Implied Volatility Surface Arbitrage: Heston Stochastic Volatility Calibration and Local Volatility Skew
Calibrating mean-reversion speed, volatility of volatility, and correlation parameters to exploit static arbitrage across equity options surfaces.
Options Surface Calibration Engine
Calibrate Heston model parameters and detect calendar spread arbitrage
Empirical Analysis & Findings
## Stochastic Volatility Modeling in Derivatives Pricing
Options market pricing deviates systematically from flat volatility assumptions. The implied volatility surface exhibits both a strike skew (smile) and a term structure. Calibrating the Heston (1993) stochastic volatility model provides an arbitrage-free pricing framework for vanilla and exotic contracts.
### 1. The Heston Coupled Stochastic Differential Equations
$$dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^S$$
$$dv_t = \kappa (\theta - v_t) dt + \xi \sqrt{v_t} dW_t^v$$
$$\text{Correlation: } dW_t^S dW_t^v = \rho dt$$
| Parameter | Symbol | Economic Meaning | Typical S&P 500 Equity Index Value |
|:----------|:------:|:-----------------|:----------------------------------:|
| Mean Reversion Speed | $\kappa$ | Rate at which variance returns to long-term mean | 1.85 to 3.20 |
| Long-Term Variance | $\theta$ | Baseline steady-state annual variance | 0.04 to 0.06 (20% to 24% IV) |
| Volatility of Volatility | $\xi$ | Kurtosis generator creating volatility smile wings | 0.45 to 0.75 |
| Correlation | $\rho$ | Asymmetric leverage driving skew steepness | -0.65 to -0.82 |
| Initial Variance | $v_0$ | Current instantaneous market variance | Market Spot State |
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Frequently Asked Questions
Why does the Black-Scholes model fail to capture the empirical volatility smile?
Black-Scholes assumes constant asset return volatility and Gaussian log-normal distributions, ignoring fat tails (kurtosis) and leverage effect skew.
What is the economic interpretation of correlation parameter rho in the Heston model?
A negative rho captures the equity leverage effect: when spot prices plummet, volatility surges, creating the pronounced downward implied volatility skew.
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