In quantitative finance, the two most common types of probability measures used are risk-neutral and real-world probabilities.
Risk-neutral probability measures are used when pricing derivatives, such as options, in the financial markets. These probabilities represent the probability of different future outcomes under an assumption that investors are indifferent to risk. In other words, in the risk-neutral world, investors only care about the expected return of an investment, and not the potential risks associated with it.
Real-world probability measures, on the other hand, represent the actual probabilities of different future outcomes. These probabilities take into account the risks associated with an investment and reflect the actual likelihood of those risks materializing.
The difference between risk-neutral and real-world probabilities is important because it influences the pricing of derivatives. When using risk-neutral probabilities, the price of a derivative is equal to its expected payout discounted at the risk-free rate. Real-world probabilities, however, may differ from risk-neutral probabilities due to risks associated with an asset, such as credit risk, liquidity risk or market risk. These risks will affect the pricing of the asset in the real world and may lead to differences between the prices obtained using real-world and risk-neutral probabilities.
To illustrate, consider the pricing of a European call option on a stock. Under the risk-neutral measure, the price of the call option is equal to the discounted expected value of the option payoff at the option expiry date. In contrast, under the real-world measure, the price of the call option will depend on the volatility of the stock price and whether the option is in or out of the money.
In conclusion, the difference between risk-neutral and real-world probabilities is important in quantitative finance, as it affects the pricing of financial instruments, particularly derivatives. Risk-neutral probabilities are used to price derivatives, while real-world probabilities reflect the actual risks and may affect the pricing of the assets in the real world.