Mathematics has played a significant role in peoples’ lives. There is a close relationship between mathematics and other areas such as technology, science, and religion. Mathematics has also interacted with poetry, art and literature. Mathematical concepts have been of immense help in financial matters. Developments in mathematics resulted to accounting, economics and finance. Economics deal with issues regarding money, and its position in society. Finance, on the other hand, has a narrow aim, and it concerns how we can invest money to make it to increase at a higher rate. Because of this narrow focus it simplifies the situation; therefore, finance is a field treated in mathematical terms.

Mathematical modelling is a technique of translating problems from real-life systems into manageable and conformable expressions. A broad variety of mathematical copies is used widely in different fields including engineering, natural sciences, besides social sciences. Mathematics plays a vital role in many fields of science such as finance. It forms a foundation of financial theory and the analysis technique (Luenberger, 2007). This is a complex procedure that uses advanced techniques of mathematical modeling.

## Modeling Stock Prices

Finance is a field that involves an application of mathematical concepts daily. It is the study of how to invest money. However to make a profit, there are risks that must be undertaken. Naturally, we would like to invest in assets whose value seems likely to increase at a faster rate. Investor must evaluate trade-offs between expected risks and profit. The term risk in finance refers to uncertainty of future occurrence as a result of your current action (Luenberger, 2007). For sample, consider the choice involving saving money in a bank account or buying shares in a company. Buying shares investment is riskier since it has more uncertainty. Solving such scenarios, we use probability mathematics in finance. The relative value of the second choice relies on the probabilities of the potential payoffs. Fluctuations in stock prices can be modeled, and assign probabilities to the likely payoffs from such an investment, and thus estimating its value to the investor.

Portfolio Management/Optimization In addition, finance involves portfolio management. Portfolio is a group of assets owned by an institution or some persons. Assets differ from stocks, bonds or other options. The portfolio optimization mathematical approach searches for an optimal way of distributing certain capital on available assets. Mathematical branch that is appropriate in such a situation is the Optimization Theory. Financial assets selection can be modeled as mathematical optimization problem as we need either to minimize or to maximize a given objective.

Assets in a portfolio may fluctuate in number in such optimization problems cannot be solved in the manual way. Therefore, optimization algorithms are used, and key algorithms are Newton’s algorithms (Sheldon, 1999). Probabilistic algorithms are applicable in such a scenario since they are independent on the derivatives such as Differential Evolution.

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Derivative Pricing Moreover, mathematical concepts are heavily relied in derivative pricing that are financial tools derived from assets. Main types of derivatives are Futures, Swaps and Options. This mathematical approach helps to determine the prices of stock. For instance, the cost of a stock option is found based on the stock. Varying price of the stock varies the price of the stock-option (Sheldon, 1999). Innovative knowledge of mathematics is vital in calculating price derivatives. For example, to value European stock options, stochastic differential equations are essential. Numerical analysis, in addition, plays a significant role, because it is not possible to have an exact resolution only approximation.

## Risk Management

Risk management in finance involves the application of mathematical skills. Risk is the unexpected variability of various asset earnings or prices. There are two main sources of risk such as financial and business. Risk management is necessary, because of the extreme market movements. In case, credit crisis, which results from the mortgage market downfall. In such a situation, risk management involves from conveying limits to situations such as stop-loss limit to the value at risk and lastly, to the stress testing.

Montecarlo simulation mockups offer a powerful mathematical tool in management of risk. The model assists to determine value at risk and for analysis of scenario. The core of Montecarlo imitation is Stochastic Calculus and Geometric Brownian Motion, which are essential concepts in mathematics. If the financial market increases, more difficult financial instruments results such as, Bermudian options and mortgage backed securities. Pricing such instruments require an advanced level of mathematics. Therefore, a firm background in mathematics definitely would help somebody to dominate quantitative and financial markets.

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## Financial Derivatives

A derivative is a contract regarding a future trade. This course requires extensive use of mathematical skills such as Calculus (integration, partial derivatives) and Probability (independent random variables, normal distribution, expectation and variance).

Combining these mathematical principles with basic principles of finance and economics such as the No-Arbitrage Principle are enough to motivate Geometric Brownian Motion and Black-Scholes Formula. However, contemporary finance goes beyond these topics and concerns considerably complicated situations. Current research into finance entails mathematical inputs from a wide range of areas such as, Measure Theory, Monte Carlo Techniques, Stochastic Differential Equations, and Game Theory (Hull, 2001)

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## Use of Mathematics in Research

In the course of his research, the lecturer tried to model price movements of various financial assets using mathematical models. He assumed stock price follows a random process then applied stochastic calculus to find the next possible value for the stock. In his research, he applied concepts of Mathematics in quantitative finance.

The lecturer also tried to measure the risk exposure in terms of convexity and duration, which can be measured as derivatives of the bond price. He utilized differential calculus to measure the risk in relation to the interest rate. Furthermore, he applied mathematics to express portfolio risks and returns as statistical mean and standard deviation. Lastly, he used optimization strategy to find the portfolio, which had largest return and risk. From his research work, the finding can form a basis of future investment decision making.

**Conclusion**

Finance is a field that applies mathematical concepts on a daily basis. Moreover, mathematical skills are applied to activities such as business valuation, cash flow forecasting, capital budgeting, risk management, financial analysis, Portfolio management and many others. A high-level mathematical concept such as modeling is a powerful technique of making a decision in financial markets. In spite of this fact, mathematical, financial models have various limitations related to the uncertainty of financial markets. Furthermore, modelling is fundamental in finance, but there might be some errors, because financial models are created by people. Such models require consideration of accuracy of data been used.