Linganisha mbinu

	Uwekaji Bei wa Crank-Nicolson	Modeli ya Hull-White	Volatilite ya Ndani (Dupire)	Mfumo wa SABR
Nyanja	Fedha za Kiidadi	Fedha za Kiidadi	Fedha za Kiidadi	Fedha za Kiidadi
Familia	Machine learning	Regression model	Regression model	Regression model
Mwaka wa asili	1947	1990	1994	2002
Mwanzilishi	John Crank and Phyllis Nicolson	John C. Hull and Alan White	Bruno Dupire	Patrick S. Hagan
Aina	PDE Solver	Interest Rate Model	Equity/FX Model	Interest Rate Model
Chanzo asilia	Crank, J., & Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Mathematical Proceedings of the Cambridge Philosophical Society, 43(1), 50-67.	Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573-592.	Dupire, B. (1994). Pricing with a smile. Risk Magazine, 7(1), 18-20.	Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). Managing smile risk. Wilmott Magazine, 1, 84-108.
Majina mbadala	CN Method, Implicit Finite Difference	Extended Vasicek, Generalized Vasicek	Deterministic Volatility Function, DVF	Stochastic Volatility Model
Zinazohusiana	3	4	4	4
Muhtasari	The Crank-Nicolson method is a widely-used implicit finite difference scheme for solving PDEs in option pricing. It provides second-order accuracy in both space and time, unconditional stability, and can efficiently price derivatives with early exercise features (American options) or complex boundary conditions.	The Hull-White model (1990) is a one-factor short-rate model with time-dependent mean reversion and volatility, designed to fit the initial yield curve exactly. It generalizes the Vasicek model to allow better calibration to observed bond and derivative prices, and is widely used for pricing interest rate exotics and managing interest rate risk.	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.	The SABR (Stochastic Alpha-Beta-Rho) model is a stochastic volatility framework introduced by Hagan et al. in 2002 for valuing interest rate derivatives. It captures the smile effect in implied volatility through correlated Brownian motions and has become industry standard for swaption and caplet pricing.