TheQuantBateman

01

INTUITION

Separate the quote from the quantity inferred from it.

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

Start from executable inputs and conventions.

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

Transform quotes without losing units or arbitrage constraints.

Formula · Definition

Smile slope

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

Local ATM slope in log-forward-moneyness coordinates.

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Formula · Full derivation

Breeden–Litzenberger density

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

Convex call prices imply a non-negative terminal density.

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Full 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.

Inputs

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

Assumptions and limits

Sparse wings are weakly identified.
Implied-vol smoothness does not guarantee price convexity.
A single expiry says nothing about calendar arbitrage.

Formula · Definition

Wing convexity

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

The fundamental butterfly-arbitrage requirement lives in price space.

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05

MODEL / PRICING

Invert, fit, and reprice the market instruments.

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.

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.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Validate domains and units at the boundary.
Keep the numerical kernel framework-free and deterministic.
Return diagnostics with values.
Test analytical limits and failure states.

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 quote and inspect every linked representation.

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%

Implied volatility by Forward moneyness K/F

Base smile: Moderate downside skew.

implied smile
ATM tangent

Forward moneyness K/F: 0.70. implied smile: 30.6%. ATM tangent: 23.8%.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

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

09MACRO CONNECTIONOpen 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.

01Risk-off demandtransmits

raises put premiums

02Downside skewtransmits

steepens left wing

03Densitytransmits

reprices tail states

04Hedgeoutput

creates spot-vol cross risk

10COMMON PITFALLSOpen the failure checklist.

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.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

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 ↗

implementation reference

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 ↗

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

Downside skew and crash insurance

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests