Boundedness of Black–Scholes Greeks in Log Spot

Summary

The document poses a mathematical question about whether derivatives of the Black–Scholes price with respect to log spot are bounded when time to maturity is positive. It seeks a bound that depends on remaining time to expiry, with a maximum magnitude controlled by a constant and a maturity exponent determined by derivative order.

The text is only a question: it gives no formula, proof, answer, or evidence establishing boundedness. It does not specify all model and payoff assumptions needed to assess such a result, so it should be read as a statement of a research problem rather than a result about option sensitivities.

Key ideas

• The question concerns higher derivatives of Black–Scholes value with respect to log spot.
• It asks whether those derivatives admit bounds for positive time to maturity.
• The proposed bound would depend on remaining maturity and derivative order.
• No proof, answer, or assumptions sufficient to establish the claim are provided.

Tags

# Are Black-Scholes Greeks bounded?


# Are Black-Scholes Greeks bounded?












For time to maturity greater than zero, has it been proved somewhere that the Black-Scholes greeks $$ \frac{\partial^n BS}{\partial x^n} $$ are bounded, where $x := \log S$ and $S$ is the current spot price.

In other words, is there an expression and corresponding proof that $$ \left| \frac{\partial^n BS}{\partial x^n} \right| \leq M(T-t)^{m(n)} $$ for some constant $M$ and $m(n)$ some function of $n$?

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