Choosing Lagged State Variables in Least Squares Monte Carlo
Summary
The document asks how to include lags when applying least squares Monte Carlo to a timing real option. The project can be started at monthly decision dates over a five-year window, while revenue follows a non-Markov ARMA process sampled hourly over a twenty-year project life. The central modeling issue is whether regression inputs should reflect decision-date observations or the hourly process history.
The quoted study on renewable energy certificates is described as testing lagged electricity-price terms in continuation-value regressions for a GARCH price process, with only fractional changes in the estimated option value across its tested lag choices. The document does not resolve how to map those regressors to the questioner's mixed time scales, nor does it give a complete LSMC procedure. Its useful lesson is that non-Markov dynamics may require regression state variables carrying relevant history, sampled consistently with the exercise decisions and information available then; future realized hours should not be treated as known at a decision point.
Key ideas
- LSMC continuation-value regressions use information available at each exercise decision.
- For a non-Markov revenue process, lagged observations can help represent relevant history in the regression state.
- The example study tests lagged price variables for a GARCH process and reports only fractional changes in option value across its tested choices.
- The document leaves unresolved how to align hourly revenue lags with monthly exercise dates.
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Full text
# Least Squares Monte Carlo with Lags
# Least Squares Monte Carlo with Lags
I would like to evaluate a timing real option using least squares Monte Carlo.
I have a project life cycle of 20 years and can decide on a monthly basis for 5 years whether the project should be implemented. The project's revenue is modeled using an ARMA 5,1, i.e., not Markov, with hourly time steps.
How exactly do I perform the evaluation with lags: Do the lags refer to the option values of the monthly decision points of the simulated paths that I have to include in the regression, or do the lags refer to my hourly revenue function, so that at each decision point I have to consider information about the following hours of each individual ARMA path? It is therefore unclear to me at which level the lags are used in the least squares Monte Carlo method.
The idea is as follows: “Deferring real options with solar renewable energy certificates" by Zhang, Assereto, and Byrne. They model the return function differently using GARCH with daily time steps.
> Longstaff and Schwartz (2001) note that using simple polynomials as basis functions provides accurate results. Since the GARCH process of electricity prices is non-Markovian, we tested the robustness of our results by adding different numbers of lags to the cross-sectional regression, as suggested by the original authors. The number of lags ranges from 1 to 10, which led to only fractional changes in the price of the real option.
The continuation value is calculated as follows ($e$, is the electricity price and $s$ is the subsidies which are not relevant to my problem): $$ {F_t,j}/{(1 + r)^{\Delta t}} = \beta_0 + \beta_{11} P^e_{t-1,j} + \beta_{12} (P^e_{t-1,j})^2 + \beta_{13} (P^e_{t-1,j})^3 + \beta_{21} P^s_{t-1,j} + \beta_{22} (P^s_{t-1,j})^2 + \beta_{23} (P^s_{t-1,j})^3 + \varsigma_{t-1} $$ For me, it is not clear whether the paper refers directly to the lags in the GARCH process or whether the lags merely refer to the monthly decision points.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.