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Option Pricing PROJ Method (Exotic/Vanilla Options)

Option pricing (exotic/vanilla derivatives) based on an efficient and general Fourier transform pricing framework - the PROJ method (short for Frame Projection). The modules are organized by Pricing Method, then by Model, and then by Contract Type. Each contract has a run script, which starts with "Script_", e.g. "Script_BarrierOptions.m". Monte Carlo and other pricing libraries are also provided to support R&D.

Pricing methods supported:

• PROJ (General Purpose Fourier Method)
• CTMC Approximation
• Monte Carlo
• Analytical
• Fourier (PROJ, Carr-Madan, CONV, Lewis, COS, Mellin Transform, Hilbert Transform)
• PDE/Finite Difference
• Lattice/Tree

Models supported:

• Diffusions (Black-Scholes-Merton)
• Multi-Dimensional Diffusions (Black-Scholes Multi-Asset)
• Jump Diffusions (Merton Jump, Kou double exponential, Mixed-Normal)
• General Levy processes (CGMY/KoBoL, Normal-Inverse-Gaussian (NIG), Variance Gamma, Meixner, FMLS, TS, Bilateral Gamma)
• SABR
• Stochastic Local Volatility (SLV)
• Regime switching jump diffusions
• Time-changed processes
• Stochastic Volatility (Heston/Bates, Hull-White, 4/2, 3/2, alpha-hypergeometric, Jacobi, Schobel-Zhu, Stein-Stein, Scott, tau/2)

Contract types supported (single underlying):

• European Options
• Barrier Options (Single/Double barrier, and rebates)
• Asian Options (Discrete/Continuous)
• Discrete Variance Swaps, Variance/Volatility Options
• Bermudan/American early-exercise Options
• Parisian Options (Cumulative and resetting Parisian barrier options)
• Cliquets/Equity Indexed Annuities (Additive/Multiplicative)
• Equity Linked Death Benefits / Guaranteed Minimum Death Benefits (GMDB)
• Forward Starting Options
• Step (Soft Barrier) Options
• Lookback/Hindsight Options
• Credit default swaps / default probabilities
• Swing Options (Fixed Rights, Linear Recovery & Constant Recovery)
• Fader/Range-Accrual Options
• Multi-Dimensional Payoffs, European/Bermudan/Barrier (Spread, Exchange, Best/Worst-of, Basket)
• Risk Measures suchs as Expected Shortfall and VaR computations

Contract types supported (multi underlying):

• European / Barrier / Bermudan Options
• Spread, Exchange, Best-of, Worst-of, Basket (Geometric/Arthmetic)

Acknowledgement: These pricing libraries have been built in collaboration with:

• Justin Lars Kirkby
• Duy Nguyen
• Zhenyu Cui
• Zhimin Zhang
• Shijie Deng
• Jean-Philippe Aguilar

Supporting Research Articles:

I) Levy Models, Jump Diffusions, Black Scholes

• Efficient Option Pricing by Frame Duality with the Fast Fourier Transform. SIAM J. Financial Math (2015)
• An Efficient Transform Method for Asian Option Pricing. SIAM J. Financial Math (2016)
• Static Hedging and Pricing of Exotic Options With Payoff Frames. Mathematical Finance (2018)
• American and Exotic Option Pricing with Jump Diffusions and Other Levy Processes. J. Computational Finance (2018)
• Robust Barrier Option Pricing by Frame Projection Under Exponential Levy Dynamics. Applied Mathematical Finance (2018)
• Robust option pricing with characteristic functions and the B-spline order of density projection, J. Compuational Finance (2017)
• Valuing Equity-Linked Death Benefits in General Exponential Levy Models. J. Comput. and Appl. Math. (2019).
• Swing Option Pricing By Dynamic Programming with B-Spline Density Projection, IJTAF, Forthcoming (2020)
• Frame and Fourier Methods for Exotic Option Pricing and Hedging. Georgia Institute of Technology (2016).

II) Stochastic Volatility, Markov Chains, and Regime Switching

• A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. European J. Operational Research (2017)
• A unified approach to Bermudan and Barrier options under stochastic volatility models with jumps. J. Econ. Dynamics and Control (2017)
• Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps. Insurance: Mathematics and Economics (2017)
• Continuous-Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing. IMA Volumes on Mathematics (2019)
• Full-Fledged SABR Through Markov Chains, Wilmott (2019)

III) Stochastic Local Volatility (SABR, Quadratic SLV, etc)

• A General Valuation Framework for SABR and Stochastic Local Volatility Models. SIAM J. Financial Mathematics (2018)

IV) Time-Changed Processes

• A General Framework for Time-Changed Markov Processes and Applications. European J. Operational Research (2018)

V) Multi-Dimensional Diffusions

• A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions (2020)