54 Dollars of Exposure for 8.76, and a Beta of 6.83

A share trades at 100. A one-year call on it, struck at 100, costs 8.76. What is that call worth tomorrow?

The confident answer is 8.76, and it comes with a reason that sounds unanswerable: an option price is a martingale, a fair game has no drift, so the best forecast of tomorrow is today. Every word of that is true under one probability measure and false under the one you actually live in. The answer is higher, and by a factor worth knowing.

A true premise attached to the wrong measure

Pricing works by discounting under an artificial probability law, chosen so that every traded asset earns the riskless rate. Under that law the discounted option price really is a martingale and its expected change really is zero. Simulating two million paths under it puts the expected price tomorrow at 8.7582 against 8.7591 today, inside two standard errors of no change.

Switch to the real law, where the share carries a risk premium, and the same two million paths give 8.7757, thirty-one standard errors above today. Nothing about the option changed. The probabilities did. So the reflex is not an arithmetic error, it is an answer to a different question, and the interviewer asked the one about the world.

Price and delta, with no cumulative normal in sight

Set the rate to zero, no dividend, strike equal to spot, a swing of 22 percent a year and one year to run. At the money with zero rates the two log-moneyness arguments become $\pm\sigma\sqrt{T}/2$ and the formula collapses to a single error function:

A 4000-step lattice with no normal distribution anywhere in it returns 8.7585 and 0.543793, so (1) is not a formula being trusted, it is a formula being confirmed.

Now read the delta as a position rather than as a derivative. The call moves 54 cents for every dollar the share moves, which is the same market exposure as owning 0.543795 of a share, and 0.543795 of a share at 100 is 54.38 dollars of exposure. You paid 8.76 for it.

The number that does all the work

Six dollars of share exposure per dollar of capital. The derivation of (2) needs no pricing model at all: for any smooth function of the share, $dc/c = (\partial c/\partial S)(S/c)\,dS/S$, so the option's instantaneous return is the elasticity times the share's return. Black-Scholes enters only to supply the two numbers in the fraction.

From leverage to market sensitivity

A market sensitivity is a covariance with the market divided by the market variance. The elasticity multiplies the option return by a constant, and a constant passes straight through a covariance, so it multiplies the sensitivity by the same constant:

A share that follows the market at 1.10 carries a call that follows it at 6.83. Apply the same multiplier to a required return rather than to a move: if the market premium is 7 percent a year then the share is expected to earn 7.7 percent, and the call is expected to earn 47.8 percent.

Measuring the multiplication instead of asserting it

Regressing the call's one-day return on the market's over two million simulated paths returns a slope of 6.834 against the 6.8292 that (3) predicts, while the share's own slope comes back at 1.100. The expected-return ratio measures 6.209 against the elasticity 6.208.

Getting that agreement took one correction worth reporting, because it is the kind of thing that silently ruins a simulation. The share was first driven by a market factor at sensitivity 1.10 on a 16 percent market swing plus an 18 percent idiosyncratic swing, which gives the share a total swing of 25.2 percent while the option was being priced on 22. A gamma term feeds on that mismatch every day and the measured ratio came out at 8.2 rather than 6.2. Setting the idiosyncratic swing to 13.2 percent, so that $\sqrt{(1.10 \times 0.16)^2 + 0.132^2} = 0.22$ exactly, brings the measurement back onto the theory. If a simulated option is priced on a volatility its simulated underlying does not have, the option is being paid for a mispricing rather than for risk.

Why the folklore figure is not reproducible

This question circulates with a stock answer of about 6.7, and it is worth being blunt about where that number comes from. It comes from a parameter set nobody states. The elasticity in (2) depends on the volatility, the time to expiry and the moneyness, and none of the three is implied by the words of the question. Change the volatility to 30 percent at the same one-year at-the-money contract and the elasticity falls; shorten the expiry and it rises. Any specific multiple is a claim about a specific contract.

So the defensible answer has two layers. The general statement is that a call has a strictly larger market sensitivity than its underlying, and the reason is the inequality below, not any arithmetic. The number is whatever the parameters you were handed produce, and if you were not handed any, say which ones you are assuming before you quote a figure. That is the difference between an answer and a memorised one.

Where 6.83 stops being the number

The general statement is the inequality, not the figure. With zero rates $c = S\Phi(d_1) - X\Phi(d_2)$ and $\Phi(d_2) > 0$, so the exposure $S\Delta$ is strictly larger than the price and $\Omega > 1$ always. A call is a levered claim on its underlying, without exception.

The size, though, is local. Far out of the money the elasticity is enormous, near 14 at a share price of 60 on this contract, and deep in the money it settles towards one as the call turns into the share. So 6.83 describes this call today. It does not describe the same call after a large move, which is why the regression above was run over one day rather than over the year.

One more caution about the word higher. It is an expectation, not a forecast. On these parameters the share has to gain about 2.2 basis points overnight for the call merely to hold its value, since time decay takes something even when nothing moves, and the share clears that bar on slightly fewer than half of all nights. So the call closes lower on a shade over half of them and closes higher on average, which is what a positive expectation on a convex payoff looks like from the inside.

Sources and further reading

• The sensitivity measure in (3) — Beta (finance)
• The quantity defined in (2), under its trading-desk name — Greeks (finance)
• The measure the reflex is silently using — Risk-neutral measure
• Where (1) comes from — Black-Scholes model
• The premium that turns 7.7 percent into 47.8 — Capital asset pricing model

Every figure here was reproduced twice: the price, delta and elasticity by a 4000-step lattice containing no normal distribution, and the sensitivity chain by least squares on two million simulated one-day returns, with both probability measures run as separate controls.