Abstract
This project develops a coherent framework for modelling the implied volatility surface by combining stochastic volatility, local volatility, and smile-consistent interpolation techniques. Starting from market conventions, risk-neutral valuation, and vanilla option pricing, the work introduces the volatility surface as a mapping from strikes and maturities to implied volatilities and formalizes the no-arbitrage constraints it must satisfy. The Heston model is then used to generate parametric volatility surfaces and to illustrate how each parameter affects skew, smile, and term structure, while the SVI family of parameterizations provides a flexible, arbitrage-aware representation of implied total variance across log-moneyness. In parallel, the Dupire local volatility framework links the implied surface to a state- and time-dependent instantaneous volatility, and the Vanna–Volga methodology is presented as a practical, market-driven correction to Black–Scholes prices that reproduces smile features from a small set of liquid quotes. Together, these tools yield a practical calibration and construction pipeline that can fit observed volatility data across asset classes, enforce static arbitrage conditions, and generate robust surfaces suitable for pricing and hedging both vanilla and first-generation exotic options.
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