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Bibliography and References

Foundational Textbooks

General Quantitative Finance

  1. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson.
  2. Standard textbook covering fundamental derivatives pricing theory

  3. Black-Scholes-Merton framework, Greeks, basic numerical methods

  4. Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer.

  5. Rigorous mathematical foundation for continuous-time finance

  6. Risk-neutral pricing, martingale measures, change of numeraire

  7. Björk, T. (2009). Arbitrage Theory in Continuous Time (3rd ed.). Oxford University Press.

  8. Complete arbitrage theory framework
  9. Term structure models, martingale methods

Advanced Topics

  1. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.
  2. Volatility smile modeling, SABR model, local volatility

  3. Market microstructure of volatility

  4. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer.

  5. Comprehensive treatment of Monte Carlo methods

  6. Variance reduction, quasi-Monte Carlo, American options

  7. Cont, R., & Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC.

  8. Lévy processes, jump-diffusion models

  9. Variance gamma, Merton, Kou models

  10. Brigo, D., & Mercurio, F. (2006). Interest Rate Models - Theory and Practice (2nd ed.). Springer.

  11. Comprehensive interest rate modeling

  12. Short rate models (Vasicek, CIR, Hull-White), HJM framework

  13. Gregory, J. (2015). The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital (3rd ed.). Wiley.

  14. CVA, DVA, FVA, MVA, KVA calculations
  15. Exposure modeling, wrong-way risk

Seminal Research Papers

Black-Scholes Framework

  1. Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 81(3), 637-654.
  2. Original Black-Scholes formula derivation

  3. Foundation of modern option pricing theory

  4. Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science, 4(1), 141-183.

    • Extended Black-Scholes framework
    • Rigorous continuous-time derivation

Stochastic Volatility

  1. Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options." Review of Financial Studies, 6(2), 327-343.

    • Heston model with semi-analytical solution
    • Characteristic function approach
    • Implementation: src/neutryx/models/heston.py

  2. Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). "Managing Smile Risk." Wilmott Magazine, September, 84-108.

    • SABR (Stochastic Alpha Beta Rho) model
    • Asymptotic expansion for implied volatility
    • Implementation: src/neutryx/models/sabr.py

  3. Bayer, C., Friz, P., & Gatheral, J. (2016). "Pricing under rough volatility." Quantitative Finance, 16(6), 887-904.

    • Rough Bergomi model with fractional Brownian motion
    • Empirical evidence for rough volatility (H ≈ 0.1)
    • Implementation: src/neutryx/models/rough_vol.py

Local Volatility

  1. Dupire, B. (1994). "Pricing with a Smile." Risk, 7(1), 18-20.

    • Dupire's local volatility formula
    • Forward Kolmogorov equation approach
    • Implementation: src/neutryx/models/dupire.py

  2. Derman, E., & Kani, I. (1994). "Riding on a Smile." Risk, 7(2), 32-39.

    • Alternative derivation of local volatility
    • Implied binomial tree construction

Jump-Diffusion Models

  1. Merton, R. C. (1976). "Option Pricing when Underlying Stock Returns are Discontinuous." Journal of Financial Economics, 3(1-2), 125-144.

    • Merton jump-diffusion model
    • Poisson jump process with lognormal jump sizes
    • Implementation: src/neutryx/models/jump_diffusion.py

  2. Kou, S. G. (2002). "A Jump-Diffusion Model for Option Pricing." Management Science, 48(8), 1086-1101.

    • Double exponential jump-diffusion model
    • Asymmetric jumps with exponential distributions
    • Implementation: src/neutryx/models/kou.py

  3. Madan, D. B., Carr, P., & Chang, E. C. (1998). "The Variance Gamma Process and Option Pricing." European Finance Review, 2(1), 79-105.

    • Variance gamma model as pure jump Lévy process
    • Gamma time change representation
    • Implementation: src/neutryx/models/variance_gamma.py

Interest Rate Models

  1. Vasicek, O. (1977). "An Equilibrium Characterization of the Term Structure." Journal of Financial Economics, 5(2), 177-188.

    • Mean-reverting Gaussian short rate model
    • Analytical bond pricing formulas
    • Implementation: src/neutryx/models/vasicek.py

  2. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). "A Theory of the Term Structure of Interest Rates." Econometrica, 53(2), 385-407.

    • CIR model with square-root diffusion
    • Non-negative interest rates, Feller condition
    • Implementation: src/neutryx/models/cir.py

  3. Hull, J., & White, A. (1990). "Pricing Interest-Rate-Derivative Securities." Review of Financial Studies, 3(4), 573-592.

    • Extended Vasicek with time-dependent parameters
    • Calibration to initial term structure
    • Implementation: src/neutryx/models/hull_white.py

Numerical Methods

  1. Carr, P., & Madan, D. B. (1999). "Option Valuation using the Fast Fourier Transform." Journal of Computational Finance, 2(4), 61-73.

    • FFT-based option pricing using characteristic functions
    • Fast computation of option prices across strikes
    • Implementation: src/neutryx/engines/fourier.py

  2. Fang, F., & Oosterlee, C. W. (2008). "A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions." SIAM Journal on Scientific Computing, 31(2), 826-848.

    • COS method for Fourier pricing
    • Superior stability and accuracy
    • Implementation: src/neutryx/engines/fourier.py

  3. Longstaff, F. A., & Schwartz, E. S. (2001). "Valuing American Options by Simulation: A Simple Least-Squares Approach." Review of Financial Studies, 14(1), 113-147.

    • LSM algorithm for American option pricing
    • Regression-based early exercise boundary
    • Implementation: src/neutryx/engines/longstaff_schwartz.py

  4. Andersen, L. (2008). "Simple and Efficient Simulation of the Heston Stochastic Volatility Model." Journal of Computational Finance, 11(3), 1-42.

    • QE (Quadratic-Exponential) scheme for Heston
    • Avoids negative variance, maintains moments
    • Implementation: src/neutryx/models/heston.py:simulate()

  5. Giles, M. B. (2008). "Multilevel Monte Carlo Path Simulation." Operations Research, 56(3), 607-617.

    • MLMC for variance reduction
    • Optimal allocation across refinement levels
    • Implementation: src/neutryx/engines/qmc.py

  6. Broadie, M., & Glasserman, P. (1996). "Estimating Security Price Derivatives Using Simulation." Management Science, 42(2), 269-285.

    • Pathwise derivative method for Greeks
    • Likelihood ratio method
    • Implementation: src/neutryx/engines/pathwise.py

Variance Reduction

  1. Glasserman, P., Heidelberger, P., & Shahabuddin, P. (1999). "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options." Mathematical Finance, 9(2), 117-152.

    • Optimal importance sampling for rare events
    • Stratified sampling strategies
    • Implementation: src/neutryx/engines/variance_reduction.py

  2. Clewlow, L., & Carverhill, A. (1994). "On the Simulation of Contingent Claims." Journal of Derivatives, 2(2), 66-74.

    • Control variate techniques
    • Moment matching methods
    • Implementation: src/neutryx/engines/variance_reduction.py

Calibration

  1. Cont, R., & Tankov, P. (2009). "Constant Proportion Portfolio Insurance in the Presence of Jumps in Asset Prices." Mathematical Finance, 19(3), 379-401.

    • Calibration of jump-diffusion models to market data
    • Regularization techniques for ill-posed problems

  2. Guyon, J., & Henry-Labordère, P. (2014). Nonlinear Option Pricing. Chapman & Hall/CRC.

    • Advanced calibration methods
    • Particle method for SLV models
    • Regularization and stability

PDE Methods

  1. Wilmott, P., Howison, S., & Dewynne, J. (1995). The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press.

    • PDE formulation of option pricing
    • Finite difference schemes
    • Implementation: src/neutryx/models/pde.py

  2. Tavella, D., & Randall, C. (2000). Pricing Financial Instruments: The Finite Difference Method. Wiley.

    • Crank-Nicolson and theta schemes
    • American options with constraint
    • Implementation: src/neutryx/models/pde.py

Risk Management

  1. Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228.

    • Axiomatic foundation of risk measures
    • CVaR (Expected Shortfall) as coherent measure

  2. Rockafellar, R. T., & Uryasev, S. (2000). "Optimization of Conditional Value-at-Risk." Journal of Risk, 2(3), 21-41.

    • CVaR optimization techniques
    • Portfolio risk management
    • Implementation: src/neutryx/valuations/risk_metrics.py

  3. Kupiec, P. H. (1995). "Techniques for Verifying the Accuracy of Risk Measurement Models." Journal of Derivatives, 3(2), 73-84.

    • VaR backtesting methodology
    • Statistical tests for model validation
    • Implementation: src/neutryx/valuations/risk_metrics.py:backtest_var()

XVA and Counterparty Risk

  1. Brigo, D., Morini, M., & Pallavicini, A. (2013). Counterparty Credit Risk, Collateral and Funding: With Pricing Cases for All Asset Classes. Wiley.

    • CVA/DVA calculation methodology
    • Collateral modeling, CSA agreements
    • Implementation: src/neutryx/valuations/xva/

  2. Pykhtin, M., & Zhu, S. (2007). "A Guide to Modeling Counterparty Credit Risk." GARP Risk Review, July/August.

    • EPE, PFE calculation
    • Wrong-way risk modeling
    • Implementation: src/neutryx/valuations/exposure.py

  3. ISDA SIMM Methodology (2021). "ISDA SIMM Methodology Version 2.4."

    • Standard Initial Margin Model
    • Risk weight calibration
    • Implementation: src/neutryx/valuations/margin/simm/

  4. Albanese, C., & Andersen, L. (2014). "Accounting for OTC Derivatives: Funding Adjustments and the Re-Hypothecation Option." Risk Magazine, January.

    • FVA (Funding Valuation Adjustment) theory
    • Asymmetric funding costs
    • Implementation: src/neutryx/valuations/xva/fva.py

Review Articles and Surveys

  1. Gatheral, J., Jaisson, T., & Rosenbaum, M. (2018). "Volatility is Rough." Quantitative Finance, 18(6), 933-949.

    • Empirical evidence for rough volatility
    • Fractional stochastic volatility models

  2. Andersen, L., & Piterbarg, V. (2010). Interest Rate Modeling (3 volumes). Atlantic Financial Press.

    • Comprehensive reference for rates derivatives
    • LMM, cross-currency models, hybrid models

  3. Rebonato, R. (2004). Volatility and Correlation: The Perfect Hedger and the Fox (2nd ed.). Wiley.

    • Market models for interest rates
    • Correlation modeling, copulas

Computational Methods

  1. Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary, C., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J., Wanderman-Milne, S., & Zhang, Q. (2018). "JAX: Composable transformations of Python+NumPy programs." http://github.com/google/jax

    • Automatic differentiation framework
    • JIT compilation, GPU/TPU support
    • Core technology: All Neutryx implementations

  2. Sobol, I. M. (1967). "On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals." USSR Computational Mathematics and Mathematical Physics, 7(4), 86-112.

    • Sobol sequences for quasi-Monte Carlo
    • Implementation: src/neutryx/engines/qmc.py

Industry Standards

  1. FpML (Financial products Markup Language) Specification Version 5.12. http://www.fpml.org/

    • Industry standard for derivatives representation
    • Implementation: src/neutryx/integrations/fpml/

  2. Basel Committee on Banking Supervision (2019). "Minimum Capital Requirements for Market Risk."

    • Regulatory framework for market risk
    • Internal models approach, standardized approach

Additional References by Topic

Greeks and Sensitivities

  1. Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas (2nd ed.). McGraw-Hill.
    • Comprehensive collection of analytical formulas
    • Greeks for various option types
    • Implementation reference: src/neutryx/valuations/greeks/

Path-Dependent Options

  1. Curran, M. (1994). "Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price." Management Science, 40(12), 1705-1711.

    • Moment matching for Asian options
    • Analytical approximations

  2. Broadie, M., Glasserman, P., & Kou, S. G. (1997). "A Continuity Correction for Discrete Barrier Options." Mathematical Finance, 7(4), 325-349.

    • Discrete monitoring bias correction
    • Barrier option adjustments

Exotic Options

  1. Lipton, A. (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach. World Scientific.

    • FX options, quanto products
    • Green's function approach

  2. Zhang, P. G. (1998). Exotic Options: A Guide to Second Generation Options (2nd ed.). World Scientific.

    • Comprehensive catalog of exotic products
    • Pricing methodologies

Software and Tools References

  1. QuantLib - Open-source library for quantitative finance. https://www.quantlib.org/

    • Reference implementation for many models
    • Industry standard C++ library

  2. Weights & Biases - MLOps platform for experiment tracking. https://wandb.ai/

    • Integration: Calibration tracking in src/neutryx/calibration/

  3. MLflow - Open-source platform for ML lifecycle. https://mlflow.org/

    • Integration: Model versioning and tracking

Citation Style

Throughout this documentation, we reference papers using the format:

[Author(s), Year] - Brief description

Implementation: file/path.py:function_name()

This links theoretical foundations to practical implementations in the Neutryx codebase.

Notes on Theoretical Rigor

The Neutryx implementation prioritizes:

  1. Mathematical Correctness: All models follow rigorous derivations from the cited literature
  2. Numerical Stability: Methods are chosen for robustness (e.g., QE scheme for Heston)
  3. Performance: JAX enables GPU acceleration while maintaining clarity
  4. Validation: Extensive test coverage against analytical solutions and benchmarks

See individual model documentation for detailed mathematical specifications and derivations.

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