Black–Scholes & the Greeks

3.1

No-arbitrage & risk-neutral pricing

The deepest idea in option pricing is also the simplest to state: you can manufacture an option out of the stock and cash. Hold a continuously adjusted position of $\Delta$ shares plus a bond, and you can build a portfolio whose value tracks the option's value instant by instant. If the replicating portfolio costs less than the option, sell the option and buy the portfolio for a riskless profit; if it costs more, do the reverse. The only price that admits no such free lunch is the cost of replication. That is the option's fair value — full stop.

The magic of the hedge is that it is locally riskless. Over an instant, the option and $\Delta$ shares move by exactly offsetting amounts, so the combined portfolio has no exposure to the stock's random moves. A riskless portfolio can only earn the riskless rate $r$ — otherwise arbitrage. Setting "what the hedged portfolio actually earns" equal to "what a riskless portfolio must earn" produces the Black–Scholes PDE (§3.3) with no reference whatsoever to the stock's expected return $\mu$.

KEY

Why $\mu$ disappears. Two traders who disagree wildly about whether a stock will rise or fall must still agree on the option's price — because both can replicate it with the same hedge at the same cost. Beliefs about direction are irrelevant; only the size of the moves (volatility) matters. This is the single most counter-intuitive fact in the chapter.

The same conclusion has a probabilistic face. There exists a risk-neutral measure $\mathbb{Q}$ — a re-weighting of the real probabilities — under which every tradable asset drifts at exactly the riskless rate and the option's price is just its discounted expected payoff:

EQ Q3.1 — RISK-NEUTRAL VALUATION $$ V_0 \;=\; e^{-rT}\, \mathbb{E}^{\mathbb{Q}}\!\big[\, \text{payoff}(S_T) \,\big], \qquad \text{with } \; \mathrm{d}S_t = r\,S_t\,\mathrm{d}t + \sigma\,S_t\,\mathrm{d}W_t^{\mathbb{Q}} $$

Under $\mathbb{Q}$ the stock drifts at $r$, not its real-world drift $\mu$. $V_0$ is the value today; $T$ the time to expiry; $\sigma$ the volatility. Pricing reduces to taking an expectation and discounting it. For a European call, $\text{payoff} = \max(S_T - K, 0)$; the expectation under lognormal $S_T$ has a closed form — that is the Black–Scholes formula. For exotic payoffs with no closed form, you estimate the same expectation by Monte-Carlo (the second runnable cell below).

This recasting — price = discounted risk-neutral expectation — is the load-bearing beam of the entire field. Binomial trees (Quant 02) compute it by backward induction; Black–Scholes computes it in closed form; interest-rate models (Quant 04) apply it under a cleverly chosen numéraire. Everything downstream is a different way to evaluate EQ Q3.1.

3.2

The model & its assumptions

Black–Scholes assumes the stock follows geometric Brownian motion (GBM): proportional returns are normal, so the log-price is a Brownian motion with drift.

EQ Q3.2 — GEOMETRIC BROWNIAN MOTION $$ \mathrm{d}S_t = \mu\,S_t\,\mathrm{d}t + \sigma\,S_t\,\mathrm{d}W_t \quad\Longrightarrow\quad S_T = S_0 \exp\!\Big[\big(\mu - \tfrac{1}{2}\sigma^2\big)T + \sigma\sqrt{T}\,Z\Big],\;\; Z \sim \mathcal{N}(0,1) $$

The integrated form follows from Itô's lemma applied to $\log S$; the $-\tfrac{1}{2}\sigma^2$ is the Itô correction (volatility drag). Because $S_T$ is an exponential of a normal, prices are lognormal — never negative, with a right-skewed distribution. The same $\sigma$ appears whether you take the real-world drift $\mu$ or the risk-neutral drift $r$; only the drift changes between measures, never the diffusion.

Stacked on GBM are four idealizations. Each is wrong in a knowable way — and §3.5–§3.6 are essentially the catalogue of those errors:

Assumption	What it says	How reality differs
Constant volatility	σ fixed for all S, t	Vol clusters, spikes in crashes, and varies by strike (the smile, §3.5)
Constant rate	r fixed and known	Rates move; matters most for long-dated options (Quant 04)
Continuous paths	no jumps in S	Gaps and crashes are jumps; fat tails the lognormal can't produce
Frictionless market	no costs, infinite liquidity, continuous hedging	Spreads, fees, and discrete rebalancing make the hedge imperfect

A working quant treats Black–Scholes less as a literal model of the world than as a coordinate system: a universally agreed map from one number everyone can argue about — implied volatility — to a price. Its assumptions being false is not a bug to be hidden but the very thing the volatility surface measures.

3.3

The Black–Scholes formula

The hedging argument of §3.1, made continuous, says any derivative value $V(S,t)$ must satisfy a parabolic PDE in which — note — $\mu$ is absent:

EQ Q3.3 — THE BLACK–SCHOLES PDE $$ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV \;=\; 0 $$

A backward heat equation: it propagates the known terminal payoff at $t = T$ back to today. The four terms are time decay, convexity (gamma), drift-at-the-riskless-rate, and discounting. Solve it with the call payoff boundary condition $V(S,T) = \max(S - K, 0)$ and you get a closed form — no simulation required.

For a European call $C$ and put $P$ on a non-dividend-paying stock, with spot $S$, strike $K$, rate $r$, volatility $\sigma$, and time to expiry $T$:

EQ Q3.4 — THE CLOSED-FORM PRICE $$ C = S\,N(d_1) - K e^{-rT} N(d_2), \qquad P = K e^{-rT} N(-d_2) - S\,N(-d_1) $$ $$ d_1 = \frac{\ln(S/K) + \big(r + \tfrac{1}{2}\sigma^2\big)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T} $$

$N(\cdot)$ is the standard normal CDF. Read the call as two pieces: $S\,N(d_1)$ is the expected value of the shares you receive if you exercise, $K e^{-rT} N(d_2)$ is the discounted cost of paying the strike — and $N(d_2)$ is precisely the risk-neutral probability the option finishes in the money. $N(d_1)$ is the call's delta (§3.4): the formula already contains its own hedge ratio.

PUT–CALL PARITY

$C - P = S - K e^{-rT}$. A model-free identity from no-arbitrage alone: a call minus a put equals a forward on the stock. Subtract EQ Q3.4's two lines and the $N(d)$ terms collapse via $N(x) + N(-x) = 1$ to exactly $S - Ke^{-rT}$. If quoted prices ever violate it, that is a pure arbitrage — and the first sanity check any desk runs.

The normal CDF $N(x)$ has no elementary closed form, so implementations evaluate it through the error function, $N(x) = \tfrac{1}{2}\big[1 + \operatorname{erf}(x/\sqrt{2})\big]$. The instruments and Python cells below use a rational approximation to $\operatorname{erf}$ accurate to ~1e-7 — more than enough for pricing.

INSTRUMENT Q3.1 — OPTION PRICER + GREEKS DASHBOARDEQ Q3.4 · LIVE · erf APPROX

SPOT S 100

STRIKE K 100

VOL σ 0.20

RATE r 0.05

EXPIRY T (yrs) 1.00

TYPE

PRICE

—

MONEYNESS

—

d₁ · d₂

—

DELTA Δ

—

GAMMA Γ

—

VEGA (per 1%)

—

THETA (per day)

—

RHO (per 1%)

—

Drag σ and watch price and vega rise together — volatility is what you are really buying. Slide S across K to see delta sweep from 0 toward 1 (a call) and the moneyness tag flip from OTM to ITM. Theta is shown per calendar day and turns red when negative — the daily rent a long option holder pays. Vega and rho are quoted per one percentage point move, as desks quote them.

PYTHON · RUNNABLE IN-BROWSER

# Black-Scholes price + all five Greeks, scipy-free (erf via math.erf)
import numpy as np, math

def N(x):  # standard normal CDF through the error function
    return 0.5 * (1.0 + math.erf(x / math.sqrt(2.0)))
def n(x):  # standard normal PDF
    return math.exp(-0.5 * x * x) / math.sqrt(2.0 * math.pi)

def bs(S, K, r, sig, T, call=True):
    d1 = (math.log(S / K) + (r + 0.5 * sig * sig) * T) / (sig * math.sqrt(T))
    d2 = d1 - sig * math.sqrt(T)
    disc = math.exp(-r * T)
    if call:
        price = S * N(d1) - K * disc * N(d2)
        delta = N(d1)
        theta = (-S * n(d1) * sig / (2 * math.sqrt(T)) - r * K * disc * N(d2))
        rho   =  K * T * disc * N(d2)
    else:
        price = K * disc * N(-d2) - S * N(-d1)
        delta = N(d1) - 1.0
        theta = (-S * n(d1) * sig / (2 * math.sqrt(T)) + r * K * disc * N(-d2))
        rho   = -K * T * disc * N(-d2)
    gamma = n(d1) / (S * sig * math.sqrt(T))
    vega  = S * n(d1) * math.sqrt(T)
    return price, delta, gamma, vega, theta, rho

S, K, r, sig, T = 100, 100, 0.05, 0.20, 1.0
for typ, c in (("CALL", True), ("PUT", False)):
    p, dl, ga, ve, th, rh = bs(S, K, r, sig, T, c)
    print(f"{typ:4}  price {p:7.4f}  delta {dl:+.4f}  gamma {ga:.4f} "
          f"vega/1% {ve/100:6.4f}  theta/day {th/365:+.4f}  rho/1% {rh/100:+.4f}")
print("\nput-call parity check  C - P  vs  S - K*e^(-rT):")
cp = bs(S,K,r,sig,T,True)[0] - bs(S,K,r,sig,T,False)[0]
print(f"  {cp:.6f}  vs  {S - K*math.exp(-r*T):.6f}")

edits are live — break it on purpose

3.4

The Greeks: a risk dashboard

The price is a single number; the Greeks are its partial derivatives, and they are where the money is managed. Each answers "if this moves by one unit, how much does my option move?" A desk does not predict the market — it measures its exposures with the Greeks and trades to flatten the ones it does not want.

EQ Q3.5 — THE FIVE GREEKS (CALL) $$ \Delta = \frac{\partial C}{\partial S} = N(d_1), \qquad \Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{N'(d_1)}{S\sigma\sqrt{T}}, \qquad \mathcal{V} = \frac{\partial C}{\partial \sigma} = S\,N'(d_1)\sqrt{T} $$ $$ \Theta = \frac{\partial C}{\partial t} = -\frac{S\,N'(d_1)\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2), \qquad \rho = \frac{\partial C}{\partial r} = KTe^{-rT}N(d_2) $$

$N'(x) = \tfrac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the normal density. Gamma and vega are identical for calls and puts (parity has no $\sigma$ or curvature). For a put, $\Delta = N(d_1) - 1 \in [-1, 0]$, $\rho = -KTe^{-rT}N(-d_2) < 0$, and theta swaps its second term to $+rKe^{-rT}N(-d_2)$. Desks quote vega and rho per 1% point (divide by 100) and theta per day (divide by 365).

Greek	Sensitivity to	Intuition	How desks hedge it
Delta Δ	spot S	Equivalent share position; ≈ probability of finishing ITM	Buy/sell $\Delta$ shares so the book is delta-neutral
Gamma Γ	spot (2nd order)	How fast delta itself moves; the curvature, peaks at-the-money	Hard to hedge with stock; trade other options to flatten Γ
Vega 𝒱	volatility σ	P&L if implied vol re-rates; largest for ATM, long-dated	Buy/sell options of similar maturity to net vega to zero
Theta Θ	time	Time decay; the rent a long option pays each day	Usually accepted, not hedged — it pays for gamma/vega
Rho ρ	interest rate r	Rate sensitivity; small for short-dated, real for LEAPS	Offset with rate instruments or longer-dated options

The central trade-off: gamma vs theta. A long-option book is long gamma (it profits from large moves in either direction — buy low, sell high as you re-hedge) but short theta (it bleeds value as time passes). The two are linked through the PDE: the daily theta you pay is, in expectation, exactly the gamma P&L you collect re-hedging at the realized volatility. A delta-hedged option is a bet that realized vol exceeds the implied vol you paid.

INSTRUMENT Q3.2 — GREEKS PROFILE EXPLORERCALL · GREEK vs SPOT S · EQ Q3.5

GREEK

VOL σ 0.20

EXPIRY T (yrs) 1.00

The dashed line marks the strike, K = 100. Switch to gamma and lower T toward expiry: the curve spikes into a tall thin peak at the money — short-dated ATM options are explosively convex, the nightmare of a hedging desk. Vega instead grows with T (more uncertainty to be exposed to). Delta sweeps a smooth S-curve from 0 to 1; raising σ smears every profile wider.

3.5

Implied volatility & the smile

Five of the six Black–Scholes inputs are observable: spot, strike, rate, expiry — and the option's market price itself. The sixth, $\sigma$, is not. So traders run the formula backwards: given a quoted price, find the $\sigma$ that reproduces it. That number is the implied volatility.

EQ Q3.6 — IMPLIED VOLATILITY $$ \sigma_{\text{imp}} : \quad C_{\text{BS}}\big(S, K, r, T;\, \sigma_{\text{imp}}\big) \;=\; C_{\text{market}} $$

Because price is monotonic and smooth in $\sigma$ (vega $> 0$ always), the inverse exists and is unique. There is no closed form, so you solve numerically — bisection is bulletproof, Newton's method (using vega as the derivative) is fast. Implied vol is the market's price quoted in the language of the model. A trader does not say "this call costs $4.12"; they say "it's trading at 22 vol."

Here is where the model indicts itself. If Black–Scholes were literally true, every option on the same underlying would imply the same $\sigma$ — volatility is a property of the stock, not the strike. It does not. Plot implied vol against strike and you get a curve, not a flat line: the volatility smile (or, in equity index markets since 1987, a downward skew — deep out-of-the-money puts trade at much higher implied vol than calls).

THE SMILE'S MESSAGE

The smile is the market disagreeing with the lognormal. By bidding up out-of-the-money puts, traders are paying for crash protection — pricing in fatter left tails and jumps than GBM allows. The shape is the error of the constant-vol, no-jump assumption, made tradeable. Quants do not discard Black–Scholes over this; they keep the formula as a quoting device and let $\sigma(K, T)$ — the volatility surface — carry all the structure the model omits.

PYTHON · RUNNABLE IN-BROWSER

# Monte-Carlo a European call by simulating GBM, compare to closed-form BS
import numpy as np, math
rng = np.random.default_rng(0)

def N(x):  # vectorized normal CDF via erf
    return 0.5 * (1.0 + np.vectorize(math.erf)(x / math.sqrt(2.0)))
def bs_call(S, K, r, sig, T):
    d1 = (math.log(S/K) + (r + 0.5*sig*sig)*T) / (sig*math.sqrt(T))
    d2 = d1 - sig*math.sqrt(T)
    return S*float(N(d1)) - K*math.exp(-r*T)*float(N(d2))

S, K, r, sig, T = 100.0, 105.0, 0.05, 0.20, 1.0
M = 200_000
Z  = rng.standard_normal(M)
ST = S * np.exp((r - 0.5*sig*sig)*T + sig*math.sqrt(T)*Z)   # risk-neutral GBM
payoff = np.maximum(ST - K, 0.0)
mc    = math.exp(-r*T) * payoff.mean()
se    = math.exp(-r*T) * payoff.std() / math.sqrt(M)        # standard error
exact = bs_call(S, K, r, sig, T)

print(f"Monte-Carlo call : {mc:.4f}  (+/- {1.96*se:.4f}, 95% CI)")
print(f"closed-form  call: {exact:.4f}")
print(f"gap              : {mc - exact:+.4f}   ({abs(mc-exact)/exact*100:.2f}% )")
plot_xy(sorted(ST), np.linspace(0, 1, M))   # empirical lognormal CDF of S_T

edits are live — break it on purpose

INSTRUMENT Q3.3 — PAYOFF & BREAK-EVEN DIAGRAMEXPIRY PAYOFF vs CURVED VALUE TODAY

TYPE

STRIKE K 100

VOL σ 0.20

EXPIRY T (yrs) 1.00

PREMIUM PAID

—

BREAK-EVEN SPOT

—

MAX LOSS

—

The mint line is the kinked payoff at expiry, net of the premium you paid; the blue curve is the option's smooth value today (the Black–Scholes price across spot). They meet only at expiry, when time value vanishes. The break-even marker is the spot at which the expiry payoff crosses zero — strike plus premium for a call, strike minus premium for a put. Raise σ and the blue curve lifts everywhere: more volatility is worth more, at every spot.

3.6

Where it breaks & what comes next

Black–Scholes is the most successful wrong model in finance. Its failures are not subtle and they are not academic — they have repeatedly arrived as market-wide losses:

Jumps. GBM has continuous paths; real prices gap. On 19 October 1987 the S&P fell ~20% in a day — a 20-sigma event under lognormal, i.e. effectively impossible, yet it happened. The permanent equity-index skew dates from that morning: the market has priced crash-jumps into puts ever since. Merton's jump-diffusion adds a Poisson jump term to restore the fat left tail.
Stochastic volatility. Vol is not constant — it clusters and spikes. The Heston model (1993) makes variance itself a mean-reverting random process correlated with the stock, generating a smile endogenously and admitting a semi-closed form via characteristic functions. SABR is the market standard for interest-rate smiles.
Hedging is not free or continuous. The replication argument assumes costless, continuous rebalancing. In practice transaction costs, discrete hedging, and liquidity gaps mean the hedge leaks — which is precisely how a delta-hedged book's P&L becomes a bet on realized-vs-implied volatility rather than a riskless lock.
Correlation and liquidity regimes. 2008 taught that diversifying assumptions fail together: correlations snap to one and liquidity evaporates exactly when hedges are needed. No single-name vol model captures this; it is a systemic, cross-asset failure.

EQ Q3.7 — HESTON: STOCHASTIC VARIANCE $$ \mathrm{d}S_t = r S_t\,\mathrm{d}t + \sqrt{v_t}\,S_t\,\mathrm{d}W_t^{S}, \qquad \mathrm{d}v_t = \kappa(\theta - v_t)\,\mathrm{d}t + \xi\sqrt{v_t}\,\mathrm{d}W_t^{v}, \qquad \mathrm{d}W^{S}\mathrm{d}W^{v} = \rho\,\mathrm{d}t $$

Variance $v_t$ mean-reverts to long-run $\theta$ at speed $\kappa$ with vol-of-vol $\xi$; the correlation $\rho < 0$ (falling stock, rising vol) produces the equity skew. Black–Scholes is the degenerate special case $\xi = 0$, $v_t \equiv \sigma^2$ — the model you reach for first, and the baseline every richer model is measured against.

We held the rate $r$ constant and known — and for short-dated equity options that is harmless. It is not harmless for bonds, swaps, and anything long-dated, where the thing being modelled is the rate. Quant 04 turns volatility loose on the yield curve itself: short-rate models (Vasicek, Hull–White, CIR), the change of numéraire that makes them tractable, and why a whole second universe of "Greeks" lives on the interest-rate desk.

3.7

References

Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3). — the original no-arbitrage derivation and the closed-form formula.
Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4(1). — the rigorous continuous-time treatment; later extended with jump-diffusion.
Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility. Review of Financial Studies 6(2). — stochastic-variance model that generates the smile endogenously (EQ Q3.7).
Hull, J. C. Options, Futures, and Other Derivatives (11th ed., 2021). Pearson. — the standard practitioner reference for the PDE, the Greeks, and implied-vol surfaces.
Rubinstein, M. (1994). Implied Binomial Trees. Journal of Finance 49(3). — an early reconstruction of the post-1987 volatility smile from market prices.