How stochastic simulation fits in

Independent from the question whether GS or SI methods are used to solve the portfolio optimization problem, it is clear that a way must be found to determine the distributions of the considered payoff functions $X_i$ (cf. Section 1, Equation (1)) as finally in any optimization routine the risk or performance measures (10), (12) and (15) must be computed. The $X_i$ (cf. (6)) were interpreted as wins or losses due to the $i$-th asset in the market where we assumed to have $n$ numbered assets. It is clear that many of our thoughts so far, especially the functions ES, ES-RORC, ES-RORAC, but also their derivatives, crucially depend on the model for future prices $V_i(t)$. The particular stochastic model we use is introduced in Section 4.

Once the model for the price processes $V_i$ (cf. (3)) is chosen and a way to get possible parameters is found, one can theoretically compute the risk and performance measures (10), (12) and (15) and their partial derivatives under budget constraint (23), (24) and (25). However, one often encounters models (also in our case) where it is not possible or quite difficult to compute these values directly. The more realistic assumption is that one succeeds in doing a stochastic simulation of the model which computes $m\in\mathbb{N}^+$ (e.g. $m = 10^3$) market scenarios, i.e. finally one has for each $i$ the numerical realizations (in increasing order)

$$x_i^1, x_i^2, \ldots, x_i^m$$ (29)

of the random variable $X_i$ defined by (6). The realizations (also in increasing order)

$$x^1, x^2, \ldots, x^m$$ (30)

for any $X=X(u)$ follow immediately.

Having these realizations, estimates for the stochastic expressions in the functions mentioned above can be used. In particular, we compute estimates using the ``empirical'' distribution given by the simulation output, e.g.

$$\widehat{\mathbf{E}}[X] = \frac{1}{m}\sum_{j=1}^{m} x^j ,$$ (31)

$$\widehat{\text{VaR}}_{\alpha}(X) = - x^{\lceil\alpha m\rceil}$$ (32)

or

$$\widehat{\text{ES}}_{\alpha}[X] = \frac{- \sum_{x^j \leq -\wideha... ...alpha}(X)} x^j} {\text{card}\{j: x^j \leq -\widehat{\text{VaR}}_{\alpha}(X)\}},$$ (33)

where $\lceil r\rceil$ denotes the smallest integer which is greater or equal the real number $r$. Of course, one can use other perhaps more sophisticated estimators. Nonetheless, replacing all stochastic expressions in (10), (12), (15), (23), (24) and (25) as suggested by (31) to (33), one obtains approximations of the respective measures and their gradients which are easy to implement in any suitable programming language.

Gradient Search methods or Swarm Intelligence optimization methods can now be executed using the obtained approximations.