Bilal Aslam

May 23, 2017

A random walk is a natural process that can be observed in many fields. Often, random walks have Markov property (that only current position is relevant). And, the observations of many random positions (or random variable) over a period of time lays the foundation of stochastic processes (also called Random processes) because the random variable changes in an uncertain way. And, by taking time limit tends to zero (making mean and variance of the random variable constant) we obtain continuous-time Markov stochastic process with standard normal distribution, called Wiener process, also called Brownian motion (Einstein's discovery that the displacement or uncertainty in a Brownian motion is proportional to the square root of time is one of his many greatest work).

Applying Markov property of Wiener process to stock prices and connecting it with weak form of market efficiency, then developing geometric Wiener process and obtaining option price formula for non-dividend paying European call option in a risk-neutral world using sophisticated Ito calculus as early as 1973 by Black, Scholes and Merton is undoubtedly an outstanding achievement. The Black-Scholes-Merton model simply invites students and researchers in Mathematics to enter the world of Finance.

However, Black-Scholes-Merton model suggests that volatility remains constant which is clearly not a case, and stock prices actually follows jumps (rather than continuous change) as suggested by Cox, Ross and Rubinstein in 1979. And, Robert C. Merton extended this approach, what is called Merton jump diffusion model (since it combines jumps with diffusion terms).

The bottom line is that Financial Mathematics is exciting interdisciplinary field with real world applications of Mathematics in Finance. That is why Mathematicians love to study modern Finance.

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