# No-Arbitrage Principle in Conic Finance

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Bid–Ask Spreads

#### 1.2. Two-Price Economy

## 2. The Model (Multi-Period)

#### 2.1. The Model Definitions

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Set of Zero-Cost Cash Flows

#### 2.3. The Characterization of No-Arbitrage

**Definition**

**3.**

**Definition**

**4.**

- (1)
- (Risk Aversion) u is strictly concave,
- (2)
- (Profit Seeking) u is strictly increasing and $\underset{t\to +\infty}{lim}u\left(t\right)=+\infty$,
- (3)
- (Bankruptcy Forbidden) for any $t<0,\phantom{\rule{4pt}{0ex}}u\left(t\right)=-\infty $.

**Theorem**

**1**

**Proof.**

## 3. FTAP for Multi-Period Model

**Theorem**

**2**

**Definition**

**5**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

- (1)
- Pricing factor is not unique. This is clear since the existence of a pricing factor in FTAP comes from the existence of a solution to the dual problem, and we know the dual problem does not necessary have a unique solution.
- (2)
- The pricing factor is related to the utility function u via the duality. In fact we saw specifically that $\overline{z}\in -\partial (-u)\left(\overline{x}\right)$.
- (3)
- The pricing factors can be used to generate prices for cash flows $c\in \mathcal{C}$ with no-arbitrage existing. We will explain this more in detail on the coming sub-section.

## 4. Price Bounds and Their Estimates

#### 4.1. Definition of the Bounds

**Remark**

**2.**

**Remark**

**3.**

#### 4.2. Computations in One-Period

**Example**

**1.**

**Theorem**

**4**

**Proof.**

#### 4.3. Two-Period Examples

#### 4.3.1. Involving Only 1-Period Bonds

**Example**

**2.**

**Example**

**3.**

#### 4.3.2. Involving 2-Period Bond

#### 4.4. Complexity of Multi-Period Model

**Lemma**

**1.**

#### 4.5. Estimate of Multi-Period Bounds (Breaking into One Periods)

${t}_{1}$ | 1.3327 | ${t}_{5}\left({B}_{11}\right)$ | 0.1371 |

${t}_{2}$ | 0.5256 | ${t}_{5}\left({B}_{12}\right)$ | 0.4847 |

${t}_{3}$ | 0.2725 | ${t}_{6}\left({B}_{11}\right)$ | 1.5377 |

${t}_{4}$ | 0.0969 | ${t}_{6}\left({B}_{12}\right)$ | 0.6964 |

#### 4.6. The General 2-Period Model

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

- (1)
- ${u}_{0}^{2}\left(II\right)\le {u}_{0}^{2}\left(I\right)$
- (2)
- ${u}_{0}^{2}\left(I\right)-{u}_{0}^{2}\left(II\right)\le \left[\left({h}_{0}^{1}\underset{\omega \in \Omega}{max}{h}_{1}^{2}\left(\omega \right)-{h}_{0}^{2}\right)+\sum _{i=1}^{M}\left(\frac{{a}_{0}^{i1}{max}_{\omega \in \Omega}{a}_{1}^{i2}\left(\omega \right)-{a}_{0}^{i2}}{{min}_{\omega \in \Omega}{S}_{2}^{i}\left(\omega \right)}\right)\right]\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)$

**Proof.**

**Remark**

**4.**

## 5. Multi-Period Case Theorem

**Theorem**

**8.**

**Proof.**

**Remark**

**5.**

## 6. Conclusions

- (a)
- using only one-period bonds,
- (b)
- involving all available bonds.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**7.**

- (1)
- Let $\mathcal{PF}\left(II\right)$ be the set of all constrains for linear programming problem in Equation (5) and similarly let $\mathcal{PF}\left(I\right)$ be the set of all constrains for linear programming problem in Equation (6). We notice that $\mathcal{PF}\left(I\right)\subset \mathcal{PF}\left(II\right)$ so that$${u}_{0}^{2}\left(II\right)=\underset{{f}_{t}^{u}\in \mathcal{PF}\left(II\right)}{sup}{\mathbf{E}}_{0}\left[{f}_{0}^{1}{c}_{1}+{f}_{0}^{2}{c}_{2}\right]\le \underset{{f}_{t}^{u}\in \mathcal{PF}\left(I\right)}{sup}{\mathbf{E}}_{0}\left[{f}_{0}^{1}{c}_{1}+{f}_{0}^{2}{c}_{2}\right]={u}_{0}^{2}\left(I\right)$$
- (2)
- We start the argument by looking in a few easier and more concrete cases.
- (a)
- Consider the case where a super-hedging bound of ${c}_{2}$ is found by only zero-coupon bonds (a 2-period bond ${h}_{0}^{2}$ or two 1-period bonds ${h}_{0}^{1},{h}_{1}^{2}$). Then the price difference would be$${h}_{0}^{1}\underset{\omega \in \Omega}{max}{h}_{1}^{2}\left(\omega \right)\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)-{h}_{0}^{2}\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)=\left({h}_{0}^{1}\underset{\omega \in \Omega}{max}{h}_{1}^{2}\left(\omega \right)-{h}_{0}^{2}\right)\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)$$
- (b)
- If we assume that there is only one asset with two options (a 2-period ${S}_{0}^{2}\ne 0$ or two 1-period ${S}_{0}^{1}and{S}_{1}^{2}$) then the difference in the hedging-price values is$$\left({a}_{0}^{1}\underset{\omega \in \Omega}{max}{a}_{1}^{2}\left(\omega \right)-{a}_{0}^{2}\right)\underset{\omega \in \Omega}{max}\left(\frac{{c}_{2}}{{S}_{2}}\left(\omega \right)\right)\le \left({a}_{0}^{1}\underset{\omega \in \Omega}{max}{a}_{1}^{2}\left(\omega \right)-{a}_{0}^{2}\right)\left(\frac{{max}_{\omega \in \Omega}{c}_{2}\left(\omega \right)}{{min}_{\omega \in \Omega}{S}_{2}\left(\omega \right)}\right)$$
- (c)
- Now if we use both bond and asset then we have an upper bound for the difference as$$\left[\left({h}_{0}^{1}\underset{\omega \in \Omega}{max}{h}_{1}^{2}\left(\omega \right)-{h}_{0}^{2}\right)+\left(\frac{{a}_{0}^{1}{max}_{\omega \in \Omega}{a}_{1}^{2}\left(\omega \right)-{a}_{0}^{2}}{{min}_{\omega \in \Omega}{S}_{2}\left(\omega \right)}\right)\right]\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)$$
- (d)
- Therefore for a super-hedging with both bonds and finite number (M) of assets we have$${u}_{0}^{2}\left(I\right)-{u}_{0}^{2}\left(II\right)\le \left[\left({h}_{0}^{1}\underset{\omega \in \Omega}{max}{h}_{1}^{2}\left(\omega \right)-{h}_{0}^{2}\right)+\sum _{i=1}^{M}\left(\frac{{a}_{0}^{i1}{max}_{\omega \in \Omega}{a}_{1}^{i2}\left(\omega \right)-{a}_{0}^{i2}}{{min}_{\omega \in \Omega}{S}_{2}^{i}\left(\omega \right)}\right)\right]\underset{\omega \in \Omega}{max}{c}_{2}\left(\omega \right)$$

**Proof**

**of**

**Theorem**

**8.**

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**MDPI and ACS Style**

Vazifedan, M.; Zhu, Q.J.
No-Arbitrage Principle in Conic Finance. *Risks* **2020**, *8*, 66.
https://doi.org/10.3390/risks8020066

**AMA Style**

Vazifedan M, Zhu QJ.
No-Arbitrage Principle in Conic Finance. *Risks*. 2020; 8(2):66.
https://doi.org/10.3390/risks8020066

**Chicago/Turabian Style**

Vazifedan, Mehdi, and Qiji Jim Zhu.
2020. "No-Arbitrage Principle in Conic Finance" *Risks* 8, no. 2: 66.
https://doi.org/10.3390/risks8020066