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Mathematics of Derivative Pricing & Risk Neutral Valuation
#Quantitative Finance
#Black-Scholes PDE
#Stochastic Calculus
#Greeks Sensitivity
#Martingale Pricing
#Monte Carlo Simulation
A comprehensive theoretical and computational investigation into continuous-time mathematical finance, asset pricing models, and risk-neutral valuation frameworks.
Mathematical Exploration & Computational Implementations
- Stochastic Calculus & Black-Scholes Formulation: Derived the Black-Scholes Partial Differential Equation (PDE) via geometric Brownian motion dynamics and dynamic replication under the absence of arbitrage. Applied the Feynman-Kac theorem to solve the PDE under risk-neutral equivalent martingale measures ((\mathbb{Q})).
- Numerical Pricing Engines:
- Implemented high-performance vectorized Monte Carlo pricing engines with antithetic variate variance reduction for exotic and path-dependent options.
- Implemented binomial lattice models (Cox-Ross-Rubinstein) and finite-difference PDE schemes for American-style early exercise boundary determination.
- Sensitivity Analysis & Payoff Engineering: Computed full analytical and numerical Greeks ((\Delta, \Gamma, \mathcal{V}, \Theta, \rho)) across implied volatility smiles and time-to-maturity surfaces.
- Hedging Strategies: Simulated dynamic delta and gamma hedging portfolios under discrete rebalancing intervals to measure residual hedging slippage and gamma risk.
- Repository: GitHub: Mathematics_Of_Pricing_Derivatives
- Project Report: Google Drive Technical Report