In finance, a price (premium) is paid or received for purchasing or selling options.
For further detail, see: Mathematical finance § Derivatives pricing: the Q world for discussion of the mathematics; Financial engineering for the implementation; as well as Financial modeling § Quantitative finance generally.
This is because the money invested by the seller can earn this risk free income in any case and hence while selling options.
For these, the result is calculated as follows, even if the numerics differ: (i) a risk-neutral distribution is built for the underlying price over time (for non-European options, at least at each exercise date) via the selected model, as calibrated to the market; (ii) the option's payoff-value is determined at each of these times, for each of these prices; (iii) the payoffs are discounted at the risk-free rate, and then averaged.
For the analytic methods, these same are subsumed into a single probabilistic result; see Black–Scholes model § Interpretation.
After the financial crisis of 2007–2008, counterparty credit risk considerations were brought into the valuation, previously using the risk-free rate to discount the payoff.