Stochastic Volatility, Jumps, and GARCH as Financial Simulation Baselines

A simulation baseline is useful when you know exactly which stylized facts it can generate and which ones it cannot.

The Intuition

Chapter 5 uses classical simulation models as baselines for learned generators. That only helps if readers can answer the right question:

what behavior does this model generate?

Not:

is this model realistic in the abstract?

Different baselines are good at different things:

• GBM gives smooth diffusion and almost nothing else
• jump-diffusion adds discontinuities
• stochastic-volatility models add persistent variance dynamics
• GARCH gives a simple data-driven volatility recursion in discrete time

The point of the baseline is not to be universally realistic. It is to be a fair comparator for the behavior you care about.

A Comparison Table

This table is more useful than a hierarchy. These models are not successive upgrades in every sense. They solve different failure modes.

GBM: the Minimal Baseline

Geometric Brownian motion is

$$ \frac{dS_t}{S_t} = \mu\,dt + \sigma\,dW_t. $$

Its role is mostly diagnostic:

• it gives continuous paths
• log returns are conditionally Gaussian
• volatility is constant

That makes GBM a weak financial model but a valuable lower bar. If a learned generator cannot beat GBM on the behaviors that matter for your task, the learned model has not earned its complexity.

Jump-Diffusion: Fat Tails from Rare Moves

A simple jump-diffusion adds a Poisson jump component:

$$ \frac{dS_t}{S_t} = \mu\,dt + \sigma\,dW_t + J\,dN_t, $$

where $dN_t$ is a jump-count increment and J is a random jump size.

This helps because pure diffusion underestimates sudden large moves. Jump-diffusion can generate:

• fatter tails
• gap risk
• event-like discontinuities

But it does not automatically solve volatility clustering. Unless jumps and volatility states are coupled, the model still misses the persistent calm-versus-stress structure seen in real returns.

Stochastic Volatility: Variance Has Dynamics Too

In a stochastic-volatility model such as Heston, price and variance are typically written together as

$$ \frac{dS_t}{S_t} = \mu\,dt + \sqrt{V_t}\,dW_t^{(s)}, $$

with variance evolving as

$$ dV_t = \kappa(\theta - V_t)\,dt + \xi \sqrt{V_t}\,dW_t^{(v)}. $$

The exact form matters less here than the idea:

volatility is itself random and persistent.

That lets the model capture:

• clustered high- and low-volatility periods
• broader return tails than constant-volatility diffusion
• richer scenario behavior over time

This is why stochastic-volatility models are often more informative baselines for synthetic finance than jump models alone. In finance, the correlation between $dW_t^{(s)}$ and $dW_t^{(v)}$ is often as important as stochastic variance itself because it helps generate leverage-style asymmetry.

GARCH: the Pragmatic Discrete-Time Baseline

GARCH(1,1) writes conditional variance recursively:

$$ \sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2. $$

Its appeal is practical:

• easy to estimate
• discrete-time
• captures volatility clustering surprisingly well

The persistence condition $\alpha + \beta < 1$ is the stationary case. When $\alpha + \beta$ is close to one, shocks to variance decay slowly, which is exactly why GARCH can mimic persistent volatility regimes without a deep generative model.

GARCH is not a full market simulator. It is a strong volatility baseline. To simulate returns, it is usually paired with a return equation such as

$$ r_t = \mu + \sigma_t z_t, $$

with an innovation law for $z_t$.

A Worked Baseline Choice

Suppose your task is to stress-test a volatility-targeting strategy.

Weak comparator

Use GBM only. You will generate smooth paths with constant variance and conclude the strategy is stable because the simulation suppresses the very clustering that breaks the strategy in practice.

Better comparator

Start with GARCH or a stochastic-volatility baseline. If jump risk is also central, add a jump component or compare against a jump-diffusion baseline too.

Now the simulation challenge is fairer:

• does the learned generator improve on volatility persistence?
• does it produce more realistic tail clustering?
• or is it merely rediscovering behaviors a classical baseline already gives you?

Calibration Versus Realism

These baselines also differ in how they are calibrated.

• GBM needs little more than drift and volatility
• jump-diffusion needs jump frequency and jump-size assumptions
• stochastic-volatility models need a variance process and often more delicate estimation
• GARCH usually fits by maximum likelihood on historical returns

The important discipline is this:

"beyond history" still means "inside model assumptions."

A jump model can invent large moves, but only in the way its jump law permits. A GARCH baseline can generate new volatility paths, but only under the recursion it learned. None of these models is a free pass to realism.

In Practice

Use these rules:

• choose the baseline according to the stylized fact your task needs
• use GBM as a floor, not as a serious crisis generator
• use GARCH when volatility persistence is the first-order concern
• use jumps when discontinuities matter materially for the strategy
• compare generators on behaviors, not on aesthetic path realism

Common Mistakes

• Treating all classical baselines as equally weak.
• Using GBM to judge a generator meant to reproduce crisis behavior.
• Calling jump-diffusion realistic just because it has fat tails.
• Forgetting that GARCH is a volatility model first, not a full structural market model.
• Confusing calibration difficulty with model quality.

Connections

This primer supports Chapter 5's classical-baseline logic for synthetic financial data. It connects directly to stylized facts, task-based evaluation of synthetic data, risk-model baselines, and later chapters that use simulated paths for stress testing rather than pretty charts.