Welcome back, my name is Han Smit. I'm a professor of Corporate Finance at the Erasmus School of Economics. In this webcast, you will learn the valuation principles behind real options theory. Option replication and risk neutral valuation. This refers to the notion that if we can set up a portfolio in the underlying asset financed by a loan that always generates the same return as the option, we can actually eliminate risk from the valuation equation. We refer to this method as risk neutral valuation. Combining this valuation method with the various options described in the previous webcasts will allow you to value <i>any</i> type of option. The basic idea enabling the pricing of options is that one can construct a portfolio that exactly replicates the future returns of the option in any state of nature. This portfolio is created by buying a particular number, N, of underlying shares and borrow against them an appropriate amount of B, against the risk-free rate. And this portfolio exactly replicates the returns of the option. Thus, we can value the option by determining the cost of constructing its equivalent replicating portfolio, that is, the cost of a <i>synthetic</i> or homemade option equivalent. Since the option and this equivalent portfolio (effectively, an appropriately levered position in the stock) would provide the same returns, it should have the same value. Let us look at the example. Suppose that the price of an underlying stock, currently valued at V is 100, will move up or down. So it can move up to 180 with a multiplicative factor of 1.8, or down to 60 with a multiplicative down factor 0.6. The probabilities are q and 1-q. The value of the option over this period would then be contingent on the price of the underlying stock. So assume that your investment in the stock, or the exercise price I, is 80, and assume that the discount rate is 4%. So the value of the call option is the maximum of the underlying value minus the exercise price or 0, and can move up, which results in a value of 180 minus 80 is 100, or can move down, which results in a value of 0. Suppose now, that we can actually construct a portfolio as described above to replicate these two payoffs. The portfolio consists of 1) buying N shares of the underlying stock at its current price, and 2) finance this by borrowing an appropriate amount of B against the risk-free rate. This can be Treasury bills. So the call option consists of buying N shares, a borrowed amount of B, at a rate r. After one period, we would need to repay the principal amount borrowed at the beginning, B, with interest, so (1 + r)B. The value of this portfolio over the next period, will thus be: either q, is the number of shares, times the up-value, minus (1+r)B, or 1 minus q, the number of shares, times the down value of the stock, minus (1+r) times B. If the portfolio is to offer the same return in each state as the option, the value should be the same. So, in the upward, we have the value equals the upward value of the option in the upward state. In the downward state, the value equals the downward value of the option. Solving these two equations, conditions of equal payoff of the option and the replica, for the two unknowns, number of shares and the bonds (B), gives you the following expression: The number of shares of the underlying asset that we need to buy to replicate one option equals one and equals the spread in option values divided by the spread in stock values. We then determine the amount to be borrowed, B, by plugging in the number of N and solving the equation. So, as you can observe, the number of N equals 100 minus 0, divided by 180 minus 60. And that's a fraction of 0.83, or 83 percent of stocks. Then the amount B equals 83% times a down value of stocks minus zero, discounted at the risk-free rate of 1.04. And that results in an amount of borrowing of $48. Following this example, we can replicate the return of the option by purchasing a fraction of 83% of the shares of the underlying stock at the current price and borrowing the amount of B of $48 at the risk-free rate. Finally, we can derive the value of our replicating portfolio as an approximation of our option value, by substituting these results for N, the number shares, and the amount of loan, back into the first equation. So the current value of that fraction of 83% of stock, minus the current loan, results in a value of $35 and that should also equal the option value. Comparing this to the original formula used for calculating the call option, one can infer that indeed it is exactly the same payoffs. So the call option is actually the risk neutral expectation of the future payoffs. So that results in 37% times 100, plus 63% times 0, discounted at the risk-free rate of 4%, results in a value corrected for rounding errors of $35. And the risk-neutral probability equals 37%. This p used in this equation, is called a transformed or risk-neutral probability, and that is the probability that would prevail in a risk-neutral world where investors were indifferent to risk. What exactly is this risk-neutral valuation? That's very important in option valuation. We're going to answer this important question. Intuitively, this equation can be rearranged. So the number of shares times the underlying value, minus the call option, equals the value of the loan. So we can buy N shares of the underlying stock, we can sell or write a call option that would give a certain amount of 50 in the next period, so one plus r times B, regardless of whether the stock moves up or whether the stock moves down. I can show you here. For the ability to construct such a risk-free hedge, that always gives certain payoff, risk can effectively be "squeezed out" of the problem, so that an investor's risk attitude doesn't matter. Therefore we can equivalently, and due to the simplicity of the calculation more conveniently, obtain the correct option valuation by pretending to be in a risk neutral world where risk is irrelevant, without having to make assumptions on the risk attitudes of the investors, because risk is important, because when exactly replicated, risk gets squeezed out of the equation. So although we can calculate the option value pretending that the world is risk neutral, in the real life investors can actually be risk-averse. In such a risk neutral world, all assets, so including stocks, options, etc., would earn the risk-free rate, and so the expected cash flows, weighted at the risk-neutral probabilities could be appropriately discounted at the risk-free rate. This risk neutral probability, p, can alternatively to the above formula, be obtained from the condition that the expected return on the stock in a risk neutral world must equal the risk-free rate. So p times the up-return, plus one minus p times the down-return, should be the expected risk-free rate. Solving this results in p is 1 + r minus d, divided by u minus d, and results in 37%. And u is 80% and the d is 40%, that our returns are in the up- and the down-state. Similarly, the expected return on the option must also equal the risk-free rate in a risk-neutral world. So p times C+, plus (1-p) times C-, divided by C minus 1, should be the risk-free rate, r. Rearranging this, we get early expression for the risk-neutral probability. Does this valuation theory also apply to real options? That's an important question. An estimation of the market value of a real option can sometimes be obtained by creating a project equivalent in the market, hence using the same replication method as discussed here in financial options. This equivalent dynamic portfolio strategy, consisting of a position of the underlying asset, partly financed with a risk-free loan, can be constructed such that in every state it has the same payoff as the real option. So it should also have the same current value. So we can observe the value of the replicating portfolio, and that's the value of the real option. Valuation of some types of real options, particularly found in natural resources, may closely resemble the no arbitrage valuation of financial options. The replication argument can be justified if a financial instrument is traded with the same risk characteristics as the project. This is the case, for example, if the underlying state variable is the price of a commodity, as in a valuation of a mining company or oil leasing valuation. The value of the license is estimated by using a financial instrument, for instance, futures in gold or futures in oil, respectively, where the probabilistic behavior is close to that of the developed project, is used to estimate the value of the mine. In other settings, option valuation by replication may be a bit stretched. For instance, valuation of an investment opportunity to build a facility to produce a new product, as nanotubes or any other new product, is less directly analogous to a financial option pricing. If a similar plant is not traded on financial markets, option valuation cannot take advantage of this traded asset or dynamic replication strategy that drives the value of flexibility of the project. This would present a fundamental problem in applying the arbitrage argument for <i>financial</i> options, but for <i>real</i> options it does not. Real options, in contradiction to financial derivatives, are not traded in arbitrage-free financial markets, but present themselves in imperfect real markets. An estimation of the real asset's value <i>as if </i>it were traded is sufficient for the valuation objectives of corporate finance, namely creation of market value for the company. Thus, real options valuation can still be applicable if you find a reliable estimate for the market value of the underlying asset. However, it is not an arbitrage relation, because we cannot replicate it, but only value creation, because we can estimate the value as if it were traded. And exactly that's what we need: the expected value creation. To summarize: in this webcast, you learned the foundation of binomial option valuation. A number of points are worth reviewing about the above call option valuation: One: option theory provides an exact formula for the value of the option in terms of its underlying value, exercise price, and risk-free rate, and the stock's volatility or spread. The motivation for pricing of the option rests with the absence of arbitrage-free opportunities. That's a very strong economic condition, a valuation condition. Financial option pricing models are typically based on arbitrage arguments in financial markets. The arbitrage relation is based on the construction of a hedge by taking a position in the underlying asset and the risk-free asset. For real options valuation, it is still applicable if you find a reliable estimate of the market value of the underlying asset. We can estimate the value as if it were traded. Three: the actual probability of up and down movements, q, does not appear in the valuation formula. Moreover, the value of the option does not depend on the investors' attitudes towards risk or on the characteristics of the other assets, it is priced only relative to the underlying asset, V. And that's why we can squeeze out risk from the valuation formula. The value of the option can be equivalently obtained in a risk-neutral world, since it's independent of the risk preferences. Actually, p is the value that probability q would have in equilibrium if investors were risk neutral. As the above valuation formula confirms, in such a risk-neutral world, where all assets are expected to earn the risk-free rate of return, the current value of the option can be obtained from its future expected value using the risk-neutral probability, discounted at the risk-free interest rate. This was a very fundamental valuation video. You probably need a drink before I see you in the next part. Thank you. Hope to see you again!
