Quant Learning Lab

Heston Stochastic Volatility

Understand why stochastic volatility matters and how the Heston framework improves option-surface intuition.

Model Overview

The Heston model allows the variance of an asset to evolve stochastically rather than staying fixed. That change helps bridge the gap between textbook constant-volatility assumptions and the skewed, smile-shaped implied volatility surfaces seen in real markets.

Intuition

Instead of assuming the market knows one stable volatility number, Heston treats volatility as a state variable with its own dynamics. The asset and variance move together through correlated shocks, which helps generate more realistic option prices across strikes and maturities.

Key Formula

dS_t = r S_t dt + sqrt(v_t) S_t dW_1

dv_t = kappa(theta - v_t)dt + sigma_v sqrt(v_t) dW_2

corr(dW_1, dW_2) = rho

Practical Use Case

A structuring team can use Heston to analyse how changes in mean reversion, long-run variance, and spot-vol correlation affect exotic payoffs or vanilla smile calibration across an equity options book.

Learning Outcome

This lesson is designed to connect quantitative theory with the exact kind of institutional workflow QuantModels.ai exposes in its pricing and analytics modules.

Static Question Bank

Work through the curated model question bank first, then generate additional mock AI question sets below.

1. What key feature does the Heston model add beyond Black-Scholes?

Deterministic interest-rate shifts only

A stochastic variance process

Zero transaction costs

Guaranteed closed-form American exercise

2. Why does correlation between price and variance matter in Heston?

It has no impact on option prices

It helps shape skew and smile behavior

It removes randomness from volatility

It guarantees variance stays constant

3. Where is Heston especially useful?

For explaining volatility smiles in equity options

Only for bond coupon amortization

For eliminating numerical methods entirely

For pricing without any calibration step

4. What does the parameter kappa represent?

Mean reversion speed of variance

Dividend payout speed

The strike scaling factor

The number of Monte Carlo paths

5. What is theta in the Heston variance process?

Option time decay

Long-run average variance level

Instantaneous spot return

Portfolio Sharpe ratio target

6. Why is sigma_v important?

It measures the volatility of variance itself

It is the risk-free rate

It fixes the maturity of the option

It controls only dividends

7. What practical issue often arises in Heston implementations?

Variance may need careful numerical treatment to remain non-negative

The model cannot use correlation at all

The asset price must remain constant

The model only works in discrete time without calibration

8. Why do practitioners calibrate Heston to an implied volatility surface?

To make the model consistent with observed option prices

To remove dependence on maturity

To avoid using spot prices

To convert it into a short-rate model

9. What does rho less than zero typically imply for equity options?

Negative skew consistent with leverage-style effects

Perfectly flat implied volatility curves

Zero variance of the underlying

A deterministic payoff

10. Why is Heston often paired with Monte Carlo or Fourier methods?

Because richer stochastic-volatility models usually need numerical techniques

Because the model cannot produce option prices at all

Because correlation forbids closed-form expressions

Because it is only a pedagogical toy model

Generate Unlimited Questions

Use the mock AI agent panel to create additional practice sets by topic and difficulty. The component is already shaped for a future API-backed generation workflow.

AI Placeholder

Model topicDifficultyNumber of questions

Generated Questions

Mock generated set for Heston at beginner difficulty.

1. What does Heston try to improve relative to Black-Scholes? (Heston · beginner · Set 1)

The treatment of changing volatility

The definition of a strike

The existence of maturity

The use of discounting

2. Why might traders care about stochastic volatility? (Heston · beginner · Set 2)

Because observed market volatility is not constant

Because spot prices never move

Because options do not depend on volatility

Because only interest rates matter

3. What kind of market pattern can Heston help explain? (Heston · beginner · Set 3)

Volatility smile or skew

Bond coupon schedules

Tax accrual rules

Settlement holidays

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