The study of diverse mathematical ideas and methods that are applied to financial markets, investment anlysis, and financial decision-making is known as the mathematics of finance. It lays the groundwork for comprehencing and simulating financial phenomena as the pricing of derivatives, asset valuation, risk control, and portfolio optimization.

Here are some essential themes and ideas in financial mathematics :-

1. Time Value and Money : Money today is worth more than the same amount of money tomorrow, according to the time value of money, a fundamental financial principle. It is used to figure out interest rates as well as present and future values. 

2. Compounding and Interest Rates : Interessst rates are important in the world of finance. Financial analysis requires a thorough understanding of various interest rate calculations, compounding times, and how interest rates affect investment returns.

3. Probability and Statistics : To analyze and comprehend uncertainities and hazards, financial institutions frequently use statistical analysis and probability theory. In investment analysis, risk  assessment, and portfolio management, concepts including probability distributions, expected values, variance, and correlation are used.

4. Portfolio Theory : In order to maximize the trade-off between risk and return, portfolip theory focuses on the distribution of investments among various assets. Diversified portfolios are built using mathematical ttechniques like mean-variance analysis and efficient frontier.

5. Capital Asset Pricing Model (CAMP) : The capital asset pricing model (CAMP) is a popular financial model thatlinks an asset's expected return to its systematic risk. It is used in asset pricing and portfolio management to provide light on the connection between risk and return.

6. Optional Pricing : Financial derivatives whose value is derived from an underlying asset. Mathematical methods, particularly stochastic calculus, are used by option pricing models like the Black-scholes model to ascertain the fair values of options.

7. Financial Derivatives : Financial instruments known as derivatives derive their value from an underlying asset or variable. Derivatives like futures, forwards, swaps, and options require a strong mathematical foundationin order to be priced and understood.

8. Risk Management : By modeling and measuring several sorts of risks, such as market risk, credit risk, and operational risk, mathematical plays a vital role in risk management. To evaluate and manages risk explosure, strategies like value at risk (VaR) and  stress testing are used.

9. Financial Engineering : To produce cutting-edge financial products are tradingmethods, financial engineering blends mathematical and quantatitive tools with finance. It makes use of computer strategies, optimization tools, and mathematical models.

10. Game Theory : Game theory is used in finance to simulate competitive strategic interactions and decision-making. It offers perceptions into market activity, auctions, and negotiation techniques.

Only a small portion of the mathematics used in finance is covered by these topics. New mathematical models and technique are constantly being created in this huge subject to adress the complexity of the financial world.