What is an option price

Question

What is an option price?

Answer 1

Option prices:

If we have the lognormal random walk for asset, and we transform the dependent variable by using a discount factor as per to

p(S, t) = e^r(T-t) V(S, t),

After that the backward equation for p becomes an equation for V that is the same to the Black-Scholes partial differential equation. The same but for one subtlety, the equation has a µ, here Black-Scholes contains r. We can conclude as the fair value of an option is the current value of the expected payoff at expiration in a risk-neutral random walk for the underlying. Risk neutral now implies that it replace µ with r.