Deriving the Black–Scholes Identity Between Normal Densities

Summary

The document addresses a Black–Scholes derivation involving the standard normal density evaluated at d1 and d2. It clarifies that the notation N′(x) refers to the normal probability density function, not to differentiating the expression with respect to maturity. Substituting the definitions of d1 and d2 into the two sides and rearranging establishes the familiar identity linking the spot and strike terms, adjusted by their dividend and interest discount factors.

The response also cautions that a separate proposed equality involving a derivative with respect to maturity is not generally valid. It checks that claim by considering the bounded maximum of the standard normal density, which makes the asserted equality impossible for sufficiently large positive maturity values. The explanation is concise and points to algebra rather than providing every rearrangement step. It assumes the usual Black–Scholes definitions and does not discuss broader model assumptions or applications to pricing and risk.

Key ideas

• In the cited identity, N′ denotes the standard normal probability density function.
• Substituting the Black–Scholes expressions for d1 and d2 allows the identity to be verified by algebra.
• The identity relates spot and strike terms after accounting for dividend and risk-free discounting.
• A separate asserted maturity-derivative equality is not generally correct.
• The bounded maximum of the normal density provides a simple way to disprove that separate equality.

Tags

# derive black scholes greeks


# derive black scholes greeks












I am reading a paper and get a problem here, the following terms are all from standard BS models. the paper says using the well known fact $$Se^{-q(T-t)}N^{'}(d1)=Ke^{-r(T-t)}N^{'}(d2)$$ here the differentiation is respect to $T$.For instance $N^{'}(T^2+1)=2T$

So anyone could give some hints to get this fact?

## Answer by pbr142 (score 4, accepted)

https://quant.stackexchange.com/a/11064

The first equality is a bit tedious to derive but straight-forward. As commented by vanguard2k, the notation $N'(x)$ is meant to denote the density function of the standard normal distribution: $$ N'(x) = n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} $$. Simply insert the $d_{i}$ terms, $$ d_1 = \frac{\log\left(\frac{S}{K}\right) + (r-q+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}} $$ $$ d_2 = \frac{\log\left(\frac{S}{K}\right) + (r-q-\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}} $$ and rearrange.

The second equality cannot be, in general, correct. Simply write out the lhs to see that. Or note that the maximum of the standard normal pdf is slighlty less than 0.4. So for T>0.2, the equality is incorrect.

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