Introduction to Stochastic Calculus

Stochastic calculus is widely used in the field of quantitative finance to create models of random asset prices. This article will provide a brief overview of the applications, particularly in cases related to the Black-Scholes model.

What is Stochastic calculus?

Stochastic calculus is a branch of mathematics that studies processes with a stochastic component, allowing for the modeling of random systems. Many stochastic processes are based on functions that are continuous but nowhere differentiable, making it impossible to use differential equations, as these require the definition of derivative terms that cannot be defined on non-smooth functions. Therefore, there is a need for a theory of integration that does not rely on the direct definition of derivatives. In quantitative finance, this theory is known as Ito Calculus.

The main application of stochastic calculus in finance

The main application of stochastic calculus in finance is the creation of a stochastic movement model for asset prices in the Black-Scholes model, utilizing geometric Brownian motion through the Wiener Process. This process is represented by a stochastic differential equation, which, despite being called a differential equation, is actually an integral equation.

The binomial model

The binomial model is one method for deriving the Black-Scholes equation. Additionally, we can use the fundamental tool of stochastic calculus known as Ito's Lemma to help us infer in another form. Ito's Lemma is akin to the chain rule of ordinary calculus, but for stochastic processes. The main difference between stochastic calculus and ordinary calculus is that stochastic calculus allows for the appearance of a random component in the derivative, determined by Brownian motion. The derivative of a random variable consists of both a deterministic component and a random component, which is normally distributed.

In the next article, we will use stochastic calculus theory to derive the Black-Scholes formula for contingent claims. We must assume that asset prices will never be negative. Vanilla equities, such as stocks, already have this property, so we cannot use the normal Brownian process because there is a chance that prices could be negative. Instead, we use the geometric Brownian motion, where the logarithm of stock prices behaves randomly.

We will create a stochastic differential equation for the movement of asset prices and solve the equation to determine the direction of stock price movement and to set the price of contingent claims. We will observe that the price of contingent claims depends on the asset price, and by intelligently constructing a portfolio of contingent claims and assets, we can eliminate the random components through cancellation. Then, we can use the no-arbitrage argument to determine the price of a European call option through the derived Black-Scholes equation.

Reference: Introduction to Stochastic Calculus

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