Bounds — what an option cannot cost

The Derivatives subject split an option premium into intrinsic value and time value. That describes a price that already exists. This subject asks where the price comes from — and the first answer needs no model at all.

The one principle this subject runs on

Everything here rests on a single idea, which the source states as the static hedging principle:

If it is possible to construct a portfolio whose value at the maturity of the European style derivative exactly matches the value of the derivative, then according to 'no free lunch' principle, the initial price of the portfolio should be the same as the price of the new derivative.

And the reason, in the same place: otherwise "some very lucrative trading strategies arise where one can make money, essentially risk-free and from nothing."

That sentence is doing more work than it looks. It does not say prices are fair, or that markets are efficient, or that anybody is right about the future. It says only that two things with the same payoff in every state must cost the same, or there is free money. That is arithmetic.

European first, and why it matters

One definition has to come before the arithmetic. A European option can be exercised only at expiry. An American option can be exercised at any time until expiry.

Indian index options are European, and the exchange says so in the contract specification: the options "shall be European styled which can be exercised only on the expiration date", with final settlement in cash against the index's closing price.

Every result in this chapter and the next assumes European exercise. The American case is different, because the right to act early is worth something and that value has to be priced. Chapter 5 handles it.

The easy bounds

Write $C$ for the price of a European call, $S$ for the underlying now, $K$ for the strike, $T$ for the time to expiry in years, and $r$ for the interest rate.

A call cannot be worth more than the underlying.

$C \le S$

The call's payoff is $\max(S_T - K, 0)$, which is never more than $S_T$. A thing that pays less in every state cannot cost more. If a 24,000 call were quoted at 24,500 you would sell the call, buy the index, and keep ₹500 whatever happened.

A call cannot be worth less than nothing.

$C \ge 0$

An option is a right, never an obligation. Nobody pays you to hold a right.

Those two are obvious. The next one is not.

The floor that sits above intrinsic value

The lower bound on a European call is

$C \ge S - K e^{-rT}$

Not $S - K$. The strike is discounted, because you do not pay it until expiry.

Why it holds. Compare two portfolios held to expiry:

	Portfolio A	Portfolio B
Now	buy the call, and lend $Ke^{-rT}$	buy the underlying for $S$
At expiry, if $S_T > K$	exercise, paying $K$ out of the matured loan → $S_T$	$S_T$
At expiry, if $S_T \le K$	let it lapse, keep the matured loan → $K$	$S_T$

A is never worse than B, and is sometimes better — when the underlying ends below the strike, A holds $K$ and B holds less. So A cannot cost less than B:

$C + Ke^{-rT} \ge S$

which rearranges to the bound. Since a call also cannot be negative, the full statement is

$C \ge \max\!\left(0,\; S - Ke^{-rT}\right)$

What that floor is worth in rupees

Take an index at 24,000, thirty days to expiry, and 6.5% a year.

Strike	Intrinsic value	No-arbitrage floor	Floor above intrinsic
24,000	₹0	₹127.88	₹127.88
23,000	₹1,000	₹1,122.55	₹122.55
22,000	₹2,000	₹2,117.22	₹117.22

Each floor is $S - Ke^{-rT}$ with $S = 24{,}000$, $T = 30/365$ and $r = 6.5\%$.

The first row is the striking one. Intrinsic value is zero, and the call still cannot trade below about ₹128 — on arithmetic alone, with no opinion whatever about volatility. The Derivatives chapter on what an option price is made of would call that ₹128 time value; this chapter shows part of it has nothing to do with time passing. It is the interest you save by not paying the strike today.

And there is the first practical consequence. A deep in-the-money call trading at intrinsic value is not cheap, it is mispriced — and the arithmetic says by how much.

The put bounds

The same argument, mirrored:

$\max\!\left(0,\; Ke^{-rT} - S\right) \le P \le Ke^{-rT}$

The upper bound is the interesting side. A put cannot be worth more than the discounted strike, because the most it can ever pay is $K$, when the underlying goes to zero. A claim to at most $K$ at time $T$ cannot cost more than $Ke^{-rT}$ now.

Note the asymmetry with the call, which has no finite ceiling other than $S$ itself. A bought put has a maximum payoff; a bought call does not. That is the asymmetry the Derivatives payoff diagrams draw, restated as a price limit.

Two bounds across strikes

Two more that need no model and that catch real quoting errors.

Monotonicity. Same expiry, lower strike, worth at least as much:

$K_1 < K_2 \implies C(K_1) \ge C(K_2)$

The lower-strike call pays at least as much in every state. If the 23,800 call were cheaper than the 24,000 call, you would buy the first and sell the second and hold a position that can only pay you.

Convexity. For three equally spaced strikes,

$C(K_1) + C(K_3) \ge 2\,C(K_2)$

The portfolio on the left minus the portfolio on the right is a butterfly, and its payoff at expiry is a triangle that is never negative. So it cannot cost less than nothing. In a 30-day chain built on the numbers above, the 23,800/24,000/24,200 butterfly comes to ₹567.11 + ₹350.35 − 2 × ₹450.57 = ₹16.31, comfortably positive.

Why these matter practically. An option chain quotes dozens of strikes at once, and these inequalities test them against each other with no model at all. Chapter 11 runs exactly this kind of check, because a chain that violates convexity is telling you something about the quotes rather than about the market.

Working the problem

Index 24,000, a 30-day 23,000 call quoted at ₹1,050, rate 6.5%.

Step 1 — compute the floor.

$S - Ke^{-rT} = 24{,}000 - 23{,}000 \times e^{-0.065 \times 30/365} = 24{,}000 - 22{,}877.45 = ₹1{,}122.55$

The quote is ₹72.55 below a floor that holds on arithmetic. It is not cheap; it is impossible.

Step 2 — the trade. Buy the cheap portfolio, sell the dear one.

Today	Cash flow
Buy the call	−₹1,050.00
Lend ₹22,877.45 for 30 days at 6.5%	−₹22,877.45
Sell the index short at 24,000	+₹24,000.00
Net received today	+₹72.55

Step 3 — check every state at expiry. The loan matures at exactly ₹23,000.

If the index ends above 23,000 — exercise the call, pay the ₹23,000 strike out of the matured loan, take the index and return it to close the short. Every leg cancels. Net zero.

If the index ends at or below 23,000 — let the call lapse. Spend $S_T$ of the matured ₹23,000 buying the index back to close the short, and keep ₹$(23{,}000 - S_T)$, which is positive.

So the position ends at zero or better in every state, having paid ₹72.55 up front. That is the free money the no-arbitrage principle says cannot exist — which is why the quote cannot either.

Three honest caveats, because this is a textbook trade and markets are not textbooks.

You must be able to short. In Indian index markets the practical short leg is a futures contract rather than a borrowed basket, and the futures price carries its own basis — chapter 12's territory.

Costs eat small discrepancies. ₹72.55 is before brokerage, exchange charges and securities transaction tax. The Derivatives chapter on costs applies in full, and chapter 12 puts current numbers on it.

Which is why the bounds hold in practice. Not because anyone enforces fairness, but because professionals with low costs watch for violations continuously. A bound is a description of what competition has already done, not an opportunity waiting for you.

The point

Before any model, arbitrage alone fixes limits: a European call sits between $\max(0, S - Ke^{-rT})$ and $S$, a put between $\max(0, Ke^{-rT} - S)$ and $Ke^{-rT}$, lower strikes cost more, and butterflies cost something. The lower bound is the one worth carrying, because it sits above intrinsic value by the interest on the strike — so a deep in-the-money call at intrinsic value is mispriced on arithmetic, with no view on volatility needed. All of it assumes European exercise, which is what Indian index options have.