My name is Han Smit, I'm a professor of Corporate Finance at Erasmus School of Economics. In this webcast, you'll learn to value an investment opportunity as a real option. In the previous webcast you learned to consider an investment opportunity as a real option. In this webcast, you learn how to value. The most simple way to value a real option is with a binomial model. To demonstrate a real options valuation, consider a specific example, as illustrated here. Suppose that there is an investment opportunity with a value of 100, and we also know something about the volatility of this value. Suppose that the project value can increase to 180, or it may decline to 60. The probability of an expected upward or downward movement is 50%. This is what we call the true probabilities. We can estimate the current value by taking the expectation of the future payoff, discounted with the risk-adjusted rate. For instance, q times V-plus, plus 1 minus q times V-minus, divided by the risk-adjusted discount rate, 1 plus k, equals the current value of this project. So using the numbers provided: 50% times 180, plus 50% times 60, divided by a risk-adjusted discount rate of 1.20, we get a current value of 100. You probably already knew this. This is just a discounted value calculation where the adjustment for risk is in the denominator. There's also another way to value this. This is called the certainty equivalent method, or risk neutral valuation. The difference is that adjustment for risk is the numerator and not in the denominator. We do this by estimating a risk-neutral probability, which we call p. In this case, p times V-plus, plus 1 minus p times V-minus, divided by 1 plus the risk-free rate, so 1 plus r (which is 4% here) should also result in a value of 100. And if we rearrange this, you will see that the risk-neutral probability, which differs from the true probability, equals 37%. So here, on the left-hand side, there is no truncation in the payoff. If there's no truncation in the payoff, you can use both valuation methods, whether the adjustment for risk is in the numerator or in the denominator, they both result in the same answer. But now we consider the option. With an option you can get a truncation in the payoff and as we have seen in the previous webcast, you can decide not to invest. So let's consider the option. If there is a favorable development, the underlying value increases to 180. The required investment equals 80, so the payoff of the option equals 100, as you can see over here. In the down-state over here, the value declines to 60. The investment outlay is still 80, so you don't make the investment. So there, as we have seen in the previous webcast, the value is truncated to 0. Now the essential point: because of this truncated payoff, we can't use the discounted value method. We can only use the certainty equivalent method. That is because with an option the risk continuously changes along the branches of the option tree. So the only right way to value the option is using risk neutral valuation. So we use risk-neutral probability p, that is 37%, times the payoff of the option in the up-state, that's 180 minus 80 is 100, plus 1 minus p times the value in the down-state, which is 0, divided by 1 plus the risk-free rate. And this gives us an option value of 36. So on the left-hand side over here, we have the value without any truncations, and without truncations we can use both methods. We can make an expectation using true probabilities and discount it with the risk-adjusted rate. Or we can make a risk neutral expectation and discount it back at the risk-free rate. On the right-hand side we have the payoff of an option, which is truncated. In this case, we can only use risk neutral or certainty equivalent valuation. Think about it: on the right-hand side, because of the truncation, the risk of the option is continuously changing each time the underlying value changes. An at-the-money or out-of-the-money option is much more risky than an in-the-money option. And because the risk continuously changes, we can only use the certainty equivalent method there. The certainty equivalent method is actually based on an arbitrage. When you consider real option valuation techniques, the risk-neutral probability is based on an option equivalent. And this option equivalent consists of the underlying value and a loan, just as we have seen in the previous webcast. So to summarize this video: with real options theory we can value an investment opportunity as an option. One: we can replicate an option payoff based on the underlying value of the investment and a loan. This comes down to risk neutral or certainty equivalent valuation, using a risk-neutral probability. With this risk-neutral probability we can calculate the current value of the future option payoff. Two: risk neutral valuation doesn't mean that you expect that the world is actually risk neutral. No, the risk-neutral probability can actually differ from the true probability in a risk-averse world. Three: in general, if the payoff is symmetrical and there are no truncations, we can use both methods: where the adjustment for risk is in the numerator or the denominator. Four: but once we face an option with a truncation in the payoff, the risk is continuously changing depending on whether the underlying value changes, and then we can only use the certainty equivalent or risk neutral valuation. Five: now you have learned the basics of options valuation. And now that you know this, you can actually value any type of option: you can look at the truncations of different options, use risk-neutral probabilities and discount it backwards with the risk-free rate. Thank you for listening and I hope you start valuing options now. You have the capabilities to do it. See you in the next webcast!