Stochastic Process

A stochastic process is a mathematical model of a quantity that changes over time in a way that involves chance. Formally it is a family of random variables, one for each point in time. A single run of the process, such as one possible history of a share price, is called a path or realization. The word stochastic simply means random.

Stochastic processes are the foundation of quantitative finance. Share prices, exchange rates, interest rates, volatility and the arrival of orders are all modelled as stochastic processes, and the choice of process determines the prices that a model assigns to derivatives and the risks it reports.

Processes are classified by how time and values are measured. In a discrete-time process the quantity is observed at separate moments, such as daily closes, and the result is a time series. In a continuous-time process it is defined at every instant. The values themselves may be discrete, as in a count of trades, or continuous, as in a price.

Several properties recur. A process has the Markov property if its future depends only on its present state and not on how it got there. A martingale is a process whose expected future value equals its current value, the mathematical expression of a fair game. A process showing stationarity has statistical behaviour that does not change over time.

• Random walk. Each value is the previous one plus an independent random step. The random walk is the basic discrete-time model of prices.
• Brownian motion. The continuous-time limit of the random walk, with normally distributed increments. Brownian motion is the building block of most continuous models.
• Geometric Brownian motion. Percentage changes, not absolute changes, follow a Brownian motion, so the price stays positive. This is the model behind the Black-Scholes-Merton model.
• Mean-reverting processes. The quantity is pulled back toward a long-run level. They are used for interest rates, volatility and price spreads that show mean reversion.
• Jump and counting processes. The Poisson process counts events arriving at random, such as defaults or orders, and jump models add sudden discontinuous moves to a price.
• Stochastic volatility models. Volatility is itself a random process, either in continuous time or in discrete form as in GARCH.

The standard model of a share price is written as a stochastic differential equation:

Its solution gives the price at a future time $T$:

• $S_t$ is the price at time $t$, and $S_0$ is the price today.
• $\mu$ is the drift, the expected rate of return per year.
• $\sigma$ is the volatility per year.
• $W_t$ is a Brownian motion; $W_T$ is normally distributed with mean zero and variance $T$.

Under this model the future price follows a lognormal distribution. Take a share at $100 with a drift of 8 percent and volatility of 20 percent. After one year the expected price is $100 \times e^{0.08} = 108.33$ dollars. The median price is lower, at $100 \times e^{0.08 - 0.02} = 100 \times e^{0.06} = 106.18$ dollars, because the term $\sigma^2 / 2 = 0.02$ is subtracted. The gap between mean and median reflects the positive skew of the lognormal distribution: a few very high outcomes pull the average above the typical result.

Louis Bachelier used a continuous random process to model prices on the Paris exchange in 1900, five years before Albert Einstein's physical theory of Brownian motion. Norbert Wiener gave the process a rigorous mathematical construction in the 1920s, and Kiyosi Itô developed the calculus for functions of such processes in the 1940s. Itô's calculus became the working language of derivatives pricing after 1973, when the option pricing theory of Black, Scholes and Merton was published.

In derivatives pricing, an assumed process for the underlying asset leads either to a formula or to a numerical method. Monte Carlo simulation generates many random paths of the process and averages the resulting payoffs, while lattice methods such as the binomial option pricing model approximate the process on a tree. Risk managers simulate portfolios forward in the same way, and market microstructure research models order flow as a stream of random arrivals.

Every process is a simplification. Geometric Brownian motion assumes constant volatility, continuous paths and normally distributed log returns, whereas real markets show changing volatility, sudden jumps and more extreme moves than the model allows. More elaborate processes fit the data better but require more parameters, which must be estimated or calibrated and may be unstable. The process chosen is therefore a modelling decision that carries its own risk, not an established fact about the market.