The Generalized Gamma Distribution as a Useful RND under Heston’s Stochastic Volatility Model

We present the Generalized Gamma (GG) distribution as a possible risk neutral (RND) for modeling European options prices under Heston’s stochastic volatility (SV) model. demonstrate that particular reparametrization, this distribution, which is member of scale-parameter family distributions with mean being forward spot price, satisfies solution and hence could be used direct risk-neutral valuation option price SV Indeed, especially useful in situations spot’s follows negatively skewed Black–Scholes-based (i.e., log-normal distribution) largely inapt. illustrate applicability GG an RND by market data on three large market-index exchange-traded funds (ETF), namely SPY, IWM QQQ well TLT (an ETF tracks index long-term US Treasury bonds). As writing paper (August 2021), chain each ETFs shows pronounced skew their ‘smile’, indicates likely distortion Black–Scholes such data. Reflective entirely different expectations, ‘smile’ appears not to exist provide thorough we have (with 15 October 2021 expiration) based compare it pricing obtained directly from well-calibrated model (both theoretically also empirically, using Monte Carlo simulations price). All exhibited distributions, are well-matched those derived RND. The inadequacy instances, involves further illustrated its impact hedging factor, delta, immediate implications retail trader. Similarly, closely related Inverse (IGG) proposed involving positively distribution. In all, utilizing RNDs valuations seen particularly traders who do numerical tools or know-how fine-calibrate