What Black-Scholes assumes

Every assumption below is false. That is not a criticism of the model — a model is a set of assumptions — and the useful question is never "is it true" but "where does the error go".

The assumptions, and what each is holding up

Assumption	What it is doing
Volatility is constant and known	supplies the one unobservable input
The price path is continuous, with no jumps	makes the hedge always adjustable
Trading is continuous and costless	makes the hedge free to run
Returns are lognormally distributed	fixes the shape of the terminal distribution
Lending and borrowing happen at one known rate	prices the financing leg
The underlying pays no dividends	keeps the carry simple
The underlying is infinitely divisible and freely shortable	makes any hedge ratio attainable
Exercise is European	removes the timing decision

They are not independent, and they are not equally damaging. Taking them in order of how much they cost you is the useful exercise.

The worst one: volatility is constant and known

This is the assumption everything else is downstream of, and it fails in the most visible possible way.

If the model held, every option on the same underlying would quote at the same volatility. Derman and Kani put the implication precisely: if Black-Scholes is correct, "all options on the index must have the same implied volatility."

They do not. From the same source:

ever since the '87 crash, the market's implied Black-Scholes volatilities for index options have shown a negative relationship between implied volatilities and strike prices — out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls.

And on what that means:

Though the exact shape and magnitude vary from day-to-day, the asymmetry persists and belies the BS theory, which assumes constant local (and therefore, constant implied) volatility for all options.

The verdict in their own words is unusually blunt for a research note. It is convenient to keep quoting prices in Black-Scholes-equivalent volatilities, they write, "but it is probably incorrect to calculate options prices using the BS formula."

Two distinct failures hide inside this one. Volatility is not constant across strikes — the smile, which chapter 11 is about. And it is not constant over time — it moves, which is why chapter 10 treats volatility as something traded rather than assumed.

The one that costs money: no jumps, and continuous costless trading

These two break together, because they are both about whether the hedge of chapters 3 to 5 can actually be run.

The hedge requires rebalancing at every node. Chapter 5 showed the holding moving from 0.554 to 1.000 or 0.000 in fifteen days. In the continuous limit it must be adjusted constantly.

Two things stop that.

Gaps. Prices do not pass through every intermediate value. An index that closes at 24,000 and opens at 23,300 never traded at 23,650, so there was no moment at which to adjust. The replication argument assumed there was. Derman and Kani note jump models as one of the standard extensions, and chapter 9 shows that a gap is precisely where a delta-hedged position stops being hedged.

Costs. Every adjustment pays a spread, brokerage, exchange charges and tax. Chapter 12 gives the current Indian rates; the aggregate is in SEBI's data, where individual traders incurred transaction costs of about ₹25,000 crore in each of FY25 and FY26 — set against FY26 net losses of ₹91,685 crore, as the Derivatives subject sets out.

The direction of this error is knowable. Costly hedging makes an option more expensive to manufacture than the formula says, so a seller who has to hedge needs more than the model price. That is one of several reasons market prices sit above model prices, and it is why implied volatility is typically a little higher than realised volatility.

The one everybody notices: lognormal returns

Real return distributions have fatter tails than a normal distribution — extreme moves are more frequent than the model allows.

Where the error goes is specific. Options far from the money are the ones whose value depends almost entirely on the tails. Understating the tails understates them, which is why the market charges more for them than the formula does — and the smile of chapter 11 is partly the market pricing a distribution the model does not have.

It matters much less for options near the money, whose value depends on the middle of the distribution where the normal approximation is decent. So "returns aren't normal" is a serious objection to pricing a 20,000 put and a weak objection to pricing a 24,000 call.

The ones that are mostly bookkeeping

One interest rate, for both lending and borrowing. You cannot borrow at the rate you can lend. The error is small over thirty days — chapter 6's whole financing term was ₹128 on a 24,000 index — and grows with maturity. For long-dated contracts it stops being negligible, which is why SEBI permits index option maturities of up to five years.

No dividends. False for stocks and for index futures. The fix is mechanical and chapter 12 covers how Indian contracts handle it, including SEBI's rule that dividends below 2% of market value are treated as ordinary and produce no strike adjustment.

European exercise. True for Indian index options, which "shall be European styled." For stock options it is a contract term to check, and chapter 5 showed what the early-exercise right is worth.

The one that is distinctively Indian: divisibility and margin

The textbook hedge holds 0.5541 units of the index. You cannot.

Contracts come in lots. SEBI requires index derivative lot sizes to be fixed so that contract value falls between ₹15 lakh and ₹20 lakh, with the lot size a multiple of 5 and not less than 10. So hedge ratios are available in steps, and for a small position the steps are coarse relative to the position.

And hedging requires capital. The model's portfolio is self-financing; a real short option position posts initial margin sized to a 99% one-day VaR, with a price scan range of "SIX standard deviations or 9.30% of the underlying value, whichever is higher", plus extreme loss margin, plus an additional 2% ELM for short options on expiry day, and since 1 February 2025 upfront collection of net option premium payable.

That capital has a cost the formula does not price, and it is the mechanism by which a correctly hedged seller can still be forced out of a position — the Financial institutions subject's forced-selling argument applied to a derivatives book.

Why the fixes are not free

The obvious response is to build a better model, and the obvious models exist: let volatility be random, or let the price jump. Derman and Kani name both — and then name the price.

since there are no securities with which to directly hedge the volatility or the jump risk, options valuation is in general no longer preference-free.

That is the sentence to understand. Chapter 3's result — that the price does not depend on anyone's beliefs or risk appetite — held because the option could be perfectly replicated. Introduce a risk you cannot hedge and the replication argument breaks, so the price starts depending on preferences again.

And the practical cost, same source: in multifactor models, "options values depend upon several additional parameters whose values must be estimated. This often makes confident option pricing difficult."

So the trade-off is real and permanent. A simple model with known failures, or a richer model with more unobservable inputs and no clean hedging argument. The industry's response was neither: it kept the formula as a quoting convention and let implied volatility absorb the discrepancy, which is chapter 11's subject.

Working the problem

A retail buyer holds a weekly index call to expiry, rebalances nothing, and pays costs.

Which assumptions that breaks — and then the more important point. Costless continuous trading, obviously. But the buyer is not attempting to hedge at all, so the honest answer is that they have broken no assumption of the pricing argument. They have stepped outside what it is about.

What the formula is the price of. Chapters 3 to 6 derived one thing: the cost of manufacturing the option by trading the underlying. That is a statement about a hedged position, whose outcome is nearly certain. It is not a statement about the distribution of outcomes for someone holding the option naked.

So the question "is the option overpriced for them?" is the wrong question. The price can be exactly right — no arbitrage available, implied volatility reasonable — and the buyer's expected outcome still be poor. Consider what they hold:

A near-certain total loss with a small chance of a large gain. Chapter 9 quantifies the decay; for a weekly out-of-the-money option the entire premium is time value, and time value goes to zero with certainty unless the index moves far enough, fast enough.

Costs on a small base. A weekly premium is a small number, and brokerage, exchange charges and securities transaction tax are a large percentage of a small number. Chapter 12 gives the rates.

A payoff shape, not an expected return. The Behavioural finance subject's finding is the relevant one: investors show a measurable preference for high idiosyncratic skewness — mostly small losses with a small chance of something enormous. A weekly out-of-the-money option is close to the purest available form of that shape.

The one-sentence answer. The option is not overpriced; it is correctly priced as a hedging instrument and is being used as a lottery ticket, and the SEBI data in the Behavioural finance subject — roughly nine in ten individual derivatives traders losing money every year from FY22 to FY26 — is what that substitution costs at scale.

Which is the right note for this chapter to end on. Listing broken assumptions is the easy part of criticising a model. The harder observation is that a model can be approximately right about the thing it prices and completely irrelevant to what you are actually doing with the contract.

The point

Volatility is not constant across strikes or over time, which the persistence of the smile establishes and which the source says makes it "probably incorrect" to price with the formula. Paths jump and trading costs money, so the replicating hedge cannot be run as specified — and because hedging is dearer than the model assumes, market prices sit above model prices. Tails are fatter than lognormal, which matters for far strikes and little near the money; rates, dividends, exercise style, lot sizes and margin are corrections of varying size. Fixing the big ones costs you the preference-free property that made the argument work, which is why the market kept the formula as a quoting language instead. And the failure that matters most to an individual is not in the list: the formula prices a hedged position, and an unhedged buyer owns a different thing.