A Call's Straight Part Crosses at the Discounted Strike, Not the Strike

Sketch a one-year call struck at 100 with a five percent rate. Deep in the money the curve straightens into a line of slope one, and that line crosses at 95.122942 rather than at 100, so drawing it through the strike is out by 4.877058 for ever. That gap is the interest saved on the strike, and it is also why an American call on a share paying no dividends is never exercised early. Plot the same option against the futures price and the crossing returns to 100 while the slope drops to 0.951229.

Sketch a European call struck at 100, with a year to run, on a share paying no dividends, at a continuously compounded rate of five percent and a twenty percent volatility. Draw it three times: the payoff at maturity against the terminal share price, the price today against the futures price, and the price today against the spot.

The first sketch nobody gets wrong. The third is where the money is, because deep in the money the curve straightens into a line of slope one and almost everyone draws that line through the strike. It crosses at 95.122942 instead, and the difference is permanent.

At maturity, and only at maturity

With no time left the option is worth what it pays, and what it pays is a hinge:

c(S_T, T) = (S_T - X)^+ = \max(S_T - 100,\, 0)

(1)

Flat along the axis up to 100, then a straight line of slope one. Two straight pieces, one corner, no discounting anywhere, since there is no time left to discount over. Every feature of this picture that people carry into the next one is a feature that only holds at maturity.

Where the straight part actually crosses

Before maturity the price is smooth, sits above the hinge everywhere, and straightens out on the right. The formula:

c = S\,\Phi(d_1) - Xe^{-r\tau}\,\Phi(d_2), \qquad \tau = T - t

(2)

Deep in the money both normal distribution functions go to one, so (2) collapses to $S - Xe^{-r\tau}$ . That is a line of slope one whose foot sits at the discounted strike:

Xe^{-r\tau} = 100\,e^{-0.05} = 95.122942

(3)

The reason is worth saying without symbols. Deep in the money the option will almost certainly be exercised, so what you are holding is the share now and the strike later. You will get the share and you will pay 100, but you will pay it in a year, and 100 in a year is 95.122942 today. Drawing the line through 100 charges you for the strike immediately, so it understates the option by the interest you save:

X\left(1 - e^{-r\tau}\right) = 100 - 95.122942 = 4.877058

(4)

Fig. 1 — The two candidate straight parts. They are parallel, so the wrong one never catches up: the gap stays at 4.877058 at every share price.

The step that carries the content

Collapsing (2) by letting two functions go to one is the right idea stated carelessly, because one of the terms it discards is a tail multiplied by a share price that is running off to infinity. Write the difference out:

c - \left(S - Xe^{-r\tau}\right) = -\,S\big(1 - \Phi(d_1)\big) + Xe^{-r\tau}\big(1 - \Phi(d_2)\big)

(5)

The second term is a constant times a tail, so it vanishes without argument. The first is the one with content, and it vanishes because a Gaussian tail decays faster than any power:

\lim_{S \to \infty} S\big(1 - \Phi(d_1(S))\big) = 0

(6)

Note that $d_1$ grows like $\ln S$ , so the tail decays like $e^{-(\ln S)^2/(2\sigma^2\tau)}$ , which beats $S$ comfortably. The limit in (6) is the whole reason the asymptote exists.

The same option against the futures price

Now change the horizontal axis. The futures price for delivery at maturity is $F = Se^{r\tau}$ , so $dS/dF = e^{-r\tau}$ and the chain rule gives the slope in the new picture:

\frac{\partial c}{\partial F} = e^{-r\tau}\,\Phi(d_1) \;\longrightarrow\; e^{-r\tau} = 0.951229

(7)

And the foot moves back to the strike, because $F = X$ and $S = Xe^{-r\tau}$ are the same event written in two currencies. So the futures picture has its corner at 100 and a straight part of slope 0.951229, while the spot picture has its corner at 95.122942 and a straight part of slope one.

Fig. 2 — Same option, same day, different axis. Here the foot returns to the strike and the straight part is the one that flattens.

One discount factor is doing both jobs. Against the spot it moves the foot and leaves the slope alone; against the futures it leaves the foot alone and takes the slope. Anyone who can say which of the two is happening has understood the sketch.

The gap has a second name

Deep in-the-money time value

The amount by which a European call exceeds its intrinsic value $S - X$ when exercise is a near certainty. From (4) it tends to $X(1 - e^{-r\tau})$ , the interest on the strike over the remaining life, and it does not go to zero until $\tau$ does.

That number, 4.877058, is also the standard reason an American call on a share paying no dividends is never exercised early. Exercising hands over the strike today rather than at maturity, which throws away exactly the quantity in (4) and buys nothing in return, since the share can be held either way. Two arguments from different directions landing on the same number is a reason to trust it.

What the sketch hides

The convergence is slower than the picture suggests. At a spot of 200 the gap to the line through the strike is 4.877724, not 4.877058, so quoting the limit there is wrong in the fourth decimal. The gap to the correct line falls below one part in a million only at a spot of 263.75 and below one part in a billion by 400. Past that point double precision runs out of room: the gap is a difference of two numbers of size four hundred, quantised at about 6e-13, and it stops decreasing strictly.

Dividends change both features at once. With a continuous yield $q$ the asymptote becomes $Se^{-q\tau} - Xe^{-r\tau}$ , so the slope against the spot falls below one and the foot moves to $Xe^{-(r-q)\tau}$ , which can land on either side of the strike. The same caveat applies to a negative rate, where the discount factor exceeds one and the foot moves to the right of 100.

The put is the mirror image and it is worth drawing once. As the share falls towards zero, a European put approaches $Xe^{-r\tau} - S$ , a line of slope minus one whose intercept is the same discounted strike. Both straight parts of both contracts are anchored at 95.122942, and the strike itself never appears on the picture until the day the option expires.

Sources and further reading

The formula behind (2) — Black-Scholes model
The straight part these curves approach — Asymptote
Why (3) is where the strike really sits — Present value
The relation used in (7) — Spot-future parity
The valuation being sketched — Valuation of options

Twenty-nine checks stand behind the figures. The gap to the correct line was evaluated out to a spot of ten million rather than argued for, the gap to the strike line was confirmed to stay above 4.87 at five hundred spots reaching to fifty thousand, and the futures slope in (7) was matched against the share count times the discount factor away from the limit as well as at it.