Reference Bibliography
References
The optimization, the implicit differentiation, the models and the Monte Carlo sensitivities the package builds on.
Optimization and implicit differentiation
- K. Levenberg, A Method for the Solution of Certain Non-Linear Problems in Least Squares, Quarterly of Applied Mathematics 2(2), 164–168, 1944.
- D. W. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, Journal of the Society for Industrial and Applied Mathematics 11(2), 431–441, 1963. The damping scaled by the diagonal of $J^\top J$.
- M. Blondel, Q. Berthet, M. Cuturi, R. Frostig, S. Hoyer, F. Llinares-López, F. Pedregosa and J.-P. Vert, Efficient and Modular Implicit Differentiation, Advances in Neural Information Processing Systems 35, 2022. Derivatives of the solution of an optimization problem from its optimality conditions.
- G. Dalle and M. Mohamed, ImplicitDifferentiation.jl, which builds the rules of
calibrated_parameters.
Models and pricing
- S. L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Review of Financial Studies 6(2), 327–343, 1993.
- H. Albrecher, P. Mayer, W. Schoutens and J. Tistaert, The Little Heston Trap, Wilmott Magazine, January 2007, 83–92.
- D. Brigo and F. Mercurio, Lognormal-Mixture Dynamics and Calibration to Market Volatility Smiles, International Journal of Theoretical and Applied Finance 5(4), 427–446, 2002.
- D. Brigo, F. Mercurio and F. Rapisarda, Smile at the Uncertainty, Risk 17(5), 97–101, 2004. Mixtures of shifted lognormals, with uncertain volatilities.
- F. Fang and C. W. Oosterlee, A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions, SIAM Journal on Scientific Computing 31(2), 826–848, 2008.
- A. L. Lewis, A Simple Option Formula for General Jump-Diffusion and Other Exponential Lévy Processes, 2001, SSRN 282110.
- L. B. G. Andersen, Simple and Efficient Simulation of the Heston Stochastic Volatility Model, Journal of Computational Finance 11(3), 1–42, 2008. The difficulty of simulating the variance when the Feller condition fails.
Monte Carlo sensitivities
- F. A. Longstaff and E. S. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach, Review of Financial Studies 14(1), 113–147, 2001.
- P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003, chapter 7. Pathwise derivatives, and why they miss the discontinuities of a payoff.