Pricing Models — Cross-Cutting Comparison
This note compares every derivatives-pricing model and numerical method referenced across the Finance library — Black-Scholes-Merton, local-vol (Dupire), stochastic-vol (Heston/SABR/SVI), jump-diffusion (Merton/Kou/Bates), Lévy processes (CGMY/VG/NIG), local-stochastic-vol (LSV), PDE solvers, Monte Carlo, binomial/trinomial trees, least-squares MC (Longstaff-Schwartz), Fourier methods (Carr-Madan), COS method (Fang-Oosterlee), quasi-Monte Carlo (Sobol/Halton), finite-difference, and deep-learning vol surfaces (Horvath-Muguruza-Tomas). Each row maps to where it sits on model complexity, calibration cost, smile fit, term-structure fit, Greek accuracy, and computational cost. Read the tables; decision tree at the end.
See also
- options-pricing-deep
- derivatives-and-quant-finance
- fixed-income-deep
- structured-products-deep
- structured-products-and-distressed-debt
- portfolio-construction-and-risk-deep
- market-microstructure-and-hft
- market-making-and-liquidity-provision-deep
- stochastic-calculus
- pde-methods
- mcmc-sampling
1. The model taxonomy
ANALYTIC / SEMI-ANALYTIC SIMULATION / NUMERICAL
Black-Scholes-Merton (1973) Monte Carlo (Boyle 1977)
Bachelier (1900, "normal model") Quasi-Monte Carlo (Sobol, Halton, Niederreiter)
Hull-White short-rate Longstaff-Schwartz LSM (2001)
Vasicek Stochastic mesh (Broadie-Glasserman 1997)
CIR Particle methods (Guyon-Henry-Labordère)
HJM (special cases) Tree (binomial Cox-Ross-Rubinstein 1979,
Black-Karasinski trinomial Boyle 1986)
Finite-difference (Crank-Nicolson, ADI)
Fourier (Carr-Madan 1999)
COS method (Fang-Oosterlee 2008)
Heath-Jarrow-Morton MC
LOCAL VOL / STOCHASTIC VOL / JUMP HYBRID / RATES + EQUITY
Dupire local vol (1994) LSV (local-stochastic vol)
Heston SV (1993) Heston + Hull-White
SABR (Hagan et al 2002, 2014 fix) LMM/BGM (LIBOR Market Model)
SVI (Gatheral 2004; eSSVI Gatheral-Jacquier 2014) Cheyette (Markov-functional HJM)
Merton jump-diffusion (1976) SABR-LMM
Kou double-exponential JD (2002)
Bates SV-jump (1996) MACHINE-LEARNING
CGMY Lévy (Carr-Geman-Madan-Yor 2002) Horvath-Muguruza-Tomas DNN (2021)
Variance Gamma (Madan-Seneta 1990) Neural SDE
Normal Inverse Gaussian (Barndorff-Nielsen 1997) Differentiable solvers
Mixed local-stochastic vol (LSV) Deep hedging (Buehler et al 2019)
Sig-payoffs (Lyons rough-path)
2. The models compared on complexity, calibration, smile fit
| Model | Year | Complexity (state dim) | Calibration cost | Smile fit | Term-structure fit | Greeks |
|---|---|---|---|---|---|---|
| Bachelier | 1900 | 1 (normal price) | trivial | none (flat smile = absent) | static | analytic |
| Black-Scholes-Merton | 1973 | 1 (S only) | trivial | none (flat) | static | analytic |
| Dupire local vol | 1994 | 1 (S, σ_loc(S, t)) | exact reproduction of smile (PDE inversion of Dupire’s equation) | exact (perfect by construction) | exact static, poor dynamics (sticky-strike implied) | numerical PDE |
| Heston SV | 1993 | 2 (S, v) | semi-closed via Carr-Madan/Lipton | good (curved smile) | static | semi-analytic |
| SABR | Hagan 2002 | 2 (F, σ) | Hagan asymptotic formula (smile arc) | excellent for short-dated FX/rates | smile-only, term separate | asymptotic |
| SVI / SSVI / eSSVI | Gatheral 2004+ | parametric (5 params/slice) | nonlinear least squares | excellent | piecewise (one slice per expiry) | numerical |
| Merton jump-diffusion | 1976 | 1 + Poisson | semi-closed (series sum) | good for short-dated (jump risk premium) | static | analytic series |
| Kou JD | 2002 | 1 + double-exp jumps | semi-closed | good | static | semi-analytic |
| Bates SV-jump | 1996 | 3 (S, v, J) | Carr-Madan | very good | static | semi-analytic |
| CGMY | 2002 | Lévy, 4 params | Carr-Madan | excellent (controls skew + kurtosis) | piecewise via term-structure | semi-analytic |
| Variance Gamma | 1990 | Lévy, 3 params | Carr-Madan | good | piecewise | semi-analytic |
| NIG | 1997 | Lévy, 4 params | Carr-Madan | good | piecewise | semi-analytic |
| LSV (Heston + local vol leverage) | 2010+ | 2 + leverage function | iterative — Heston SV + Dupire layer | excellent | good (mixed dynamics) | numerical |
| Hull-White | 1990 | 1 (r) | bond curve + cap/swaption surface | n/a | yes, by construction | semi-analytic |
| LMM / BGM | 1996/1997 | n (one rate per tenor) | swaption surface + cap surface | yes (rate smile) | yes | MC primarily |
| Cheyette (Markov-functional HJM) | 1992 | low-dim Markov state | swaption surface | yes | yes | semi-analytic |
| Horvath-Muguruza-Tomas DNN (2021) | 2021 | parametric (DNN weights) | DNN fitted once to model output | excellent (matches Heston/SABR/etc.) | depends on training set | autograd |
| Neural SDE | 2020+ | learned drift + diffusion | adjoint sensitivity | trained to data | trained | autograd |
3. Numerical methods — when to use which
| Method | Pros | Cons | When to use |
|---|---|---|---|
| Closed-form (BSM, Bachelier, Black ‘76) | instant, exact, analytic Greeks | restrictive assumptions | vanilla European on simple dynamics |
| Binomial tree (Cox-Ross-Rubinstein 1979) | intuitive, handles American, early-exercise | slow convergence (1/N), only one risk factor easily | teaching, simple American |
| Trinomial tree (Boyle 1986) | better convergence than binomial | still 1D | American w/ better Greeks |
| Implicit / Crank-Nicolson finite-difference | stable, accurate for low-dim PDE | curse of dimensionality (≤ 3D practical) | Heston/local-vol PDE, exotic w/ barriers |
| ADI / Strang-splitting | multi-dim FD | requires sparse linear algebra | LSV, Heston-HW hybrid |
| Monte Carlo | any dim, any payoff, path-dependent native | slow convergence (1/√N) | exotics, path-dependent, high-dim |
| Quasi-Monte Carlo (Sobol, Halton) | 1/N convergence rate (vs MC’s 1/√N) for smooth integrands | breaks for non-smooth (some barrier options) | high-dim smooth |
| Multi-level MC (Giles 2008) | variance reduction via telescoping | extra implementation | SDEs with strong-convergence |
| Longstaff-Schwartz LSM (2001) | American on MC (polynomial regression on continuation value) | basis function choice matters | Bermudan / American on complex dynamics |
| Stochastic mesh (Broadie-Glasserman 1997) | exact bounds for American | expensive | high-fidelity American |
| Fourier transform (Carr-Madan 1999) | exploit characteristic function (Lévy, affine) | requires affine model | Heston, Bates, Lévy, CGMY, VG |
| COS method (Fang-Oosterlee 2008) | faster than Carr-Madan, cleaner truncation | requires bounded characteristic | same as Fourier |
| Particle methods (Guyon-Henry-Labordère) | LSV with exact LV matching | implementation-heavy | LSV in production |
| Deep PDE solvers / PINNs | high-dim PDE | requires training | research |
| Differentiable MC (autograd) | Greeks for free | requires PyTorch/JAX rewrite | growing |
| GPU MC | 100× speedup for large product books | CUDA / Triton skill | production dealer books |
4. The Fourier / COS family — the workhorse for affine + Lévy
For models with a known characteristic function (BSM, Heston, Bates, Merton JD, CGMY, VG, NIG):
- Carr-Madan 1999 — option price via Fourier transform of damped call: with the characteristic function of . FFT-able; one model evaluation gives the entire strike range.
- COS method (Fang-Oosterlee 2008) — Fourier-cosine series on a truncated domain. Same idea, better convergence and easier error control. The de-facto standard 2010s+ for Lévy and Heston vanilla pricing.
- Lewis 2001 — slightly different transform; cleaner for some structured products.
For non-vanilla payoffs (barriers, Asians) one combines these with Monte Carlo or path-Fourier methods.
5. The exotic ladder — what works for what
| Payoff | Best model + numerical method |
|---|---|
| Vanilla European | BSM + closed-form OR Heston + Carr-Madan |
| Vanilla American (equity) | Heston + LSM or Crank-Nicolson PDE |
| Vanilla American (rates) | LMM + LSM or Markov-functional |
| Barrier (knock-in/out) | local vol + PDE; LSV + MC; closed-form BSM for simple |
| Asian (arithmetic) | MC with control variate; Curran approximation for closed-form quasi-analytic |
| Lookback (max/min) | PDE in 2D (S, M) or MC; Goldman-Sosin-Gatto formula for closed-form BSM |
| Cliquet / ratchet | MC with LSV; vega-sensitive |
| Forward-start | Heston + Carr-Madan; vol-of-vol matters |
| Volatility swap, variance swap | static replication via vanilla strip (Carr-Madan 2001) |
| VIX future, VIX option | Heston / Bates + MC; or fit VIX vol surface directly |
| Autocallable | LSV + MC; LSM for early-redemption; key for retail structured note (Asia, Europe) |
| Accumulator (“I-kill-you-later”) | LSV + MC; massive vega-of-vega risk |
| Reverse convertible | BSM or Heston + closed-form (short put-like) |
| Principal-protected note | rates model + closed-form on call piece |
| CMS spread | LMM w/ smile + MC; vega-of-vega + correlation; Brigo-Mercurio |
| Bermudan swaption | LMM + LSM or Markov-functional + PDE |
| Range accrual | Hull-White or LMM + MC |
| TARN (Target Redemption Note) | LMM + MC; early-termination tail risk |
| Catastrophe bond | Poisson + lognormal severity + MC |
| Mortgage prepayment | structural prepayment model + interest-rate MC; OAS engine |
| CDO / CLO | factor copula (Gaussian, Student-t, Marshall-Olkin) + MC; base-correlation surface |
| CDS index tranche | base-correlation surface + Gaussian copula + MC |
| Compound option (call on call) | Geske 1979 closed-form BSM; otherwise MC |
| Quanto | dual-currency + correlation; quanto adjustment via Girsanov |
| Multi-asset basket | Gaussian or Student-t copula + MC; PDE for low-dim |
| Best-of / worst-of | LSV + MC; correlation skew matters |
6. Calibration — the fitting problem
| Model | What to fit to | Fitting method |
|---|---|---|
| BSM | a single ATM vol point | trivial |
| Local vol | the entire implied-vol surface | Dupire formula gives σ_loc(K, T) by inversion |
| Heston | smile + term structure (5 params: v0, θ, κ, ξ, ρ) | nonlinear least squares on vanilla prices via Carr-Madan |
| SABR | per-slice smile (4 params: α, β, ρ, ν) | Hagan formula + nonlinear LS |
| SVI / eSSVI | per-slice smile (5 params SVI / 2-3 SSVI) | nonlinear LS; arbitrage-free constraints |
| Bates | smile + skew (Heston + jump) | nonlinear LS via Carr-Madan |
| CGMY / VG / NIG | smile + term | Carr-Madan inversion |
| Hull-White | bond curve + cap/swaption ATM | calibration to swaption surface |
| LMM/BGM | swaption surface + cap surface | global LS on calibrated swaption + cap |
| LSV | local-vol leverage to vanilla + SV dynamics to exotic | iterative (Heston seed → LV adjustment) |
| Horvath DNN | offline-trained on model output | one-time training; inference is fast |
The 2024 frontier: Joint calibration of vol + rates (Brigo-Mercurio book is the bible), machine-learning-accelerated calibration (Horvath et al 2021 Quant Finance; Liu-Oosterlee-Bohte 2019), differentiable calibration via PyTorch/JAX gradients.
7. Smile-fit quality on the SPX vol surface
Empirical observation (Gatheral Volatility Surface 2nd ed; Sepp & Yashin 2022 Quant Finance):
| Model | Front-month smile (1W–1M) | Mid (3M–6M) | Long-dated (1Y+) | Behavior in extreme strikes |
|---|---|---|---|---|
| BSM | flat (none) | flat (none) | flat (none) | n/a |
| Local vol | exact | exact | exact | sticky-strike (unrealistic for surface dynamics) |
| Heston | OK (smile too symmetric) | good | good | poor in wings |
| SABR | excellent | good | needs hump correction | OK |
| Bates | excellent | excellent | good | good |
| LSV (Heston + local) | excellent | excellent | excellent | good |
| CGMY / VG | excellent | good | good | excellent in wings |
Local vol is the right answer for today’s smile; LSV / SV is the right answer for how the smile evolves. Vanna-volga (Castagna-Mercurio 2007) is a pragmatic interpolation for FX desks.
8. Computational cost — order-of-magnitude
| Operation | BSM | Heston (Carr-Madan) | SABR (asymptotic) | LSV PDE | LSV MC | LSV deep-hedging |
|---|---|---|---|---|---|---|
| One vanilla price | µs | ms | µs | 10 ms | 100 ms | µs (after training) |
| One Greek (delta) | µs | ms | µs (perturb) | 10 ms | 100 ms (pathwise) | µs |
| One vega | µs | ms | µs | 10 ms (perturb) | 100 ms (likelihood ratio) | µs |
| One barrier option | ms (analytic) | 100 ms | n/a | 100 ms | 1 s | µs (trained) |
| One Bermudan (LSM) | n/a | n/a | n/a | 100 ms PDE | 10 s LSM | n/a |
| One TARN | n/a | n/a | n/a | n/a | 1 min | n/a (specialized) |
| One CDO tranche | n/a | n/a | n/a | n/a | 1 s | n/a |
| Daily revaluation of dealer book (10⁵ trades) | n/a | n/a | n/a | minutes | hours | minutes |
A modern dealer’s overnight grid (revalue tens of thousands of trades) typically uses GPU Monte Carlo for exotics, PDE for European/American on small dynamics, and Carr-Madan / COS for vanillas in batch.
9. Fixed income — separate world
Equity-options models (Heston, Dupire, etc.) do not apply directly. Rates products are priced via:
| Model | When | Notes |
|---|---|---|
| Vasicek | toy / teaching | mean-reverting; can hit negative rates |
| CIR | toy / teaching | non-negative; affine |
| Hull-White (extended Vasicek) | swap/cap/swaption with rate curve | analytic for bond, cap; semi for swaption |
| Black-Karasinski | cap, swaption | non-negative |
| HJM (in general) | research | curve modeled directly |
| LIBOR Market Model (BGM 1996/1997) | cap + swaption surface; modern industry standard | calibrate to swaption surface; MC pricing |
| SABR-LMM | smile-extended LMM | rates smile + LMM dynamics |
| Cheyette / Markov-functional HJM | low-dim Markov state | semi-analytic, dealer-friendly |
| Two-factor short-rate (G2++) | hybrid | flexible |
| Multi-curve framework (post-2008 OIS / IBOR split) | mandatory for valuing collateralized | discount on OIS, project IBOR forwards |
| SOFR transition models (post-2021 LIBOR cessation) | post-LIBOR | term SOFR (CME), backward-looking compound SOFR |
See fixed-income-deep for the full rates stack.
10. Credit pricing
| Model | When |
|---|---|
| Merton structural (1974) | corporate credit risk; equity = call on firm value |
| Black-Cox (1976) | barrier extension |
| Reduced-form intensity (Jarrow-Turnbull 1995, Lando 1998, Duffie-Singleton 1999) | CDS pricing; default = Poisson w/ intensity |
| CreditMetrics | portfolio loss; transition matrix-based |
| CreditRisk+ (CSFB) | actuarial Poisson |
| Gaussian copula (Li 2000) | CDO pricing — infamous for 2008 |
| Marshall-Olkin / random factor loading | post-2008 alternative copulas |
| Base-correlation surface | dealer-grade CDO tranche |
| Random-recovery model (Andersen-Sidenius-Basu 2003) | CDO w/ recovery uncertainty |
| Hawkes processes | clustered defaults |
11. Machine-learning pricing models — the 2020–2026 wave
| Method | When |
|---|---|
| Horvath-Muguruza-Tomas 2021 “Deep learning vol surfaces” | offline-train DNN to take (model params) → (vol surface); 1000× faster inference for calibration |
| Liu-Oosterlee-Bohte 2019 | DNN as a fast surrogate for Heston Carr-Madan |
| Buehler-Gonon-Teichmann-Wood 2019 “Deep hedging” | learn hedging strategy directly; no PDE; handles transaction cost natively |
| Becker-Cheridito-Jentzen-Welti 2019 | DNN for American option pricing |
| Sirignano-Spiliopoulos 2018 “Deep Galerkin” | DNN solves high-dim PDE |
| Han-Jentzen-E 2018 “Deep BSDE” | backward stochastic differential equation solver |
| Lyons rough-path / signature payoffs (Lyons, Salvi-Pannier-Lyons 2021) | path-dependent payoffs as functions of signatures |
| Neural SDE (Kidger-Foster-Li-Lyons 2021) | learn drift + diffusion from price data |
| Differentiable Monte Carlo | autograd through MC for fast Greeks |
| Differentiable PDE solvers (JAX-FD, jax-pde) | gradient w.r.t. boundary conditions, params |
12. The post-2008 valuation-adjustment (xVA) stack
For OTC derivatives, the all-in price includes:
| Adjustment | What it covers |
|---|---|
| CVA (Credit Valuation Adjustment) | counterparty default risk |
| DVA (Debit VA) | your own default risk (controversial; subtract from CVA) |
| FVA (Funding VA) | funding cost of uncollateralized position |
| ColVA (Collateral VA) | imperfect collateralization |
| KVA (Capital VA) | regulatory capital required for the trade |
| MVA (Margin VA) | initial margin funding cost |
| AVA (Additional VA) | prudent valuation adjustment under EBA |
Each is a high-dim expectation over rates + credit + equity scenarios, typically computed by GPU MC on a global netting set. The dealer-grade xVA engine (Murex, Numerix, Quantifi, Calypso) costs millions in licensing.
13. Decision tree — pick by product + accuracy + budget
What's the product?
├─ Vanilla European, liquid
│ → BSM closed-form (1 µs) or Heston + Carr-Madan if smile matters
├─ Vanilla American, equity
│ ├─ Quick → Cox-Ross-Rubinstein binomial tree
│ ├─ Better → Crank-Nicolson PDE
│ └─ Complex dynamics → Longstaff-Schwartz LSM
├─ Barrier (knock-out / knock-in)
│ ├─ Simple → BSM closed-form (Rich)
│ ├─ Smile → local vol + PDE
│ └─ Smile + dynamics → LSV + MC
├─ Asian (path-dependent average)
│ ├─ Geometric → closed-form
│ └─ Arithmetic → MC + control variate (Kemna-Vorst geometric)
├─ Bermudan / autocallable / TARN
│ └─ LSV + MC + Longstaff-Schwartz LSM
├─ VIX option / variance swap
│ ├─ Variance swap → static replication via vanilla strip (Carr-Madan 2001)
│ └─ VIX option → Heston/Bates + MC
├─ FX option (short-dated)
│ └─ SABR + Hagan asymptotic
├─ FX exotic (long-dated, smile-sensitive)
│ └─ LSV + MC or Vanna-Volga
├─ Rates cap, floor, swaption
│ ├─ ATM only → Black '76 + Hull-White
│ └─ Smile → SABR-LMM + MC
├─ CMS spread, range accrual, callable bond
│ └─ LMM + MC + LSM for early exercise
├─ Mortgage-backed security
│ └─ structural prepayment + Hull-White rate model + MC; OAS engine
├─ CDO / CLO tranche
│ └─ base-correlation surface + Gaussian copula + MC
├─ Catastrophe bond
│ └─ Poisson frequency + lognormal severity + MC
├─ ADC / weather derivative
│ └─ custom Poisson / mean-reverting OU + MC
├─ Crypto option (high vol, jumps)
│ └─ Bates SV-jump or Lévy + Carr-Madan
└─ Production xVA engine (dealer book)
└─ GPU MC on Heston/Hull-White hybrid; xVA accounting layer
14. Anti-patterns
- BSM for short-dated equity options near earnings — vol smile + jumps mandate at least Heston, often Bates.
- Local vol for forward-vol-sensitive products — sticky-strike dynamics give wrong forward vol; use SV or LSV.
- Constant-vol Heston for long-dated — calibrate term structure of v0, θ; or use term-structured Heston.
- MC without variance reduction for vanilla — use control variate or QMC.
- PDE for high-dim — > 3 dim is impractical; use MC.
- Pricing a Bermudan with European MC — use LSM or PDE.
- Ignoring xVA for OTC trades — material in modern post-2010 P&L.
- Calibrating to OTM with poor data — wings are noisy; use SSVI / eSSVI for arbitrage-free interpolation.
- Calibrating to today’s smile only — without time-series fit, dynamics will be wrong.
- Using a Gaussian copula for tail-risk credit — 2008 lesson; use t-copula or Marshall-Olkin.
15. The 2024–2026 frontier
- GPU + JAX/PyTorch differentiable pricing — Cantor, Goldman, JP Morgan, BlackRock all moving toward differentiable MC for daily P&L attribution.
- Rough-vol models (Gatheral-Jaisson-Rosenbaum 2018 “Volatility is rough”) — vol is best modeled as fractional Brownian motion with H ≈ 0.1; rough Bergomi, rough Heston.
- rough Heston / rough Bergomi — improved smile fit short-dated, especially equity.
- Sig-payoffs — Lyons rough path signatures as universal approximators of path-dependent payoffs.
- Deep hedging (Buehler et al 2019) at Bank of America, JP Morgan — learn hedging strategy under realistic constraints (frictions, capital).
- PINN / neural-PDE for high-dim — Sirignano-Spiliopoulos, Han-E.
- xVA at the cell level (Murex, Numerix, Adaptiv) — netting-set level XVA, post-trade analytics in seconds.
- SOFR transition models — post-LIBOR rates with overnight compounding base curve.
- Crypto derivatives modeling — Bates / Lévy plus weekend-effect dummies; high-frequency funding-rate models.
- AI-aided model validation — replicate independent pricing by transformer-based DNN; back-test for arbitrage.
Adjacent
- Options theory — options-pricing-deep for full BSM derivation, Greeks, exotic taxonomy.
- Derivatives general — derivatives-and-quant-finance for forwards, futures, swaps, basic exotic stack.
- Fixed income — fixed-income-deep for HW, LMM, swaption smile, MBS prepayment.
- Structured products — structured-products-deep, structured-products-and-distressed-debt for CDO/CLO/ABS pricing.
- Portfolio risk — _compare_risk-measures for VaR/CVaR (relies on pricing).
- Market microstructure — market-microstructure-and-hft for the bid-ask + price-discovery side underlying mid-price quoting.
- Stochastic calculus — stochastic-calculus for Itô, Girsanov, BSDE, martingales.
- PDE — pde-methods for the FD / Crank-Nicolson / ADI methods.
- Sampling — mcmc-sampling for MC primitives and quasi-MC.
- Optimization — _compare_optimization-methods for the calibration step (nonlinear LS).
When to pick what
The fastest narrowing: liquid vanilla → BSM; smile matters → Heston + Carr-Madan; smile + path dependence → LSV + MC; Bermudan → LSM; barrier → local vol + PDE; rates → Hull-White or LMM; CDO → base-correlation + Gaussian copula (with t-copula for tails); xVA → GPU MC. The single biggest practical lesson of the last 25 years (post-1998 LTCM, post-2008 GFC, post-2020 COVID) is calibrate to today’s surface but stress against alternative dynamics — local vol is correct for today, wrong for tomorrow, and a model that fits perfectly today is not a model of forward dynamics. Pick the simplest model that captures the relevant risk; price under the chosen model; stress-test under alternatives.