Derivative Pricing and Structuring

Derivative Pricing represents the classical sell-side quantitative domain, founded upon stochastic calculus, partial differential equations, and martingale measure theory. Its primary mandate is valuing complex non-linear financial contracts (options, swaps, structured exotics) and engineering self-financing dynamic hedges that insulate trading desks from market risk.

This pillar is organised into fifteen topic folders, each a self-contained hub with six sub-pages. Follow them in the order below - each assumes the vocabulary of the ones before it.

Core Pricing & Structuring Topics

1. Options, Futures & Markets: The pillar’s entry point - option contracts, payoff diagrams, put-call parity, moneyness, and how vanilla derivatives trade. No prerequisites; start here if you have never priced a derivative.
2. No-Arbitrage & the Binomial Model: Law of one price, discrete dynamic replication, risk-neutral measure, and Cox–Ross–Rubinstein (CRR) trees.
3. Black–Scholes–Merton: Delta-neutral hedging, the PDE, Feynman–Kac conditional expectations, and the closed-form European pricing formula.
4. Volatility Surfaces & Smiles: Implied-volatility root finding, skew/smile dynamics, sticky-strike vs sticky-delta rules, and Dupire local volatility.
5. Advanced Volatility - Heston, SABR & Stochastic-Vol Dynamics: Stochastic-variance processes, the Feller condition, characteristic functions, and smile fitting for exotics.
6. American Options & Optimal Stopping: Early-exercise frontiers, free-boundary problems, dynamic programming, and tree / PDE / Monte Carlo valuation of American claims.
7. Exotic & Path-Dependent Options: Barriers, Asians, lookbacks, and structured exotics - valuation under the full toolkit of analytic, PDE, and simulation methods.
8. Numerical Methods: Finite-difference schemes, Monte Carlo simulation, and the FFT - the computational engines that price models without closed forms.
9. Interest Rate & Term Structure: Yield-curve bootstrapping, short-rate dynamics (Vasicek, CIR, Hull–White), and HJM / market models.
10. Counterparty Risk & xVA: CVA/DVA/FVA, collateral, credit exposure, and how counterparty credit risk reprices a derivative.
11. Calibration & Market Practice: Objective-function design, model selection, parameter fitting to market quotes, and model-risk governance on the desk.
12. Rough Volatility & Fractional Models: Fractional Brownian motion with $H\approx 0.1$, the rough Bergomi model, Hurst estimation, and why an $T^{H-1/2}$ skew is where Markovian SV models fail structurally.
13. Local-Stochastic Volatility (LSV): The leverage function, Gyöngy’s Markovian projection, Dupire consistency, and the Guyon–Henry-Labordère particle method - how to keep stochastic-vol dynamics and fit the smile exactly.
14. Deep Hedging & BSDEs: Convex-risk minimisation in incomplete markets, the quadratic-BSDE / nonlinear Feynman–Kac backbone, and Deep BSDE / Deep Galerkin solvers.
15. Path Signatures & Rough Paths: Iterated integrals, Chen’s identity and the shuffle product, the log-signature and free Lie algebra, and rough path theory - the language rough volatility is written in.

Reading Path (Zero to Expert)

Before this pillar (foundations): read Stochastic Calculus · Probability & Measure Theory first - see the Math Foundations hub for the full consumption order.

A guided route through the fifteen folders, in three stages (the last of which is the frontier).

• Start (foundations): Options, Futures & Markets → No-Arbitrage & the Binomial Model → Black–Scholes–Merton. This builds the vocabulary and the no-arbitrage/replication core that everything else extends.
• Intermediate (volatility & exotics): Volatility Surfaces & Smiles → SABR → American Options & Optimal Stopping → Exotic & Path-Dependent Options → Numerical Methods. Here you move from single-model pricing to smiles, early exercise, path dependence, and the numerics that make them computable.
• Expert (institutional pricing): Interest Rate & Term Structure → Counterparty Risk & xVA → Calibration & Market Practice. These cover rates desks, credit-adjusted pricing, and the live calibration discipline of a production trading floor.
• Frontier (research-grade): Rough Volatility & Fractional Models → Path Signatures & Rough Paths → Local-Stochastic Volatility → Deep Hedging & BSDEs. Where the current literature is: rough vol for the skew term structure, signatures for path-dependent learning, LSV for exact smile calibration, and deep BSDEs for hedging when the market is incomplete.

The Derivative Engineering Pipeline