Excerpt

## Abstract

According to previous research, the issue on the applicability of the original Black-Scholes model to the inverse quantity of price can be formulated as the argument of the symmetry between price and its inverse, whether there exists the set of real numbers as the drift and the volatility about the inverse quantity satisfying a certain system of stochastic differential equations. As the result of solving the equations in terms of such real numbers, it is revealed that there exist symmetries between not only them but also the coefficients of two equations. The aim of this article is to reveal in which cases these symmetries exist in the generalized Black-Scholes model, where the coeficients are deterministic or stochastic processes.

Math. Subject Classification: 39A50, 91B70, 91G30.

**Keywords:** Black-Scholes model, foreign currency, applicability, Ito lemma.

## 1 Introduction

According to Black and Scholes1, the price of stocks or any financial instruments *S* is described by a stochastic differential equation2 as follows:

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The model is referred to as the Black-Scholes model3. We can consider ß and *a* in the model as the drift and volatility of the infinitesimal increase ratio of S in infinitesimal time respectively4.

According to previous research5, the applicability of the model to foreign currency can be generalized as the symmetry between the price S and its inverse quantity *ß*:= S-1 in the model, and this issue can be formulated as the argument whether there exists the set *(ß ' , a ' ) E* R2 that satisfies the system of equations below:

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These real numbers can be determined uniquely and in addition, the relation between them and constants in the system i.e. (ß, a) and *(ß ' , a ')* can be described by a very beautiful symmetric expression as follows:

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After Black and Scholes, research considering the coefficients of the model as deterministic or stochastic processes have been done by Merton6, Engle7, etc.8. The generalization of these models seems to be expressed as follows:

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where drift *ß* and volatility *a* are deterministic or stochastic processes depending on time, price, element of the sample space, or any variables. The issue of in which cases this generalized model has the same symmetries between the price and its inverse, and the two sets of coefficients, as the original Black-Scholes model with constant coefficients, seems to be extremely interesting. Below I will argue on it.

## 2 Preparation

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On the other hand, coefficient comparison method for stochastic differential equations is needed too, which is enabled by the uniqueness of coefficients2:

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Using these lemmas, we can argue the cases which the symmetries on the generalized Black-Scholes model holds.

## 3 Argument

The inverse quantity of price is defined as follows:

**Definition 3.**

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Let *S* be the price of foreign currency from the view of domestic currency, ß express the price of the later seen from the former as mentioned in previous research. This is why the issue on the applicability can be fomulated as the argument on the symmetry between *S* and *ß* in which cases there exists the set of processes *(g ' , a ')* satisfying the system of equations below:

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Due to (2) in the system and **Definition 3.**, by substituting *x* = *ß*, *f* = *g z ß*, *g* = *o z ß* and *h* = *S* = ß-1 into **Lemma 1.**,

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By comparing coefficients of (1) and this, validated by **Lemma 2.**, Processes (gz, oz) satisfying this exists uniquely in all cases. Therefore there exists the symmetry between S and ß and the generalized Black-Scholes model is applicable to the inverse quantity of price too, as same as the original Black-Scholes model. In addition to this, the relation between sets of variable coefficients can be described by the same expression of it in the model which the coefficients are constants.

## 4 Conclusion and Future Work

As mentioned above, it is proved that the generalized Black-Scholes model is applicable to the inverse quantity of price and the relation between two sets of coefficients of two equations is expressed in the same symmetric form as in the original model, by the same way of argument on the original model. Therefore to make progress in the research focusing on this is becoming even more important and necessary.

**[...]**

- Quote paper
- Anonymous, 2020, On the symmetries in the generalized Black-Scholes model with variable coefficients, Munich, GRIN Verlag, https://www.grin.com/document/917156

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