Certain Upside, and the Twelve Dollars of Insurance You Do Not Need

A foreign share trades at 100. You know, by assumption and without qualification, that it finishes the year at 120. Interest rates are zero, there are no dividends, and you may only take a long position: buy the share, enter a forward, buy a future, or buy a call struck at 100. Which one?

The call attracts most people because options are where leverage lives. The right answer is the forward, and the argument that gets you there uses no distribution, no volatility and no pricing model. It is one algebraic identity applied to a payoff.

A call is a forward with a put stapled to it

Write $s$ for the terminal price and $X$ for the strike. For every value of $s$ whatsoever,

This is not an approximation and it is not a theorem about markets. Split into the two cases and it reduces to $s - X = s - X$ both times. Now read the same identity from the other direction. A long forward struck at $X$ pays $s - X$, so

as portfolios of payoffs, not as an analogy. Buying the call means buying the forward and, in the same transaction, buying a put. If you genuinely know the share rises, that put pays nothing and you have bought it anyway.

With rates at zero the forward price is today's price, $F = S_0 e^{rT} = 100$, and the strike was set at today's price too. So $F - X = 0$ and the parenthesis vanishes:

The at-the-money call and the at-the-money put cost exactly the same. Every penny of the call's premium is the price of a put, which is to say the price of protection against a fall. You have assumed the fall away. You are paying full price for insurance on a house you have declared fireproof.

What it costs to be wrong about this

For a size, put a volatility on the share. At the money with zero rates the standard European formula loses its cumulative normal entirely, because $d_1$ and $d_2$ collapse to $\pm\sigma\sqrt T/2$:

At $\sigma = 30\%$ and one year that is $11.9235$. The forward costs nothing to enter and pays the whole move, so the two profits at 120 are

The volatility fixed the size of that gap and had no say in its direction, because (3) does not contain $\sigma$. Drop the volatility to 10 percent and the call costs 3.9878, so it pays 16.01 and the gap narrows to a factor of 1.25. Raise it to 60 percent and the call costs 23.5823, at which point a certain rise to 120 loses you 3.58. The forward wins at every level, which is the mark of an answer that came from an identity rather than from arithmetic.

Why the share loses too, for a different reason

Notice from Fig. 1 that the share and the forward have identical profit lines. With zero rates they pay the same 20, so the argument against buying the share cannot be about payoff. It is about the balance sheet. Buying the share requires 100 today and returns 120, a 20 percent return on capital committed. The forward requires nothing, so its return on capital is not a smaller number than 20 percent. It is not a number at all.

That is the whole content of "maximum leverage", once you insist on a definition. Leverage measured per dollar of capital committed puts the forward first with no competition, because the denominator is zero and the position size is therefore unbounded. Leverage measured per dollar of premium paid is a different quantity, and on that measure a far out-of-the-money call can show a bigger multiple. But it only shows it on outcomes the question has excluded by assumption, so it is a comparison about a world you were told does not exist.

Buying the call is not a mistake in the sense of losing money. It makes 8.08 on a certainty. It is dominated, which is a different and slightly humiliating verdict: another legal position beats it in every state of the world you admit.

Futures, and the assumption doing the quiet work

A future on the same share with deterministic rates carries the same price as the forward, so its payoff is identical. What separates them is margin. A future is marked daily and calls for variation margin, which ties up cash on the way to 120 even though the destination is known. Under the problem's own assumptions the forward has no margin at all, so it wins by exactly the amount of that funding.

Which is where the honest reader should get suspicious. A forward with no margin is a counterparty extending you unlimited credit on your word. Real desks charge for that with initial margin and a collateral agreement, and the moment they do, the forward stops committing zero capital and the unbounded leverage becomes very bounded. The answer survives, but it survives because a modelling assumption was granted, not because leverage is free.

Where the reasoning stops holding

Positive rates break (3) rather than (1). With $r > 0$ the forward price $F = S_0e^{rT}$ sits above the strike, so $C - P = e^{-rT}(F - X) > 0$and the call is worth strictly more than the put. Part of its premium is then forward value rather than insurance, and the sentence "the whole premium is protection" becomes false. A forward struck at $F$ is still free to enter and still pays $120 - F$, so the ranking does not change, but the reason has to be restated against the forward instead of against today's price. A dividend yield does the same thing with the opposite sign.

Certainty is the load-bearing assumption. Downgrade it to a strong view and the put stops being worthless to you. The call becomes preferable as soon as the probability you assign to a fall, weighted by its size, is worth more than 11.9235, which for these parameters is exactly the market's own view. The whole exercise is a statement about how expensive disagreement with the market is.

And the argument assumes capital is the binding constraint. If your limit is written in notional rather than in cash, all three instruments give you the same 20 per share, the forward's zero capital buys you nothing, and the correct answer collapses back to a tie between the forward and the share. A candidate who says so has understood the problem better than one who only says "forward".

Sources and further reading

• The relation in the definition box, and its derivation from (1) — Put-call parity
• Forward prices, and why zero rates put them at today's price — Forward contract
• The margin mechanics that separate a future from a forward — Futures contract
• The formula in (4) — Black-Scholes model
• The general principle that (2) is an instance of — Rational pricing

The identity in (1) was checked symbolically and then at 20 001 grid points to machine precision. Equation (3) was checked on five terminal laws with mean 100, a lognormal, a two-point law, a uniform, a Student t with four degrees of freedom and an exponential, each with 400 000 draws, and the call and the put agreed to within four standard errors every time. A deliberate control with mean 120 separated them by 19.97, which is its own mean minus the strike, exactly as parity predicts. The 11.9235 was reproduced by a 4000-step binomial lattice at 11.9228, so nothing above depends on a normal distribution function.