Black–Scholes — Pricing and Hedging

Takeaway

Under no-arbitrage with geometric Brownian motion and continuous hedging, option prices satisfy the Black–Scholes PDE; the delta hedge replicates payoffs and eliminates risk.

The problem (before → after)

Before: How to price a derivative without guessing expected returns?
After: Replicate the option with a self-financing portfolio; absence of arbitrage pins down price via risk-neutral valuation.

Mental model first

Imagine you’re shadowing the option with a tiny adjustable stock position and cash. If you always track its small moves, the leftover risk vanishes and the cost of your shadow becomes the option’s fair price.

Just-in-time concepts

GBM: dS = μ S dt + σ S dW.
Itô’s lemma and delta: Δ = ∂V/∂S.
Risk-neutral measure: Discounted asset becomes a martingale; μ → r in pricing.

First-pass solution

Apply Itô to V(S,t); choose Δ to cancel dW term; self-financing portfolio earns risk-free rate → Black–Scholes PDE. Solve with boundary condition for European call to get closed form.

Greeks quantify sensitivities; hedging error arises with discrete rebalancing.
Implied volatility smiles reveal model misspecification.
Extensions: Local/stochastic vol, jumps, transaction costs.

Code as a byproduct (call price)

import math
from mpmath import quad, erfc

def norm_cdf(x):
    return 0.5 * erfc(-x / math.sqrt(2))

def bs_call(S, K, T, r, sigma):
    from math import log, sqrt
    d1 = (math.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*sqrt(T))
    d2 = d1 - sigma*sqrt(T)
    return S*norm_cdf(d1) - K*math.exp(-r*T)*norm_cdf(d2)

Principles, not prescriptions

Price by replication, not by subjective expected returns.
Hedging assumptions matter; discrete trading introduces risk and bias.

Common pitfalls

Using historical μ in pricing; risk-neutral uses r.
Ignoring dividends, rates, or early exercise (requires adjustments).

Connections and contrasts

See also: [/blog/kelly-criterion] (growth vs hedging), [/blog/rough-volatility] (beyond constant σ).

Quick checks

Why μ disappears? — Replication removes risk → risk-neutral valuation.
What is Δ? — Shares per option to hedge small moves.
Why smiles? — Market-implied σ varies with strike/maturity.