main assumption of analytical model is given by the following system of stochastic differential equations, which describe evolution of discounted prices of assets :


In the above expression drift vector (further referred to as the Mu vector), the vector of
continuously compounded
dividend yields and volatility matrix  consist of constant values, while elements of represent independent Wiener processes.

Definition. For the sake of convenience the vector , will be further referred to as the Excess Mu vector.
Definition. Matrix is called Covariance Matrix.

Note.
Readers, who are not familiar with stochastic differential equations in continuous time, may interpret
as very short time range, as
simple return in -th asset over  and as
normally distributed random vector, whose elements are independent of each other (and of components of
other vectors ), have zero mean and variance .



Denote by diagonal matrix with elements of vector at the main diagonal. At the same time for diagonal matrix denote column vector , whose elements are equal to diagonal elements of .
In these notations the above system of stochastic differential equations reduces to:


A simple solution is presented below:

where components of the volatility vector are calculated by means of the formula
Using compact notation, this solution is written as


Definition. Components of vector


are called Expected Excess Growth Rates.

The above definition arises from the fact that for any


where symbol denotes mathematical expectation.
On the other hand, it can be shown that



Note. Elements of are also called Expected Instantaneous Rates of Return, reflecting the notion of expected simple rate of return over an infinitesimally short period of time. Indeed, it is easy to check that for
any the following chain of equalities holds:  when  approaches zero.










 
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