Stochastic Calculus: A brief note


I interject part II of the correlation series to pen a brief note on the ubiquitous nature and pertinence of stochastic calculus in finance. It is most synonymous with derivatives pricing among other applications as normal calculus doesn’t work well when describing randomness and uncertainty.

By definition, this branch of mathematics is used to make continuous-time models based on probability distributions. It differs from normal calculus by virtue of its application of Ito’s Lemma to compute derivatives versus the chain rule that; a primary tenet of calculus. Additionally, it performs quadratic variation by summing the squares of the variation while calculus simply sums the variation and models continuous change via means and standard deviations.

In the mix are some important terms viz. Markov property, Martingale property, Wiener process, Itô’s process (a generalized Wiener process).  Markov property describes the path of a variable as memory-less i.e. in order to predict future prices, all we need is the information today. Its rationale is based on the weak-form efficient market hypothesis. The martingale property is a zero-drift stochastic process; the mathematical equivalent of the random walk theory.

dθ = σdz; where dz is a Weiner process

Effectively a variable has an equal chance of increasing and decreasing in value at any time and successful price changes are independent. Thus a variable’s expected value at any future time will equal its value today.

E(θT)=θ0 ; θ0and θT denote the values of the variable in times zero and T, respectively

A Wiener process is a process describing the evolution of a normally distributed variable. The drift of the process is 0 and the variance rate is 1.0 per unit time. The Ito’s Lemma is a way of calculating the stochastic process followed by a function of a variable from the stochastic process followed by the variable itself.

The above properties fall under the umbrella of the standard Brownian motion which is among the simplest of the continuous-time stochastic processes. An extension of this is the Geometric (Exponential) Brownian motion that forms the basis of the common Black-Scholes option pricing model.

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