Replication, and why a price is not a forecast

Chapters 1 and 2 bounded and tied prices together without ever pricing one. This chapter prices one — and the result is strange enough that, as the source notes, "the pricing formulas were suspected before the seminal paper of Black and Scholes was published and even the authors first doubted their findings."

The simplest possible market

Strip everything away. An index stands at 24,000. In one month it will be either 25,200 or 22,800 — nothing in between, nothing outside. The one-month interest rate is 0.5%, so ₹1 lent becomes ₹1.005.

A 24,000 call on this index pays ₹1,200 if the index rises and ₹0 if it falls.

Now the question: what is the call worth today?

The instinct is to reach for probabilities. Resist it for a page.

Rebuild the option out of other things

Buy $\Delta$ units of the index and borrow $B$ rupees. The portfolio is worth $\Delta \times 24{,}000 - B$ today, and at expiry:

$\text{up: } \Delta \times 25{,}200 - 1.005B \qquad \text{down: } \Delta \times 22{,}800 - 1.005B$

Choose $\Delta$ and $B$ so that both match the call. Two equations, two unknowns.

Subtracting the down case from the up case removes $B$ entirely:

$\Delta \times (25{,}200 - 22{,}800) = 1{,}200 - 0 \quad\Longrightarrow\quad \Delta = \frac{1{,}200}{2{,}400} = 0.5$

Then from the down case, where the call pays nothing:

$0.5 \times 22{,}800 - 1.005B = 0 \quad\Longrightarrow\quad B = \frac{11{,}400}{1.005} = ₹11{,}343.28$

Check the up case: $0.5 \times 25{,}200 - 1.005 \times 11{,}343.28 = 12{,}600 - 11{,}400 = ₹1{,}200$. It matches.

So half a unit of the index, funded with ₹11,343.28 of borrowing, pays exactly what the call pays, in both states. What does that recipe cost?

$0.5 \times 24{,}000 - 11{,}343.28 = ₹656.72$

That is the price of the call. Not an estimate of it. By the static hedging principle of chapter 1, anything else is free money for somebody.

Where the probability went

Look back over that calculation. The probability of the rise never appeared. Not as an assumption, not as a parameter, not anywhere.

This is the single most counter-intuitive fact in option pricing, and it is worth making concrete. Suppose you are convinced the index rises with probability 0.8:

Your probability of a rise	Expected payoff	Discounted	Arbitrage-free price
0.20	₹240	₹238.81	₹656.72
0.50	₹600	₹597.01	₹656.72
0.80	₹960	₹955.22	₹656.72
0.90	₹1,080	₹1,074.63	₹656.72

The right-hand column does not move. Four investors with wildly different views on the index agree on the price of the option, because the price is the cost of a recipe and the recipe does not care.

Derman and Kani state the general principle:

The primary feature of the theory is that it is preference-free — the values of contingent claims do not depend upon investors' risk preferences. Therefore, you can value an option as though the underlying stock's expected return is riskless. This risk-neutral valuation is allowed because you can hedge an option with stock to create an instantaneously riskless portfolio.

The hedge is what does it. Because the seller can neutralise the option with the underlying, the seller's opinion about direction stops being relevant to what the option must cost.

The same answer by a different route

There is a second way to get ₹656.72, and it is the one most texts lead with — which is unfortunate, because it looks like a probability calculation and is not.

Ask: what probability of a rise would make the index itself fairly priced if investors demanded no risk premium? The index must then be expected to grow at the interest rate:

$q \times 25{,}200 + (1-q) \times 22{,}800 = 1.005 \times 24{,}000 = 24{,}120$

$q = \frac{24{,}120 - 22{,}800}{25{,}200 - 22{,}800} = \frac{1{,}320}{2{,}400} = 0.55$

Price the call with that number:

$\frac{0.55 \times 1{,}200 + 0.45 \times 0}{1.005} = \frac{660}{1.005} = ₹656.72$

Identical, and necessarily so — it is the replication arithmetic rearranged. The 0.55 is called the risk-neutral probability.

It is not a forecast and not a belief. The source is blunt about what it is: the authors prefer to work with tradeable securities "instead of fictitious risk-neutral probabilities." The 0.55 is a bookkeeping weight that makes a discounted-expectation calculation reproduce a replication cost. Nobody thinks the chance of a rise is 55%.

Why this matters for reading market prices

This is where the chapter earns its title, because a specific confusion is everywhere.

"The options market is pricing a 30% chance of a crash." Sentences of that form are published constantly, and they are a misreading. What option prices contain is a risk-neutral distribution, which differs from any real-world distribution by exactly the risk premium investors demand for bearing that risk. A high price on a crash put reflects both the probability of a crash and how desperate people are to avoid one — and the price cannot separate them.

So implied probabilities systematically overstate bad outcomes, because insurance against bad outcomes is something people pay extra for. Chapter 11 returns to this: the shape of implied volatility across strikes is partly a distribution and partly a price of fear, and no amount of arithmetic will tell you which part is which.

The honest reading of an option price is therefore: this is what it costs to transfer this risk, not this is how likely the event is.

What the price does depend on

If not the direction, then what? Everything in the recipe:

The spread between the two outcomes. Widen them to 26,400 and 21,600 and the call is worth more — not because a rise is likelier but because the recipe needs more of the index and more borrowing. That spread is volatility, and it is the one input that genuinely moves the price. Chapters 4 to 6 build this out properly.

The interest rate, because the recipe is financed.

The strike and the time, because they set the payoff.

Not the drift. The source analyses this directly and calls the irrelevance of the trend parameter $\mu$ "a bit of a paradox" — showing that the effect of $\mu$ on prices vanishes as the time step shrinks, so that the parameter "does not appear in the Q-distribution which is used in pricing options."

Working the problem

You believe the rise has probability 0.8. The expected payoff, discounted, is ₹955. The call costs ₹656.72. Is the call cheap?

Why ₹656.72 is still the right price. The seller does not take your bet. They sell you the call for ₹656.72, immediately buy 0.5 units of the index with ₹11,343.28 of borrowed money, and have no exposure left to hedge. At expiry their position settles to exactly what they owe you, in both states. Their profit does not depend on which state occurs, so they will sell at ₹656.72 however confident you are. Your conviction has no counterparty to persuade.

And if the market did price at ₹955. Then anyone — including you — could sell the call at ₹955, build the ₹656.72 replicating portfolio, and pocket ₹298.28 with no exposure at all. That is the free money chapter 1 rules out, and it is why ₹955 cannot be a market price.

What your view actually is. You believe the index will probably rise. That is a view about the index, and the instrument that expresses it is the index. Buying the call at ₹656.72 expresses the same directional view plus two things you have not formed a view on: the size of the move, and leverage. If the index rises 5% to 25,200 you make ₹543 on a ₹657 outlay — but a 1% rise to 24,240 pays you nothing at all in this two-state world, where the index position would have paid ₹240.

So what you should do. If the view is directional, take it in the underlying, or sized deliberately in the future. If the view is specifically that the move will be larger than the option price implies, then the option is the right instrument — but that is a different view, and you should be able to say what move size you are disagreeing with. An option is not a leveraged opinion about direction; it is an opinion about distribution.

The Behavioural finance subject adds the uncomfortable coda. Confidence of the "80% chance" kind is the overconfidence that chapter measures, and the SEBI data there is what happens to people who express directional views through options at scale.

The point

Choose a holding of the underlying and a loan that pay exactly what an option pays in every state, and the option's price is the cost of that portfolio — in the two-state example, 0.5 units of the index funded by ₹11,343.28 of borrowing, costing ₹656.72. The probability of a rise never enters, because the seller hedges and so has no exposure to direction; the theory is preference-free, and the "risk-neutral probability" of 0.55 is bookkeeping, not a forecast. What moves the price is the spread between outcomes, the rate, the strike and the time — never the drift. So an option price is the cost of transferring a risk, not an estimate of how likely the event is.