The Hedging Argument: Pricing Without Forecasting

Pricing an option before expiry looks like it needs a forecast of the underlying. It does not, and the reason reuses the trick from put-call parity: find something with an identical payoff.

The difference is that no static portfolio replicates an option. A call's sensitivity to the underlying changes as the underlying moves, from nearly zero when far out of the money to nearly one when deep in. Any fixed holding is wrong almost immediately.

The insight is to allow the portfolio to be rebalanced. Hold some quantity of the underlying, funded partly by borrowing, and adjust that quantity continuously as the price moves. If the adjustments can be chosen so the portfolio's value matches the option's in every scenario, then the two are the same instrument, and the option must cost what the strategy costs.

Fischer Black and Myron Scholes published this in "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, volume 81, issue 3, 1973, pages 637 to 654. Robert Merton developed the argument in parallel in "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, volume 4, 1973, pages 141 to 183.

The quantity to hold has a name. Delta is the rate at which the option's value changes with the underlying: hold delta units of the asset and the portfolio's small moves match the option's.

Holding delta of the underlying against a short option produces a position whose value does not change for small moves in the price. That is a delta-hedged position, and it is the manufacturing process.

The hedge is only instantaneously correct. Delta itself depends on the price, so once the price moves, the required holding has changed and the position must be adjusted. Rebalancing means buying more as the price rises and selling as it falls.

Notice what that trading pattern is: buying high and selling low, in small amounts, repeatedly. It loses money on average, and the amount it loses is exactly the cost of manufacturing the option. The premium a buyer pays is compensation for running that losing strategy on their behalf.

Here is the result that surprises everyone who meets it, and it is the real content of the theory.

The expected return of the underlying does not appear in the option price. Two traders who violently disagree about whether a stock will rise or fall must still agree on what its options are worth, provided they agree on volatility.

The reason follows from the hedge. The replicating portfolio is constructed so that its value does not depend on which way the price moves. Once the directional exposure is removed, the drift has nothing to attach to. Whether the asset tends upward or downward, the hedged position behaves identically.

What survives is volatility, because the cost of rebalancing depends on how much the price moves, not on where it ends up. A wilder path forces more buying high and selling low, and therefore costs more to hedge.

So an option is not a bet on direction that has been priced. It is a position in the magnitude of movement, and that is what the buyer is paying for.

Read the two branches as the whole theory. The left one removes something, the right one is what remains.

The left branch is why the model needs no view on the asset. Once the hedge cancels directional exposure, the drift cannot influence the price, so no forecast is required and no disagreement about prospects can show up in the quote.

The right branch is why volatility is the traded quantity. It is the one input that the hedge does not cancel, and consequently the one thing an option position is really exposed to.

This is also the practical reading of an option trade. Buying a call is not a way to be long the stock cheaply. It is a way to be long its volatility, with a directional exposure that a hedger can remove and a speculator chooses to keep.

Solving the replication problem under the model's assumptions gives a closed form.

import math
from statistics import NormalDist

N = NormalDist().cdf

def black_scholes(S, K, r, sigma, T, kind="call"):
    if T <= 0:
        return max(S - K, 0.0) if kind == "call" else max(K - S, 0.0)
    d1 = (math.log(S / K) + (r + sigma ** 2 / 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    if kind == "call":
        return S * N(d1) - K * math.exp(-r * T) * N(d2)
    return K * math.exp(-r * T) * N(-d2) - S * N(-d1)

black_scholes(100, 100, 0.04, 0.20, 1.0)              # about 9.925
black_scholes(100, 100, 0.04, 0.20, 1.0, "put")       # about 6.004

Five inputs: spot, strike, rate, volatility and time. Four are observable or contractual. Only volatility is not, and that single fact drives the rest of the subject.

A quick check on the two results: parity says call minus put should equal spot minus discounted strike, which is 100 - 100*exp(-0.04) or about 3.921. The difference above is 9.925 - 6.004, which matches.

The derivation is often summarised as valuing the option as if investors were risk neutral, discounting the expected payoff at the risk-free rate. That summary invites a misunderstanding worth heading off.

It is not a claim that anyone is risk neutral, or that the asset will grow at the risk-free rate. It is a computational device. Because the hedge removed directional exposure, the answer must be the same under any assumption about drift, so you may as well choose the one that makes the mathematics easiest. Risk neutrality is that choice.

The probabilities that appear are consequently not forecasts. They are risk-neutral probabilities: numbers that reproduce observed prices when used for discounting. They already contain whatever risk premium the market demands.

This matters when people read an option's implied probability of finishing in the money as a real-world forecast. It is not one, and for index puts the two differ substantially and systematically.

The argument requires the replication to actually work. Every assumption is a condition for that, and every one fails somewhere.

The jump row is qualitatively different from the others. The first four degrade the hedge; a jump defeats it. Between the last trade before a gap and the first after, delta was constant while the price moved a long way, and no amount of diligence helps.

That is why the market does not treat this model as true. It treats it as a language.

Given that list, it is fair to ask why anyone still uses it. The answer is that it is not being used as a description of the world.

Its real function is as a change of variables. The formula maps volatility to price, and it is monotonic, so it can be inverted: given a market price, solve for the volatility that reproduces it. That number is the implied volatility, and it is how options are actually quoted and compared.

This works even though everyone involved knows the model is wrong, because both sides use the same wrong model to translate. Quoting a price in implied volatility strips out strike, spot, rate and time, leaving a number comparable across contracts in a way raw premiums are not.

So the model survives as a unit of account rather than a theory of value. Its predictions are violated daily, and the violations are informative precisely because everyone agrees on the yardstick they are measured against.

What those violations look like, and what they reveal, is the last lesson. Before that, the next one covers what a hedged position is actually exposed to once you accept the hedge is imperfect.

Three things are now fixed, and they are worth separating from the formula that gets remembered.

First, an option is manufacturable. It is not an independent instrument whose price reflects opinion; it is a package that can be assembled from the underlying and cash, and the price is the assembly cost.

Second, the assembly cost depends on volatility and not on direction. That is why option markets are volatility markets, and why traders discuss vol rather than premium.

Third, the manufacturing is imperfect in practice, and each imperfection is a real exposure someone must hold. Discreteness, costs, changing volatility and jumps do not cancel; they are risks transferred to whoever is short the option.

Managing those exposures individually, rather than as one lump, is what the sensitivities in the next lesson are for.