Mathematics behind financial derivatives
Financial derivatives are complex financial instruments that derive their value from an underlying asset, such as a stock, bond, commodity, or currency. These instruments are used extensively in financial markets for a variety of purposes, such as hedging against risk, speculating on future price movements, or simply for trading.
The mathematics behind financial derivatives involves a range of mathematical concepts, including calculus, probability theory, and stochastic calculus. The basic idea is to use mathematical models to describe the behavior of the underlying asset and to use these models to determine the value of the derivative instrument.
One of the key concepts in the mathematics of financial derivatives is the notion of the derivative itself, which is a mathematical concept that describes the rate of change of a function with respect to its input. In finance, the derivative represents the rate of change of the value of the derivative instrument with respect to the value of the underlying asset.
Another important concept in the mathematics of financial derivatives is probability theory, which is used to model the random fluctuations in the price of the underlying asset. This is particularly important for options, which are financial derivatives that give the holder the right, but not the obligation, to buy or sell the underlying asset at a specified price and time in the future.
Stochastic calculus is a branch of calculus that deals with random processes, and it is used extensively in the mathematics of financial derivatives. Stochastic calculus is used to model the random fluctuations in the price of the underlying asset, and it is used to derive equations that describe the value of the derivative instrument as a function of the underlying asset price.
Overall, the mathematics behind financial derivatives is complex and involves a range of mathematical concepts and models. However, it is an essential tool for understanding and valuing these complex financial instruments, and it is used extensively in financial markets around the world
