In their article "Gas Storage Valuation in Incomplete Markets" Nils Löhndorf and David Wozabal propose a multistage stochastic programming model for extrinsic valuation and optimization of natural gas storages. They show that rolling intrinsic valuation is an inconsistent pricing rule when markets are incomplete and that computing the extrinsic value requires analysis of a combined control problem of storage operation and portfolio optimization that takes risk preferences into account.
The analyses in the article are based on Monte Carlo simulations of term structure dynamics of monthly futures contracts which are calibrated using historical futures and options price data from NYMEX Henry Hub natural gas futures.
In this blog post, we report how the proposed policies from the paper perform in a real trading situation when evaluated against real prices instead of simulations. In particular, we are interested in a mutual comparison of the trading strategy derived from the stochastic programming model and a rolling intrinsic trading strategy.
To do so, we collected data of historical 12-month price futures curves of NYMEX Henry Hub natural gas futures from 1991 to 2015. We compute profits of a gas storage unit that accumulate over a 12-month time horizon with zero interest rate from March to February. To estimate the parameters of the MGBM price process, we used data from the respective previous 5 years for each evaluation period. Calculations are based on a storage with 1.0 mmBtu capacity, and injection (withdrawal) rates of 0.15 mmBtu/month (0.3 mmBtu/month).
We investigate the decisions of a hypothetical gas trader who purchases a storage contract each year and then manages the contract over the course of the year. This requires to solve the stochastic programming problem over a receding horizon and update model parameters accordingly. Starting in 1996, each year began with a clean slate and a 12-month planning horizon. After one problem was solved, the immediate profit from implementing the first-stage decision was recorded and the final resource state (storage, portfolio of futures contracts) was passed on as initial state to the subsequent 11-month problem. Before solving the subsequent problem, MGBM parameters were reestimated and a new scenario lattice was built. This process was repeated over a receding horizon until only 1 month was left. Then, a new futures contract was purchased. The resulting sample of records contains 20 annual and 240 monthly reward observations.
The results of the backtest for different policy parameters are summarized in the following Table:
| Nodes | Iterations | VaR | Risk aversion | Avg Profit | Std Dev | Difference to Rolling Intrinsic | p-value |
|---|---|---|---|---|---|---|---|
| 100 | 200 | 0.05 | 0.50 | 0.851 | 0.843 | 0.075 | 0.011 |
| 100 | 200 | 0 | 1.084 | 1.579 | 0.308 | 0.184 | |
| 100 | 200 | 0.01 | 0.50 | 0.836 | 0.874 | 0.060 | 0.043 |
| 100 | 200 | 0.10 | 0.50 | 0.884 | 0.922 | 0.109 | 0.046 |
| 100 | 200 | 0.05 | 1.00 | 0.808 | 0.821 | 0.033 | 0.062 |
| 100 | 200 | 0.05 | 0.10 | 1.023 | 1.170 | 0.247 | 0.136 |
| 10 | 100 | 0.05 | 0.50 | 0.844 | 0.866 | 0.069 | 0.081 |
| 1000 | 1000 | 0.05 | 0.50 | 0.846 | 0.852 | 0.071 | 0.007 |
The rolling intrinsic strategy which serves as benchmark achieves to an average annual profit of 0.775 ($/mmBtu) with a standard deviation of 0.815. The table reports the difference in average profits between the rolling intrinsic strategy and the approximate optimal policies for different parameters, along with the p-values of a two-sided t-test for equal means.
The results show that all trading strategies based stochastic programming result in higher profits than the rolling intrinsic strategy. In line with the findings from the simulation study, we observe a clear trade-off between reward and risk. The difference to the rolling intrinsic value is in the range of 8-40% and is more sensitive towards the risk preferences than to algorithmic parameters.
Although the difference is greatest for the model with zero risk aversion (risk-neutral case), the result is not significant at the 90% confidence level, due to the high variability of the reward distribution. Setting the risk aversion parameter greater than 0.5 decreases risk as expected, and p-values decrease accordingly, so that for these instances differences become statistically significant.
Varying the number of lattice nodes per stage had almost no effect. This confirms the result of the simulation study from the published article which shows that coarse discretizations already provide quality solutions.
The results demonstrate that the rolling intrinsic strategy is not optimal for a risk-neutral decision maker and more profit can be made by solving the underlying multistage stochastic programming problem. Futures trading for storages of natural gas clearly benefits from using stochastic programming models and outperform the classic risk-free pricing approach based on rolling intrinsic valuation.