Appreciate different stochastic models for security prices and how and when they can be applied

Welcome to Stochastic Models for Security Prices!

In financial and actuarial economics, security prices fluctuate over time in response to unpredictable market news and economic forces. A stochastic model uses probability theory to model these dynamic, uncertain movements across continuous time. In this chapter, we develop the continuous-time framework for security prices, exploring standard Brownian motion, stochastic differential equations, Ito's Lemma, Geometric Brownian Motion, and the mean-reverting Ornstein-Uhlenbeck process.

1. Standard Brownian Motion (The Wiener Process)

At the heart of continuous-time stochastic modelling is standard Brownian motion, also called a Wiener process, denoted by \( W_t \) (or \( B_t \)) for \( t \ge 0 \).

Basic Properties of \( W_t \):

1. Starting Value: \( W_0 = 0 \) with probability 1.
2. Continuous Sample Paths: The path \( t \mapsto W_t \) is continuous everywhere with probability 1.
3. Independent Increments: For any times \( 0 \le s < t \le u < v \), the increment \( W_v - W_u \) is independent of \( W_t - W_s \). The process possesses the Markov property: future increments depend only on the current state, not on past history.
4. Stationary, Normally Distributed Increments: For any \( s < t \), the increment \( W_t - W_s \sim N(0, t - s) \). In particular, \( E[W_t] = 0 \) and \( \operatorname{Var}(W_t) = t \).
5. Non-Differentiability: Although paths are continuous everywhere, they are nowhere differentiable with probability 1 and exhibit infinite total variation over any finite interval.

2. Stochastic Differential Equations (SDEs) and the Ito Integral

Because sample paths of Brownian motion are non-differentiable, ordinary calculus cannot be used. We model the dynamics of a continuous-time process \( X_t \) using a Stochastic Differential Equation (SDE) written in differential form as:
\( dX_t = μ(t, X_t) dt + σ(t, X_t) dW_t \)

In rigorous integral form, this means:
\( X_t = X_0 + \int_0^t μ(s, X_s) ds + \int_0^t σ(s, X_s) dW_s \)

Here:
- \( \int_0^t μ(s, X_s) ds \) is a standard Riemann (or Lebesgue) integral representing the deterministic drift.
- \( \int_0^t σ(s, X_s) dW_s \) is an Ito integral with respect to Brownian motion, representing the random diffusion.
- A key property of the Ito integral is that it is a martingale (when \( \sigma \) satisfies standard integrability conditions), meaning its expected future change given current information is zero.

Diffusion Processes vs Mean-Reverting Processes

- A general diffusion process has drift and diffusion functions \( \mu(t, X_t) \) and \( \sigma(t, X_t) \) where the drift might lead to open-ended wandering or growth (such as exponential drift in equity prices).
- A mean-reverting process features a drift term that pulls the process back toward a long-term central level whenever it deviates, making it well-suited for economic variables that do not grow indefinitely.

3. Ito's Lemma

The cornerstone tool for manipulating functions of stochastic processes is Ito's Lemma. It accounts for the non-zero quadratic variation of Brownian motion, where informally \( (dW_t)^2 = dt \), while \( dt \cdot dW_t = 0 \) and \( (dt)^2 = 0 \).

One-Dimensional Ito's Formula:

If \( dX_t = \mu_t dt + \sigma_t dW_t \) and \( f(t, x) \) is a twice continuously differentiable function, then \( Y_t = f(t, X_t) \) satisfies:
\( dY_t = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (dX_t)^2 \)

Substituting \( (dX_t)^2 = \sigma_t^2 dt \):
\( dY_t = \left( \frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma_t \frac{\partial f}{\partial x} dW_t \)

4. Geometric Brownian Motion (The Continuous-Time Lognormal Model)

The standard model for equity and security prices is Geometric Brownian Motion (GBM), governed by the SDE:
\( dS_t = \mu S_t dt + \sigma S_t dW_t \)

where \( \mu \) is the constant expected return (drift rate) and \( \sigma > 0 \) is the constant volatility.

Solving the SDE with Ito's Lemma:

Let \( f(S_t) = \ln S_t \). Then:
\( \frac{\partial f}{\partial S} = \frac{1}{S_t} \), \( \frac{\partial^2 f}{\partial S^2} = -\frac{1}{S_t^2} \), and \( \frac{\partial f}{\partial t} = 0 \).

Applying Ito's Lemma:
\( d(\ln S_t) = \frac{1}{S_t} dS_t + \frac{1}{2} \left(-\frac{1}{S_t^2}\right) (dS_t)^2 \)
\( d(\ln S_t) = \frac{1}{S_t} (\mu S_t dt + \sigma S_t dW_t) - \frac{1}{2 S_t^2} (\sigma^2 S_t^2 dt) = \left(\mu - \frac{1}{2}\sigma^2\right) dt + \sigma dW_t \)

Integrating from 0 to \( t \):
\( \ln S_t - \ln S_0 = \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t \)
\( S_t = S_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t \right) \)

Properties of the Lognormal Model:

1. Strict Positivity: Since the exponential function is strictly positive, \( S_t > 0 \) for all \( t \) as long as \( S_0 > 0 \). A security with limited liability cannot take negative values.
2. Distribution of Returns: Continuously compounded returns \( \ln(S_t / S_0) \) are normally distributed with mean \( (\mu - \frac{1}{2}\sigma^2)t \) and variance \( \sigma^2 t \). Therefore, security prices follow a lognormal distribution.
3. Expected Value and Variance:
\( E[S_t] = S_0 e^{\mu t} \)
\( \operatorname{Var}(S_t) = S_0^2 e^{2\mu t} (e^{\sigma^2 t} - 1) \)

Empirical Evidence For and Against the Lognormal Model:

Evidence For:
- Prices remain strictly non-negative, respecting limited liability.
- Percentage changes, rather than absolute changes, are modelled, reflecting how investors assess returns.
- Mathematical tractability: yields analytical solutions, such as the Black-Scholes formula.

Evidence Against:
- Heavy/Fat Tails (Kurtosis): Real-world return distributions exhibit fatter tails than the normal distribution; market crashes and large jumps occur much more frequently than GBM predicts.
- Volatility Clustering and Non-Constant Volatility: Real market volatility varies over time and forms clusters (periods of high volatility followed by high volatility), contradicting constant \( \sigma \).
- Volatility Smile / Skew: Implied volatilities from option markets vary across strikes, demonstrating that the market does not price derivatives under a single constant volatility.
- Continuity / Jump Risk: Real security prices jump discontinuously upon major news announcements, whereas Brownian motion paths are strictly continuous.

5. The Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck (OU) process is a continuous-time stochastic process that exhibits mean reversion. Its SDE is given by:
\( dX_t = \alpha(\mu - X_t) dt + \sigma dW_t \)

where:
- \( \alpha > 0 \) is the speed of mean reversion.
- \( \mu \) is the long-term mean (the reversion level).
- \( \sigma > 0 \) is the constant volatility.
- \( W_t \) is a standard Brownian motion.

How the Drift Operates:

- When \( X_t > \mu \), the drift \( \alpha(\mu - X_t) < 0 \), exerting downward pressure back toward \( \mu \).
- When \( X_t < \mu \), the drift \( \alpha(\mu - X_t) > 0 \), pulling \( X_t \) upward toward \( \mu \).

Properties and Explicit Solution:

By applying Ito's Lemma to \( e^{\alpha t} X_t \), the exact solution is:
\( X_t = \mu + (X_0 - \mu) e^{-\alpha t} + \sigma \int_0^t e^{-\alpha(t - s)} dW_s \)

Because the integral of a deterministic integrand with respect to Brownian motion is normally distributed, \( X_t \) is normally distributed at all times:
- Expectation: \( E[X_t] = \mu + (X_0 - \mu) e^{-\alpha t} \), which tends to \( \mu \) as \( t \to \infty \).
- Variance: \( \operatorname{Var}(X_t) = \frac{\sigma^2}{2\alpha} (1 - e^{-2\alpha t}) \), which tends to the stationary variance \( \frac{\sigma^2}{2\alpha} \) as \( t \to \infty \).

6. How and When to Apply These Models

In actuarial practice and financial engineering, model selection depends on the underlying economic characteristics of the asset:

When to Apply Geometric Brownian Motion (GBM):

- Ordinary Equities and Share Indices: Assets that have no natural ceiling or anchor and are expected to grow exponentially over the long term with economic productivity.
- Limited Liability Securities: Assets where values cannot drop below zero.
- Benchmark Derivative Pricing: Closed-form options valuation (Black-Scholes-Merton framework).

When to Apply the Ornstein-Uhlenbeck Process:

- Short-Term Interest Rates: The OU process forms the basis of the Vasicek model for the term structure of interest rates, where macroeconomic policy and market forces pull rates back toward an equilibrium target.
- Commodity Prices and Energy Markets: Prices of oil, gas, and agricultural products are anchored to long-run marginal costs of production; elevated prices encourage increased supply, pulling prices back down.
- Stochastic Volatility Models: Modelling volatility itself as a mean-reverting process around an average market volatility level.

Limitation to remember for OU: Because \( X_t \) is normally distributed, it can take negative values with non-zero probability. While negative interest rates have been observed historically, it is not suitable as a direct model for assets that cannot have negative prices without modification.