This thesis explores the mathematical foundations of arbitrage-free pricing in financial markets, from linear programming to derivative pricing for European options. Beginning with linear programming fundamentals, we establish the equivalence between primal and dual optimization problems to derive the Arbitrage Theorem, which demonstrates that risk-neutral probability measures exist if and only if markets are free from arbitrage opportunities. Then, we analyze optimal portfolio construction strategies, including efficient frontiers, Capital Market Line derivation, and the Capital Asset Pricing Model (CAPM), showing how investors can balance expected returns against risk in arbitrage-free markets. Finally, this thesis derives and solves the Black-Scholes equation for European option pricing, where we employ techniques from stochastic calculus and partial differential equations to obtain closed-form pricing formulas.
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