Introduction to Stochastic Calculus

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems. Many stochastic processes are based on functions which are continuous, but *nowhere differentiable*. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions. Instead, a theory of integration is required where integral equations do not need the direct definition of derivative terms. In quantitative finance, the theory is known as *Ito Calculus*.

The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model. The physical process of *Brownian motion* (in particular, a *geometric Brownian motion*) is used as a model of asset prices, via the *Weiner Process*. This process is represented by a *stochastic differential equation*, which despite its name, is in fact an integral equation.

The Binomial Model provides one means of deriving the Black-Scholes equation. A fundamental tool of stochastic calculus, known as Ito's Lemma, allows us to derive it in an alternative manner. Ito's Lemma is a stochastic analogue of the chain rule of ordinary calculus. The fundamental difference between stochastic calculus and ordinary calculus is that stochastic calculus allows the derivative to have a random component determined by a Brownian motion. The derivative of a random variable has both a deterministic component and a random component, which is normally distributed.

In the subsequent articles, we will utilise the theory of stochastic calculus to derive the Black-Scholes formula for a contingent claim. For this we need to assume that our asset price will never be negative. A vanilla equity, such as a stock, always has this property. A standard Brownian motion cannot be used as a model here, since there is a non-zero probability of the price becoming negative. A geometric Brownian motion is used instead, where the logarithm of the stock price has stochastic behaviour.

We will form a stochastic differential equation for this asset price movement and solve it to provide the path of the stock price. In order to price our contingent claim, we will note that the price of the claim depends upon the asset price and that by clever construction of a portfolio of claims and assets, we will eliminate the stochastic components by cancellation. We can then finally use a no-arbitrage argument to price a European call option via the derived Black-Scholes equation.

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