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	Szybka transformata Fouriera Carr-Madana	Model Batesa	Wycena w mierze neutralnej względem ryzyka
Dziedzina	Finanse ilościowe	Finanse ilościowe	Finanse ilościowe
Rodzina	Machine learning	Regression model	Regression model
Rok powstania	1999	1996	1979
Twórca	Peter Carr and Dilip B. Madan	David S. Bates	John Harrison and David Kreps
Typ	Valuation Algorithm	Equity/FX Model	Fundamental Principle
Źródło pierwotne	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.
Inne nazwy	FFT Pricing, Characteristic Function Method	SVJ Model, Jump Diffusion	Risk-Neutral Measure, Q-Measure
Pokrewne	3	4	4
Podsumowanie	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.