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Reference Bibliography

References

Fourier pricing

  • P. Carr and D. B. Madan, Option Valuation Using the Fast Fourier Transform, Journal of Computational Finance 2(4), 61–73, 1999. The damped call transform and its FFT (CarrMadanMethod).
  • A. L. Lewis, A Simple Option Formula for General Jump-Diffusion and Other Exponential Lévy Processes, 2001, SSRN 282110. The integral along the line $\operatorname{Im} u = 1/2$ (LewisMethod, CarrMadanLewisMethod).
  • 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. The COS method (CosMethod, CosDensityMethod), and the rule of the cumulants for its range that the package replaces with a Chernoff bound (Truncation range).
  • C. W. Clenshaw, A Note on the Summation of Chebyshev Series, Mathematical Tables and Other Aids to Computation 9(51), 118–120, 1955. The recurrence that sums the cosine series of a smile, and of a density, at one sincos per point.

Models

  • R. C. Merton, Option Pricing when Underlying Stock Returns Are Discontinuous, Journal of Financial Economics 3(1–2), 125–144, 1976.
  • S. G. Kou, A Jump-Diffusion Model for Option Pricing, Management Science 48(8), 1086–1101, 2002.
  • D. B. Madan, P. Carr and E. C. Chang, The Variance Gamma Process and Option Pricing, European Finance Review 2(1), 79–105, 1998.
  • O. E. Barndorff-Nielsen, Processes of Normal Inverse Gaussian Type, Finance and Stochastics 2(1), 41–68, 1998.
  • 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. The form of the Heston characteristic function that stays continuous in the maturity, used by the package.
  • L. B. G. Andersen and V. V. Piterbarg, Moment Explosions in Stochastic Volatility Models, Finance and Stochastics 11(1), 29–50, 2007. The time at which the moments of Heston explode, which bounds its truncation range.

Tail bounds

  • H. Chernoff, A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations, Annals of Mathematical Statistics 23(4), 493–507, 1952. The exponential bound on the tails (Chernoff bounds).
  • H. Cramér, Sur un nouveau théorème-limite de la théorie des probabilités, Actualités Scientifiques et Industrielles 736, 5–23, 1938. The rate of large deviations, the Legendre transform of the cumulant generating function.
  • W. Squire and G. Trapp, Using Complex Variables to Estimate Derivatives of Real Functions, SIAM Review 40(1), 110–112, 1998. The complex step that gives the cumulant generating function and its derivative in one evaluation (Finding the minimum).
  • R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, 1973, Chapter 4. The root finder of the truncation range.

Calibration

  • 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 $J^\top J$ (Levenberg–Marquardt).
  • J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006, Chapter 10: nonlinear least squares, Gauss–Newton and Levenberg–Marquardt.
  • P. Jäckel, Let's Be Rational, Wilmott 2015(75), 40–53, 2015. The implied volatility of blsimpv in FinancialToolbox.jl, used for the model and quoted volatilities.

Implicit function theorem

  • W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Theorem 9.28.
  • S. G. Krantz and H. R. Parks, The Implicit Function Theorem: History, Theory, and Applications, Birkhäuser, 2002.

It gives the derivatives of the model volatilities (Derivatives of the model volatilities) and the sensitivities of the calibrated parameters to the quotes (Sensitivity of the calibrated parameters to the quotes) without differentiating a solver.

Sensitivities from densities

  • M. Broadie and P. Glasserman, Estimating Security Price Derivatives Using Simulation, Management Science 42(2), 269–285, 1996. The likelihood ratio method, whose weights are the scores of a density.
  • M. B. Giles, Vibrato Monte Carlo Sensitivities, in Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer, 2009, 369–382.