Linganisha mbinu

	Modeli ya Bates	Volatilite ya Ndani (Dupire)	Uthamini Usio na Hatari (Risk-Neutral Valuation)
Nyanja	Fedha za Kiidadi	Fedha za Kiidadi	Fedha za Kiidadi
Familia	Regression model	Regression model	Regression model
Mwaka wa asili	1996	1994	1979
Mwanzilishi	David S. Bates	Bruno Dupire	John Harrison and David Kreps
Aina	Equity/FX Model	Equity/FX Model	Fundamental Principle
Chanzo asilia	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.	Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408.
Majina mbadala	SVJ Model, Jump Diffusion	Deterministic Volatility Function, DVF	Risk-Neutral Measure, Q-Measure
Zinazohusiana	4	4	4
Muhtasari	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.	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.