TheQuantBateman

01

INTUITION

Build the state before the equation.

A smile is the strike slice of a surface at one expiry. Its slope and curvature summarize relative option premiums, but only option-price derivatives—not a visual curve—determine arbitrage consistency.

01

Equity index skew is often downside-heavy rather than symmetric.

02

Forward log-moneyness removes much of the carry distortion.

03

A static fit and a rule for how the smile moves are different model claims.

02

WHY MARKETS CARE

The product exists before the model.

Smile shape drives relative-value trades, barrier exposure, delta conventions and every model calibrated beyond ATM.

INSTRUMENTS

puts and calls by strike

risk reversals

butterflies

digitals and barriers

QUOTE CONVENTION

Use k=ln(K/F_T) unless a market-native delta convention is required. State premium inclusion, forward and option type.

03

MATHEMATICS

Notation, units and exact claims.

k=ln(K/F_T): forward log-moneynessw(k,T)=σ²T: total variance∂K C: digital information∂KK C: density information

QUANT NOTESYNTHETIC · EDUCATIONAL

Smile slope

Local ATM slope in log-forward-moneyness coordinates.

\operatorname{Skew}(T)=\left.\partial_k\sigma_{imp}(k,T)\right|_{k=0}

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Breeden–Litzenberger density

Convex call prices imply a non-negative terminal density.

f_{S_T}^{Q}(K)=D(0,T)^{-1}\partial_{KK}C(K,T)

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Wing convexity

The fundamental butterfly-arbitrage requirement lives in price space.

\partial_{KK}C(K,T)\ge 0

Read the equation together with its financial domain and convention.

04

DERIVATION

Do not jump to the final expression.

DERIVATION

From strike prices to a terminal density

Differentiate a continuum of call payoffs with respect to strike.

01
Write the call as an integral
A discounted call is the integral of terminal payoff against the risk-neutral density.
$C(K)=D\int_K^\infty(s-K)f(s)ds$
02
Differentiate once
The first strike derivative is minus the discounted tail probability.
$\partial_K C=-D\,Q(S_T>K)$
03
Differentiate twice
The second derivative recovers discounted density.
$\partial_{KK}C=Df(K)$
04
Translate prices to implied vol
Inverting each premium through Black–Scholes produces the displayed smile; arbitrage checks must still be performed on reconstructed prices.
05
Specify smile dynamics
Sticky-strike, sticky-delta, local-vol and stochastic-vol rules imply different P&L when spot moves.
NUMERICAL CHECKNever infer dynamics from one static slice.

The smile is a quote coordinate for relative option prices; convex prices and explicit dynamics determine whether it is usable.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

METHOD

Clean one expiry, convert to forward moneyness, fit total variance or prices with shape constraints, and validate reconstructed premiums.

CALIBRATION

Weight liquid nodes by spread or vega, constrain wings, and separate interpolation from extrapolation.

LIMITATIONS

Sparse wings are weakly identified.

Implied-vol smoothness does not guarantee price convexity.

A single expiry says nothing about calendar arbitrage.

MODEL COMPARISON

Static fit is not dynamics.

Question	Black–Scholes	Local volatility	Heston
Volatility state	One constant σ	σ(S,t) deterministic	vₜ stochastic
Fits today’s surface	No	Exactly, in ideal theory	Approximately by calibration
Forward dynamics	Flat smile	Spot-driven	Variance + correlation driven
Primary strength	Transparent baseline	Vanilla-consistent diffusion	Richer smile dynamics
Primary failure	No smile	Often unrealistic forward skew	Parameter and calibration instability
Compute	Low	Medium: PDE/MC	Medium–high: Fourier/PDE/MC
Hedge implication	Greeks at one σ	State-localized vol hedge	Variance and vol-of-vol risk

Implementation with current QuantLib

Current QuantLib separates market structures, processes, instruments, engines and calibration helpers. Use those abstractions only after the lesson’s conventions, domains and numerical checks are explicit.

API authority: upstream QuantLib reference pinned in the source registry.

06

PYTHON LAB

Theory → implementation → checks.

PYTHON 3 · NUMPY / SCIPY

Finite-difference density check

Recover a non-negative density proxy from a convex call-price slice.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import numpy as np

04

05def density_proxy(strikes: np.ndarray, calls: np.ndarray, discount: float) -> np.ndarray:

06 if strikes.ndim != 1 or calls.shape != strikes.shape or not np.allclose(np.diff(strikes), np.diff(strikes)[0]):

07 raise ValueError("use an aligned uniform strike grid")

08 h = strikes[1] - strikes[0]

09 density = (calls[:-2] - 2.0 * calls[1:-1] + calls[2:]) / (h * h * discount)

10 return density

11

12k = np.arange(80.0, 125.0, 5.0)

13c = np.array([22.0, 18.0, 14.5, 11.5, 9.0, 7.0, 5.5, 4.5, 3.8])

14f = density_proxy(k, c, 0.98)

15assert np.isfinite(f).all() and np.all(f >= -1e-12)

16print(np.round(f, 5))

07

INTERACTIVE LAB

Move the state. Challenge the equation.

STRIKE GEOMETRY AND DENSITY

Smile geometry lab

Move spot, skew and curvature; inspect the smile, risk-neutral density proxy and arbitrage flags.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

ATM20.00%

ATM skew-0.126

70% wing30.60%

08

FRONT OFFICE

Where the model meets the book.

ON THE DESK

“A wing quote is cheap until the extrapolator makes it expensive.”

VISIBLE INPUTS

forward

strike premiums

discount factors

spread weights

wing rules

CALIBRATION

Weight liquid nodes by spread or vega, constrain wings, and separate interpolation from extrapolation.

RISK

skew delta

wing vega

digital risk

smile dynamics

DAILY WORKFLOW

normalize strikes
invert premiums
fit slice
reprice
check convexity

Production failure modes

stale forward
mixed call/put sides
unconstrained splines
unstable wing extrapolation

09

MACRO CONNECTION

Map the transmission channel.

MACRO CONNECTION

Downside skew and crash insurance

Leverage, jump risk and demand for protection concentrate risk-neutral mass in the downside tail.

Risk-off demand

raises put premiums

Downside skew

steepens left wing

Density

reprices tail states

Hedge

creates spot-vol cross risk

10

COMMON PITFALLS

Most failures begin outside the formula.

01

Treating a smooth line as arbitrage-free.

02

Comparing raw strikes across expiries.

03

Using mid quotes in illiquid wings without uncertainty bands.

04

Assuming sticky strike by default.

11

SOURCES / FURTHER READING

Attribution with implementation authority.

researchBSD-3-Clause

Volatility, Monte Carlo and stochastic-volatility lectures

Research map for the mathematical progression and numerical experiments; prose, examples and code are original.

Source: Computational Finance Course
Author: L. A. Grzelak
Ref: main

OPEN ORIGINAL SOURCE ↗LICENSE ↗

implementation referenceQuantLib permissive license

Current volatility structures, processes, calibration helpers and tests

Implementation reference for production abstractions and validation patterns.

Source: QuantLib upstream
Author: QuantLib contributors
Ref: v1.42.1

OPEN ORIGINAL SOURCE ↗LICENSE ↗

Smile slope

Breeden–Litzenberger density

Wing convexity

From strike prices to a terminal density

Write the call as an integral

Differentiate once

Differentiate twice

Translate prices to implied vol

Specify smile dynamics

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests