Calibrating Heston Stochastic Volatility Parameters to Call Prices

Summary

The document describes calibrating the Heston stochastic volatility model to European call prices. It lists the model's mean-reverting variance process and the parameters to estimate: the variance reversion speed and long-run level, volatility of variance, correlation between price and variance shocks, and initial variance. Calibration uses a weighted least-squares objective that compares market call prices with model prices, with weights inversely related to bid–ask spread.

The author reports applying the procedure across 170 days of prices and asks whether the resulting empirical distributions for several parameters have plausible magnitudes. Parameter bounds are given, but the document provides no fitted values, diagnostics, or answer to the question. The setup is informative as an example of an objective function and constrained calibration; it does not establish calibration quality or general parameter ranges.

Key ideas

• The Heston model represents variance as a mean-reverting stochastic process correlated with stock-price shocks.
• Calibration estimates five parameters by fitting model prices to observed European call prices.
• The objective weights squared pricing errors inversely by each option's bid–ask spread.
• The calibration was repeated across 170 days, but no parameter estimates or validation results are supplied.
• The stated parameter bounds constrain the optimization but do not demonstrate that the fitted model is reliable.

Tags

# Values for Heston Model Parameters


# Values for Heston Model Parameters












Under the Heston model, the stock price and volatility follow the processes \begin{align*} dS & = \mu S dt + \sqrt{V} S dW^1, \\ dV & = \kappa (\theta - V)dt + \sigma \sqrt{V} dW^2, \\ dW^1 dW^2 & = \rho dt. \end{align*}

The parameters to be calibrated are $\kappa$, $\theta$, $\sigma$, $\rho$, and $V_0$, which appears in the pricing formula. Below, I've used MATLAB's lsqnonlin function to minimize the objective function $$ \sum_{i=1}^N \frac{1}{ask_i - bid_i}(C_i - \hat{C}_i)^2, $$ where $C_i$ is the $i^{th}$ European call's market price and $\hat{C}_i$ is the corresponding Heston value. After running this calibration to 170 separate days of prices, I've obtained the following empirical distributions for $\kappa$, $\theta$ and $\sigma$, and was curious if the orders of magnitude seem about right?

I should mention I placed the following restrictions on the admissible parameter values during calibration, following the paper by Moodley: $$ \kappa > 0 \\ \theta \in [0, 1] \\ \sigma \in [0, 5] \\ \rho \in [-1, 0] \\ V_0 \in [0, 1] $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.