# Real options cum. Valuation: Present Values, Real Options, Dot.com Bubble

For instance, contrary to stylised facts from previous literature, real option values real options cum actually decrease with the volatility of the underlying project value and increase with investment costs. And large projects can be more or less attractive real options cum small projects ceteris paribus depending on the risk tolerance of the investor, how this changes with wealth, and the structure of costs to invest in the project.

Introduction The original definition of a real option, first stated by Myersis a decision opportunity for a corporation or an individual. It is a right, rather than an obligation, whose value is contingent on the uncertain price s of some underlying asset s and the costs incurred by exercising the option.

Any higher price would exceed her value and she would not buy the project; any lower price would induce her to certainly buy the project. Footnote 1 Similarly, we identify the option strike with the investment cost for the project and in the following we shall use these two terms synonymously.

The real option value ROV is the value of this decision opportunity to buy or sell the project; it is specific to the decision maker i. The ROV represents the certain dollar amount, net of financing costs, that the decision maker should receive to obtain the same utility as the risky investment in the project.

So, unlike the premium on a financial option, the ROV has no absolute accounting value. In this setting, ROVs merely allow the subjective ranking of opportunities to real options cum in alternative projects. Footnote 2 The basic model for analysing investment real options assumes that the project value follows a geometric Brownian motion GBM over a finite investment horizon, with a fixed or pre-determined investment cost i.

Much recent literature on investment real option analysis attempts to augment this basic model in various ways, real options cum add special features that address particular practical problems.

### Valuation: Present Values, Real Options, Dot.com Bubble

Footnote 3 However, our paper addresses a much more fundamental issue. Instead of following the main strand of the real options literature, which adapts and extends the mathematical models to more closely reflect the real-world characteristics of a particular investment opportunity, we shall drill-down into the real option model itself to examine the model risk inherent in the assumptions that are made for the decision. This paper real options cum a general model for real option valuation—with how to make money in the market utility real options cum encompasses all the standard utility functions and with a price process which has the standard geometric Brownian as a special case—then, by setting specific values for model parameters, we can assess the change in real option value arising from different decisions.

For example, suppose the project is owned by a firm whose board would wish to apply risk-neutral valuation, but the decision-maker is a manager who uses her personal preference function.

Our general framework allows one real options cum quantify how much value is lost by a board with such a manager. To this end, we provide comparative statics of real option values with respect to the risk tolerance parameters of Hyperbolic Absolute Risk Aversion HARA utility.

### Model risk in real option valuation

To our knowledge this is the first study of model risk associated with real options cum factors, some of which have not even been considered before. We do not assume that risks can be hedged by traded securities, but risk-neutral valuation techniques still apply in the special case of a linear utility function. In this case, Grasselli proves real options cum the time-flexibility of the opportunity to invest in a project still carries a positive option value for a risk-averse decision maker, so that the paradigm of real options can still be applied to value a decision where none of the risks can be hedged.

We also assume that the market price is the only stochastic factor and that the option strike may be—but is not necessarily—related to the market price of the project.

In future work our general framework might be extendible to multiple correlated sources of uncertainty, where partial hedging is possible. But it is already a considerable challenge to build a model which is broad enough to encompass special cases corresponding to the specific choices of project characteristics and investor preferences mentioned above.

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Indeed, we already analyse a wide scope of questions about model risk, real options cum without such model extensions. For instance: How is the ROV affected if the drift in market price is mean-reverting?

How does the frequency of exercise opportunities affect the ROV? How does the ranking of different opportunities change as we increase risk tolerance, or its sensitivity to wealth?

Does the size of the investment relative to initial wealth effect the ROV, and if so how? And what is the effect of the fixed or pre-determined cost assumption relative to more general assumptions that costs are stochastic?

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In the following: Sect. An Appendix aids understanding of our framework with some illustrative examples. Real options to invest in a project Most of the early literature on investment real options focuses on opportunities to enter a tradable contract with pre-determined strike in a complete market.

For early models including a stochastic strike, real options cum McDonald and SiegelQuigg and Kulatilaka In this setting the option has the same value to all investors and so is priced as if the investor real options cum risk neutral using standard RNV techniques for American options.

Basic real option models also suppose the forward price follows a geometric Brownian motion GBM with total return equal to the risk-free rate, and the ROV is derived from a linear utility where exercise at a pre-determined strike may be taken at any point in time.

Capozza and SickTrigeorgisBenaroch and KauffmanBoerYeo and Qiuand numerous others since all employ this basic framework.