# Call Option on the Square of a Log-Nomral Asset

I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors).I cannot understand its answer well and really appreciate your advice:

Here is the question: suppose you have a call option on the square of a log-normal asset. What equation does the price satisfy?

The answer says "the price still satisfy the Black-Scholes equation", I'm confused with Black Scholes formula {ex. call option premium = SN(d1)-Kexp(-rt)N(d2)} and Black Scholes PDE {ex. dC/dt+rS*dC/dS+(1/2)*sigma^2*S^2*d^2C/dS^2 = r*C}. So what is "Black Scholes equation" (from the answer) referring to? The Black Scholes PDE or Black Scholes formula? My understanding about the difference between the two is: PDE has boundary condition that can be used to price almost all options(ex.European,American,Asian), but Black Scholes formula can be used to price European option, is that right? In other words, PDE can be used to price an option whose value is dependent on past prices since we can solve PDE backwards and take those past prices into account, but for Black Scholes formula, it can only price an option whose value is dependent on maturity, is that right?

• Yes it satisfies the BMS pde. – Frido Rolloos Nov 14 '19 at 17:21