# Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Assumption**

**1.**

- The process S is square integrable and adapted to the filtration ${\mathbb{F}}^{0}$.
- The process X is a square integrable martingale with respect to the filtration $\mathbb{F}$.
- The process $\overline{X}$ that is defined by ${\overline{X}}_{t}:=\mathbb{E}\left[{X}_{t}\right|{\mathcal{F}}_{t}^{0}]$ for $0\le t\le T$ is a square integrable martingale with respect to the filtration $\mathbb{F}$.

**Theorem**

**1.**

## 3. A Game of a Major and Minor Agents

**Definition**

**2**

- i.
- ${\varphi}^{*}$ and ${\varphi}^{0*}$ are admissible strategies for, respectively, the representative minor and the major players, the consistency condition ${\overline{\varphi}}_{t}^{*}=\mathbb{E}\left[{\varphi}_{t}^{*}\right|{\mathcal{F}}_{t}^{0}]$ is satisfied for all $t\in [0,T]$ and for any other admissible strategy for the representative minor player ϕ,$${J}^{MF}(\varphi ,{\overline{\varphi}}^{*},{\varphi}^{0*})\le {J}^{MF}({\varphi}^{*},{\overline{\varphi}}^{*},{\varphi}^{0*})$$
- ii.
- For any other triple $(\varphi ,\overline{\varphi},{\varphi}^{0})$ satisfying condition i.,$$\begin{array}{c}\hfill {J}^{MF,0}({\varphi}^{0},\overline{\varphi})\le {J}^{MF,0}({\varphi}^{0*},{\overline{\varphi}}^{*}).\end{array}$$

**Assumption**

**2.**

**Proposition**

**1**

**Proof.**

**Lemma**

**1.**

- i.
- For every ${\mathbb{F}}^{0}$-adapted square integrable process ν,$$\mathbb{E}\left(\right)open="["\; close="]">{\int}_{0}^{T}{\nu}_{t}\{\alpha \left(t\right){\dot{\overline{\varphi}}}_{t}^{*}+{S}_{t}+{a}_{0}{\varphi}_{t}^{0*}+a{\overline{\varphi}}_{t}^{*}\}dt+\lambda ({\overline{\varphi}}_{T}^{*}-{\overline{X}}_{T}){\int}_{0}^{T}{\nu}_{t}dt$$
- ii.
- For every other couple $({\varphi}^{0},\overline{\varphi})$ satisfying the condition i, the inequality (7) holds true.

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**2**

**Remark**

**1.**

**Remark**

**2.**

**Proof.**

**Corollary**

**1.**

- i.
- Assume that the fundamental price process S is a martingale. Subsequently, the equilibrium mean field position of minor agents and the position of the major agent satisfy$${\Xi}_{t}=-{\Pi}_{0,t}{(I+D{\Pi}_{0,T})}^{-1}\left(\right)open="("\; close=")">\begin{array}{c}\hfill {S}_{0}-{\lambda}_{0}{X}_{0}^{0}\\ \hfill 0\\ \hfill {S}_{0}-\lambda {\overline{X}}_{0}\end{array}.$$
- ii.
- In the limit of infinite terminal penalty (when $\lambda ,{\lambda}_{0}\to \infty $), the equilibrium mean field position of minor agents and the position of the major agent satisfy,$${\Xi}_{t}\to {{\rm Y}}_{t}-{\Pi}_{0,t}{\Pi}_{0,T}^{-1}({\tilde{{\rm Y}}}_{0}-{D}_{\infty}{\mathcal{X}}_{0})-{\int}_{0}^{t}{\Pi}_{s,t}{\Pi}_{s,T}^{-1}(d{\tilde{{\rm Y}}}_{s}-{D}_{\infty}d{\mathcal{X}}_{s}),$$almost surely for all $t\in [0,T]$, where$${D}_{\infty}=\left(\right)open="("\; close=")">\begin{array}{ccccccc}& & \hfill 1& & \hfill 0& & \hfill 0\\ & & \hfill 0& & \hfill 0& & \hfill 0\\ & & \hfill 0& & \hfill 0& & \hfill 1\end{array}$$When the fundamental price process S is a martingale, in the limit of infinite terminal penalty, the strategies do not depend on the fundamental price and we have,$${\Xi}_{t}\to {\Pi}_{0,t}\phantom{\rule{0.166667em}{0ex}}{\Pi}_{0,T}^{-1}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\begin{array}{c}\hfill {X}_{0}^{0}\\ \hfill 0\\ \hfill {\overline{X}}_{0}\end{array}.$$
- iii.
- In the absence of terminal penalties (when $\lambda ={\lambda}_{0}=0$), the equilibrium mean field position of minor agents and the position of the major agent satisfy,$$\begin{array}{cc}\hfill {\Xi}_{t}& ={{\rm Y}}_{t}-{\Pi}_{0,t}{(I+{D}_{0}{\Pi}_{0,T})}^{-1}{D}_{0}{\tilde{{\rm Y}}}_{0}-{\int}_{0}^{t}{\Pi}_{s,t}{(I+{D}_{0}{\Pi}_{s,T})}^{-1}{D}_{0}d{\tilde{{\rm Y}}}_{s}.\hfill \end{array}$$

**Proof.**

**Corollary**

**2**

**Proof.**

## 4. Approximate Nash Equilibrium in the N-Player Stackelberg Game

**Assumption**

**3.**

- The process S is square integrable and adapted to the filtration ${\mathbb{F}}^{0}$.
- The demand forecast ${X}^{0}$ of the major agent is a square integrable ${\mathbb{F}}^{0}$-martingale.
- The processes ${\left({X}^{i}\right)}_{i=1}^{\infty}$ are square integrable $\mathbb{F}$-martingales.
- There exists a square intergrable $\mathbb{F}$-martingale $\overline{X}$, such that for all $i\ge 1$, and all $t\in [0,T]$, almost surely, $\mathbb{E}\left[{X}_{t}^{i}\right|{\mathcal{F}}_{t}^{0}]={\overline{X}}_{t}$.
- The processes ${\left({\stackrel{\u02c7}{X}}^{i}\right)}_{i=1}^{\infty}$ that are defined by ${\stackrel{\u02c7}{X}}_{t}^{i}={X}_{t}^{i}-{\overline{X}}_{t}$ for $t\in [0,T]$, are orthogonal square integrable $\mathbb{F}$-martingales, such that the expectation $\mathbb{E}\left[{\left({\stackrel{\u02c7}{X}}_{T}^{i}\right)}^{2}\right]$ does not depend on i.

**Definition**

**3**

- i.
**Deviation of a minor player:**for any other admissible strategy ${\varphi}^{i}$ for the minor player i, $i=1,\dots ,N$,$${J}^{N,i}({\varphi}^{i},{\varphi}^{-i*})-\epsilon \le {J}^{N,i}({\varphi}^{i*},{\varphi}^{-i*}).$$- ii.
**Deviation of the major player:**for any other set of admissible strategies $\left({\varphi}^{i}\right),i=0,\dots ,N$, such that ${\varphi}^{1},\dots ,{\varphi}^{N}$ are optimal responses of minor players to the major player strategy ${\varphi}^{0}$, we have,$$\begin{array}{c}\hfill {J}^{N,0}({\varphi}^{0},{\varphi}^{-0})-\epsilon \le {J}^{N,0}({\varphi}^{0*},{\varphi}^{-0*}).\end{array}$$

**Proposition**

**3.**

**Remark**

**3.**

**Proof.**

**Step 1.**

**Step 2.**

## 5. Numerical Illustration

#### 5.1. Model Specification

#### 5.2. Equilibrium Price and Market Impact

#### 5.3. Volatility and Price-Forecast Correlation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Aïd, René, Andrea Cosso, and Huyên Pham. 2020. Equilibrium price in intraday electricity markets. arXiv arXiv:2010.09285. [Google Scholar]
- Aïd, René, Pierre Gruet, and Huyên Pham. 2016. An optimal trading problem in intraday electricity markets. Mathematics and Financial Economics 10: 49–85. [Google Scholar] [CrossRef][Green Version]
- Alasseur, Clémence, Imen Ben Taher, and Anis Matoussi. 2020. An extended mean field game for storage in smart grids. Journal of Optimization Theory and Applications 18: 644–70. [Google Scholar] [CrossRef][Green Version]
- Bensoussan, Alain, Michael Chau, and Phillip Yam. 2016. Mean field games with a dominating player. Applied Mathematics & Optimization 74: 91–128. [Google Scholar]
- Bensoussan, Alain, and Sheung Chi Phillip Yam. 2019. Mean field approach to stochastic control with partial information. arXiv arXiv:1909.10287v3. [Google Scholar]
- Bouchard, Bruno, Masaaki Fukasawa, Martin Herdegen, and Johannes Muhle-Karbe. 2018. Equilibrium returns with transaction costs. Finance and Stochastics 22: 569–601. [Google Scholar] [CrossRef][Green Version]
- Cardaliaguet, Pierre, Marco Cirant, and Alessio Porretta. 2020. Remarks on Nash equilibria in mean field game models with a major player. Proceedings of the American Mathematical Society 148: 4241–55. [Google Scholar] [CrossRef]
- Carmona, René, and François Delarue. 2018. Probabilistic Theory of Mean Field Games with Applications I. Berlin/Heidelberg: Springer. [Google Scholar]
- Carmona, Rene, and Xiuneng Zhu. 2016. A probabilistic approach to mean field games with major and minor players. The Annals of Applied Probability 26: 1535–80. [Google Scholar] [CrossRef]
- Casgrain, Philippe, and Sebastian Jaimungal. 2020. Mean-field games with differing beliefs for algorithmic trading. Mathematical Finance 30: 995–1034. [Google Scholar] [CrossRef]
- Donier, Jonathan, Julius Bonart, Iacopo Mastromatteo, and Jean-Philippe Bouchaud. 2015. A fully consistent, minimal model for non-linear market impact. Quantitative Finance 15: 1109–21. [Google Scholar] [CrossRef][Green Version]
- Evangelista, David, and Yuri Thamsten. 2020. Finite population games of optimal execution. arXiv arXiv:2004.00790. [Google Scholar]
- Féron, Olivier, Peter Tankov, and Laura Tinsi. 2020. Price formation and optimal trading in intraday electricity markets. arXiv arXiv:2009.04786. [Google Scholar]
- Fu, Guanxing, and Ulrich Horst. 2020. Mean-field leader-follower games with terminal state constraint. SIAM Journal on Control and Optimization 58: 2078–113. [Google Scholar] [CrossRef]
- Fujii, Masaaki, and Akihiko Takahashi. 2020. A mean field game approach to equilibrium pricing with market clearing condition. CARF Working Paper CARF-F-473. arXiv arXiv:2003.03035. [Google Scholar]
- Huang, Minyi. 2010. Large-population LQG games involving a major player: The Nash certainty equivalence principle. SIAM Journal on Control and Optimization 48: 3318–53. [Google Scholar] [CrossRef]
- Huang, Minyi, Roland P. Malhamé, and Peter E. Caines. 2006. Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the nash certainty equivalence principle. Communications in Information & Systems 6: 221–52. [Google Scholar]
- Kiesel, Rüdiger, and Florentina Paraschiv. 2017. Econometric analysis of 15-minute intraday electricity prices. Energy Economics 64: 77–90. [Google Scholar] [CrossRef][Green Version]
- Lacker, Daniel. 2020. On the convergence of closed-loop Nash equilibria to the mean field game limit. Annals of Applied Probability 30: 1693–761. [Google Scholar] [CrossRef]
- Lasry, Jean-Michel, and Pierre-Louis Lions. 2007. Mean field games. Japanese Journal of Mathematics 2: 229–60. [Google Scholar] [CrossRef][Green Version]
- Lasry, Jean-Michel, and Pierre-Louis Lions. 2018. Mean-field games with a major player. Comptes Rendus Mathematique 356: 886–90. [Google Scholar] [CrossRef]
- Nourian, Mojtaba, and Peter E. Caines. 2013. ϵ-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM Journal on Control and Optimization 51: 3302–31. [Google Scholar] [CrossRef][Green Version]
- Shrivats, Arvind, Dena Firoozi, and Sebastian Jaimungal. 2020. A mean-field game approach to equilibrium pricing, optimal generation, and trading in solar renewable energy certificate (srec) markets. arXiv arXiv:2003.04938. [Google Scholar]
- Tan, Zongjun, and Peter Tankov. 2018. Optimal trading policies for wind energy producer. SIAM Journal on Financial Mathematics 9: 315–46. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**(

**Left**) volatility of simulated prices for different market shares of the major agent. (

**Right**) volatility for different delivery hours, estimated empirically from EPEX spot intraday market data of January 2017 for the Germany delivery zone.

**Figure 3.**Correlation between the price increments and the major player renewable production increments vs. the correlation between the price increments and the total renewable production increments.

Parameter | Value | Parameter | Value |
---|---|---|---|

${S}_{0}$ | 40 €/MWh | a | 1 €/MWh${}^{2}$ |

${\sigma}^{S}$ | 10 €/MWh.h${}^{1/2}$ | $\lambda $ | 100 €/MWh${}^{2}$ |

${\overline{X}}_{0}$ | 0 MWh | ${\lambda}_{0}$ | 100 €/MWh${}^{2}$ |

$\overline{\sigma}$ | 73 MWh/h${}^{1/2}$ | $\alpha $ | 0.3 €/h·MW${}^{2}$ |

${\stackrel{\u02c7}{X}}_{0}^{i}$ | 0 MWh | ${\alpha}_{0}$ | 0.3 €/h·MW${}^{2}$ |

${\sigma}^{X},{\sigma}^{0}$ | 73 MWh/h${}^{1/2}$ | $\beta $ | 0.1 €/MW${}^{2}$ |

N | 100 | ${\beta}_{0}$ | 0.1 €/MW${}^{2}$ |

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**MDPI and ACS Style**

Féron, O.; Tankov, P.; Tinsi, L.
Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player. *Risks* **2020**, *8*, 133.
https://doi.org/10.3390/risks8040133

**AMA Style**

Féron O, Tankov P, Tinsi L.
Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player. *Risks*. 2020; 8(4):133.
https://doi.org/10.3390/risks8040133

**Chicago/Turabian Style**

Féron, Olivier, Peter Tankov, and Laura Tinsi.
2020. "Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player" *Risks* 8, no. 4: 133.
https://doi.org/10.3390/risks8040133