Stochastic Calculus for Finance

Stochastic calculus provides the mathematical framework for modeling random phenomena evolving over time. It is the foundation of modern derivatives pricing and risk management, enabling us to work with processes that have continuous but non-differentiable sample paths.

Brownian Motion (Wiener Process)

Definition

A Brownian motion (or Wiener process) $W_t$ is a continuous-time stochastic process with the following properties:

1. $W_0 = 0$
2. $W_t$ has independent increments: For $0 < s < t$, $W_t - W_s$ is independent of $W_s$
3. $W_t - W_s \sim N(0, t-s)$ (normally distributed increments)
4. Sample paths are continuous but nowhere differentiable

Key Properties

• $E[W_t] = 0$
• $Var(W_t) = t$
• $Cov(W_s, W_t) = \min(s, t)$
• Quadratic variation: $[W]_t = t$ (the sum of squared increments converges to $t$)

Simulation

For discrete time step $\Delta t$: $W_{t+\Delta t} = W_t + \sqrt{\Delta t} \cdot Z$ where $Z \sim N(0, 1)$.

Geometric Brownian Motion (GBM)

Stock prices cannot be negative and exhibit proportional rather than absolute changes. The geometric Brownian motion model addresses this: $dS_t = \mu S_t dt + \sigma S_t dW_t$

This is the standard model for stock price dynamics in the Black-Scholes framework.

Solution (using Ito's lemma): $S_t = S_0 \exp\left[(\mu - \frac{\sigma^2}{2})t + \sigma W_t\right]$

Note that $S_t$ is log-normally distributed, ensuring positive prices.

Stochastic Differential Equations (SDEs)

General Form

An SDE describes a process $X_t$ as: $dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dW_t$

• Drift term $\mu(X_t, t)$: Deterministic tendency
• Diffusion term $\sigma(X_t, t)$: Random fluctuations

Important Processes in Finance

Ornstein-Uhlenbeck (mean-reverting): $dX_t = \theta(\mu - X_t)dt + \sigma dW_t$ Used for interest rate models (Vasicek) and pairs trading.

Cox-Ingersoll-Ross (CIR): $dr_t = \kappa(\theta - r_t)dt + \sigma\sqrt{r_t}dW_t$ Ensures non-negative interest rates.

Heston (stochastic volatility): $dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^1$ $dv_t = \kappa(\theta - v_t)dt + \xi\sqrt{v_t}dW_t^2$ where $dW_t^1$ and $dW_t^2$ are correlated.

Ito's Lemma

The Fundamental Result

For ordinary calculus, if $f$ is a function of $x$, we have $df = f'(x)dx$. For stochastic processes, the situation is more complex because $(dW_t)^2 = dt$ (not zero!).

Ito's Lemma (one-dimensional): If $X_t$ follows $dX_t = \mu dt + \sigma dW_t$, and $f(X_t, t)$ is twice differentiable, then: $df = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}dW_t$

The crucial extra term $\frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}$ arises from the quadratic variation of Brownian motion.

Application: Deriving GBM Solution

Let $Y_t = \ln(S_t)$ where $dS_t = \mu S_t dt + \sigma S_t dW_t$.

Applying Ito's lemma with $f(S) = \ln(S)$:

• $\frac{\partial f}{\partial S} = \frac{1}{S}$
• $\frac{\partial^2 f}{\partial S^2} = -\frac{1}{S^2}$

$dY_t = \frac{1}{S}(\mu S dt + \sigma S dW_t) - \frac{1}{2}\frac{\sigma^2 S^2}{S^2}dt$ $dY_t = (\mu - \frac{\sigma^2}{2})dt + \sigma dW_t$

Integrating: $Y_t - Y_0 = (\mu - \frac{\sigma^2}{2})t + \sigma W_t$

Therefore: $S_t = S_0 e^{(\mu - \sigma^2/2)t + \sigma W_t}$

Risk-Neutral Valuation

Change of Measure

Under the risk-neutral measure $\mathbb{Q}$, all assets earn the risk-free rate in expectation. This is achieved by Girsanov's theorem, which changes the drift of Brownian motion.

Under $\mathbb{Q}$: $dS_t = rS_t dt + \sigma S_t d\tilde{W}_t$

where $\tilde{W}_t$ is a Brownian motion under $\mathbb{Q}$.

Derivative Pricing Formula

The price of a derivative with payoff $\Phi(S_T)$ is: $V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\Phi(S_T)]$

This expectation under the risk-neutral measure can be:

• Computed analytically (Black-Scholes for European options)
• Approximated via Monte Carlo simulation
• Solved via PDE methods (Feynman-Kac)

Stochastic Integrals

Ito Integral

The stochastic integral $\int_0^T f(t) dW_t$ is defined as a limit of sums evaluated at the left endpoint: $\int_0^T f(t) dW_t = \lim_{n\to\infty} \sum_{i=0}^{n-1} f(t_i)(W_{t_{i+1}} - W_{t_i})$

Key properties:

• $E[\int_0^T f(t) dW_t] = 0$
• $E[(\int_0^T f(t) dW_t)^2] = \int_0^T E[f(t)^2] dt$ (Ito isometry)

Stratonovich vs. Ito

Stratonovich integrals evaluate at midpoints, giving standard calculus rules but losing the martingale property. Finance uses Ito calculus because:

• Ito integrals are martingales
• Compatible with no-arbitrage pricing
• Naturally handles discrete trading approximations

Numerical Methods for SDEs

Euler-Maruyama Scheme

The simplest discretization: $X_{t+\Delta t} = X_t + \mu(X_t, t)\Delta t + \sigma(X_t, t)\sqrt{\Delta t}Z$

Has strong convergence order 0.5.

Milstein Scheme

Adds a correction term for better accuracy: $X_{t+\Delta t} = X_t + \mu\Delta t + \sigma\sqrt{\Delta t}Z + \frac{1}{2}\sigma\sigma'(\Delta t Z^2 - \Delta t)$

Has strong convergence order 1.0.

Choosing the right scheme depends on the application: weak convergence (distribution accuracy) vs. strong convergence (path accuracy).