Monte Carlo Simulation

Abstract

This chapter discusses another major numerical method, namely simulation. It has proven extremely flexible and useful for addressing uncertainty in the price of assets and portfolios of assets. This notwithstanding, it is relatively inefficient when the number of state variables is low. Improved sampling methods are one attempt to making convergence faster. Regarding the valuation of derivative assets, simulation was initially confined to European-type options and path-dependent options given its forward-looking nature. Nonetheless, significant advances have taken place during the last decade or so. As a consequence, simulation has also extended to the valuation of American-type options, which require backward induction. Here we show how to generate random paths for a single stochastic process (be it either non-stationary or stationary) and for several (correlated or independent) processes. We then show how to apply Monte Carlo simulation to the valuation of investment options. We can also follow this approach for assessing the risk profile of a number of energy projects.

Appendices

Appendix

Below we introduce several models that can be (and have been) used for valuing investment options in different energy contexts.

Two IGBMs (One of Them a Two-Factor GBM)

Assume that we want to evaluate a base load Natural Gas Combined Cycle (NGCC) power plant. We can consider uncertain gas prices with regard to both the current level and the long-run equilibrium level. The current electricity price can be similarly assumed to be stochastic. Specifically, we assume that the three variables follow an Inhomogeneous Geometric Brownian Motion (IGBM). Thus, the time-t price of natural gas evolves according to:

$$ dG_{t} = k_{G} (L_{t} - G_{t} )dt + \sigma_{G} G_{t} dW_{t}^{G} , $$

(6.34)

where the long-term equilibrium vule $L_{t}$ in turn follows another IGBM process:

$$ dL_{t} = k_{L} \left( {L_{G} - L_{t} } \right)dt + \sigma_{L} L_{t} dW_{t}^{L} , $$

(6.35)

with $L_{G}$ acting as an anchor value. Besides, the electricity price is also an IGBM:

$$ dE_{t} = k_{E} \left( {L_{E} - E_{t} } \right)dt + \sigma_{E} E_{t} dW_{t}^{E} . $$

(6.36)

The model in a risk-neutral world would be:

$$ dG_{t} = \left[ {k_{G} \left( {L_{t} - G_{t} } \right) - \lambda_{G} \sigma_{G} G_{t} } \right]dt + \sigma_{G} G_{t} dW_{t}^{G} , $$

(6.37)

$$ dL_{t} = \left[ {k_{L} \left( {L_{G} - L_{t} } \right) - \lambda_{L} \sigma_{L} L_{t} } \right]dt + \sigma_{L} L_{t} dW_{t}^{L} $$

(6.38)

$$ dE_{t} = \left[ {k_{E} \left( {L_{E} - E_{t} } \right) - \lambda_{E} \sigma_{E} E_{t} } \right]dt + \sigma_{E} E_{t} dW_{t}^{E} . $$

(6.39)

We assume that $\rho_{{W^{G} W^{L}}} = \rho_{{W^{L} W^{E}}} = 0$ while $\rho_{{W^{G} W^{E}}} = \rho$. In addition, $\lambda_{G}$ is the market price of risk stemming from current natural gas price (assumed to be constant); similar interpretation applies to both $\lambda_{L}$ and $\lambda_{E}$.

In our Monte Carlo simulations below, we shall use the following discretization of the last three equations:

$$ \Updelta G_{t} = \left[ {k_{G} L_{t} - G_{t} (k_{G} + \lambda_{G} \sigma_{G} )} \right]\Updelta t + \sigma_{G} G_{t} \sqrt {\Updelta t} \varepsilon_{t}^{G} , $$

(6.40)

$$ \Updelta L_{t} = \left[ {k_{L} L_{G} - L_{t} (k_{L} + \lambda_{L} \sigma_{L} )} \right]\Updelta t + \sigma_{L} L_{t} \sqrt {\Updelta t} \varepsilon_{t}^{L} , $$

(6.41)

$$ \Updelta E_{t} = \left[ {k_{E} E_{t} - E_{t} (k_{E} + \lambda_{E} \sigma_{E} )} \right]\Updelta t + \sigma_{E} E_{t} \sqrt {\Updelta t} \varepsilon_{t}^{E} . $$

(6.42)

The disturbances $\varepsilon_{t}^{G}$, $\varepsilon_{t}^{L}$, and $\varepsilon_{t}^{E}$ are standard Normal variates; $\Updelta t$ is measured in yearly terms. Whereas $\varepsilon_{t}^{G}$ and $\varepsilon_{t}^{L}$ are assumed to be independent, just like $\varepsilon_{t}^{L}$ and $\varepsilon_{t}^{E}$ (so $\rho_{GL} = \rho_{LE} = 0$), the correlation coefficient between electricity and gas prices $\rho_{GE}$ may be different from zero.

Regarding natural gas, it can be seen in Eqs. (6.40) and (6.41) that generating a simulation path requires knowledge of the state variable $k_{G} L_{t}$ on each day t, the three composites ($k_{L} L_{G}$, $k_{G} + \lambda_{G} \sigma_{G}$, $k_{L} + \lambda_{L} \sigma_{L}$), and the two volatilities $\sigma_{G}$ and $\sigma_{L}$ in the actual (as opposed to risk neutral) world.

Any simulation run fits the discretized equations Eqs. (6.40–6.42). Depending on the specific values of the correlation coefficients, the Monte Carlo simulation technique may require the generation of two or more correlated Normal variates.

The series obtained for $G_{t}$, $L_{t}$ and $E_{t}$ allow to compute at any time the value $V_{t}$ of an investment at that time, taking into account the evolution of electricity and gas prices, as well as the behavior of the equilibrium gas price in the short term ($L_{t}$). Given the values of $V_{t}$ at any moment and in each path, the LSMC approach is used. At the last moment (T), the value of the investment in each path is:

$$max\left( {V\left( {G_{T} ,L_{T} ,E_{T} } \right) - I;0} \right)$$

(6.43)

At earlier moments, the method is based on the computation of a series of parameters that allow construct a linear combination of basic functions. This combination allows estimate the continuation value at each step. The specification adopted consists of a second-order expected continuation value function with 10 regressors (since there are 3 sources of risk), namely:

$$ \begin{aligned} E_{t}^{Q} \left[ {e^{ - r\Updelta t} V_{t + 1} \left( {G_{t + 1} ,L_{t + 1} ,E_{t + 1} } \right)} \right] \approx & \,a_{1} + a_{2} G_{t} + a_{3} G_{t}^{2} + a_{4} L_{t} + a_{5} L_{t}^{2} \\ & \, + a_{6} E_{t} + a_{7} E_{t}^{2} + a_{8} G_{t} L_{t} + a_{9} G_{t} E_{t} + a_{10} L_{t} E_{t} . \\ \end{aligned} $$

(6.44)

At any time, considering the paths that are in-the-money and by applying ordinary least squares, we can get the value of the coefficients $a_{1}$, …, $a_{10}$.

One GBM and Two IGBMs

Assume that we want to evaluate an investment to enhance energy efficiency in a coal-fired power plant that operates under the EU Emissions Trading Scheme (ETS). As a matter of fact, natural gas-fired power plants usually set the price in electricity markets, or their bid price is very close to the actual marginal price. Therefore, we consider three stochastic processes: natural gas price, coal price, and carbon allowance price. Gas price and carbon price contribute to determining the electricity price and therefore the expected revenues of coal-fired plants; instead, their costs are determined by coal price and carbon price.

The risk-neutral behavior of natural gas price is assumed to be governed by the following IGBM stochastic process with seasonality:

$$ dG_{t} = df\left( t \right) + \left[ {k_{G} G_{m} - (k_{G} + \lambda_{G} )\left( {G_{t} - f(t)} \right)} \right]dt + \sigma_{G} \left( {G_{t} - f(t)} \right)dW_{t}^{G} . $$

(6.45)

In this setting, $G_{m}$ denotes the level to which natural gas price tends in the long run. $f\left(t \right)$ is a deterministic time function. Since we are interested in reflecting the seasonal pattern on the gas price time series throughout the year, we resort to a sinusoidal function like the cosine function: $\left(t \right) = \gamma \cos \left({2\pi (t + \varphi)} \right)$. Here ${ \cos }$ stands for the cosine function measured in radians, and $\gamma$ is a constant parameter (Lucia and Schwartz 2002). The cosine function has annual periodicity, hence the time is measured in years. At time $t = - \varphi$ we have $f\left( {t = - \varphi } \right) = \gamma$ and seasonality is highest.

Regarding coal price we adopt a stochastic process that is similar to that for natural gas but does not display seasonality:

$$ dC_{t} = \left[ {k_{C} \left( {C_{m} - C_{t} } \right) - \lambda_{C} C_{t} } \right]dt + \sigma_{C} C_{t} dW_{t}^{C} . $$

(6.46)

The notation runs akin to that for the dynamics in gas price.

The price of the emission allowance in a risk-neutral world $A_{t}$ is assumed to follow a standard GBM:

$$ dA_{t} = \left( {\alpha - \lambda_{A} } \right)A_{t} dt + \sigma_{A} A_{t} dW_{t}^{A} ; $$

(6.47)

$\lambda_{A}$ is the market price of carbon price risk.

Correlated (deseasonalised) random variables are generated according to the scheme:

$$ C_{t + \Updelta t} \cong \frac{{k_{C} C_{m} }}{{k_{C} + \lambda_{C} }}\left( {1 - e^{{ - (k_{C} + \lambda_{C} )\Updelta t}} } \right) + C_{t} e^{{ - (k_{C} + \lambda_{C} )\Updelta t}} + \sigma_{C} C_{t} \sqrt {\Updelta t} \varepsilon_{t}^{1} , $$

(6.48)

$$ A_{t + \Updelta t} \cong A_{t} e^{{(\alpha - \lambda_{A} )\Updelta t}} + \sigma_{A} A_{t} \sqrt {\Updelta t} \left[ {\varepsilon_{t}^{1} \rho_{CA} + \varepsilon_{t}^{2} \sqrt {1 - \rho_{CA}^{2} } } \right], $$

(6.49)

$$ \begin{aligned} G_{t + \Updelta t}\, \cong \,& f\left({t + \Updelta t} \right) + \frac{{k_{G} G_{m}}}{{k_{G} + \lambda_{G}}}\left({1 - e^{{- (k_{G} + \lambda_{G})\Updelta t}}} \right) + \left({G_{t} - f(t)} \right)e^{{- (k_{G} + \lambda_{G})\Updelta t}} \\ & + \sigma_{G} \left({G_{t} - f(t)} \right)\sqrt {\Updelta t} \left[{\varepsilon_{t}^{1} \rho_{CG} + \varepsilon_{t}^{2} \frac{{\rho_{AG} - \rho_{CG} \rho_{CA}}}{{\sqrt {1 - \rho_{CA}^{2}}}} + \varepsilon_{t}^{3} \sqrt {1 - \rho_{CG}^{2} - \frac{{\left({\rho_{AG} - \rho_{CG} \rho_{CA}} \right)^{2}}}{{1 - \rho_{CA}^{2}}}}} \right]. \\ \end{aligned} $$

(6.50)

$\varepsilon_{t}^{1}$, $\varepsilon_{t}^{2}$, and $\varepsilon_{t}^{3}$ are standardized Gaussian white noises with zero correlation. If samples from a standardized bivariate normal distribution are required, an appropriate procedure is the one shown above, where $\rho_{GC}$, $\rho_{GA}$, and $\rho_{CA}$ are the correlation coefficients between the variables in the multivariate distribution.

One GBM and Two Ornstein–Uhlenbeck Processes

We could aim to study value and risk involved in coal stations operating under the EU ETS after Kyoto Protocol’s expiration. We accomplish this by means of simulation techniques. Since our aim is to derive values of the Earnings at Risk (EaR), simulation must use real parameters and not risk-neutral parameters (Wilmott 2006).

We adopt the simplest mean-reverting stochastic process (also known as an Ornstein–Uhlenbeck process or O-U process) for the Clean Spark Spread (the first ingredient to the Clean Dark Spread as defined in Abadie and Chamorro 2009):

$$ dS_{t} = k_{S} \left({S_{m} - S_{t}} \right)dt + \sigma_{S} dW_{t}^{S}. $$

(6.51)

The current value $S_{t}$ tends to the level $S_{m}$ in the long term at a speed of reversion $k_{S}$. Besides, $\sigma_{S}$ is the instantaneous volatility, and $dW_{t}^{S}$ stands for the increment to a standard Wiener process. This model allows $S_{t}$ to take on negative and positive values.

Next we adopt the notation in Kloeden and Platen (1992). The homogeneous equation is:

$$ \frac{{dS_{t}}}{{S_{t}}} = - k_{S} dt. $$

Therefore, its fundamental solution is ${{\Upphi}}_{{{\text{t}},{\text{t}}_{0}}} = {\text{e}}^{{- {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0})}}$. By making ${\text{Y}}_{\text{t}} \equiv {{\Upphi}}_{{{\text{t}},{\text{t}}_{0}}}^{- 1} S_{t} = {\text{e}}^{{{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0})}} S_{t}$, derivatives can be computed:

$$ \frac{{dY_{t} }}{{dS_{t} }} = {\text{e}}^{{{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} ,\frac{{{\text{d}}^{2} Y_{t} }}{{dS_{t}^{2} }} = 0,\frac{{dY_{t} }}{dt} = k_{S} {\text{e}}^{{{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} S_{t} . $$

By Ito’s Lemma:

$$ dY_{t} = d\left( {\Upphi_{{{\text{t}},{\text{t}}_{0} }}^{ - 1} S_{t} } \right) = k_{S} S_{m} {\text{e}}^{{{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} dt + {\text{e}}^{{{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} \sigma_{S} dW_{t}^{S} . $$

Hence we deduce that:

$$ S_{t} = S_{0} {\text{e}}^{{ - {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} + {\text{e}}^{{ - {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} k_{S} \int\limits_{{t_{0} }}^{T} {{\text{e}}^{{{\text{k}}_{\text{S}} ({\text{s}} - {\text{t}}_{0} )}} } ds + {\text{e}}^{{ - {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} \sigma_{S} \int\limits_{{t_{0} }}^{T} {{\text{e}}^{{{\text{k}}_{\text{S}} ({\text{s}} - {\text{t}}_{0} )}} } dW_{s}^{S} . $$

The first moment is:

$$ E\left( {S_{t} } \right) = S_{0} {\text{e}}^{{ - {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} + S_{m} \left[ {1 - {\text{e}}^{{ - {\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} } \right]; $$

(6.52)

therefore: $E\left({S_{\infty}} \right) = S_{m}$. The variance is given by:

$$ Var\left( {S_{t} } \right) = {\text{e}}^{{ - 2{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} \sigma_{S}^{2} \int_{{t_{0} }}^{t} {{\text{e}}^{{2{\text{k}}_{\text{S}} ({\text{s}} - {\text{t}}_{0} )}} } ds = \frac{{\sigma_{S}^{2} }}{{2{\text{k}}_{\text{S}} }}\left[ {1 - {\text{e}}^{{ - 2{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} } \right]. $$

(6.53)

Since both mean and variance remain finite as $t \to \infty$, this process is stationary.

Equation (6.51) is the continuous-time version of a first-order autoregressive process AR(1) in discrete time:

$$ S_{t + \Updelta t} = S_{m} \left[ {1 - {\text{e}}^{{ - {\text{k}}_{\text{S}} \Updelta t}} } \right] + S_{t} {\text{e}}^{{ - {\text{k}}_{\text{S}} \Updelta t}} + \varepsilon_{t + \Updelta t}^{S} = a_{S} + b_{S} S_{t} + \varepsilon_{t + \Updelta t}^{S} , $$

(6.54)

where $\varepsilon_{t + \Updelta t}^{S} \sim N\left({0,\sigma_{\varepsilon}^{S}} \right)$, and the following notation holds:

$$ a_{S} \equiv S_{m} \left[ {1 - b_{S} } \right] \to S_{m} = \frac{{a_{S} }}{{1 - b_{S} }}, $$

(6.55)

$$ b_{S} \equiv {\text{e}}^{{ - {\text{k}}_{\text{S}} \Updelta t}} \to {\text{k}}_{\text{S}} = - \frac{{\ln b_{S} }}{\Updelta t}. $$

(6.56)

Also, as shown in Abadie and Chamorro (2009, Appendix A):

$$ \left( {\sigma_{\varepsilon }^{S} } \right)^{2} = \frac{{\sigma_{S}^{2} }}{{2{\text{k}}_{\text{S}} }}\left[ {1 - {\text{e}}^{{ - 2{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} } \right] \to \sigma_{S}^{2} = \frac{{2{\text{k}}_{\text{S}} \left( {\sigma_{\varepsilon }^{S} } \right)^{2} }}{{1 - {\text{e}}^{{ - 2{\text{k}}_{\text{S}} ({\text{t}} - {\text{t}}_{0} )}} }} = \frac{{2\ln b_{S} \left( {\sigma_{\varepsilon }^{S} } \right)^{2} }}{{\Updelta t\left[ {b_{S}^{2} - 1} \right]}}. $$

(6.57)

Equations (6.55–6.57) will allow us to recover the continuous-time process parameters (${\text{k}}_{\text{S}}$, $S_{m}$, $\sigma_{S}$) upon estimation of the regression coefficients ($a_{S}$, $b_{S}$) and the standard deviation of the regression residuals ($\sigma_{\varepsilon}^{S}$).

Now we turn to the second term in the Clean Dark Spread. Again we adopt an Ornstein–Uhlenbeck process for the difference $G_{t}/0.55 - C_{t}/0.40$. We have another AR(1) process as its counterpart in discrete time:

$$ D_{t + \Updelta t} = D_{m} \left[ {1 - {\text{e}}^{{ - {\text{k}}_{\text{D}} \Updelta t}} } \right] + D_{t} {\text{e}}^{{ - {\text{k}}_{\text{D}} \Updelta t}} + \varepsilon_{t + \Updelta t}^{\text{D}} = a_{\text{D}} + b_{\text{D}} {\text{D}}_{t} + \varepsilon_{t + \Updelta t}^{\text{D}} , $$

(6.58)

where $D_{t}$ denotes the price gap at time t, $D_{m}$ is the level of the gap in the long term, and ${\text{k}}_{\text{D}}$ stands for the speed of reversion. The remainder of the notation goes as before.

During the current period we assume that carbon price ${\text{A}}_{\text{t}}$ (in €/tCO₂) follows a GBM:

$$ dA_{t} = \alpha A_{t} dt + \sigma_{A} A_{t} dW_{t}^{A}. $$

Therefore, the expected value for the allowance price in the near future is:

$$ E\left( {A_{t} } \right) = A_{0} e^{\alpha t} ,\quad for\,t < 5. $$

(6.59)

At the end of this period we assume there will be a sudden jump J in price, which would push the expected value upwards:

$$ \begin{aligned} t =\; & 5^{-} :\;E\left({A_{t}} \right) = A_{0} e^{5\alpha}; \\ t =\; & 5^{+} :E\left({A_{t}} \right) = A_{0} e^{5\alpha} + J. \\ \end{aligned} $$

From then on, we assume allowance scarcity is just right as an environmental policy measure and price evolves once again following a GBM:

$$ E\left( {A_{t} } \right) = A_{0} e^{\alpha t} + Je^{{\alpha \left( {t - 5} \right)}} \quad for\,t > 5. $$

(6.60)

No further jumps are assumed in subsequent years for the sake of simplicity. Though environmental policy is conceivably expected to become stricter and push allowance prices to new heights at the end of this period, it is hard to foresee what will happen then.

According to Ito’s Lemma, the transformed variable $X_{t} \equiv \ln A_{t}$ follows a stochastic process:

$$ dX_{t} = \left({\alpha - \frac{{\sigma_{A}^{2}}}{2}} \right)dt + \sigma_{A} dW_{t}^{A}. $$

In discrete time:

$$ y_{t} \equiv \Updelta \ln A_{t} = \ln A_{t} - \ln A_{t - \Updelta t} = \left( {\alpha - \frac{{\sigma_{A}^{2} }}{2}} \right)\Updelta t + \sigma_{A} \sqrt {\Updelta t} \varepsilon_{t}^{3} , $$

(6.61)

where $\varepsilon_{t}^{3}$ is a standard Gaussian white noise.

Now, let $e_{1}$, $e_{2}$, and $e_{3}$. be uncorrelated standard normal deviates. Random samples of correlated variables can be generated as follows:

$$ x_{1} = f_{11} e_{1}, $$

(6.62)

$$ x_{2} = f_{21} e_{1} + f_{22} e_{2}, $$

(6.63)

$$ x_{3} = f_{31} e_{1} + f_{32} e_{2} + f_{33} e_{3}, $$

(6.64)

where $E\left({x_{i}} \right) = 0$, and $Cov\left({x_{i},x_{j}} \right) = \rho_{ij}$, with $i,j = 1,2,3,i \ne j$.

Random deviates with this correlation structure must satisfy the conditions:

$$ \begin{aligned} E\left({x_{1}^{2}} \right) = & 1 \to f_{11} = 1, \\ E\left({x_{1} x_{2}} \right) = & \rho_{12} \to f_{21} = \rho_{12}, \\ E\left({x_{2}^{2}} \right) = & 1 = f_{21}^{2} + f_{22}^{2} \to f_{22} = \sqrt {1 - \rho_{12}^{2}}, \\ E\left({x_{1} x_{3}} \right) = & \rho_{13} = f_{11} f_{31} \to f_{31} = \rho_{13}, \\ E\left({x_{2} x_{3}} \right) = & \rho_{23} = f_{21} f_{31} + f_{22} f_{32} \to f_{32} = \frac{{\rho_{23} - \rho_{12} \rho_{13}}}{{\sqrt {1 - \rho_{12}^{2}}}}, \\ E\left({x_{3}^{2}} \right) = & 1 = f_{31}^{2} + f_{32}^{2} + f_{33}^{2} = \rho_{13}^{2} + \frac{{\rho_{23} - \rho_{12} \rho_{13}}}{{1 - \rho_{12}^{2}}} + f_{33}^{2} \to \\ f_{33} = & \sqrt {1 - \rho_{13}^{2} - \frac{{\left({\rho_{23} - \rho_{12} \rho_{13}} \right)^{2}}}{{1 - \rho_{12}^{2}}}}. \\ \end{aligned} $$

Therefore,

$$ x_{1} = e_{1}, $$

(6.65)

$$ x_{2} = e_{1} \rho_{12} + e_{2} \sqrt {1 - \rho_{12}^{2}}, $$

(6.66)

$$ x_{3} = e_{1} \rho_{13} + e_{2} \frac{{\rho_{23} - \rho_{12} \rho_{13} }}{{\sqrt {1 - \rho_{12}^{2} } }} + e_{3} \sqrt {1 - \rho_{13}^{2} - \frac{{\left( {\rho_{23} - \rho_{12} \rho_{13} } \right)^{2} }}{{1 - \rho_{12}^{2} }}} . $$

(6.67)

Correlated random variables are thus generated according to the above scheme:

$$ S_{t + \Updelta t} = S_{m} \left[ {1 - {\text{e}}^{{ - {\text{k}}_{\text{S}} \Updelta t}} } \right] + S_{t} {\text{e}}^{{ - {\text{k}}_{\text{S}} \Updelta t}} + \sigma_{S} \sqrt {\frac{{1 - {\text{e}}^{{ - 2{\text{k}}_{\text{S}} \Updelta t}} }}{{2{\text{k}}_{\text{S}} }}} \varepsilon_{t}^{1} , $$

(6.68)

$$ D_{t + \Updelta t} = D_{m} \left[ {1 - {\text{e}}^{{ - {\text{k}}_{\text{D}} \Updelta t}} } \right] + D_{t} {\text{e}}^{{ - {\text{k}}_{\text{D}} \Updelta t}} + \sigma_{D} \sqrt {\frac{{1 - {\text{e}}^{{ - 2{\text{k}}_{\text{D}} \Updelta t}} }}{{2{\text{k}}_{\text{D}} }}} \left[ {\varepsilon_{t}^{1} \rho_{SD} + \varepsilon_{t}^{2} \sqrt {1 - \rho_{SD}^{2} } } \right], $$

(6.69)

$$ \begin{aligned} \ln A_{t + \Updelta t} = & \ln A_{t} + \left({\alpha - \frac{{\sigma_{A}^{2}}}{2}} \right)\Updelta t \\ & + \sigma_{A} \sqrt {\Updelta t} \left[{\varepsilon_{t}^{1} \rho_{SA} + \varepsilon_{t}^{2} \frac{{\rho_{DA} - \rho_{SA} \rho_{SD}}}{{\sqrt {1 - \rho_{SD}^{2}}}} + \varepsilon_{t}^{3} \sqrt {1 - \rho_{SA}^{2} - \frac{{\left({\rho_{DA} - \rho_{SA} \rho_{SD}} \right)^{2}}}{{1 - \rho_{SD}^{2}}}}} \right], \\ \end{aligned} $$

(6.70)

where $\varepsilon_{t}^{1}$, $\varepsilon_{t}^{2}$ and $\varepsilon_{t}^{3}$ are standardized Gaussian white noises with zero correlation. The first expression above is derived after replacing $\sigma_{\varepsilon}^{S}$ in terms of $\sigma_{S}$. Similarly in the second expression. At the same time, if samples from a standardized bivariate normal distribution are required, an appropriate procedure is the one shown above, where $\rho_{SD}$, $\rho_{SA}$, and $\rho_{DA}$ are the correlation coefficients between the variables in the multivariate distribution.