Long Maturity

Consider an at-the-money call option on a stock with an initial price of $190 and maturity $$T=1000$$ years. Determine its price at $$t=0$$ under Black–Scholes.

Answer

We start from the BS formula for a non-dividend stock:

C = S\,N(d_1)\;-\;K\,e^{-rT}N(d_2).

As $T\to\infty$ the second term $K\,e^{-rT}N(d_2)\to0$ , so

C \approx S\,N(d_1).

i.e. approximate to the stock itself times the $\Delta$ of the option.

Now, at very short maturities an ATM call’s delta $$N(d_1)$$ behaves like a digital (jumping between 0 and 1), but as $$T$$ grows the $\Delta$ – $$S$$ curve "flattens" and the ATM delta rises toward 1 (see the questions above for this). Hence,

C\approx S = \$190.

Notes and comments

Comment 1: By the same logic, an ATM put would have $\;P = K\,e^{-rT}N(-d_2) - S\,N(-d_1)\approx0 - 0=0,$ so essentially worthless.