Risk-Neutral World: Why All Assets Earn Risk-Free Rate?

Imagine you are playing a game, a very simplified game of investing. In this game, everyone is considered ‘risk-neutral’. What does that even mean? It’s a bit like saying everyone in this game is indifferent to taking risks. Think of it like this: if you offered someone two choices, either a guaranteed return of 5% or a risky investment that might return 10% but also might return 0%, a risk-neutral person would only care about the average expected return, not the potential ups and downs.

This idea of a risk-neutral world isn’t how people actually behave in real life. Most of us are ‘risk-averse’. We generally prefer a sure thing over a gamble with the same average payoff. However, in the world of option pricing, especially when we are using models like Black-Scholes, we often step into this theoretical risk-neutral world. It’s a mathematical trick, a simplification that allows us to calculate fair prices for options. It doesn’t mean we believe everyone is risk-neutral, but it’s a useful framework for pricing these complex financial instruments.

Now, the question is why, in this risk-neutral game, must all assets, even the underlying stock an option is based on, be expected to earn only the risk-free rate of return? Let’s break that down. The risk-free rate is the return you can get on an investment that is considered completely safe, like a government bond in a stable country. It’s the baseline return, the minimum you should expect to earn for simply lending your money over time.

In a risk-neutral world, the concept of ‘risk premium’ disappears. A risk premium is the extra return you demand for taking on additional risk. Think about it like this: if you are asked to climb a slightly wobbly ladder versus a very stable staircase to reach the same height, you would probably expect a little extra reward for taking the riskier ladder option, right? That extra reward is the risk premium.

But in our risk-neutral game, nobody cares about the wobbly ladder versus the staircase. They only care about reaching the height. So, if both the stock and the risk-free bond are going to get you to the same ‘height’ in terms of expected return, and nobody cares about the ‘wobbliness’ or risk, then logically, they must offer the same expected return.

If a stock, in a risk-neutral world, was expected to earn more than the risk-free rate, say 10% when the risk-free rate is 5%, it would create an immediate problem. Risk-neutral investors, who only care about expected returns, would flock to buy this stock. They would see it as a free lunch, a higher return with no additional risk penalty in their eyes. This increased demand would drive up the stock price, and as the price goes up, the expected return would naturally decrease. This process would continue until the expected return of the stock falls down to the risk-free rate.

Conversely, if a stock was expected to earn less than the risk-free rate, say 3% when the risk-free rate is 5%, risk-neutral investors would avoid it. They could get a guaranteed 5% elsewhere with no perceived additional risk. This lack of demand would push the stock price down, and a lower stock price increases the expected return. This would continue until the expected return of the stock rises to match the risk-free rate.

This constant adjustment, driven by the actions of risk-neutral participants, ensures that in this theoretical world, all assets, including the underlying stock, must be expected to earn the risk-free rate. It’s an equilibrium condition. If it weren’t true, there would be arbitrage opportunities, chances to make risk-free profit in a world where risk is irrelevant to decision-making. These opportunities would be quickly exploited, pushing asset prices back into alignment and ensuring everything earns the risk-free rate in expectation.

So, while it might seem counterintuitive that a stock, which is inherently risky in the real world, should only be expected to earn the risk-free rate in a risk-neutral world, remember this is a theoretical construct designed for pricing options. It’s a simplification that allows us to focus on the relationships between option prices, underlying asset prices, time, and volatility, without getting bogged down in complex risk preferences. It’s a powerful tool, even if it operates in a world that’s a bit different from our everyday investment experience.