Stochastic Volatility Models: Random Variance, Risk Premia, and Smile Dynamics

For educational purposes only; not investment advice.

Direct answer

A stochastic volatility model makes an asset’s instantaneous variance a random state variable instead of a constant or a deterministic function of spot and time. In a generic continuous two-factor specification:

dS_t = μ_t S_t dt + √v_t S_t dW_t^S

dv_t = a(v_t,t)dt + b(v_t,t)dW_t^v, with dW_t^S dW_t^v = ρ_t dt

The extra variance shock can create time-varying volatility, clustering, mean reversion, skew, and richer smile dynamics. The model does not make volatility “known”; it specifies a probability law whose parameters and risk-neutral version must be estimated or calibrated.

This is a model family, not one formula. Heston uses a mean-reverting square-root variance process, SABR evolves a forward and its volatility, forward-variance models model the variance term structure, and rough-volatility models use non-Markovian memory. Each choice fits and hedges different features and introduces different numerical and model risks.

Physical dynamics are not pricing dynamics

Historical returns are modeled under the physical measure P; no-arbitrage option valuation uses a risk-neutral measure Q. With a non-traded variance shock, stock and cash generally cannot span every source of uncertainty. The market is incomplete, so no-arbitrage alone does not uniquely determine the variance risk premium or the Q drift of variance.

Schematically, the same variance state may have:

dv_t = a_P(v_t,t)dt + b(v_t,t)dW_t^{v,P}

under P, but

dv_t = a_Q(v_t,t)dt + b(v_t,t)dW_t^{v,Q}

under Q. The difference between a_P and a_Q embeds compensation for volatility risk. Estimating P parameters from returns and inserting them unchanged into an option pricer silently assumes away that distinction.

Correlation ρ links return and variance shocks. In equity models, a negative fitted risk-neutral correlation often helps generate richer downside Puts. Volatility of volatility controls how dispersed future variance can become and influences smile curvature. Mean reversion shapes the term structure. These interpretations are conditional: parameters interact, and multiple parameter sets can produce similar vanilla prices.

Static fit versus dynamics

A calibration minimizes errors across strikes and expirations at one time. A risk model must also specify how that fitted surface changes after spot, time, and volatility shocks. Two models can match today’s vanillas and produce different barrier values, forward skews, hedge ratios, and next-day P/L. Smile dynamics and out-of-sample hedging are therefore separate validation targets from calibration error.

Example: one current variance, two measures

Assume current variance is v₀ = 0.04, so current instantaneous volatility is √0.04 = 20%. For illustration, suppose variance is mean reverting under each measure:

E[v_T] = θ + (v₀ − θ)e^(−κT)

Use these hypothetical parameters:

Measure	`κ`	`θ`	Six-month `E[v_0.5]`
Physical `P`	`1`	`0.04`	`0.04`
Risk-neutral `Q`	`2`	`0.0625`	`0.05422`

Under P:

0.04 + (0.04 − 0.04)e^(−1×0.5) = 0.04

Under Q:

0.0625 + (0.04 − 0.0625)e^(−2×0.5) = 0.05422

The square roots are 20% and 23.29%, respectively. This difference is not an option-profit forecast and does not identify the risk premium from one observation. It shows why a risk-neutral variance term structure calibrated from options cannot be relabeled as expected realized volatility.

Also, √E[v_T] is not generally equal to E[√v_T]. The example reports the square root of expected variance, not expected volatility. A full option value depends on the distribution and path of variance, its correlation with returns, and the payoff, not only this mean.

Model selection and validation checklist

State the stochastic equations, measure, state variables, parameter units, boundary behavior, and correlation structure.
Separate parameters estimated from returns under P from those calibrated to option prices under Q.
Use synchronized executable quotes, forwards, discount factors, dividends, and contract terms.
Weight calibration errors by economic relevance such as Bid-Ask or Vega, and inspect each expiry and wing.
Run multiple initial guesses and report near-equivalent parameter sets; one optimizer result can be non-identifiable.
Check positivity, moment conditions, simulation bias, discretization, transform integration, and branch handling.
Reprice all calibration instruments and compare errors with market spreads, not only IV RMSE.
Validate future surface moves, hedge P/L, and parameter stability out of sample.
Stress price jumps, event variance, correlation breaks, volatility-of-volatility spikes, and liquidity loss.
Compare Heston, local volatility, stochastic-local volatility, jump, and rough alternatives for the actual payoff.
Treat American exercise, discrete dividends, barriers, and path dependence with suitable numerical methods.
Include transaction costs, discrete rehedging, funding, assignment, and unavailable hedge instruments.
Record which risks the model deliberately omits; a precise calibration does not shrink omitted risk.

Common misconceptions

“Stochastic volatility predicts volatility.” It defines conditional dynamics after model and parameters are chosen.
“Historical variance parameters price options.” Pricing generally needs risk-neutral dynamics and a volatility risk premium.
“A perfect vanilla fit proves the model.” Static fit does not identify realistic paths or hedge behavior.
“More factors always improve accuracy.” Flexibility can increase instability, overfitting, and unhedgeable parameter risk.
“Stochastic volatility includes crashes.” A continuous variance model can still omit discontinuous price and variance jumps.
“Risk-neutral expected variance is a market forecast.” It is a pricing object containing risk premia and depends on conventions.
“One calibrated parameter has one economic meaning.” Parameters compensate for each other and can be weakly identified.