Cox and Ross (1976), Harrison and Kreps (1979), and Harrison and Pliska (1981) show that the absence of arbitrage implies the existence of a probability distribution, such that securities are priced at their discounted (at the risk-free rate) expected cash flows under these risk-neutral or risk-adjusted probabilities. Moreover, these probabilities are unique if markets are complete–all risks can be hedged. If, on the other hand, markets are not complete, their probabilities are not unique, but any of them can be used for pricing.11. The risk-neutral valuation approach can be generalized to include stochastic discount rates:=∫-VEeX[]Qrt dtT0( )Tf0

166LATIN AMERICAN JOURNAL OF ECONOMICS|Vol. 50 No. 2 (Nov, 2013), 163–177 There are three cases to consider in real option risk-neutral valuation. The first case is when the risk-neutral distribution is known, as in the Black-Scholes framework; unfortunately, the only pure example of this case in the real world are gold mines. In such case, the futures prices are Fo,T=So(1+rf)T.The second case is when the risk-neutral distribution is unknown but can be obtained from futures prices or other traded assets (e.g., copper mines and oil deposits). In Section 4 this topic will be explored further.The last case is when the risk-neutral distribution is unknown and futures prices do not exist. In this case the risk-neutral distribution can be obtained by using an equilibrium model, such as the CAPM. This is the most common case in R&D projects, Internet companies, and information technology where no futures prices exist. Thus, using the risk-neutral framework to value investment projects allows for use of all information contained in futures prices when these prices exist, to take into account all flexibilities/options the projects may have and use the powerful analytical tools that have been developed in contingent claims analysis. 3. Solution procedures to option valuation problems There are three main solution methods for solving option valuation problems: the dynamic programming approach, partial dif ferential equations, and the simulation approach. The first approach uses dynamic programming techniques to lay out possible future outcomes and folds back the value of the optimal future strategy using risk-neutral distributions. The binomial method is a dynamic programming approach widely employed to value simple options. It can also be used to price American-type options. However, this solution method becomes inadequate when there are multiple factors af fecting the value of the option or when there are path dependencies. The second method directly solves the partial dif ferential equations (PDE) that result from most option pricing problems. This approach leads to closed-form solutions in very few cases, such as the Black-Scholes equation for European call options. In most option valuation problems the PDE has to be solved numerically. This is a very flexible method, and it is appropriate for valuing American options. Finding a solution however becomes extremely complicated when there are

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- Real options analysis, Mathematical finance, Eduardo Schwartz