Stochastic Calculus Basics for Traders
This blog post summarizes fundamental concepts in stochastic calculus, particularly as they relate to modeling stock prices and derivative pricing.
1. Stochastic Process
We start by modeling the stock price as a time-dependent random variable, i.e., a stochastic process. This is essentially a sequence of random variables indexed by time (continuous in this case).
We assume:
- Price change over short intervals is normally distributed with mean 0 and variance Δt
- Changes over non-overlapping intervals are independent
This simplifies real-world behavior and helps build mathematical intuition.
1.1 Brownian Motion
Brownian motion (also called Wiener process) satisfies:
- $B(0) = 0$
- $B(t) - B(s) \sim \mathcal{N}(0, t - s)$
- Increments over non-overlapping intervals are independent
1.2 Properties of Brownian Motion
Non-differentiability
$B(t)$ is not differentiable:
$\lim_{h \to 0} \frac{B(t + h) - B(t)}{h}$ does not exist.
Despite appearing smooth, Brownian paths are too erratic to define a slope.
Quadratic Variation
For a well-behaved function $f(x)$:
\[\sum (f(t_{i+1}) - f(t_i))^2 \to 0\]But for Brownian motion:
\[\sum (B(t_{i+1}) - B(t_i))^2 = T\]i.e., $\Delta B^2 = \Delta t$
1.3 Ito’s Calculus
Because $B(t)$ isn’t differentiable, we can’t apply standard calculus. Instead, we use Ito’s calculus to analyze functions of stochastic processes.
Taylor Expansion Analogy
Regular calculus:
\[\Delta f(x) = f'(x)\Delta x\]Stochastic version:
\[\Delta f(B_t) = f'(B_t)\Delta B_t + \frac{1}{2}f''(B_t)\Delta t\]As $\Delta B_t \sim \mathcal{N}(0, \Delta t)$, we treat $\Delta B_t^2 = \Delta t$.
Thus, Ito’s formula becomes:
\[df = f'(B)\,dB + \frac{1}{2}f''(B)\,dt\]1.3.1 Ito’s Lemma for $f(t, X)$
If $X$ is stochastic and $t$ is deterministic:
\[df(t, X) = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial X}\,dX + \frac{1}{2}\frac{\partial^2 f}{\partial X^2}\,dX^2\]If $dX = dB$, then $dX^2 = dt$, and we get:
\[df = \left(\frac{\partial f}{\partial t} + \frac{1}{2}\frac{\partial^2 f}{\partial X^2}\right)dt + \frac{\partial f}{\partial X}dB\]1.3.2 Application to Geometric Brownian Motion (GBM)
Stock price modeled as:
\[\frac{dX}{X} = \mu\,dt + \sigma\,dB\]Applying Ito’s lemma:
\[df(t, X) = \left(\frac{\partial f}{\partial t} + \mu X \frac{\partial f}{\partial X} + \frac{1}{2}\sigma^2 X^2 \frac{\partial^2 f}{\partial X^2}\right)dt + \sigma X \frac{\partial f}{\partial X}dB\]1.4 Conclusion
These results lay the foundation for pricing models like Black-Scholes. Ito’s product and quotient rules extend regular calculus:
- Product: \(d(XY) = X\,dY + Y\,dX + dX\,dY\)
- Quotient: \(d\left(\frac{X}{Y}\right) = \frac{1}{Y}dX - \frac{X}{Y^2}dY + \frac{X}{Y^3}dY^2 - \frac{1}{Y^2}dX dY\)