# Approximating the conditional expectation in simulations

I am simulating stock returns, which are governed by the following equations

$r_t = \mu + \delta r_{t-1} + \varepsilon_t$

$\sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma^2_{t-1}$

$\varepsilon_t = \sigma_t z_t$

$z_t \sim N(0, 1)$

I am trying to solve an asset allocation problem by calculating utility for each simulation path and then I need to approximate the expected utility from all the simulations. I know that some methods involve simply calculating the across-path mean, however, I think that it is not possible in this case, because each path has different predictor variables, that is f.e. the previous period return or the conditional volatility. Another mean to do so is using across-path OLS regression. This would be simpler in case of a specification like this $y = \alpha x + z$ because $E[z] = 0$ from the OLS assumptions. Then I would use the fitted values for a proxy of the expectation. However, I do not know how to do in case of my specification. The problem is mainly with the fact that it is $z_r$ that is distributed with zero mean, not $\varepsilon_t$, which is a multiplication of a Gaussian zero mean random number and volatility obtained from a GARCH model. Does it mean that I should take expectation $E[r_t] = E[\mu + \delta r_{t-1}] + E[\sigma_t]E[z_t] = E[\mu + \delta r_{t-1}]$? As $E[\sigma_t]E[z_t] = E[\sigma_t] * 0$.

Any help with this problem would be great!

• What is different between the paths are the innovations; predictor variables are different only because they are functions of innovations; so your concern does not seem relevant. – Richard Hardy Mar 6 '16 at 13:20
• I edited the equation as per your suggestion, but unless I am missing something this does not change anything? – Masher Mar 6 '16 at 13:23
• @RichardHardy So you are suggesting I use the across-path mean to approximate the conditional expectation? I remember a case from my studies where there was a VAR model where both dependent variables $r_{t+1}$ and $x_{t+1}$ where constructed as $\alpha_0 + \alpha_1 x_{t} + \varepsilon_{t+1}$, where $\varepsilon \sim N(0, \Sigma)$ with $x$ being a predictor variable and I had a fixed starting point for generating simulations of $x$. And in that case I had to use the across-path regression to approximate the conditional expectation. – Masher Mar 7 '16 at 12:56
• Well, maybe I am missing something. I have never worked on anything quite like this. – Richard Hardy Mar 7 '16 at 14:51