ATM call value with a thousand-year maturity

Question

Assuming Black–Scholes dynamics, what is an at-the-money call worth today if it expires in $1000$ years and the stock price is $190$ ? Answer without a calculator.

Short answer

About $190$ . As maturity becomes huge, $d_1\to\infty$ and $d_2\to-\infty$ (and the discounted strike vanishes when rates are positive), so $C\to S_0$ . The call is effectively the stock itself.

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Lessons behind this question

The Black-Scholes PDE, Formula and Implied Volatility
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
Option Trading Strategies: Spreads, Combinations, and Payoff Engineering

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