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3.3.1 Wiener process and martingales $ \spadesuit $

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Although it is not possible to predict with any certainty the spot price $ X$ of an asset in an efficient market, sect.2.1.1 demonstrated in that possible realizations can be simulated with their probability of occurrence by summing price increments $ dX$ over small steps in time $ dt$ . Separating the deterministic $ (\mu dt)$ from the remaining random component $ (\sigma dW)$ , the evolution of the random variable $ X$ satisfies a stochastic differential equation

$\displaystyle \frac{dX}{X^\kappa} = \mu dt +\sigma dW(t)$ (3.3.1#eq.1)

where $ \kappa$ =0 (alt. 1) chooses between the normal (alt. log-normal) distribution of increments discussed previously in (2.1.1#eq.1). Integrating over time, the first term yields a uniform (alt. exponential) growth with a rate $ \mu$ that accounts for example for a continuous payment of a fixed dividend (alt. a compounded interest rate). The second term reproduces a random walk proportional to the market volatility $ \sigma$ , using a so-called Wiener process that has the following properties
  1. The Wiener increment $ dW(t)=W_{t+dt}-W_t$ over a time step $ dt$ is a random variable drawn from a normal distribution with zero mean and a time-step variance $ \mathcal{N}[0,dt]$ (1.4#eq.5). Extensions to other distributions (e.g. 2.1.1#eq.3) are possible.
  2. The Wiener increment is independent of the past. Using the definition (1.4#eq.3) for the correlation, this is satisfied when $ E[dW(t_1)dW(t_2)]=0, \forall t_1<t_2$ .
A convenient way of writing this is

$\displaystyle dW(t) = \zeta \sqrt{dt} \hspace{1cm}\mathrm{with} \hspace{1cm} \zeta\in\mathcal{N}(0,1)$ (3.3.1#eq.2)

where different realizations of the random variable $ dW(t)$ are generated using random numbers $ \zeta$ that are normally distributed. This construction provides the mathematical foundation for the Monte-Carlo simulations, where the mean value of the increment

$\displaystyle E[dX] = E[X^\kappa (\mu dt +\sigma dW(t))] = \mu X^\kappa dt$ (3.3.1#eq.3)

and the variance

$\displaystyle \mathrm{Var}[dX] = E[dX^2] - E[dX]^2 = E[(X^\kappa \sigma dW(t))^2] = \sigma^2 X^{2\kappa} dt$ (3.3.1#eq.4)

are matched with historical data to forecast possible evolutions into the future. Third or even higher order moments of the probability distribution can in principle also be matched; experience, however, shows that little is to be gained from such a procedure. A better description of the market is obtained by performing a principal components analysis [19], where several imperfectly correlated random variables are identified and then superposed to drive increments of the form

$\displaystyle dX=\sum_i X_i^\kappa [\mu_i dt +\sigma_i dW_i(t)]$ (3.3.1#eq.5)

$\displaystyle E[dW_i(t_1) dW_i(t_2)] =$ 0 $\displaystyle \hspace{1cm} t_1<t_2$  
$\displaystyle E[dW_i(t) dW_j(t)] =$ $\displaystyle \rho_{ij} dt$   (3.3.1#eq.6)

where $ \rho_{ij} \in [-1;1]$ is a correlation factor. The first component typically accounts for 80-90% (and the first three for up to 95-99%) of the variance observed in the market price of forward rates [19].
Note that the forecast value for the price of a share $ (S+dS)$ or an interest rate $ (r+dr)$ constructed using Wiener increments depends only the present value: this independence from the past is known in mathematics as the Markov property. Also note that a suitable choice of a numeraire can always be found to normalize random variables and make them risk-neutral by scaling out the drift observed in the real world: called martingales, such variables play an important role in the construction of financial models.

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