In this blog on derivative basics, we are going to look at the concept of risk-neutrality and risk-neutral pricing. Risk-neutrality is a powerful concept that lies at the heart of the pricing of financial derivatives and its proper understanding is necessary in order to be able to understand derivative pricing theory.
Before delving into risk-neutral pricing, lets examine the concept of risk. Every investor expects a return on their investment and the possibility of the investment giving less return than expected is classified as the risk. Different types of assets have different risk profiles. The higher the risk, the higher is the reward expected by the investor. Financial assets can be classified into three broad categories based on their risk profiles:
The following graph shows the risk-reward profiles for various types of assets:
It is important to note that the any two assets in the graph above can be combined in a portfolio to replicate the risk profile of the third asset. For example, one can create a portfolio that holds a certain amount of derivatives and stock to replicate risk-free cash portfolio. Similarly, a portfolio consisting of cash and stock can be used to create a portfolio that has similar risk profile as that of a portfolio containing only derivatives. This ability to replicate risk profiles of assets by combination of other types of assets is widely used in derivatives pricing and we shall see examples of this in derivative pricing blogs.
Having explained risk, lets now examine risk-neutrality. A risk-neutral investor is one that is agnostic to risk and focuses solely on the expected return on the investment. This is in contrast to a risk-averse investor, who will take into account the risk whilst making the investment. For example, consider two stocks. One offers a return of 10% with 100% probability, whereas the other offers 20% return but with only 50% probability. Although the expected returns are the same in both cases (10%), the risk-averse investor will choose the first stock as it has the lower risk of the two. The risk-neutral investor on the other hand, has got no preference as both offer the same expected return.
Why is risk-neutrality important to the pricing of derivatives? This is because risk-neutrality takes the concept of risk out of the picture and allows us to focus solely on the expected outcomes. Different investors have different risk appetites, and their view of the ‘price’ or ‘value’ of a derivative is based on their risk appetite. A risk seeking investor will be willing to pay higher for an expected outcome (say, 60% chance of a 10% return) as opposed to a risk-averse investor. Since the risk appetites vary among investors, it is not possible to find a single and agreed-upon price of a derivative that encompasses the risk appetites of all of the players in the market.
Risk-neutrality allows us to price a derivative from the perspective of a risk-neutral investor that sits in the middle (average) of the spectrum with risk-averse and risk-seeking investors at the extreme ends. This means that we are not taking any risk appetites into consideration. Instead of first calculating the expected outcomes and then adjusting the expectations with risk profiles to compute the derivative prices, the risk-neutral pricing calculates the probabilities that are adjusted with the average of risk profiles of all investors. The average here is the risk-neutral profile. Once the risk-neutral probabilities are computed, the price of the derivative is simply the present value of the expected payoff.
Since we have taken the notion of risk out of the picture, the rate of return of assets under risk neutral measure is equivalent to risk-free rate. This is because the excess return (the return offered by a risky asset on top of risk-free return) is zero. This allows us to discount the future payoff of the derivative (expected outcome) using the risk-free rate to compute its present value.
It is important to note that the risk-neutral pricing is based on the assumptions that the market that is complete and has no arbitrage opportunities. A complete market is a market where the transaction costs are negligible and all players have complete and uniform picture of the market. Also, the no-arbitrage rule is important as it prevents players from making risk-free money that is above the risk-free rate (the excess return has to be zero for risk-neutral pricing).
In this blog we have learnt about