What is Risk neutral probability measure?

Answer 1

A probability measure allocates a non-negative probability to each possible outcome. All individual probabilities together add up to 1.

The "risk-neutral probability measure" is used in mathematical finance. Generally, risk-neutral probabilities are used for the arbitrage-free pricing of assets for which replication strategies exist. This is about relative pricing, based on possible replication strategies. The first argument is that a complete and arbitrage-free market setting is characterised by unique state prices. A state price is the price of a security which has a payoff of 1 unit only if a particular state is reached (these securities are called Arrow securities). In a complete market, every conceivable Arrow security can be traded. It is more easy to visualise these securities in terms of discrete scenarios. (On a continuous range of scenarios we would have to argue in terms of state price density.) The arbitrage-free price of every asset is the sum (over all scenarios) of the scenario-payoff weighted with its state price. Any pricing discrepancy with regards to an implicit state price would enable arbitrage in a complete market. The assumption is that the pursuit of such opportunities drives the prices towards the arbitrage-free levels. Hence the state prices are unique.

Since the whole set of Arrow securities is the same as a risk-free bond (sure payoff of 1 unit at maturity), the price of the whole set of Arrow securities must be e^(-rt) (assuming we are now at maturity minus t). Risk-neutral probabilities can then be defined in terms of state prices, or vice versa. A probability measure has to fulfil the condition that the sum of all individual probabilities adds up to 1. Therefore, if we want to create an artificial probability distribution based on the state price distribution, we have to multiply each state price with e^(rt) in order to obtain its probability equivalent. It is not surprising then that any expectation taken under the risk-neutral probability measure grows at the risk-free rate. This is an artificial probability measure, why should we create such a construct? This connection allows us to exploit mathematical tools in probability theory for the purpose of arbitrage-free pricing. The main difficulty about risk-neutral probabilities is that the probability concepts used have not initially been developped for the purpose of financial pricing, therefore, two different languages are used, which can easily be confusing. The economic interpretation of a risk-neutral probability is a state price compounded at the risk-free rate. Anything that has an effect on a state price (preferences, real probability, ...), has an effect on the risk-neutral probability.

So now we have a bridge to go from state prices to risk-neutral probabilities and back again. What is this good for? According to the second argument, we can, under certain conditions, specify the unique risk-neutral probability distribution of an underlying asset price with the help of an only incomplete specification of its real probability distribution, thanks to the Girsanov Theorem. If the innovation in the price of the underlying asset is driven by a Brownian motion, then all we need to obtain the risk-neutral probability distribution is the volatility parameter. What can we now do with this risk-neutral probability distribution? We can use the first argument to convert the obtained risk-neutral probability distribution back to a state price distribution, and the state price distribution applied to the payoff distribution (i.e. taking the sum over all scenarios) leads to the arbitrage-free price. These arguments save us a lot of trouble when trying to calculate the arbitrage-free price of an asset. They allow us to avoid the estimation of risk premia, by implicitly using those incorporated in the underlying asset price. The arbitrage-free price is, however, NOT independent of risk-premia. The price of the underlying asset is part of the pricing equation, and the risk-premia are inherent in this price, but because the price of the underlying asset is known to us, we obviously do not need estimate it.

It is important to emphasise that the risk-neutral valuation approach only works if the asset to be priced can be perfectly replicated. This is often not true in reality, especially when dynamic replication strategies are involved.

Paper explaining risk-neutral probabilities:

http://ssrn.com/abstract=1395390