Market scenario generation

As described in Subsection 3.4, we carry out a stochastic simulation to obtain an ``empirical'' distribution of the considered random payoffs.

A simulation requires discretization. We consider points of time $t_m$ $(m\in\mathbb{N})$. The width of the time step is the constant $\Delta$, e.g. one day, one month etc., i.e. $t_{m+1} = t_m+\Delta$ and $t_0 = 0$. Increments

$$\delta W_{i,m} := W_i(t_{m}) - W_i(t_{m-1})$$ (46)

of the $(d+2)$ brownian motions have to be simulated. For fixed $m$, the $\delta W_{i,m}$ are correlated by the covariance matrix $\Delta\Sigma$ (cf. (34) and (35)). For fixed $i$, the increments $\delta W_{i,m}$ are independent normally distributed random variables with variance $\Delta$ and expectation 0.

Hence, all discretized dynamics is driven by a series of standard normally distributed random variables $N_{i,m}$ $(i=1,\ldots,d+2; m\in\mathbb{N}^+)$, where for each $m$ the random variables $(N_{i,m})_{i=1,\ldots,d+2}$ are correlated by the covariance matrix $\Sigma$ which will later be estimated from real data.