Mathematics of Derivative Pricing: Theory and Computational Implementation
Abstract
This technical monograph explores the mathematical foundations and computational realization of modern derivative pricing. Beginning with discrete binomial lattices and Brownian motion kinematics, we derive the fundamental Black-Scholes partial differential equation under non-arbitrage principles and Feynman-Kac representations. We formulate numerical schemes using both vectorized Monte Carlo methods with antithetic variate variance reduction and finite-difference PDE solvers for early-exercise American options. Complete Python workflows evaluate Greeks surfaces and delta-hedging performance under realistic volatility and transaction frictions.
Publication & Pre-print Details
- SSRN Pre-print: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6439720 (Abstract ID:
6439720) - SSRN Author Profile: Vatsh Van on SSRN (Author ID: 10485196)
- Author ORCID: 0009-0001-7236-8213
- Primary Affiliation: Department of Mechanical Engineering & Centre for Machine Intelligence and Data Science (CMInDS), Indian Institute of Technology Bombay
- Full Technical Report (PDF): View on Google Drive
- Open-Source Code: GitHub Repository
Mathematics_Of_Pricing_Derivatives
Theoretical Architecture & Mathematical Formulations
1. Stochastic Calculus & Itô’s Lemma
Under a filtered probability space ((\Omega, \mathcal{F}, (\mathcal{F}t){t \ge 0}, \mathbb{P})), the asset price (S_t) follows geometric Brownian motion (GBM): $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ Applying Itô’s formula to a twice continuously differentiable derivative pricing function (V(S_t, t)): $$ dV = \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt + \sigma S \frac{\partial V}{\partial S} dW_t $$
2. Risk-Neutral Valuation & Black-Scholes PDE
By constructing a self-financing delta-hedged portfolio (\Pi_t = V_t - \Delta_t S_t) with (\Delta_t = \frac{\partial V}{\partial S}), the stochastic (dW_t) term vanishes: $$ \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0 $$
3. Numerical Implementations
- Vectorized Monte Carlo Simulation: Antithetic variates variance reduction achieving (\mathcal{O}(1/\sqrt{N})) convergence across 1,000,000 paths.
- Finite-Difference Solvers: Crank-Nicolson implicit-explicit discretization for American put options with free boundary conditions.
- Greeks Surfaces: Exact closed-form and finite-difference evaluations of Delta ((\Delta)), Gamma ((\Gamma)), Vega ((\nu)), Theta ((\Theta)), and Rho ((\rho)).