# An Application of Symmetry Approach to Finance: Gauge Symmetry in Finance

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## Abstract

**:**

## 1. Introduction

## 2. Basic Assumption and Pricing Equations of Options

**Theorem**

**1.**

## 3. Numéraire Transformation

## 4. Geometric Framework of Pricing Equation and Symmetry

#### 4.1. Model of Fiber Bundle for Options

#### 4.2. Covariant Form of Pricing Equation and Gauge Symmetry

**Theorem**

**2.**

## 5. Connection and Curvature

## 6. Unification of Covariant Derivatives

**Theorem**

**3.**

## 7. Conclusions

## Acknowledgements

## References and Notes

- Ilinski, K. Physics of Finance: Gauge Modeling in Non-equilibrium Pricing; John Wiley and Sons Ltd.: New York, NY, USA, 2001. [Google Scholar]
- Young, K. Foreign Exchange Market as a Lattice Gauge Theory. Am. J. Phys.
**1999**, 67, 862–868. [Google Scholar] [CrossRef] - Henry-Labordère P. made a very similar research in the part of geometric framework, see his paper: Solvable local and stochastic volatility models: supersymmetric methods in option pricing, SSRN
**2005**, available at SSRN: http://ssrn.com/abstract=773568. His geometric framework is about the backward Kolmogorov equation of the conditional probability density for options and is used to making the asymptotic expansion for his model. - Etheridge, A. A Course in Financial Calculus; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- In fact, we have ${\rho}^{ij}={\mathrm{lim}}_{h\to 0}\mathbb{E}\left\{\frac{1}{h}({W}^{i}(t+h)-{W}^{i}(t))({W}^{j}(t+h)-{W}^{j}(t))|{\mathcal{F}}_{t}\right\}$.
- de Jong, F.J. Dimensional Analysis for Economists; North-Holland: Amsterdam, The Netherlands, 1967. [Google Scholar]
- Frankel, T. The Geometry of Physics: An Introduction, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]

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**MDPI and ACS Style**

Zhou, S.; Xiao, L.
An Application of Symmetry Approach to Finance: Gauge Symmetry in Finance. *Symmetry* **2010**, *2*, 1763-1775.
https://doi.org/10.3390/sym2041763

**AMA Style**

Zhou S, Xiao L.
An Application of Symmetry Approach to Finance: Gauge Symmetry in Finance. *Symmetry*. 2010; 2(4):1763-1775.
https://doi.org/10.3390/sym2041763

**Chicago/Turabian Style**

Zhou, Shipeng, and Liuqing Xiao.
2010. "An Application of Symmetry Approach to Finance: Gauge Symmetry in Finance" *Symmetry* 2, no. 4: 1763-1775.
https://doi.org/10.3390/sym2041763