Call Option on the Square of a Log-Nomral Asset

Question

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?

Answer 1

Generally the Black Scholes equation is used to refer to the Black Scholes PDE (PD equation). And the formula refers to the analytical formula, usually cover both call and put versions.

The extension of the BS to the square or power of S is frequently covered in the textbooks and tests; however, it could be tricky in an interview situation. When using Black Scholes logic, they can question you on self financing portfolio or martingale measure or change of numeraire. As you can see, justifying the extension of Black Scholes arguments to these payoffs is by no means trivial. Hence these pay offs are a good exercise to learn about the machinery behind the Black Scholes, and when to use or not to use the BS PDE.

For the path dependent options, please see the discussion here: Can we use Black-Scholes to price path dependent options?