Linganisha mbinu

	FFT ya Carr-Madan	Modeli ya Bates	Volatilite ya Ndani (Dupire)
Nyanja	Fedha za Kiidadi	Fedha za Kiidadi	Fedha za Kiidadi
Familia	Machine learning	Regression model	Regression model
Mwaka wa asili	1999	1996	1994
Mwanzilishi	Peter Carr and Dilip B. Madan	David S. Bates	Bruno Dupire
Aina	Valuation Algorithm	Equity/FX Model	Equity/FX Model
Chanzo asilia	Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(4), 61-73.	Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9(1), 69-107.	Dupire, B. (1994). Pricing with a smile. Risk Magazine, 7(1), 18-20.
Majina mbadala	FFT Pricing, Characteristic Function Method	SVJ Model, Jump Diffusion	Deterministic Volatility Function, DVF
Zinazohusiana	3	4	4
Muhtasari	The Carr-Madan Fast Fourier Transform (1999) is a highly efficient method for computing option prices across a range of strikes using characteristic functions and FFT. It enables rapid pricing of European options under any model with a known characteristic function (Heston, Merton jumps, Variance Gamma), with computational complexity that scales logarithmically in the number of strikes.	The Bates model (1996) combines stochastic volatility and jump diffusion to capture both the volatility smile and the implied volatility skew observed in equity and currency option markets. It extends the Heston model by adding a Poisson jump component to returns, making it suitable for pricing options when sudden price moves are expected.	Dupire's local volatility model (1994) is a deterministic framework that extracts a term and strike-dependent volatility function from market option prices. Unlike constant volatility, local volatility perfectly fits the observed implied volatility smile and is implemented via finite difference methods for European and American option pricing.