Linganisha mbinu

	FFT ya Carr-Madan	Modeli ya Bates	Uthamini Usio na Hatari (Risk-Neutral Valuation)
Nyanja	Fedha za Kiidadi	Fedha za Kiidadi	Fedha za Kiidadi
Familia	Machine learning	Regression model	Regression model
Mwaka wa asili	1999	1996	1979
Mwanzilishi	Peter Carr and Dilip B. Madan	David S. Bates	John Harrison and David Kreps
Aina	Valuation Algorithm	Equity/FX Model	Fundamental Principle
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.	Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408.
Majina mbadala	FFT Pricing, Characteristic Function Method	SVJ Model, Jump Diffusion	Risk-Neutral Measure, Q-Measure
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.	Risk-neutral valuation (1979) is the fundamental principle that derivative prices equal the expected payoff discounted at the risk-free rate, computed under a risk-neutral probability measure (Q-measure). This principle, formalized by Harrison and Kreps, eliminates the need to estimate risk premia and is the foundation of modern derivatives pricing.