Assessing a Product Inequality Involving Black–Scholes Probabilities

Summary

The document poses a sign question for a difference of products of standard normal cumulative probabilities, where the arguments depend on two prices, volatility, and time. It sets the interest rate to zero and defines four Black–Scholes-style terms, then asks whether the expression is always positive under the stated ordering of the prices.

The author notes that the expression approaches zero at very small and very large time-volatility scales and considers proving its sign through derivatives or local approximations. The discussion gives no proof, counterexample, numerical evidence, or final conclusion. It is best read as an open mathematical problem about option-pricing expressions, rather than an established identity. Endpoint behavior alone does not determine the sign between the endpoints, and the document does not specify all parameter restrictions needed to assess the claim.

Key ideas

• The proposed inequality combines products of normal cumulative distribution values with arguments derived from Black–Scholes quantities.
• The setup simplifies the interest rate to zero and assumes an ordering between the two prices.
• The author reports zero limits at extreme time-volatility scales but does not establish the sign between them.
• No proof, counterexample, or parameter-complete conclusion is supplied.

Tags

# Any Simple Way to Prove Black Scholes Type Identies?


# Any Simple Way to Prove Black Scholes Type Identies?












A certain complicated option pricing formula results in products of Black Scholes $N$ components like this:

$-p_1N(d_1)N(d_6)+p_sN(d_2)N(d_5)>?0$ where $p_s>p_1$

Trying to find a simple way to prove if this identity is true or not without having to use differential geometry on a multi variable function. Taylor series don't work because the Ln's and $a*t^{1/2}$ part can be very big. interest set to zero for simplicity

$\begin{align} d_1 &= \frac{1}{\alpha\sqrt{t}}\left[\ln\left(\frac{p_1}{p_s}\right) + \left(0 + \frac{\alpha^2}{2}\right)t\right] \\ d_2 &= \frac{1}{\alpha\sqrt{t}}\left[\ln\left(\frac{p_1}{p_s}\right) + \left(0 - \frac{\alpha^2}{2}\right)t\right] \\ d_5 &= \frac{1}{\alpha\sqrt{t}}\left[ \left(0 + \frac{\alpha^2}{2}\right)t\right] \\ d_6 &= \frac{1}{\alpha\sqrt{t}}\left[\left(0 - \frac{\alpha^2}{2}\right)t\right] \\ \end{align}$

Trying to prove this with differentials requires taking the derivative of a complicated function with respectto the time/vol variable and then the p_s an p_1 variables to determine if the manifold has a maximum above zero

At the endpoints $at \to 0$ and $at \to infinity $ it equals zero, so the function is a frown or a smile that has positive or negative curvature. This is an extremely hard problem because linear approximations don;t work

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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