Heston Stochastic Volatility & Kelly Tail-Risk Terminal

Heston Call Fair Value

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vs Black-Scholes benchmark

99% Value-at-Risk (VaR)

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1st percentile terminal drawdown

99% Expected Shortfall (CVaR)

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Conditional tail expectation

Half-Kelly Optimal Allocation

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Variance-penalized capital fraction

Feller Boundary ($2\kappa\theta / \xi^2$)

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Variance positivity condition

SDE System Parameters

Initial Spot ($S_0$) & Strike ($K$) $100 / $105

Annualized Drift ($\mu$) 11.0%

Initial & Long-Run Vol ($\sqrt{\theta}$) 24.0%

Mean Reversion Speed ($\kappa$) 2.40

Vol-of-Vol ($\xi$) 0.42

Spot-Variance Correlation ($\rho$) -0.65

Monte Carlo Trajectories ($N$) 2,000 paths

Coupled Spot Price Trajectories ($S_t$) & Interdecile Fan ($T = 1.0\text{ yr}$) dt = 1/252 Trading Days

Terminal Log-Return Density vs. Log-Normal Black-Scholes Reference Left-Skew & Kurtosis Comparison