 # Is Algebra Used in Finance?

Is Algebra Used in Finance?

#### Stochastic process is the most used abstract algebra in finance

Stochastic processes are useful for many aspects of quantitative financing, including but not limited to derivatives pricing, risk management, and investment management

When pricing derivatives, most quantum will use one of two approaches. They will either solve (or find a solution) the Black Scholes model for their priced derivatives, or they will use simulation methods to estimate the value of the derivatives. Both techniques rely heavily on the use of stochastic processes to simulate the underlying.

#### 1. The Black Scholes model

Black Scholes model prices a particular type of derivative contract under a set of assumptions. Moreover, under the assumptions, the famous Black Scholes partial differential equation becomes derivable. The derivation of the Black Scholes formula, along with various closed-form option pricing formulas, is the main reason for the massive growth of derivatives exchanges over the past three decades.

#### 2. Simulation method

Given the limitations and assumptions implied by the Black Scholes formula. Monte Carlo methods (simulations) will generate fewer simplifying assumptions and thus become more attractive.

These two options trade off between computational complexity and time. Using the simulation method is more computationally complex every time you want to price a derivative, but instead of a stochastic process, deriving the “equivalence” of the Black Scholes partial differential equation is more time consuming, and then still having to find closed-form derivative pricing Mode. Therefore, most quantum uses analog methods.

However, in daily price forecasting, we seldom directly use stochastic processes, because the derivative prices based on stochastic differential equations is full of instability even with the aid of simulation methods, and it is difficult to obtain accurate formulas. We usually take stochastic processes as the theoretical basis to establish a fixed model based on the Black Scholes equation without mathematical correction, which makes us often ignore the application of abstract algebra in finance.

In conclusion, Abstract algebra, especially the stochastic processes. Are useful for describing stochastic processes found in the world around us. And lastly play an important role in predicting likely returns in financial markets and changes in interest rates over time.

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#### Written by Jiahui Zhou

Is Algebra Used in Finance?