# Can you explain the Black-Scholes fair option equation with RND?

I am trying to learn Black-Scholes risk-neutral densities with only prior knowledge of fundamental B-S equations (not the derivation). Sorry if this was asked already or if I sound completely clueless.

Basically I do not understand what lowercase x stands for in the integral formula below. It is then subsequently used for the lognormal density. X stands for exercise price of a European call option as usual.

"Let $f_{Q}$ denote the risk-neutral density of $S_{T}$. Then

$c=e^{-rT}E^{Q}[max(S_{T}-X,0)]=e^{-rT}\int_{X}^{\infty }(x-X)f_{Q}(x)dx$

If you have any recommendations on sources to read about this more thoroughly, feel free to link them, thanks.

• Formally $x$ is the integration variable, which starts at $x=X$ and ends at $x=\infty$. The integral represents the area under the curve $(x-X)f_Q(x)$ as $x$ varies over this range. $f_Q$ represents the lognormal density. – noob2 May 19 at 13:12
• In financial terms $x$ represents all the possible stock price values at time $T$ that have a positive call option payoff (i.e $x\equiv S_T>=X$) – noob2 May 19 at 13:24
• @noob2 thank you very much for confirmation! – br0323 May 19 at 13:30
• Yeah, if the person who wrote this equation had used a consistent symbol (either $x$ or $S_T$) on both sides of the equal sign instead of mixing them it would have avoided misunderstanding. – noob2 May 19 at 14:10