# Hedging on Betting Markets

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Terminology and Previous Results

**Example**

**1.**

**Example**

**2.**

**Lemma**

**1.**

**Proof.**

**Example**

**3.**

**Lemma**

**2.**

**Proof.**

## 3. The General Case

**Example**

**4.**

**Proposition**

**1.**

**Proof.**

**Example**

**5.**

**Example**

**6.**

**Proposition**

**2.**

**Proof.**

## 4. Hedging on a Betting Exchange

**Example**

**7.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Example**

**8.**

**Proposition**

**4.**

**Proof.**

## 5. Hedging without a Lay Option

**Example**

**9.**

**Proposition**

**5.**

**Proof.**

**Example**

**10.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 2

**Proof.**

#### Appendix A.2. Full Derivation of Proposition 3

#### Appendix A.3. Proof of Proposition 6

**Proof.**

#### Appendix A.4. Proof of Proposition 7

**Proof.**

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1 | Another possible method is the parimutuel method, where the odds are determined only after people have bet. This is however a different problem as the bettors are unaware of their expected return when placing bets (Ottaviani and Sørensen 2005, 2009). |

2 | We approximately follow the notation of Axén and Cortis (2019) here as the paper studies a similar problem using the same inputs. |

3 | |

4 | We tossed a coin and use the male pronoun throughout the paper. |

5 | See Axén and Cortis (2019) for the full derivation. |

6 | An equivalent result can be derived for Dutching over lay bets, but it is not necessary for our purposes. |

7 | We can also frame this as the problem of buying a cash flow of $1 if i occurs at the price of $\frac{1}{{\alpha}_{i}}$ and then selling a cash flow of $1 if i occurs at the price of $\frac{1}{{\beta}_{i}^{\prime}}$, which yields the return of investment as $(\mathrm{Price}\phantom{\rule{4.pt}{0ex}}\mathrm{sold})/(\mathrm{Price}\phantom{\rule{4.pt}{0ex}}\mathrm{bought})-1={\alpha}_{i}/{\beta}_{i}^{\prime}-1$. |

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**MDPI and ACS Style**

Axén, G.; Cortis, D.
Hedging on Betting Markets. *Risks* **2020**, *8*, 88.
https://doi.org/10.3390/risks8030088

**AMA Style**

Axén G, Cortis D.
Hedging on Betting Markets. *Risks*. 2020; 8(3):88.
https://doi.org/10.3390/risks8030088

**Chicago/Turabian Style**

Axén, Gustav, and Dominic Cortis.
2020. "Hedging on Betting Markets" *Risks* 8, no. 3: 88.
https://doi.org/10.3390/risks8030088