Abstract
This dissertation investigates the valuation of reload options by replacing the traditional Black-Scholes (BS) framework with a Bilateral Gamma (BG) Lévy process. While the BS model assumes log-normal returns, empirical evidence demonstrates that asset returns exhibit skewness and leptokurtosis (heavy tails), leading to significant valuation errors for path-dependent derivatives. Using ten years of S&P 500 (SPY) data, this study calibrates a BG model to capture these non-Gaussian dynamics, employing a deterministic approximation of the first-passage-time (FPT) distribution to overcome the analytical intractability of the stationary BG process.
The empirical analysis establishes that the BG model provides a statistically superior static fit to market data compared to the Normal distribution, reducing the Akaike Information Criterion (AIC) by more than 480 points. Dynamically, the BG framework predicts barrier-hitting times with significantly higher accuracy, reducing the mean prediction error from 49.8% under the BS model to 13.3%. Under a risk-neutral measure derived via the Esscher transform, these faster and more accurate hitting times result in reload option valuations that are 16–20% higher than standard BS prices. These findings indicate that the continued use of Gaussian models materially understates the economic cost of executive compensation and corporate liabilities.