#
A Robust Approach to Hedging and Pricing in Imperfect Markets^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Analytical Setup

- S1.
- Normality if $0\in \mathcal{Y}$;
- S2.
- Positive homogeneity if $\lambda \mathcal{Y}\subseteq \mathcal{Y}$, for all $\lambda >0$;
- S3.
- Translation-invariance if $\mathbb{R}+\mathcal{Y}\subseteq \mathcal{Y}$;
- S4.
- Sub-additivity if $\mathcal{Y}+\mathcal{Y}\subseteq \mathcal{Y}$;
- S5.
- Convexity if $\lambda \mathcal{Y}+(1-\lambda )\mathcal{Y}\subseteq \mathcal{Y}$.

#### 2.1. Risk Measures

- R1.
- $\varrho (0)=0$;
- R2.
- $\varrho (\lambda x)=\lambda \varrho (x)$, for all $\lambda >0$ and $x\in \mathcal{D}$;
- R3.
- $\varrho (x+c)=\varrho (x)-c$, for all $x\in \mathcal{D}$ and $c\in \mathbb{R}$;
- R4.
- $\varrho (x)\le \varrho (y)$, for all $x,y\in \mathcal{D}$ and $x\ge y$;
- R5.
- $\varrho (x+y)\le \varrho (x)+\varrho (y),\forall x,y\in \mathcal{D};$
- R6.
- $\varrho (\lambda x+(1-\lambda )y)\le \lambda \varrho (x)+(1-\lambda )\varrho (y).$

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Definition**

**2.**

#### 2.2. Pricing Rules

- P1.
- $\pi (0)=0$;
- P2.
- $\pi (\lambda x)=\lambda \pi (x)$, for all $\lambda >0$ and $x\in \mathcal{X}$;
- P3.
- $\pi (x+c)=\pi (x)+c$, for all $x\in \mathcal{X}$ and $c\in \mathbb{R}$ (cash-invariance);
- P4.
- $\pi (x)\le \pi (y)$, for all $x,y\in \mathcal{X}$ and $x\le y$;
- P5.
- $\pi (x+y)\le \pi (x)+\pi (y)$, for all $x,y\in \mathcal{X}$;
- P6.
- $\pi (\lambda x+(1-\lambda y))\le \lambda \pi (x)+(1-\lambda )\pi (y)$.

**Definition**

**3.**

#### 2.3. Projection

**Remark**

**1.**

**Remark**

**2.**

## 3. Main Theoretical Results

#### 3.1. Market Principles

**Proposition**

**1.**

- ${\pi}_{\varrho}$ and ${\mathcal{X}}_{\varrho}$ are positive homogeneous if ϱ and π are.
- ${\pi}_{\varrho}$ and ${\mathcal{X}}_{\varrho}$ are translation-invariant if ϱ and π are.
- ${\pi}_{\varrho}$ and ${\mathcal{X}}_{\varrho}$ are sub-additive if ϱ and π are.
- ${\pi}_{\varrho}$ and ${\mathcal{X}}_{\varrho}$ are convex if ϱ and π are.Furthermore,
- ${\pi}_{\varrho}$ is monotone if ϱ and π are.

**Proof.**

**Normality (N).**${\pi}_{\varrho}(0)=0$.

**No Good Deal Assumption (NGD).**There is no financial position x such that

**Consistency Principle (CP).**For any member $x\in \mathcal{X}$, $\pi $ and ${\pi}_{\varrho}$ are consistent, i.e.,

**Compatibility (C).**For a risk measure $\varrho $ and a pricing rule $\pi $, (12) has a finite infimum.

#### 3.2. Positive-Homogeneous and Monotone Risk and Pricing Rules

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

#### 3.3. Positive-Homogeneous and Sub-Additive Risk and Pricing Rules

**Proposition**

**2.**

**Proof.**

**Theorem**

**3.**

- ${\mathcal{R}}_{\varrho ,\pi}=({\mathsf{\Delta}}_{\varrho}-{\mathcal{D}}^{\perp})\cap ({\mathcal{R}}_{\pi}+{\mathcal{X}}^{\perp})\ne \Phi \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\forall \varrho \in \mathbb{D},\forall \pi \in \mathbb{M}$Furthermore, if condition 3 holds for π and ϱ, these statements are equivalent to
- There is no Good Deal in the market.

**Proof.**

**Corollary**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 4. An Application to Hedging Economic Risk

#### 4.1. Estimation Problem

**Remark**

**8.**

#### 4.2. Data Description

#### 4.3. Results

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs of Propositions and Theorems

#### Appendix A.1. Proof of Theorem 1

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

#### Appendix A.3. Proof of Theorem 2

#### Appendix A.4. Proof of Proposition 2

#### Appendix A.5. Proof of Theorem 3

**Proposition**

**A1.**

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1 | All of the results can be easily extended to a probability space with no atoms in an appropriate space—for instance, ${L}^{2}(\mathrm{\Omega})$. |

2 | For technical reasons, we use $-z$ instead of $z.$ |

**Table 1.**Hedging of Inflation. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | 0.0026 | 0.0022 | |||

(0.0001) | (0.0002) | ||||

RF | 0.0072 | −0.6737 | −0.7844 | 0.0058 | 0.0027 |

(0.0558) | (0.0705) | (0.1106) | (0.0041) | (0.0019) | |

MARKET | −0.0048 | −0.0072 | −0.0123 | −0.0038 | −0.0066 |

(0.0029) | (0.0030) | (0.0043) | (0.0013) | (0.0023) | |

SMB | −0.0008 | 0.0131 | 0.0383 | −0.0008 | −0.0003 |

(0.0030) | (0.0048) | (0.0065) | (0.0005) | (0.0002) | |

HML | 0.0022 | 0.0013 | −0.0009 | 0.0038 | 0.0031 |

(0.0042) | (0.0013) | (0.0006) | (0.0016) | (0.0023) | |

UMD | 0.0015 | 0.0006 | 0.0002 | 0.0019 | 0.0023 |

(0.0027) | (0.0006) | (0.0001) | (0.0012) | (0.0015) | |

TERM | 0.0084 | −0.1427 | −0.1440 | 0.0025 | 0.0104 |

(0.0109) | (0.0162) | (0.0263) | (0.0018) | (0.0070) | |

DEF | −0.0265 | 0.1063 | 0.1370 | −0.0246 | −0.0227 |

(0.0242) | (0.0209) | (0.0369) | (0.0080) | (0.0117) | |

Price | 0.0023 | 0.0077 | 0.0093 | 0.0025 | 0.0037 |

(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |

**Table 2.**Hedging of Real Interest Rate. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | 0.0021 | 0.0035 | |||

(0.0001) | (0.0002) | ||||

RF | −0.0304 | −0.1473 | −0.6805 | −0.0147 | −0.0296 |

(0.0563) | (0.0621) | (0.1000) | (0.0063) | (0.0380) | |

MARKET | 0.0049 | −0.0020 | 0.0094 | 0.0048 | 0.0038 |

(0.0028) | (0.0017) | (0.0045) | (0.0015) | (0.0032) | |

SMB | 0.0013 | −0.0009 | 0.0042 | 0.0009 | 0.0013 |

(0.0029) | (0.0007) | (0.0033) | (0.0003) | (0.0020) | |

HML | −0.0029 | −0.0142 | 0.0188 | −0.0031 | −0.0031 |

(0.0042) | (0.0042) | (0.0072) | (0.0012) | (0.0037) | |

UMD | −0.0008 | −0.0346 | −0.0011 | −0.0007 | −0.0007 |

(0.0027) | (0.0031) | (0.0008) | (0.0003) | (0.0013) | |

TERM | −0.0167 | −0.0664 | −0.1187 | −0.0284 | −0.0166 |

(0.0109) | (0.0123) | (0.0239) | (0.0065) | (0.0133) | |

DEF | 0.0205 | 0.1095 | 0.2810 | 0.0222 | 0.0226 |

(0.0244) | (0.0194) | (0.0305) | (0.0073) | (0.0186) | |

Price | 0.0023 | 0.0075 | 0.0110 | 0.0027 | 0.0035 |

(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |

**Table 3.**Hedging of Term Spread. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | −0.0000 | −0.0000 | |||

(0.0012) | (0.0018) | ||||

RF | 0.3958 | 0.3782 | 0.3696 | 0.3886 | 0.3883 |

(0.0922) | (0.0877) | (0.1094) | (0.0526) | (0.0898) | |

MARKET | 0.0011 | −0.0023 | −0.0016 | 0.0018 | 0.0006 |

(0.0043) | (0.0061) | (0.0095) | (0.0005) | (0.0006) | |

SMB | 0.0034 | 0.0047 | 0.0033 | 0.0026 | 0.0028 |

(0.0048) | (0.0092) | (0.0138) | (0.0008) | (0.0016) | |

HML | 0.0071 | 0.0010 | 0.0013 | 0.0091 | 0.0078 |

(0.0053) | (0.0227) | (0.0230) | (0.0021) | (0.0042) | |

UMD | −0.0038 | −0.0017 | −0.0015 | −0.0065 | −0.0051 |

(0.0042) | (0.0069) | (0.0099) | (0.0018) | (0.0026) | |

TERM | 0.1346 | 0.1055 | 0.1024 | 0.1277 | 0.1532 |

(0.0163) | (0.0194) | (0.0242) | (0.0126) | (0.0251) | |

DEF | −0.0691 | −0.0789 | −0.0756 | −0.0926 | −0.0571 |

(0.0296) | (0.0307) | (0.0341) | (0.0138) | (0.0268) | |

Price | 0.0033 | 0.0057 | 0.0082 | 0.0029 | 0.0044 |

(0.0000) | (0.0001) | (0.0002) | (0.0001) | (0.0002) |

**Table 4.**Hedging of Default Spread. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | 0.0010 | 0.0011 | |||

(0.0000) | (0.0001) | ||||

RF | −0.0616 | 0.0358 | −0.0017 | −0.0572 | −0.0359 |

(0.0335) | (0.0214) | (0.0010) | (0.0119) | (0.0133) | |

MARKET | −0.0008 | 0.0002 | 0.0115 | −0.0011 | −0.0005 |

(0.0018) | (0.0001) | (0.0026) | (0.0003) | (0.0002) | |

SMB | −0.0015 | 0.0030 | −0.0090 | −0.0016 | −0.0027 |

(0.0014) | (0.0015) | (0.0036) | (0.0005) | (0.0011) | |

HML | −0.0005 | 0.0091 | 0.0177 | −0.0004 | −0.0001 |

(0.0023) | (0.0015) | (0.0040) | (0.0001) | (0.0001) | |

UMD | −0.0003 | 0.0026 | 0.0097 | −0.0002 | −0.0000 |

(0.0012) | (0.0010) | (0.0026) | (0.0001) | (0.0000) | |

TERM | −0.0147 | −0.0153 | −0.0169 | −0.0135 | −0.0005 |

(0.0049) | (0.0051) | (0.0082) | (0.0025) | (0.0003) | |

DEF | 0.0503 | 0.1078 | 0.0965 | 0.0498 | 0.0882 |

(0.0125) | (0.0063) | (0.0158) | (0.0052) | (0.0079) | |

Price | 0.0011 | 0.0036 | 0.0048 | 0.0009 | 0.0016 |

(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |

**Table 5.**Hedging of Dividend Yield. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | 0.0007 | 0.0011 | |||

(0.0001) | (0.0001) | ||||

RF | −0.0735 | −0.0323 | −0.0794 | −0.0774 | −0.0736 |

(0.0148) | (0.0291) | (0.0352) | (0.0074) | (0.0104) | |

MARKET | −0.0309 | −0.0313 | −0.0343 | −0.0293 | −0.0313 |

(0.0010) | (0.0015) | (0.0017) | (0.0012) | (0.0016) | |

SMB | −0.0000 | 0.0004 | 0.0000 | −0.0000 | −0.0000 |

(0.0012) | (0.0007) | (0.0000) | (0.0000) | (0.0000) | |

HML | −0.0028 | −0.0026 | −0.0020 | −0.0029 | −0.0032 |

(0.0014) | (0.0021) | (0.0019) | (0.0004) | (0.0005) | |

UMD | −0.0012 | 0.0016 | −0.0028 | −0.0013 | −0.0015 |

(0.0008) | (0.0012) | (0.0015) | (0.0002) | (0.0002) | |

TERM | −0.0036 | 0.0146 | −0.0163 | −0.0035 | −0.0040 |

(0.0029) | (0.0069) | (0.0067) | (0.0006) | (0.0008) | |

DEF | −0.0038 | −0.0152 | 0.0276 | −0.0037 | 0.0010 |

(0.0044) | (0.0083) | (0.0105) | (0.0008) | (0.0002) | |

Price | 0.0007 | 0.0018 | 0.0028 | 0.0007 | 0.0010 |

(0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) |

**Table 6.**Hedging of Consumption Growth. The table reports the estimates and their corresponding bootstrap errors (based on 400 bootstrap replications) for different risk measures (mean-variance MV, conditional value-at-risk CVaR, and value-at-risk VaR). The bold font represents statistical significance (at the 5% nominal level) of individual coefficients except for the last row where the bold font signifies a statistically different price from $E(y)$.

Securities\risk | MV | ${\mathbf{CVaR}}_{0.9}$ | ${\mathbf{CVaR}}_{0.95}$ | ${\mathbf{VaR}}_{0.9}$ | ${\mathbf{VaR}}_{0.95}$ |
---|---|---|---|---|---|

Intercept | 0.0059 | 0.0080 | |||

(0.0003) | (0.0005) | ||||

RF | −0.2533 | −0.0510 | 0.0696 | −0.2882 | −0.1159 |

(0.1112) | (0.0718) | (0.1184) | (0.0672) | (0.0788) | |

MARKET | 0.0082 | 0.0079 | −0.0067 | 0.0074 | 0.0048 |

(0.0056) | (0.0062) | (0.0102) | (0.0023) | (0.0037) | |

SMB | 0.0256 | 0.0315 | 0.0467 | 0.0288 | 0.0237 |

(0.0080) | (0.0097) | (0.0169) | (0.0051) | (0.0094) | |

HML | 0.0132 | 0.0079 | −0.0096 | 0.0034 | 0.0108 |

(0.0080) | (0.0089) | (0.0141) | (0.0020) | (0.0075) | |

UMD | −0.0034 | 0.0334 | 0.0546 | −0.0032 | −0.0032 |

(0.0052) | (0.0073) | (0.0120) | (0.0015) | (0.0028) | |

TERM | −0.0509 | −0.0581 | −0.0070 | −0.0835 | −0.0819 |

(0.0241) | (0.0223) | (0.0160) | (0.0165) | (0.0263) | |

DEF | −0.0568 | 0.0005 | 0.1904 | −0.0562 | −0.0802 |

(0.0312) | (0.0007) | (0.0648) | (0.0178) | (0.0332) | |

Price | 0.0054 | 0.0159 | 0.0223 | 0.0064 | 0.0083 |

(0.0000) | (0.0001) | (0.0002) | (0.0001) | (0.0002) |

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**MDPI and ACS Style**

Assa, H.; Gospodinov, N.
A Robust Approach to Hedging and Pricing in Imperfect Markets. *Risks* **2017**, *5*, 36.
https://doi.org/10.3390/risks5030036

**AMA Style**

Assa H, Gospodinov N.
A Robust Approach to Hedging and Pricing in Imperfect Markets. *Risks*. 2017; 5(3):36.
https://doi.org/10.3390/risks5030036

**Chicago/Turabian Style**

Assa, Hirbod, and Nikolay Gospodinov.
2017. "A Robust Approach to Hedging and Pricing in Imperfect Markets" *Risks* 5, no. 3: 36.
https://doi.org/10.3390/risks5030036