Stochastic Differential Equations

Stochastic Differential Equations

The previous article on Brownian motion and the Wiener Process introduced the standard Brownian motion, as a means of modeling asset price paths. However, a standard Brownian motion has a non-zero probability of being negative. This is clearly not a property shared by real-world assets - stock prices cannot be less than zero. Hence, although the stochastic nature of a Brownian motion for our model should be retained, it is necessary to adjust exactly how that randomness is distributed. In particular, the concept of geometric Brownian motion (GBM) will now be introduced, which will solve the problem of negative stock prices.

However, before the geometric Brownian motion is considered, it is necessary to discuss the concept of a Stochastic Differential Equation (SDE). This will allow us to formulate the GBM and solve it to obtain a function for the asset price path.

Stochastic Differential Equations

Now that we have defined Brownian motion, we can utilise it as a building block to start constructing stochastic differential equations (SDE). We need SDE in order to discuss how functions $f = f(S)$ and their derivatives with respect to $S$ behave, where $S$ is a stock price determined by a Brownian motion.

Some of the rules of ordinary calculus do not work as expected in a stochastic world. We need to modify them to take into account both the random behaviour of Brownian motion as well as its non-differentiable nature. We will begin by discussing stochastic integrals, which will lead us naturally to the concept of an SDE.

Definition (Stochastic Integral)
A stochastic integral of the function $f=f(t)$ is a function $W=W(t)$, $t\in[0,T]$ given by:
\begin{eqnarray*} W(t) = \int^t_0 f(s) dB(s) = \lim_{n\rightarrow \infty} \sum_{k=1}^N f(t_{k-1})\left(B(t_k)-B(t_{k-1})\right) \end{eqnarray*}

where $t_k = \frac{kt}{N}$.

Note that the function $f$ is non-anticipatory, in the sense that it is evaluated within the summation sign at time $t_{k-1}$. This means that it has no information as to what the random variable at $X(t_k)$ is. Supposing that $f$ represented some portfolio allocation based on $B$, then if it were not evaluated at $t_{k-1}$, but rather at $t_k$, we would be able to anticipate the future and modify the portfolio accordingly.

The previous expression provided for $W(t)$ is an integral expression and thus is well-defined for a non-differentiable variable, $B(t)$, due the property of finiteness as well as the chosen mean and variance. However, we wish to be able to write it in differential form:

\begin{eqnarray*} dW = f(t)dB \end{eqnarray*}
It should be stressed that this is shorthand notation for the integral form. Indeed, to divide through by $dB$ would necessitate the definition of $\frac{dW}{dB}$ - a differential operator on a non-smooth function $W$.

One can consider the term $dB$ as being a normally distributed random variable with zero mean and variance $dt$. The formal definition is provided:

Definition (Stochastic Differential Equation)
Let $B(t)$ be a Brownian motion. If $W(t)$ is a sequence of random variables, such that for all $t$, \begin{eqnarray*} W(t+\delta t)-W(t)-\delta t \mu (t, W(t)) - \sigma(t, B(t)) (B(t+\delta t)-B(t)) \end{eqnarray*} is a random variable with mean and variance that are $o(\delta t)$, then: \begin{eqnarray*} d W = \mu(t, W(t)) dt + \sigma(t, W(t)) dB \end{eqnarray*} is a stochastic differential equation for $W(t)$.

A sequence of random variables given by the above is termed an Ito drift-diffusion process, or simply an Ito process or a stochastic process.

It can be seen that $\mu$ and $\sigma$ are both functions of $t$ and $W$. $\mu$ has the interpretation of a non-stochastic drift coefficient, while $\sigma$ represents the coefficient of volatility - it is multiplied by the stochastic $dB$ term. Hence, stochastic differential equations have both a non-stochastic and stochastic component.

In the following section on geometric Brownian motion, a stochastic differential equation will be utilised to model asset price movements.