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| The [[Brownian motion]] models for [[financial markets]] are based on the work of [[Robert C. Merton]] and [[Paul A. Samuelson]], as extensions to the one-period market models of [[Harold Markowitz]] and [[William Forsyth Sharpe|William Sharpe]], and are concerned with defining the concepts of financial [[assets]] and [[financial markets|markets]], [[Portfolio (finance)|portfolios]], [[gains]] and [[wealth]] in terms of continuous-time [[stochastic processes]].
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| Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes.<ref>{{cite journal|last=Tsekov|first=Roumen|year=2013|url=http://cpl.iphy.ac.cn/EN/Y2013/V30/I8/088901|title=Brownian Markets|accessdate=July 29, 2013}}</ref> This model requires an assumption of perfectly divisible assets and a [[frictionless market]] (i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in [[jump diffusion]] models.
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| ==Financial market processes==
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| Consider a financial market consisting of <math>N + 1 </math> financial assets, where one of these assets, called a ''[[Bond (finance)|bond]]'' or ''[[money market]]'', is [[risk]] free while the remaining <math>N</math> assets, called ''[[stocks]]'', are risky.
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| ===Definition===
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| A ''financial market'' is defined as <math>\mathcal{M} = (r,\mathbf{b},\mathbf{\delta},\mathbf{\sigma},A,\mathbf{S}(0)) </math>:
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| # A probability space <math>(\Omega, \mathcal{F}, P)</math>
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| # A time interval <math>[0,T]</math>
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| # A <math>D</math>-dimensional Brownian process <math>\mathbf{W}(t) = (W_1(t) \ldots W_D(t))', \; 0 \leq t \leq T</math> adapted to the augmented filtration <math> \{ \mathcal{F}(t); \; 0 \leq t \leq T \} </math>
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| # A measurable risk-free money market rate process <math>r(t) \in L_1[0,T] </math>
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| # A measurable mean rate of return process <math>\mathbf{b}: [0,T] \times \mathbb{R}^N \rightarrow \mathbb{R} \in L_2[0,T] </math>.
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| # A measurable dividend rate of return process <math>\mathbf{\delta}: [0,T] \times \mathbb{R}^N \rightarrow \mathbb{R} \in L_2[0,T] </math>.
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| # A measurable volatility process <math>\mathbf{\sigma}: [0,T] \times \mathbb{R}^{N \times D} \rightarrow \mathbb{R}</math> such that <math> \sum_{n=1}^N \sum_{d=1}^D \int_0^T \sigma_{n,d}^2(s)ds < \infty </math>.
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| # A measurable, finite variation, singularly continuous stochastic <math> A(t)</math>
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| # The initial conditions given by <math>\mathbf{S}(0) = (S_0(0),\ldots S_N(0))'</math>
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| ===The augmented filtration===
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| Let <math>(\Omega, \mathcal{F}, p)</math> be a [[probability space]], and a
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| <math>\mathbf{W}(t) = (W_1(t) \ldots W_D(t))', \; 0 \leq t \leq T</math> be
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| D-dimensional Brownian motion [[stochastic process]], with the [[Filtration (mathematics)|natural filtration]]:
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| :<math> \mathcal{F}^\mathbf{W}(t) \triangleq \sigma\left(\{\mathbf{W}(s) ; \; 0 \leq s \leq t
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| \}\right), \quad \forall t \in [0,T]. </math>
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| If <math>\mathcal{N}</math> are the [[measure (probability)|measure]] 0 (i.e. null under
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| measure <math>P</math>) subsets of <math>\mathcal{F}^\mathbf{W}(t)</math>, then define
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| the ''augmented'' filtration:
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| :<math> \mathcal{F}(t) \triangleq \sigma\left(\mathcal{F}^\mathbf{W}(t) \cup
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| \mathcal{N}\right), \quad \forall t \in [0,T] </math>
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| The difference between <math> \{ \mathcal{F}^\mathbf{W}(t); \; 0 \leq t \leq T \}
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| </math> and <math> \{ \mathcal{F}(t); \; 0 \leq t \leq T \} </math> is that the
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| latter is both [[continuous function#Directional continuity|left-continuous]], in the sense that:
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| :<math> \mathcal{F}(t) = \sigma \left( \bigcup_{0\leq s <t} \mathcal{F}(s)
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| \right),</math>
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| and [[continuous function#Directional continuity|right-continuous]], such that:
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| :<math> \mathcal{F}(t) = \bigcap_{t < s \leq T} \mathcal{F}(s),</math>
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| while the former is only left-continuous.<ref>{{Cite book | author=Karatzas,
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| Ioannis; Shreve, Steven E. | authorlink= | coauthors= | title=Brownian motion and stochastic calculus | year=1991 | publisher=Springer-Verlag | location=New
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| York | isbn=0-387-97655-8 | pages=}}</ref>
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| ===Bond===
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| A share of a bond (money market) has price <math>S_0(t) > 0</math> at time
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| <math>t</math> with <math>S_0(0)=1</math>, is continuous, <math> \{ \mathcal{F}(t); \; 0 \leq t \leq T \} </math> adapted, and has finite [[Bounded variation|variation]]. Because it has finite variation, it can be decomposed into an [[Absolute continuity|absolutely continuous]] part <math>S^a_0(t)</math> and a singularly continuous part <math>S^s_0(t)</math>, by [[Lebesgue's decomposition theorem]]. Define:
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| :<math>r(t) \triangleq \frac{1}{S_0(t)}\frac{d}{dt}S^a_0(t), </math> and
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| :<math> A(t) \triangleq \int_0^t \frac{1}{S^s_0(s)}dS_0(s), </math>
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| resulting in the [[Stochastic differential equation|SDE]]:
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| :<math>dS_0(t) = S_0(t)[r(t)dt + dA(t)], \quad \forall 0\leq t \leq T, </math>
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| which gives:
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| :<math>S_0(t) = \exp\left(\int_0^t r(s)ds + A(t)\right), \quad \forall 0\leq t \leq T. </math>
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| Thus, it can be easily seen that if <math>S_0(t)</math> is absolutely continuous (i.e. <math>A(\cdot) = 0 </math>), then the price of the bond evolves like the value of a risk-free savings account with instantaneous interest rate <math>r(t)</math>, which is random, time-dependent and <math>\mathcal{F}(t) </math> measurable.
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| ===Stocks===
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| Stock prices are modeled as being similar to that of bonds, except with a randomly fluctuating component (called its [[Volatility (finance)|volatility]]). As a premium for the risk originating from these random fluctuations, the mean rate of return of a stock is higher than that of a bond.
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| Let <math> S_1(t) \ldots S_N(t) </math> be the strictly positive prices per share of the <math> N</math> stocks, which are continuous stochastic processes satisfying:
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| :<math> dS_n(t) = S_n(t)\left[b_n(t)dt + dA(t) + \sum_{d=1}^D \sigma_{n,d}(t)dW_d(t)\right] , \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N. </math>
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| Here, <math>\sigma_{n,d}(t), \; d=1\ldots D</math> gives the volatility of the <math>n</math>-th stock, while <math>b_n(t)</math> is its mean rate of return.
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| In order for an [[arbitrage]]-free pricing scenario, <math>A(t)</math> must be as defined above. The solution to this is:
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| :<math> S_n(t) = S_n(0)\exp\left(\int_0^t \sum_{d=1}^D \sigma_{n,d}(s)dW_d(s) + \int_0^t \left[b_n(s) - \frac{1}{2}\sum_{d=1}^D \sigma^2_{n,d}(s)\right]ds + A(t)\right), \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N, </math>
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| and the discounted stock prices are:
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| :<math> \frac{S_n(t)}{S_0(t)} = S_n(0)\exp\left(\int_0^t \sum_{d=1}^D \sigma_{n,d}(s)dW_d(s) + \int_0^t \left[b_n(s) - \frac{1}{2}\sum_{d=1}^D \sigma^2_{n,d}(s)\right]ds )\right), \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N. </math>
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| Note that the contribution due to the discontinuites in the bond price <math> A(t) </math> does not appear in this equation.
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| ===Dividend rate===
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| Each stock may have an associated [[dividend]] rate process <math>\delta_n(t)</math> giving the rate of dividend payment per unit price of the stock at time <math>t</math>. Accounting for this in the model, gives the ''yield'' process <math>Y_n(t) </math>:
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| :<math> dY_n(t) = S_n(t)\left[b_n(t)dt + dA(t) + \sum_{d=1}^D \sigma_{n,d}(t)dW_d(t) + \delta_n(t)\right] , \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N. </math>
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| ==Portfolio and gain processes==
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| ===Definition===
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| Consider a financial market <math>\mathcal{M} = (r,\mathbf{b},\mathbf{\delta},\mathbf{\sigma}, A,\mathbf{S}(0)) </math>.
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| A ''portfolio process'' <math>(\pi_0, \pi_1, \ldots \pi_N) </math> for this market is an <math>\mathcal{F}(t) </math> measurable, <math>\mathbb{R}^{N+1} </math> valued process such that:
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| :<math>\int_{0}^T | \sum_{n=0}^N\pi_n(t)| \left[|r(t)|dt + dA(t) \right] < \infty </math>, [[almost surely]],
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| :<math>\int_{0}^T |\sum_{n=1}^N\pi_n(t)[b_n(t) + \mathbf{\delta}_n(t) - r(t)]| dt < \infty </math>, almost surely, and
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| :<math>\int_{0}^T \sum_{d=1}^D|\sum_{n=1}^N\mathbf{\sigma}_{n,d}(t)\pi_n(t)|^2 dt < \infty </math>, almost surely.
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| The ''gains process'' for this porfolio is:
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| :<math>G(t) \triangleq \int_0^t \left[\sum_{n=0}^N\pi_n(t)\right]\left(r(s)ds + dA(s)\right) + \int_0^t \left[\sum_{n=1}^N\pi_n(t)\left(b_n(t) + \mathbf{\delta}_n(t) - r(t)\right)\right]dt + \int_{0}^t \sum_{d=1}^D\sum_{n=1}^N\mathbf{\sigma}_{n,d}(t)\pi_n(t) dW_d(s) \quad 0 \leq t \leq T</math>
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| We say that the porfolio is ''[[Self-financing portfolio|self-financed]]'' if:
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| :<math>G(t) = \sum_{n=0}^N \pi_n(t) </math>.
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| It turns out that for a self-financed portfolio, the appropriate value of <math>\pi_0</math> is determined from <math>\pi =(\pi_1, \ldots \pi_N) </math> and therefore sometimes <math>\pi</math> is referred to as the portfolio process. Also, <math>\pi_0 < 0</math> implies borrowing money from the money-market, while <math>\pi_n < 0</math> implies taking a [[short position]] on the stock.
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| The term <math>b_n(t) + \mathbf{\delta}_n(t) - r(t)</math> in the SDE of <math>G(t)</math> is the ''[[risk premium]]'' process, and it is the compensation received in return for investing in the <math>n</math>-th stock.
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| ===Motivation===
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| Consider time intervals <math>0 = t_0 < t_1 < \ldots < t_M = T </math>, and let <math>\nu_n(t_m) </math> be the number of shares of asset <math>n = 0 \ldots N </math>, held in a portfolio during time interval at time <math>[t_m,t_{m+1} \; m = 0 \ldots M-1 </math>. To avoid the case of [[insider trading]] (i.e. foreknowledge of the future), it is required that <math>\nu_n(t_m) </math> is <math>\mathcal{F}(t_m) </math> measurable.
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| Therefore, the incremental gains at each trading interval from such a portfolio is:
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| :<math> G(0) = 0, </math>
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| :<math> G(t{m+1}) - G(t_m) = \sum_{n=0}^N \nu_n(t_m) [Y_n(t_{m+1}) - Y_n(t_m)] , \quad m = 0 \ldots M-1, </math>
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| and <math>G(m)</math> is the total gain over time <math>[0,t_m]</math>, while the total value of the portfolio is <math>\sum_{n=0}^N \nu_n(t_m)S_n(t_m)</math>.
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| Define <math>\pi_n(t) \triangleq \nu_n(t) </math>, let the time partition go to zero, and substitute for <math>Y(t)</math> as defined earlier, to get the corresponding SDE for the gains process. Here <math>\pi_n(t) </math> denotes the dollar amount invested in asset <math>n </math> at time <math>t </math>, not the number of shares held.
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| ==Income and wealth processes==
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| ===Definition===
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| Given a financial market <math>\mathcal{M}</math>, then a ''cumulative income process'' <math>\Gamma(t) \; 0 \leq t \leq T </math> is a [[semimartingale]] and represents the income accumulated over time <math>[0,t]</math>, due to sources other than the investments in the <math>N+1</math> assets of the financial market.
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| A ''wealth process'' <math>X(t)</math> is then defined as:
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| :<math>X(t) \triangleq G(t) + \Gamma(t) </math>
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| and represents the total wealth of an investor at time <math>0 \leq t \leq T</math>. The portfolio is said to be ''<math>\Gamma(t)</math>-financed'' if:
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| :<math>X(t) = \sum_{n=0}^N \pi_n(t).</math>
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| The corresponding SDE for the wealth process, through appropriate substitutions, becomes:
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| <math>dX(t) = d\Gamma(t) + X(t)\left[r(t)dt + dA(t)\right]+ \sum_{n=1}^N \left[ \pi_n(t) \left( b_n(t) + \delta_n(t) - r(t) \right) \right] + \sum_{d=1}^D \left[\sum_{n=1}^N \pi_n(t) \sigma_{n,d}(t)\right]dW_d(t)</math>.
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| Note, that again in this case, the value of <math>\pi_0</math> can be determined from <math>\pi_n, \; n = 1 \ldots N</math>.
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| ==Viable markets==
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| The standard theory of mathematical finance is restricted to viable financial markets, i.e. those in which there are no opportunities for [[arbitrage]]. If such opportunities exists, it implies the possibility of making an arbitrarily large risk-free profit.
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| ===Definition===
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| In a financial market <math>\mathcal{M}</math>, a self-financed portfolio process <math>\pi(t)</math> is considered to be an ''[[arbitrage]] opportunity'' if the associated gains process <math>G(T)\geq 0</math>, almost surely and <math> P[G(T) > 0] > 0</math> strictly. A market <math> \mathcal{M}</math> in which no such portfolio exists is said to be ''viable''.
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| ===Implications===
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| In a viable market <math>\mathcal{M}</math>, there exists a <math>\mathcal{F}(t)</math> adapted process <math>\theta :[0,T] \times \mathbb{R}^D \rightarrow \mathbb{R}</math> such that for almost every <math> t \in [0,T]</math>:
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| :<math>b_n(t) + \mathbf{\delta}_n(t) - r(t) = \sum_{d=1}^D \sigma_{n,d}(t) \theta_d(t)</math>.
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| This <math>\theta</math> is called the ''market price of risk'' and relates the premium for the <math>n</math>-the stock with its volatility <math>\sigma_{n,\cdot}</math>.
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| Conversely, if there exists a D-dimensional process <math>\theta(t)</math> such that it satisfies the above requirement, and:
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| :<math> \int_0^T \sum_{d=1}^D |\theta_d(t)|^2 dt < \infty</math>
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| :<math>\mathbb{E}\left[ \exp\left\{ -\int_0^T \sum_{d=1}^D \theta_d(t)dW_d(t) - \frac{1}{2}\int_0^T \sum_{d=1}^D |\theta_d(t)|^2 dt \right\} \right] = 1 </math>,
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| then the market is viable.
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| Also, a viable market <math>\mathcal{M}</math> can have only one money-market (bond) and hence only one risk-free rate. Therefore, if the <math>n</math>-th stock entails no risk (i.e. <math>\sigma_{n,d}=0, \; d = 1 \ldots D</math>) and pays no dividend (i.e.<math>\delta_n(t)=0</math>), then its rate of return is equal to the money market rate (i.e. <math>b_n(t) = r(t)</math>) and its price tracks that of the bond (i.e. <math>S_n(t) = S_n(0)S_0(t)</math>).
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| ==Standard financial market==
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| ===Definition===
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| A financial market <math>\mathcal{M}</math> is said to be ''standard'' if:
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| :(i) It is viable.
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| :(ii) The number of stocks <math>N</math> is not greater than the dimension <math>D</math> of the underlying Brownian motion process <math>\mathbf{W}(t)</math>.
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| :(iii) The market price of risk process <math>\theta</math> satisfies:
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| ::<math>\int_0^T \sum_{d=1}^D |\theta_d(t)|^2 dt < \infty</math>, almost surely.
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| :(iv) The positive process <math>Z_0(t) = \exp\left\{ -\int_0^t \sum_{d=1}^D \theta_d(t)dW_d(t) - \frac{1}{2}\int_0^t \sum_{d=1}^D |\theta_d(t)|^2 dt \right\} </math> is a [[Martingale (probability theory)|martingale]].
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| ===Comments===
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| In case the number of stocks <math>N</math> is greater than the dimension <math>D</math>, in violation of point (ii), from linear algebra, it can be seen that there are <math>N-D</math> stocks whose volatilies (given by the vector <math>(\sigma_{n,1} \ldots \sigma_{n,D})</math>) are linear combination of the volatilities of <math>D</math> other stocks (because the rank of <math>\sigma</math> is <math>D</math>). Therefore, the <math>N</math> stocks can be replaced by <math>D</math> equivalent mutual funds.
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| The ''standard martingale measure'' <math>P_0</math> on <math> \mathcal{F}(T)</math> for the standard market, is defined as:
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| :<math>P_0(A) \triangleq \mathbb{E}[Z_0(T)\mathbf{1}_A], \quad \forall A \in \mathcal{F}(T)</math>.
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| Note that <math>P</math> and <math>P_0</math> are [[absolutely continuous]] with respect to each other, i.e. they are equivalent. Also, according to [[Girsanov's theorem]],
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| :<math>\mathbf{W}_0(t) \triangleq \mathbf{W}(t) + \int_0^t \theta(s)ds </math>,
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| is a <math>D</math>-dimensional Brownian motion process on the filtration <math> \{\mathcal{F}(t); \; 0 \leq t \leq T\}</math> with respect to <math>P_0</math>.
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| ==Complete financial markets==
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| A complete financial market is one that allows effective [[hedge (finance)|hedging]] of the risk inherent in any investment strategy.
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| ===Definition===
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| Let <math>\mathcal{M}</math> be a standard financial market, and <math>B</math> be an <math> \mathcal{F}(T)</math>-measurable random variable, such that:
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| :<math>P_0\left[\frac{B}{S_0(T)} > -\infty \right] = 1 </math>.
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| :<math> x \triangleq \mathbb{E}_0\left[\frac{B}{S_0(T)} \right] < \infty </math>,
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| The market <math>\mathcal{M}</math> is said to be ''complete'' if every such <math>B</math> is ''financeable'', i.e. if there is an <math>x</math>-financed portfolio process <math>(\pi_n(t); \; n = 1 \ldots N)</math>, such that its associated wealth process <math>X(t)</math> satisfies
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| :<math>X(t) = B</math>, almost surely.
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| ===Motivation===
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| If a particular investment strategy calls for a payment <math>B</math> at time <math>T</math>, the amount of which is unknown at time <math>t=0</math>, then a conservative strategy would be to set aside an amount <math>x = \sup_\omega B(\omega)</math> in order to cover the payment. However, in a complete market it is possible to set aside less capital (viz. <math>x</math>) and invest it so that at time <math>T</math> it has grown to match the size of <math>B</math>.
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| ===Corollary===
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| A standard financial market <math>\mathcal{M}</math> is complete if and only if <math>N=D</math>, and the <math>N \times D</math> volalatily process <math> \sigma(t)</math> is non-singular for almost every <math>t \in [0,T]</math>, with respect to the [[Lebesgue measure]].
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| ==Notes==
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| <references/>
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| ==See also==
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| *[[Black-Scholes model]]
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| *[[Martingale pricing]]
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| *[[Mathematical finance]]
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| *[[Monte Carlo method]]
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| ==References==
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| {{Cite book | author=Karatzas, Ioannis; Shreve, Steven E. | authorlink= |
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| coauthors= | title=Methods of mathematical finance | year=1998 |
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| publisher=Springer | location=New York | isbn=0-387-94839-2 | pages=}}
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| | |
| {{Cite book | author=Korn, Ralf; Korn, Elke | authorlink= | coauthors= |
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| title=Option pricing and portfolio optimization: modern methods of financial mathematics | year=2001 | publisher=American Mathematical Society |
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| location=Providence, R.I. | isbn=0-8218-2123-7 | pages=}}
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| | |
| {{Cite jstor|1926560}}
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| {{Cite journal | url = http://www.math.uwaterloo.ca/~mboudalh/Merton1971.pdf
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| | title = Optimum consumption and portfolio rules in a continuous-time model
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| | year = 1970
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| | author = Merton, R.C.
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| | journal = Journal of Economic Theory
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| | pages =
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| | volume = 3
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| | issue =
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| | doi =
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| | format = w
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| | accessdate = 2009-05-29
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| }} {{Dead link|date=November 2010|bot=H3llBot}}
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| {{DEFAULTSORT:Brownian Model Of Financial Markets}}
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| [[Category:Finance theories]]
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