0) The Core Idea of Option Pricing
An option pricing model estimates the fair theoretical value of an option. Key drivers: underlying price $S$, strike $K$, time to expiry $T$, volatility $\sigma$, risk‑free rate $r$, and dividends/carry $q$.
Inputs
- Underlying Price ($S$)
- Strike ($K$)
- Time ($T$)
- Volatility ($\sigma$)
- Risk‑free ($r$)
- Dividends/Carry ($q$)
Model Families
- Analytical (closed‑form)
- Numerical (trees, PDE, Monte Carlo)
1) Analytical Models (Closed‑Form)
Black‑Scholes‑Merton (BSM)
Core idea: replicate and hedge to a risk‑free portfolio; no arbitrage implies the BSM PDE and its solution.
European Call: $C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2)$
European Put: $P = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1)$
with $d_1 = \dfrac{\ln(S/K) + (r - q + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}$, $\;d_2 = d_1 - \sigma\sqrt{T}$.
- Assumptions: constant $\sigma$ and $r$, lognormal returns, frictionless continuous trading, etc.
- Use: benchmark pricing and Greeks for European options; quote via implied vol.
Black (1976)
Use forward/futures $F$ as underlying for options on futures and rate products.
$C = e^{-rT}\big(F N(d_1) - K N(d_2)\big)$ (analogous put), using forward‑based $d_1,d_2$.
Bachelier (Normal) Model
Linear/normal moves; useful when underlyings or forwards can be near/under zero (rates). Supports normal vols.
2) Numerical Models (Computational)
Binomial Options Pricing (CRR)
Build an up/down lattice, value at expiry, and backward‑induct with risk‑neutral probabilities. Natural support for American exercise and discrete dividends.
Finite Difference Methods (PDE)
Discretize the PDE on a space‑time grid (Explicit/Implicit/Crank–Nicolson). Accurate for barriers/locals, but requires care with stability and boundaries.
Monte Carlo Simulation
Simulate many risk‑neutral paths (GBM or richer dynamics), average discounted payoffs. Excellent for path‑dependent payoffs; use LSM for early exercise.
Quick Summary Table
| Model | Type | Key Feature | Best For | Limitation |
|---|---|---|---|---|
| BSM | Analytical | Closed‑form | European, Greeks | Flat vol, lognormal tails |
| Binomial | Numerical | Lattice/backward | American, dividends | Slower; tuning needed |
| Monte Carlo | Numerical | Path sampling | Asian/lookback/exotics | Slow; early exercise hard |
| Finite Difference | Numerical | PDE grid | Barriers, locals | Implementation complexity |
Beyond the Basics
- Volatility smile/skew: implied vol varies with strike and maturity $\Rightarrow$ need a surface.
- Stochastic volatility: e.g., Heston captures skew term‑structure and more realistic smile dynamics.
- Local volatility: $\sigma(S,t)$ fits the whole surface exactly (static) via Dupire; watch dynamics.
- Jump‑diffusion: Merton/Bates add jumps to handle gaps/fat tails.
Conclusion (Foundations)
Pick the model by payoff and purpose: BSM for European quoting/Greeks; Lattices/PDE for Americans; Monte Carlo for path‑dependence; advanced models for smile‑consistent pricing and dynamics.
Preliminaries That Matter in Practice
- Carry & Parity: cost‑of‑carry $b=r-q$. Put–call parity with carry: $C-P = e^{-qT}S - e^{-rT}K$.
- Dividends: prefer discrete ex‑dates for single stocks; continuous $q$ is a proxy for indices.
- Volatility means implied: we quote/use risk‑neutral implied vol, not realized.
- Surface (not a number): work with $\sigma(K,T)$ and its clean param (e.g., SVI) + evolution rules.
Semi‑Analytical / Transform Models
Heston (Stochastic Vol)
Vol is mean‑reverting; semi‑closed pricing via characteristic functions/FFT or COS. Parameters $(\kappa,\theta,\xi,\rho,v_0)$.
Bates / Merton (Jumps)
Add jumps to address gaps/fat tails; useful for earnings and jump‑prone commodities.
Exponential‑Lévy (VG, NIG, CGMY)
Richer tails/asymmetry; efficient via FFT/COS.
SABR (Rates/FX)
Stochastic vol on forwards (normal/lognormal). Great for smile interpolation/extrapolation in rates/FX.
Local Vol + SVI
SVI fits a clean, no‑arb surface; Dupire local vol reproduces it exactly (static). Dynamics tend to sticky‑strike.
Bachelier (Normal)
Handles negative underlyings/forwards and absolute‑move regimes (near‑zero rates).
Numerical Details (Advanced)
- Lattices: CRR/JR/Tian/Leisen–Reimer; use trinomial + Richardson for faster, smoother convergence.
- PDE: Crank–Nicolson with careful far‑field BCs; American exercise via free‑boundary methods.
- MC & Quasi‑MC: antithetic/control variates; Sobol + Brownian bridge; LSM for early exercise.
Americans & Early Exercise
- American calls on non‑dividend equities: early exercise is never optimal.
- American puts/dividend calls: early‑exercise regions exist → trees/PDE or fast approximations (BA‑Whaley, Bjerksund–Stensland).
Greeks — What They Are and Why They Matter
Greeks are sensitivities of price to risk factors under the risk‑neutral measure. Manage level (Δ), curvature (Γ), time (Θ), vol level (Vega), rates/carry (ρ, dividend‑rho), and smile dynamics (Vanna/Volga).
1) First‑Order Greeks (BSM with yield $q$)
Delta (Δ)
Call: $\Delta_c = e^{-qT}N(d_1)$; Put: $\Delta_p = -e^{-qT}N(-d_1)$.
Hedge with underlying; mind spot vs forward delta conventions.
Vega (ν)
$\nu = e^{-qT}S\,\phi(d_1)\sqrt{T}$ (same for calls/puts). Bucket by expiry.
Theta (Θ)
Call: $\Theta_c = -\frac{e^{-qT}S\phi(d_1)\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2) + qSe^{-qT}N(d_1)$.
Put: $\Theta_p = -\frac{e^{-qT}S\phi(d_1)\sigma}{2\sqrt{T}} + rKe^{-rT}N(-d_2) - qSe^{-qT}N(-d_1)$.
Rho (ρ)
Call: $\rho_c = KTe^{-rT}N(d_2)$; Put: $\rho_p = -KTe^{-rT}N(-d_2)$.
Dividend‑Rho (carry, $\psi$)
Call: $\psi_c = -TSe^{-qT}N(d_1)$; Put: $\psi_p = TSe^{-qT}N(-d_1)$.
2) Second‑Order & Smile Greeks
Gamma (Γ)
$\Gamma = \dfrac{e^{-qT}\phi(d_1)}{S\sigma\sqrt{T}}$ — curvature of price in spot; largest near ATM, short‑dated.
Vanna
Cross‑sensitivity of spot & vol: $\partial^2 V / (\partial S\,\partial \sigma)$. Drives PnL when spot and skew co‑move.
Volga (Vomma)
Curvature in vol: $\partial^2 V / \partial \sigma^2$. Exposed in wings/long‑dated vega books.
Charm, Veta, Speed
Time‑decay of Δ (Charm), time‑decay of ν (Veta), change of Γ with spot (Speed). Relevant for intraday hedging and barriers.
3) Surface vs Model Greeks
Model Greeks use primitives $(S,\sigma,r,q,...)$. Surface Greeks use smile parameters (e.g., SVI; or RR/BF buckets). Hedge/attribute PnL using the convention your desk follows (e.g., sticky‑delta intraday).
4) Smile Dynamics
- Sticky‑Strike: IV at each strike is fixed as spot moves → may overstate vanna PnL in equities.
- Sticky‑Delta: IV fixed for a given moneyness (delta) → closer to practice intraday.
- Local vs Stoch Vol: Local vol is often too sticky‑strike; Heston/SABR give more realistic dynamics.
5) Hedging & PnL Decomposition
For a delta‑hedged option over $dt$:
$$\mathrm{PnL} \approx -\Theta\,dt + \tfrac12\,\Gamma\,S^2\,d\langle \ln S\rangle + \nu\,d\sigma + \text{(higher‑order)}.$$
BSM (constant $\sigma$): $d\langle \ln S\rangle \approx \sigma^2 dt$ ⇒ short‑gamma/long‑theta wins if realized < implied.
6) Hedge Building Blocks
Δ‑hedge
Underlying/futures; prefer forward‑delta for clean carry.
Γ/Θ
Manage with verticals/calendars near ATM.
Vega term
Calendars/diagonals; bucket by tenor (1w/1m/3m/1y).
Skew (Vanna)
Risk‑reversals balance opposite wings.
Curvature (Volga)
Butterflies/condors concentrate curvature.
7) Outside BSM
- Local Vol: Greeks from PDE with $\sigma(S,t)$; Δ/Γ often larger in wings.
- Heston/Bates: extra sensitivities to $(\xi,\kappa,\theta,\rho)$; state‑dependent Greeks.
- SABR: Delta depends on normal/lognormal convention; parameters map to RR/BF risks.
- Americans/Barriers: Greeks can be discontinuous near boundaries → use trees/PDE or smoothed MC.
8) Numerical Greek Estimation
- Finite differences with common random numbers in MC.
- Pathwise for smooth payoffs; LR for discontinuities (digitals/barriers).
- Adjoint/Auto‑diff for large books (risk in one sweep).
9) Conventions & Pitfalls
- Discrete dividends cause Greek jumps across ex‑dates unless modeled explicitly.
- Know FX delta conventions (premium‑adjusted vs not; spot vs forward delta).
- Be consistent: surface‑based hedging requires surface‑based Greeks (e.g., SVI + Dupire), not plain BSM.
- $\Delta_c=e^{-qT}N(d_1)$, $\Delta_p=-e^{-qT}N(-d_1)$; $\Gamma=\dfrac{e^{-qT}\phi(d_1)}{S\sigma\sqrt{T}}$; $\nu=e^{-qT}S\phi(d_1)\sqrt{T}$.
- Full $\Theta,\rho,\psi$ as above; $\Delta_c-\Delta_p=e^{-qT}$.
Calibration Workflow
- Data hygiene: clean the IV surface; enforce no‑arb (no butterfly/calendar arb). SVI is standard.
- Choose model by book: vanillas only vs barriers/Americans; need for smile dynamics.
- Objective: fit vanilla surface (and sometimes dynamics) with stability/regularization.
- Validation: out‑of‑sample options, Greeks stability, daily PnL explain (Δ/Γ/ν/ρ) and stress tests.
Model Selection Playbook
- European vanillas; speed: BSM/Black/Bachelier with per‑strike IVs.
- Americans: Trinomial/CRR or PDE; BA‑Whaley for fast approximations.
- Smile‑consistent vanillas: Heston (Bates for jumps) or Local‑Vol from SVI when exact static fit is needed.
- Path‑dependent: MC (GBM/Heston); add LSM for early exercise.
- Rates/FX: SABR (normal/lognormal variants); Bachelier near zero/negative regimes.
- Event risk: Jump‑diffusion/Hybrid SVJ.
Gotchas & Tips
- Handle discrete dividends and borrow costs properly; otherwise parity and Greeks break.
- Be cautious with wings (short‑dated and far OTM) — impose no‑arb slopes/convexity.
- Regularize Heston parameters (avoid extreme $\rho,\xi$); prefer robust optimizers.
- Speed vs accuracy: COS/FFT for transform models; Leisen–Reimer/Richardson for lattices; Andersen QE for simulating Heston.
- Smooth the surface before computing surface Greeks to reduce noise.
Mini Reference Sheet
- BSM Greeks (spot‑delta): as summarized in the Greeks section.
- Forward measure tip: forwards $F=Se^{(r-q)T}$ simplify many payoffs under discount $e^{-rT}$.
Expanded Comparison (Daily Feel)
| Model / Method | Best For | Smile Fit | Americans | Path‑Dependence | Speed | Calibration |
|---|---|---|---|---|---|---|
| BSM / Black / Bachelier | Vanilla quoting, fast Greeks | Per‑strike only | ✗ | ✗ | ★★★★★ | ★ |
| Local Vol (SVI→Dupire) | Exact static surface fit | Exact | ✓ (PDE/tree) | △ | ★★★ | ★★★ |
| Heston | Smile with realistic dynamics | Good | ✗ | △ | ★★★★ | ★★★★ |
| Bates / Jumps | Gap‑prone assets | Very Good | ✗ | △ | ★★★ | ★★★★ |
| SABR | Rates/FX surfaces | Very Good | ✗ | △ | ★★★★ | ★★★ |
| Binomial/Trinomial | Americans, dividends | Surface via node σ | ✓ | △ | ★★ | ★★ |
| PDE (CN) | Americans, barriers, locals | Via coefficients | ✓ | △ | ★★ | ★★★ |
| MC (+LSM) | Path‑dependent, Americans | Path‑wise | ✓ (LSM) | ✓ | ★★ | ★★★ |
Minimal Model‑Choice Checklist
- Payoff: European / American / Path‑dependent / Barriers?
- Market quirks: Negative rates/forwards? Discrete dividends? Borrow?
- Smile needs: Static fit only or realistic dynamics too?
- Compute budget: Real‑time risk vs EoD batch?
- Governance: Calibration stability, explainability, no‑arb.