Synthetic American Option Pricing via Jump-HMM-Driven Heston Implied Volatility

Valuing American options along simulated stock paths requires an implied volatility (IV) for every option on every date. A stock-return model alone does not provide it. We built a simulator that assigned IV to American options on simulated dates for any chosen stock model. We fitted parametric and neural IV surfaces to vendor option quotes for 31 tickers. Each surface predicted IV from moneyness and time to expiration. Stock paths came from a jump hidden Markov model with capped daily returns. Each day, every option's implied variance moved partway toward its surface prediction, following the form of the Heston variance equation. Random shocks tended to raise IV when the stock fell. A binomial tree converted IV into option values and price sensitivities. Neural surfaces fitted by sector or ticker matched the quotes more closely than one parametric surface. Their errors still varied by date and carried into dollar prices. Repricing identical stock paths under five ways of updating IV changed option values before expiration, and the worst simulated short-position losses at a fixed horizon. Payoffs at expiration did not change. In forecasts of later Goldman Sachs and Eli Lilly option prices, stochastic IV did little better than fixed IV. Rerunning the forecasts with the realized stock paths pointed to stock prediction as a major source of option-price error. The hidden Markov model and an adaptive-volatility model predicted stock prices about as accurately as assuming no change, and tuning found no gain from predicting direction. The adaptive-volatility model improved predicted price ranges for Eli Lilly but not Goldman Sachs. The simulator supported reproducible comparisons of IV and stock assumptions but did not forecast better than simpler alternatives. Complete option histories and prices consistent across strikes are needed before simulated prices can replace market data.