Risk-seeking investors give away a risk premium because they enjoy taking risk. Risk-averse investors expect a risk premium to compensate for the risk. Risk-neutral investors neither give nor receive a risk premium because they have no feelings about risk.

Risk-neutral pricing: Suppose you want to price a derivative. The payoff of this derivate can be replicated using the underlying asset and risk-free rate. The market price of this derivative and the replicating strategy must be exactly the same under the principle of no arbitrage, regardless of risk preferences.

The derivative price is calculated assuming that the investor is risk-neutral. That's why the derivatives pricing is sometimes called risk-neutral pricing. That means the expected payoff of the derivative can be discounted at the risk-free rate rather than the risk-free rate plus a risk premium.

To obtain the derivative price we should assume the investor is risk-neutral, because an investor's risk aversion is not a factor in determining the derivative price. Risk can be eliminated by dynamic hedging in a situation where there is no arbitrage possible. Once risk is eliminated in this way the expected return becomes equal to the risk-free rate for all investors. Assets can be assumed to grow at the risk-free rate and also discounted at the risk-free rate.

Arbitrage-Free Pricing

If the calculated price of the derivative is different from the market price, then there is an arbitrage opportunity. The arbitrage transactions will bring back the price to where it should be, and that price will eliminate any arbitrage opportunities. This process of pricing derivatives by arbitrage and risk neutrality is called arbitrage-free pricing.