Financial Mathematics · Lecture 12 of 12

Greeks, Hedging, Monte Carlo, and Where Black–Scholes Breaks

How traders actually use the model: sensitivities, discrete hedging P&L, pricing by simulation, implied volatility, and the smile that tells you the assumptions are wrong.

November 15, 2026By Diganta Bhattacharya5 min read

The Greeks

Partial derivatives of the price, each a risk you can hedge:

$$\Delta=\frac{\partial V}{\partial S}=N(d_1),\quad \Gamma=\frac{\partial^2 V}{\partial S^2}=\frac{n(d_1)}{S\sigma\sqrt T},\quad \nu=\frac{\partial V}{\partial\sigma}=S\,n(d_1)\sqrt T,$$ $$\Theta=\frac{\partial V}{\partial t}=-\frac{S\,n(d_1)\sigma}{2\sqrt T}-rKe^{-rT}N(d_2),\qquad \rho=\frac{\partial V}{\partial r}=KTe^{-rT}N(d_2)\quad\text{(calls)}.$$

Here $n$ is the standard normal density. Gamma and vega peak at the money and grow as expiry approaches (gamma) or as maturity lengthens (vega). The calculator in Lecture 11 prints them live.

Hedging in discrete time

Black–Scholes assumes continuous rebalancing. In reality you rebalance daily or less. Over each interval the hedged P&L is approximately (Lecture 4):

$$\delta\Pi\approx\tfrac12\Gamma S^2\big(\sigma_{\text{realised}}^2-\sigma_{\text{implied}}^2\big)\delta t+\text{noise}.$$

Two consequences show up in the simulation below. (1) The P&L of a hedged short option is not zero: its spread shrinks roughly like $1/\sqrt{\text{number of rebalances}}$. (2) If realised volatility exceeds the volatility you sold at, you lose on average. An option trader is really trading volatility.

Pricing by Monte Carlo

Risk-neutral valuation says price = discounted average payoff under $\mathbb Q$. So simulate it (Lectures 1 and 7):

Z  = rng.standard_normal(M)
ST = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)
price = np.exp(-r * T) * np.maximum(ST - K, 0).mean()
stderr = np.exp(-r * T) * np.maximum(ST - K, 0).std() / np.sqrt(M)

The error falls like $1/\sqrt M$, so 100× more paths buys 10× more accuracy. That's slow for a vanilla call, which has a formula. But Monte Carlo works unchanged for path-dependent payoffs (Asians, barriers, mortgage-backed securities) and for many underlyings, where trees and PDEs struggle. Variance-reduction tricks (antithetic variates, control variates using the BS price) speed it up a lot.

Implied volatility and the smile

Every input to Black–Scholes is observable except σ. Turn the formula around: given a market price, find the σ that reproduces it. This is the implied volatility, solved by Newton's method using vega:

$$\sigma_{k+1}=\sigma_k-\frac{BS(\sigma_k)-C_{\text{mkt}}}{\nu(\sigma_k)}.$$

If the model were right, implied vol would be the same for every strike and maturity. It isn't. Equity index options show a skew: low strikes (crash protection) trade at higher implied vol. FX and many single stocks show a smile. Markets quote options in implied vol, using Black–Scholes as a translation layer rather than a truth.

Where the assumptions break

Volatility isn't constant. It clusters and mean-reverts, and it rises when prices fall. Fixes: local volatility (Dupire), stochastic volatility (Heston, SABR).
Prices jump. Returns have fat tails, and a 5σ day is far more common than the normal predicts. Fix: jump-diffusion (Merton), which can't be hedged perfectly with stock alone.
Trading isn't free or continuous. Transaction costs make perfect hedging infinitely expensive. Real hedges use bands and accept residual risk.
Rates and dividends vary: use forwards (Black's formula) and term structures.

The lasting ideaEven where the formula fails, the method survives: find a hedge, price by no-arbitrage, and evaluate an expectation under a martingale measure. Every model in modern derivatives pricing, from Heston to HJM (see the Finance in Banks posts), follows the path laid out in these twelve lectures.

The road we took

Probability (1–2) → finance and no-arbitrage (3) → Taylor series (4) → differential equations (5) → Brownian motion (6) → SDEs (7) → Itô's lemma (8) → martingales and Q (9) → binomial replication (10) → the PDE and formula (11) → using and stress-testing it (12).

Key takeaways

Greeks are sensitivities: Δ = N(d₁), Γ and vega peak at the money; theta pays for gamma.
Discrete hedging leaves residual P&L ≈ ½ΓS²(σ²_real − σ²_impl)δt: options trade volatility.
Monte Carlo prices any payoff with error ~1/√M; implied vol and the smile show where BS is wrong.

Exercises

Write a Newton solver for implied volatility and recover σ = 20% from the price 10.4506 (S = K = 100, T = 1, r = 5%).
Price an arithmetic Asian call by Monte Carlo and use the geometric Asian (closed form) as a control variate.
In the hedging widget, set realised σ = 30%. Explain the sign of the mean P&L using gamma.