Stochastic Calculus & Probability

Stochastic Calculus & Probability

Stochastic calculus is a branch of mathematics that analyses random processes and their effects on systems. It is used in various fields, including physics, engineering, and finance.

Concept of Probability:

At its core, stochastic calculus is concerned with the concept of probability. Probability is a measure of the likelihood of a particular event occurring. In the context of stochastic calculus, probability is used to model the behaviour of a system over time.

It is helpful first to understand the concept of a stochastic process to understand the role of probability in stochastic calculus. A stochastic process is a mathematical model of a system that changes over time randomly or unpredictably. For example, the price of a stock can be modelled as a stochastic process, as it is constantly changing due to various factors.

The behaviour of a stochastic process:

Stochastic calculus allows us to analyze the behaviour of a stochastic process over time. This is done using various mathematical tools, including differential equations and integration. These tools allow us to calculate the probability of different events occurring in a system, such as the likelihood of a stock reaching a specific price at a particular time.

Pricing a Derivative:

Pricing financial derivatives is a practical application of stochastic calculus in the stock market. Financial derivatives are instruments derived from the value of other assets, such as stocks or commodities. Examples of derivatives include options and futures.

To determine the value of a derivative, we must first determine the probability of different events occurring, such as the stock reaching a specific price at a particular time. Stochastic calculus allows us to do this by providing a framework for analyzing the behaviour of a stock over time.

Overall, stochastic calculus is a powerful tool for analyzing the behaviour of complex systems over time. In finance, it is used to analyze the behaviour of stocks and other financial instruments and determine the value of financial derivatives.

Python Code : Implementation of Black-Scholes for pricing derivatives:

Simple implementation of the Black-Scholes model for pricing a European call option:

This function takes the following inputs:

• S: The current price of the underlying asset.
• K: The strike price of the option.
• T: The time to expiration of the option, in years.
• r: The risk-free interest rate.
• sigma: The volatility of the underlying asset.
• option: The type of option, either "call" or "put".

It returns the theoretical price of the option. This function assumes that the log-returns of the underlying asset are normally distributed.

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