Why a Black–Scholes Call Approaches the Stock Price at Extreme Volatility

Summary

The document poses an apparent contradiction in the Black–Scholes model. As volatility rises without bound, the probability that a geometric Brownian motion finishes above the strike tends toward zero, which seems to imply a worthless call. Yet the Black–Scholes call price tends toward the current stock price. The central issue is that an option’s value is not simply the probability of finishing in the money.

A call payoff depends on how far the terminal stock price exceeds the strike, and under the model an increasingly dispersed distribution can produce rare, very large payoffs. Those outcomes can sustain value even as the probability of ending above the strike falls. The prompt provides no derivation or answer, so it does not discuss assumptions such as discounting, dividends, or the risk-neutral measure. The limiting claim should therefore be understood within the Black–Scholes setup rather than as a general prediction about real options markets.

Key ideas

• A call price depends on expected payoff, not only on the chance of finishing above the strike.
• Extreme volatility spreads terminal outcomes and can create rare, very large call payoffs.
• A falling probability of finishing in the money does not by itself imply a falling option value.
• The document states the apparent limit but does not provide a derivation or examine model assumptions.

Tags

# Paradox for call option price when vol goes infinity


# Paradox for call option price when vol goes infinity












if stock price follow geometric brownian motion, then at time T the probability of stock price to surpass strike price K , is

so from above formula, if volatility increase, the probability of stock price surpass K is decreasing, when vol goes infinity, the probability goes to 0, so the call option price should be 0

However from bs formula, if vol goes infinity, the call option price goes to S0

I am puzzled by the contradiction, can anyone explain please? thanks

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.