TheQuantBateman

01

INTUITION

Read the contract before deriving its price.

Black–Scholes is not merely a formula. It is an equivalence between a replicating strategy, a parabolic PDE, and a discounted risk-neutral expectation under a precise market model.

01

The physical drift disappears because replication, not forecasting, pins the price.

02

Boundary and terminal conditions are part of the pricing problem.

03

The closed form is a benchmark and quote coordinate even when smile dynamics require richer models.

02

WHY MARKETS CARE

Fix cash flows, conventions, and hedge instruments.

Vanilla option quotes, implied volatility, desk Greeks, hedging diagnostics and model validation all use Black–Scholes as a common coordinate system.

INSTRUMENTS

European calls and puts

equity and FX vanillas

option hedges

implied-volatility quotes

QUOTE CONVENTION

S and K are currency amounts per underlying unit; r and q are continuous annual rates; σ is decimal annualized volatility; T is year fraction; prices are present values.

03

MATHEMATICS

From dynamics and replication to valuation.

Formula · Full derivation

Black–Scholes PDE

V_t+\tfrac12\sigma^2S^2V_{SS}+(r-q)SV_S-rV=0,\qquad V(T,S)=\Phi(S)

No-arbitrage fixes the value once dynamics, carry, terminal payoff and boundary behavior are specified.

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Full derivation

From self-financing replication to the pricing PDE

Apply Itô to the option, cancel the Brownian exposure with stock, and impose that the locally riskless portfolio earns the funding rate.

01
State risk-neutral-compatible spot dynamics
Use continuous dividend yield q and volatility σ; the replication result does not require the physical drift.
$dS_t=(\mu-q)S_tdt+\sigma S_tdW_t^{\mathbb P}$
02
Apply Itô to the option
The option inherits drift, delta exposure and the gamma correction from the same Brownian driver.
$dV=(V_t+(\mu-q)SV_S+\tfrac12\sigma^2S^2V_{SS})dt+\sigma SV_SdW$
03
Build the delta-hedged portfolio
Hold one option and short Δ=V_S stock units; include dividend cash flow on the stock position.
$\Pi=V-V_SS,\qquad d\Pi=dV-V_SdS-qSV_Sdt$
04
Cancel diffusion risk
Substitution removes both dW and the unknown physical drift μ.
$d\Pi=(V_t+\tfrac12\sigma^2S^2V_{SS}-qSV_S)dt$
05
Impose no-arbitrage funding
A locally riskless, self-financing portfolio must earn r; rearranging yields the PDE.
$d\Pi=r(V-SV_S)dt\Longrightarrow V_t+\tfrac12\sigma^2S^2V_{SS}+(r-q)SV_S-rV=0$
Numerical checkThe PDE contains r−q but no μ; put–call parity and intrinsic limits provide independent checks.

Feynman–Kac maps this terminal-value PDE to V_t=E^Q[e^{-r(T-t)}Φ(S_T)|F_t]; the closed form is the European call/put solution for a lognormal terminal law.

Inputs

V(t,S): derivative value
τ=T−t: time to expiry
N(·), n(·): standard normal CDF and density
Δ=∂V/∂S: spot units held in the hedge

Assumptions and limits

Constant volatility and lognormal tails contradict observed smiles.
Continuous frictionless hedging omits gaps, liquidity and transaction costs.
European exercise excludes early exercise and path dependence.

Formula · Full derivation

European call

C=S_0e^{-qT}N(d_1)-Ke^{-rT}N(d_2),\quad d_{1,2}=\frac{\log(S_0/K)+(r-q\pm\tfrac12\sigma^2)T}{\sigma\sqrt T}

Discounted spot and strike terms are weighted by the model's exercise probabilities in different numeraires.

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Formula · Short derivation

Put–call parity

C-P=S_0e^{-qT}-Ke^{-rT}

A static replication identity that must hold independently of volatility in the model domain.

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Formula · Short derivation

Call delta

\Delta_C=e^{-qT}N(d_1)

The instantaneous stock units in the continuous-time replication; desk convention and premium adjustment may differ by asset class.

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05

MODEL / PRICING

Reconcile PDE, expectation, and closed form.

METHOD

Use closed form for European vanillas, implied-volatility inversion and analytical Greeks; use the PDE/expectation equivalence as a benchmark for numerical engines.

CALIBRATION

Black–Scholes has one volatility per quote. A surface is a market coordinate assembled from many inversions, not one globally calibrated constant-σ model.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

One validated Black–Scholes kernel shared by price, IV and Greeks
Analytical parity and limit checks
Desk-unit conversion outside the raw kernel

PYTHON 3 · NUMPY / SCIPY

Call, put and parity reference

Compute European prices from one state and verify put–call parity to numerical tolerance.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import math

04from statistics import NormalDist

05

06N = NormalDist()

07

08

def black_scholes(spot: float, strike: float, time: float, rate: float, dividend: float, vol: float) -> tuple[float, float]:

09 if min(spot, strike, time, vol) <= 0:

10 raise ValueError("positive spot, strike, time and vol required")

11 root_t = math.sqrt(time)

12 d1 = (math.log(spot/strike)+(rate-dividend+0.5*vol*vol)*time)/(vol*root_t)

13 d2 = d1-vol*root_t

14 call = spot*math.exp(-dividend*time)*N.cdf(d1)-strike*math.exp(-rate*time)*N.cdf(d2)

15 put = strike*math.exp(-rate*time)*N.cdf(-d2)-spot*math.exp(-dividend*time)*N.cdf(-d1)

16 return call, put

17

18s, k, t, r, q, vol = 100.0, 105.0, 1.25, 0.04, 0.015, 0.22

19call, put = black_scholes(s, k, t, r, q, vol)

20parity = s*math.exp(-q*t)-k*math.exp(-r*t)

21assert abs((call-put)-parity) < 1e-12

22assert 0.0 <= call <= s*math.exp(-q*t)

23print(f"call={call:.6f} put={put:.6f} parity_error={(call-put-parity):.2e}")

07

INTERACTIVE LAB

Move the state. Challenge the equation.

RISK RESPONSE LAB

Black–Scholes: replication, PDE & pricing

Move spot, volatility and horizon. Prices, desk-unit Greeks and hedge residuals share one pricing state.

SYNTHETIC · EDUCATIONAL

Option value9.8421Black–Scholes

Hedge residual-0.6098after transaction costs

Turnover charge0.640010.0 bp

P&L (currency units) by hedge progress

Synthetic cumulative hedge P&L after transaction costs.

cumulative hedge P&L

hedge progress: 0.0%. cumulative hedge P&L: 0.0000.

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

08

FRONT OFFICE

Translate the derivation into a hedge.

ON THE DESK

“Black–Scholes is the common language; the hedge residual tells you where the language stopped describing the market.”

VISIBLE INPUTS

spot/forward and curves

strike, expiry and exercise style

option premium and volatility convention

hedge frequency and costs

CALIBRATION

Invert each liquid premium to implied volatility with consistent curves, timestamps, bid/offer and solver diagnostics.

RISK

delta/gamma/vega

gap and discrete-hedging error

smile dynamics

DAILY WORKFLOW

normalize contract and carry
price and invert volatility
compute desk-unit Greeks
reconcile hedge P&L

Production failure modes

mixed day counts or dividend inputs
stale volatility quotes
uncontrolled near-expiry Greeks

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

From policy and earnings to the option coordinate

Rates and dividends change forward carry; uncertainty and protection demand change quoted implied volatility. The formula maps those inputs to price but does not explain their economic cause.

01Rates/dividendstransmits

move forward carry

02Event risktransmits

moves option premium

03Implied volatilitytransmits

normalizes the quote

04Greeksoutput

translate price into hedge

10COMMON PITFALLSOpen the failure checklist.

01

Presenting the closed form without the replication assumptions.

02

Using historical volatility as an implied quote.

03

Ignoring dividends or inconsistent discount curves.

04

Treating continuous delta hedging as a realizable guarantee.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

Ch. 3 §§3.1–3.3, printed pp. 51–78 (PDF pp. 70–96)

The formulation was checked against the cited sections; prose, examples and code are original to the Academy.

Source: Mathematical Modeling and Computation in Finance
Author: Cornelis W. Oosterlee and Lech A. Grzelak
Ref: First edition (2020)

OPEN ORIGINAL SOURCE ↗

From self-financing replication to the pricing PDE

State risk-neutral-compatible spot dynamics

Apply Itô to the option

Build the delta-hedged portfolio

Cancel diffusion risk

Impose no-arbitrage funding

From policy and earnings to the option coordinate

Ch. 3 §§3.1–3.3, printed pp. 51–78 (PDF pp. 70–96)