This information we can now use to value our option! Now we know that there is a 55% probability of ending up in the high scenario and a 45% probability of ending up in the low scenario. With these additional steps, we could back out the risk neutral probabilities of the future state of the world from the current asset prices.
If the risk neutral probability of a value increase is 55%, the corresponding probability of a value decline is 45%. Without risk neutrality, we could therefore not directly use the probabilities that are implied in the underlying asset value to price the option. Why risk neutrality? The reason is that the "underlying asset" and the option that is written on this underlying asset generally do not have the same risk (the option is generally riskier than the underlying asset). It is the probability that a risk neutral investor would attach to the scenario of a value increase. Solving this expression for π implies that the probability of a value increase is 55%. Because we have assumed risk neutral investors (discounting at the risk-free rate of return), this probability is called the risk neutral probability. PV(Expected value of underlying asset 1) = \( \frac \). Therefore, the expected value of the underlying asset in 1 year is:Įxpected value of underlying asset 1 = π × 150 + (1 - π) × 50Īssuming that investors are risk neutral, we can discount this expected value at the risk-free rate of return (R) to get its present value: Consequently, the probability that the value of the underlying asset drops to 50 is (1 - π). If the asset pays no dividend, we know that the current value of the underlying asset (100) must correspond to the present value of the expected future value of the underlying asset (either 150 or 50 with unknown probabilities)! Let us denote the probability that the value of the underlying asset increases to 150 with the greek letter pi (π).
The future value of the underlying asset is either 150 or 50.The current value of the underlying asset is 100.But we do not know the probabilities of these future state of the world! There, the graph with the value of the underlying asset comes in handy, as it implies the following:
REAL OPTIONS VALUATION EXAMPLE HOW TO
The following graph summarizes the values of the investment opportunity that we have just derived.Īs in the previous section, the question now is how to value this tree? We know that the value of the investment opportunity will be either 30 or 0 in 1 year. Consequently, in this scenario, the value of our right to invest will be 0. If we have the right to invest, but not the obligation, we will not invest and simply walk away. In contrast, if the value of the underlying asset drops to 50 million in 1 year, it will not make sense to realize the project, as the NPV would be -70 million.Consequently, the NPV of the project in this particular state of the world will be 30 million in 1 year. If the value of the underlying asset goes up in 1 year, the investment of 120 million will unlock a future cash flow stream with a value of 150 million.How does this option to postpone the investment affect the analysis? While it does not make sense to realize the investment opportunity today, there is one future state of the world where the value of the underlying asset exceeds the necessary investment of 120 million:
The following graph summarizes this value development: Let us assume that the value of the underlying asset can either increase to 150 or drop to 50. It can wait for 1 year and observe how the market for the product in question evolves. However, the company does not have to invest today. NPV (investment today) = -Investment + PV(Future FCF) = -120 + 100 = -20 million. Let's refer to this present value of all future FCF the "value of the underlying asset." Consequently, it does not make sense to invest today, as the NPV of the investment is -20 million: As of today, the present value of all future FCF the firm could generate with the investment is 100 million. Suppose a company has an investment opportunity that it can realize within the next year.
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