How is the risk-neutral probability calculated?

The risk-neutral probability is calculated using the concept of the risk-neutral measure, which assumes that investors are indifferent to risk and only consider the expected returns of different investment options.

To calculate the risk-neutral probability, we need to follow these steps:

1. Identify the possible outcomes or states of the world associated with the investment. For example, if we are considering investing in a stock, the possible outcomes could be an increase in stock price, a decrease in stock price, or no change in stock price.

2. Determine the expected returns of each outcome. This involves estimating the probability of each outcome occurring and multiplying it by the corresponding return. For example, if there is a 40% chance of the stock price increasing by 10% and a 60% chance of the stock price decreasing by 5%, the expected return would be (0.4 * 0.1) + (0.6 * -0.05) = 0.04 - 0.03 = 0.01 or 1%.

3. Calculate the risk-free rate of return. This is the rate of return that an investor can earn with certainty, typically by investing in a risk-free asset such as government bonds.

4. Adjust the expected returns by subtracting the risk-free rate of return. This adjustment is made to account for the risk associated with the investment. The risk-neutral probability is calculated by dividing the adjusted expected return of each outcome by the sum of all adjusted expected returns.

For example, if the risk-free rate of return is 2% and the adjusted expected returns for the three outcomes are 3%, -1%, and 2%, the risk-neutral probabilities would be (0.03 - 0.02) / (0.03 - 0.02 + -0.01 - 0.02 + 0.02 - 0.02) = 0.01 / 0.02 = 0.5 or 50% for the increase in stock price, (-0.01 - 0.02) / (0.03 - 0.02 + -0.01 - 0.02 + 0.02 - 0.02) = -0.03 / 0.02 = -1.5 or -150% for the decrease in stock price, and (0.02 - 0.02) / (0.03 - 0.02 + -0.01 - 0.02 + 0.02 - 0.02) = 0 / 0.02 = 0 or 0% for no change in stock price.

In summary, the risk-neutral probability is calculated by adjusting the expected returns of different outcomes by subtracting the risk-free rate of return and then dividing each adjusted expected return by the sum of all adjusted expected returns.