Black–Scholes as replication
M2 — Quantitative finance fundamentals
M0.8 asserted that a three-month at-the-money call on a £100 stock costs about £5, and openly deferred the question of where that number comes from. Here it is — and the honest route to it does not start with the famous formula. It starts with a much better idea.
First, the intuition that fails
The natural guess is that an option should be worth its expected payoff, discounted. Take the M0.8 setup, simplified to one period and zero interest, and let the stock go to either 110 or 90:
S = 100. In three months it is 110 or 90.
Call, strike 100: pays 10 if up, 0 if down.
if the stock rises with probability p …
p = 0.5 → E[payoff] = 5
p = 0.9 → E[payoff] = 9 ← a bullish stock, dearer option?
p = 0.1 → E[payoff] = 1
Reasonable, and wrong. The price is £5 in all three cases. A stock everyone is certain will rise does not carry a more expensive call, and understanding why is the whole lesson.
The idea that works: build the thing
Forget forecasting. Ask a construction question instead:
Can I build a portfolio out of the stock and cash that pays exactly what the option pays, in every state of the world?
If I can, then that portfolio and the option are the same object with different names, and no-arbitrage says they must cost the same. Otherwise buy the cheap one, sell the dear one, and collect the difference with no risk.
Let the portfolio hold Δ shares and B in cash, and demand it match the option in both states:
up state: Δ·110 + B = 10
down state: Δ· 90 + B = 0 (r = 0, so cash doesn’t grow)
───────────────────────────────────────────────────────────────
subtract: Δ·(110 − 90) = 10 − 0
f_up − f_down 10 − 0
Δ = ───────────────── = ─────────── = 0.5
S_up − S_down 110 − 90
back-substitute: 0.5·90 + B = 0 ⟹ B = −45
So: hold half a share, borrow £45. That portfolio pays 10 if the stock rises and 0 if it falls — precisely the option. Its cost today is
price = Δ·S + B = 0.5 × 100 − 45 = £5
There is M0.8’s £5, and it isn’t a plausible-sounding number any more. It is the cost of manufacturing the option out of parts.
Notice what Δ = 0.5 is: the number of shares you must hold to track the option. It is the hedge ratio — and it is the same delta that becomes the first of the greeks in M2.3. Delta was never fundamentally a sensitivity or a probability; it is how much stock the replication requires.
Where the probability went
Now rearrange the same answer to expose what happened. Define
S − S_down 100 − 90
q = ────────────── = ───────── = 0.5
S_up − S_down 110 − 90
price = q·f_up + (1−q)·f_down = 0.5 × 10 + 0.5 × 0 = £5 ✓
The price looks like a discounted expected payoff again — but under q, not under the real probability p. And p appears nowhere. It cancelled, exactly, and not by luck.
Why it must cancel: the replicating portfolio matches the option in every state. A hedged position has no exposure to which state occurs, so how likely each state is cannot enter its cost. What you are paying for is not a bet on direction — it is the cost of running the hedge, and that depends on how far apart the states are, not on their probabilities.
This is the single most counter-intuitive true thing in derivatives pricing, and it is worth being able to say in one sentence: the option price doesn’t depend on where you think the stock is going, because the seller doesn’t have to guess — they can hedge.
The risk-neutral measure, demystified
q is called the risk-neutral probability, which is an unhelpful name that has confused generations. Three clarifications:
- It is not a belief. Nobody thinks the stock rises with probability q. It is the weight that makes the arithmetic of replication come out right.
- It is a change of measure. Under q, the discounted price of every tradeable asset is a martingale — its expected future value equals its value now. Pricing by "expectation under q" is a computational device that packages the replication argument, nothing more.
- It is why the name says risk-neutral. In a world where investors genuinely didn't care about risk, prices would be discounted expected payoffs. Replication lets us borrow that world's easy arithmetic without believing we live in it.
The deep version of this is the Fundamental Theorem of Asset Pricing: no arbitrage ⟺ a risk-neutral measure exists, and completeness (every payoff replicable) ⟺ it is unique. That equivalence is what M2 is quietly built on.
From two states to Black–Scholes
The rest is limits, not new ideas. Chain the one-period tree into many periods, rebalancing Δ at each node; let the steps shrink and the number of them grow. The binomial price converges, and its limit is the Black–Scholes formula:
C = S·N(d₁) − K·e^(−rT)·N(d₂)
ln(S/K) + (r + σ²/2)T
d₁ = ─────────────────────── , d₂ = d₁ − σ√T
σ√T
Two readings worth carrying:
- N(d₂) is the risk-neutral probability the option finishes in the money — the q of the two-state model, grown up.
- N(d₁) is delta, the hedge ratio. The formula's structure is exactly the binomial one — hold N(d₁) shares, borrow the discounted strike times N(d₂) — which is to say the whole thing is still "cost of the replicating portfolio."
Now look at the inputs:
S spot ......................... observable
K strike ....................... in the contract
T time to expiry ............... a calendar
r interest rate ................ observable
σ volatility ................... ✗ NOT OBSERVABLE
Exactly one input is unknown, and it is the one describing the future. That asymmetry is the reason options markets talk in volatility rather than price, and it is the entire subject of M2.4.
Which assumptions actually bite
Black–Scholes assumes continuous costless trading, a constant known σ, and lognormal prices. All false. But they fail with very different severity, and knowing which is which is the difference between using the model and believing it:
assumption reality how bad
──────────────────────────────────────────────────────────────────
no transaction costs spread + impact (M0.4, M1.5) bad — continuous
rehedging is
infinitely
expensive
constant σ volatility clusters and moves bad — this is
what the smile
is telling you
(M2.4)
lognormal returns fat tails, jumps, crashes WORST — the
Gaussian core
understates
extremes (M3)
continuous trading discrete, and gappy overnight moderate
Which is why the standard line — that Black–Scholes is the wrong model that everyone uses correctly — is not a joke but a description of practice. What it is used for is not predicting the price. It is used as an invertible, universally agreed translation between price and volatility. M2.4 is about what people say once they have that language.
Source: Mark Joshi, The Concepts and Practice of Mathematical Finance, ch.3–5 — take the option-pricing intuition and martingale-measure chapters and skip what is already obvious to you. Hull ch.13–15 for the formula and its mechanics. If you do one thing by hand, extend the binomial above to three periods and watch the price converge.