Projects

Advanced Derivative Pricing and Risk Management using the SABR Model

Published: February 25, 2026

Abstract: This project provides a robust implementation and comparative analysis of the SABR (Stochastic Alpha, Beta, Rho) model—a sophisticated stochastic volatility framework designed to capture the “volatility smile” and “skew” prevalent in modern financial markets. While the traditional Black-Scholes model assumes constant volatility, this work demonstrates how the SABR model mathematically accounts for the dynamic relationship between an asset’s price and its volatility, offering a more accurate representation of market reality.

Trinomial Tree Option Pricing: A Discussion

Published: January 21, 2026

Abstract: The study begins with the Binomial model, demonstrating how its step-by-step backward induction allows for the valuation of early exercise features in American options—a capability lacking in the standard Black-Scholes formula. It further investigates the Trinomial Tree model, which introduces a “stay” node to improve convergence stability and computational efficiency. Key comparative analyses include:

• Early Exercise Incentives: Evaluating how dividend yields ($q$) and interest rates ($r$) create price premiums for American options compared to their European counterparts.
• Put-Call Parity: Demonstrating that while strict parity holds for European options, American options are governed by specific price inequalities.
• The Greeks: Implementing methodologies to calculate price sensitivities, including Delta, Gamma, and Theta via internal tree nodes, and Vega and Rho using numerical finite difference (bumping) methods.

Results confirm that the Trinomial model provides a smoother convergence to Black-Scholes-Merton values as the number of time steps ($N$) increases, establishing it as a robust industry standard for complex exotic and American options.

Modelling USDEUR FX Swaptions: A Comparative Analysis

Published: November 13, 2025

Abstract: This project investigates the valuation of a USDEUR Cross-Currency FX Swaption by comparing the widely used closed-form pricing solution, the Black ’76 model (Benchmark), against the advanced Merton Jump-Diffusion (JD) model (Alternative). The goal was to quantify the pricing impact of assuming fat-tailed (leptokurtic) distributions over the Black model’s standard log-normal assumption. Key financial engineering procedures, including yield curve bootstrapping and Vanna-Volga surface interpolation, were performed. The primary finding is that the Merton JD model yields a price that is 8.06% higher than the Black model, accurately capturing a Jump Risk Premium necessary for a more robust and prudent valuation in volatility-sensitive markets.