Assumptions in Derivative Pricing

Payoff: the gross outcome of an investment or trade. The amount you earned from that trade, regardless of commissions, extraneous costs, etc.
Profits: the gross outcome of an investment or trade, including commissions, extraneous cost
- → the distinction happens in Options (Finance). Payoffs don’t consider the option premium, while the profits do.
Return: short for “rate of return”. The percentage of profit (not payoff!) per original investment. Quoted in percentage (%) or log-returns.

Risk-Neutral Assumption

A risk-neutral world is where there is no risk premium on return. This implies:

All Present Value Calculations are done at the risk-free rate
All assets have risk-free rate of return (=zero riAssumptions in Derivative Pricingitrage Condition]] holds
& This is not really realistic, but simplifies calculations a lot.

No arbitrage condition means without any risk, there cannot be excess return (=return above risk-free rate).

Law of One Price states that:

let security $A$ have payoff $p_{A}$ , cost $c_{A}$ and $B$ payoff $p_{B}$ , cost $c_{B}$ .
If $p_{A} = p_{B}$ then $c_{A} = c_{B}$