Dynamic hedging for the real option management of hydropower production with exchange rate risks

Abstract

We study the risk management problem of a hydropower producer that hedges risk by trading currency and power futures contracts. The model considers three types of risks: operational risk due to supply uncertainty, profit risk due to power price variability, and exchange rate risk when operation and trading take place in different currencies. We cast the problem as a Markov decision process and propose a sequential solution approach that separates operational management from trading. To solve the problem, we first reduce the high-dimensional Markovian process that models inflows, exchange rates, and future curve dynamics to a scenario lattice and then employ stochastic dual dynamic programming under a risk measure. We find that dynamic hedging leads to significant risk reduction and that it performs better than static hedge ratios that are often used in practice. We also find that a sequential approach leads to better outcomes than an integrated approach across various metrics, which supports the functional separation of operation and hedging that is common practice in most power companies.

Appendices

Appendix A: Cash flows and balancing of electricity contracts

The following details how positions in electricity futures are tracked, depending on trading, listing and de-listing, and varying contract delivery periods. We also explain the cash flows from this trading, including the effect of transaction costs, product listing, settlement before and during delivery, and taxes.

Let us introduce decision variables for short positions in power futures. Denote $u_{t,Mi}$, $u_{t,Qj}$, and $u_{t,Y1}$ [MWh] the total short position at stage t in futures contracts with delivery in i months, j quarters and 1 year. The last index denotes contracts that have not yet entered delivery. We use $u_{t,M}$, $u_{t,Q}$ and $u_{t,Y}$ [MWh] to denote short positions of contracts that are currently in delivery. Variables $w_{t,Mi}$, $w_{t,Qj}$, and $w_{t,Y1}$ [MWh] denote new short positions that enter at stage t. If the delivery period of a contract exceeds the model horizon $\hat{T} = 49$ semi-months, then the corresponding decision variable ($w_{t,Mi}$, $w_{t,Qj}$ or $w_{t,Y1}$) is set to zero to guarantee that no trading takes place.

Balance constraints for short positions in power futures depend on whether time stage t represents the first or second part of a month, the beginning of a new quarter or the beginning of a new year. If t represents the second part of a month, then the total short position is given by the previous stage value plus new short positions for contracts not yet in delivery.

$$\begin{aligned} \begin{aligned} u_{t,M} = u_{t-1,M}, \quad u_{t,Q} = u_{t-1,Q}, \quad u_{t,Y} = u_{t-1,Y} \\ u_{t,Mi} = u_{t-1,Mi} + w_{t,Mi}, \quad i = [1,\ldots ,6] \\ u_{t,Qj} = u_{t-1,Qj} + w_{t,Qj}, \quad j = [1,\ldots ,8] \\ u_{t,Y1} = u_{t-1,Y1} + w_{t,Y1} \end{aligned} \end{aligned}$$

(A.1)

When t represents the first part of a month, the contract that was 1 month ahead (M1) in $t-1$ goes into delivery, M2 becomes M1, M3 becomes M2, etc. A new contract is introduced for delivery in six months (M6). If t represents the first part of the month but not a new quarter, then balance constraints for month contracts are given by (A.2). The remaining relationships in (A.1) do not change, that is, they remain valid for all stages t.

$$\begin{aligned} \begin{aligned} u_{t,Mi} = u_{t-1,Mi+1} + w_{t,Mi}, \quad i = [1,\ldots ,5] \\ u_{t,M6} = w_{t,M6} \\ \end{aligned} \end{aligned}$$

(A.2)

Using the same logic, balance constraints for quarter contracts for stages marking the beginning of a quarter, but not a new year, are given by (A.3).

$$\begin{aligned} \begin{aligned} u_{t,Qj} = u_{t-1,Qj+1} + w_{t,Qj}, \quad j = [1,\ldots ,7] \\ u_{t,Q8} = w_{t,Q8} \\ \end{aligned} \end{aligned}$$

(A.3)

If t represents the beginning of a year, then balance constraints for annual contracts are given by (A.4).

$$\begin{aligned} \begin{aligned} u_{t,Y1} = 0 \end{aligned} \end{aligned}$$

(A.4)

Having established balance constraints for short positions in power futures, let us define variables and restrictions for the portfolio of power futures. Denote $y_{t,t}^F$ as cash flow from power futures trading in t that can take positive and negative values, and is part of the value function of the hedging problem (13). The currency spot rate at which cash flows occur is not known in advance. $y_{t,t}^F$ is therefore denoted in EUR and $y_{t,t}^C$ is denoted in NOK.

Variable $y_{t,T}^F$ tracks the committed, positive part of the cash flows from power futures trading that will occur at time T. Variable $y_{t,T}^F$, where $T > t$ is not part of the value function. It is only used to store the positive part of the cash flows that will occur in subsequent periods. In (A.11), the negative part of the cash flows is added to the positive flows to obtain time t cash flows $y_{t,t}^F$. Variable $y_{t,t+1}^F$ stores two types of cash flows. The first is related to changes in the value of the portfolio of contracts not yet in delivery. All contracts are marked-to-market regularly. We therefore need to store the forward prices in stage t to calculate the price changes in stage $t+1$. The second type of cash flow is associated with contracts in delivery. Variable $y_{t,T}^F$ for $T > t + 1$ only stores the positive part of cash flows associated with contracts in delivery. The longest delivery period spanned by any of the available contracts is 24 semi-months. It is therefore only necessary to define $y_{t,T}^F$ for $T = [t+1,\ldots ,t+24]$.

We first consider the case where t represents the first part of a month. The cash flow balances will then be given by

$$\begin{aligned} \begin{aligned} y_{t,t+1}^F&= y_{t-1,t+1}^F + (1 - \gamma _\textrm{c})\left( \sum _{i=1}^{6} u_{t,Mi} F_{t,Mi} + \sum _{j=1}^{8} u_{t,Qj} F_{t,Qj} + u_{t,Y1} F_{t,Y1} \right) \\&y_{t,T}^F = y_{t-1,T}^F, \quad t+2 \le T \le t+23\\&y_{t, T}^F = 0, \quad \quad \quad \quad \quad \quad T = t+24 \end{aligned} \end{aligned}$$

(A.5)

Only $y_{t,t+1}^F$ is updated in this case. This is because the next stage is in the same month as the current stage, so that no new contracts go into delivery.

If t is the second part of a month, then multiple contracts can potentially enter into delivery in the upcoming stage ($t+1$). The price and position of the futures contracts that go into delivery must thus be stored. The cash flows for the next 2, 6, or 24 periods must also be stored. How long they will be stored depends on whether the contract is monthly, quarterly, or yearly. We introduce the indicator functions ${\mathbb {I}}_Q$ and ${\mathbb {I}}_Y$ to make the formulation more compact. Function values are equal to 1 if the next stage ($t+1$) marks the beginning of a new quarter and year, respectively, and 0 otherwise. With this definition, cash flow balances are given by

$$\begin{aligned} y_{t,t+1}^F&= y_{t-1,t+1}^F + (1- \gamma _\textrm{c}) \Big ( \frac{u_{t,M1}F_{t,M1}}{2} +{\mathbb {I}}_Q \frac{u_{t,Q1}F_{t,Q1}}{6} + {\mathbb {I}}_Y \frac{u_{t,Y1}F_{t,Y1}}{24} \\ & \quad + \sum _{i=2}^{6} u_{t,Mi} F_{t,Mi} + (1 - {\mathbb {I}}_Q) u_{t,Q1} F_{t,Q1} \\ &\quad + \sum _{j=2}^{8} u_{t,Qj} F_{t,Qj} + (1 - {\mathbb {I}}_Y) u_{t,Y1} F_{t,Y1} \Big ) \end{aligned}$$

(A.6)

$$\begin{aligned} y_{t,t+2}^F&= y_{t-1,t+2}^F + (1- \gamma _\textrm{c}) \Big ( \frac{u_{t,M1}F_{t,M1}}{2} + {\mathbb {I}}_Q \frac{u_{t,Q1}F_{t,Q1}}{6} + {\mathbb {I}}_Y \frac{u_{t,Y1}F_{t,Y1}}{24} \Big ) \end{aligned}$$

(A.7)

$$\begin{aligned} y_{t,t+i}^F&= y_{t-1,t+i}^F + (1- \gamma _\textrm{c}) \Big ({\mathbb {I}}_Q \frac{u_{t,Q1}F_{t,Q1}}{6} + {\mathbb {I}}_Y \frac{u_{t,Y1}F_{t,Y1}}{24} \Big ), \quad i = [3,\ldots ,6] \end{aligned}$$

(A.8)

$$\begin{aligned} y_{t,t+i}^F&= y_{t-1,t+i}^F + (1- \gamma _\textrm{c}) {\mathbb {I}}_Y \frac{u_{t,Y1}F_{t,Y1}}{24}, i = [7,\ldots ,23] \end{aligned}$$

(A.9)

$$\begin{aligned} y_{t,t+24}^F&= (1- \gamma _\textrm{c}) {\mathbb {I}}_Y \frac{u_{t,Y1}F_{t,Y1}}{24} \end{aligned}$$

(A.10)

In (A.11), we formulate the stage t cash flows from power trading ($y_{t,t}^F$). As with currency forwards, the expression consists of the committed, positive cash flows saved in $y_{t-1,t}^F$ and all negative cash flows. Note that we must subtract $w_{t,Mi}, w_{t,Qj}$ and $w_{t,Y1}$ from positions $u_{t,Mi}, u_{t,Qj}$, and $u_{t,Y1}$ to obtain the negative part of the cash flows associated with price changes in contracts prior to delivery, because these are based on previous positions. We also include variable transaction costs, $c_\textrm{F}$ [EUR/MWh].

$$\begin{aligned} y_{t,t}^F&= y_{t-1,t}^F + (1-\gamma _\textrm{c}) \left[ - \left( \frac{u_{t,M}}{2}+\frac{u_{t,Q}}{6}+\frac{u_{t,Y}}{24}\right) F_{t,t} \right. \\ &\quad \left. - \left( \sum _{i = 1}^{6} (u_{t,Mi}-w_{t,Mi}) F_{t,Mi} +\sum _{j=1}^{8} \big (u_{t,Qj}-w_{t,Qj}\big ) F_{t,Qj} + \big (u_{t,Y1}-w_{t,Y1}\big ) F_{t,Yi} \right) \right. \\ &\quad \left. - c_\textrm{F} \left( \sum _{i=1}^{6} w_{t,Mi} + \sum _{j=1}^{8} w_{t,Qj} + w_{t,Y1}\right) \right] \end{aligned}$$

(A.11)

The cash flows from power trading (A.11) enters the value function for hedging.

Appendix B: Coefficients and parameter values

For all simulations, we used coefficient and parameter values given in Table 7. The tax rates $\gamma _\textrm{r}$ and $\gamma _\textrm{c}$ (Thorvaldsen et al. 2018) and transaction costs $c_\textrm{F}$ (Nasdaq Oslo ASA and Nasdaq Clearing AB 2018) are correct as of 2018. As in Dupuis et al. (2016), the variable transaction costs for trading at NASDAQ are given by the sum of the market trading fee (0.0045 EUR/MWh) and clearing fee (0.0099 EUR/MWh). This clearing fee is applicable if the total quarterly volume cleared by the firm is below 3 TWh. The time-dependent discount factor $\beta _t$ is given by $\beta _t = \exp {(-r\Delta t)}$, where $\Delta t$ denotes the length of the semi-month t. The energy coefficient $\kappa$ is based on the empirical relationship between production and water dispatch, and calculations considering the mean empirical reservoir level and turbine/generator efficiency rate. It has been found to be slightly lower than the one currently used by the plant.

Table 7 Coefficients and constants used for numerical studies

Appendix C: Sensitivity analysis of hedging results

In this section, we present some results illustrating the sensitivity of using 500 forward-backward passes and $10^5$ iterations. This has been done by performing six separate runs of the hedging model with $\alpha = 0.1$ and trading in all contracts, and comparing the mean and standard deviation of the main statistical measures included in e.g. Table 1. These results are summarized in Table 8. The computational time of each run is approximately 4.5 hours, and its memory usage is close to the maximum capacity.

Table 8 Sensitivity of statistical measures based on six separate runs

As Table 8 shows, the obtained results are subject to a certain degree of uncertainty. The standard deviations indicate that all statistical measures are stable in the first two digits, while there is some uncertainty in the third digit. This indicates that the results are sufficiently precise to assess and compare the general risk performance of the model variants, but increasing the number of passes and simulations would result in more precise results.