- Email: [email protected]
Modelling a Bio-Pharmaceutical Company as a Compound Option
ELSEVIER
Copyright@IFAC Modeling and Control of Economic Systems. Klagenfurt. Austria. 2001
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
MODELLING A BIO-PHARMACEUTICAL COMPANY AS A COMPOUND OPTION Stefan Worner*. Boryana Racheva-Iotova**, Stoyan Stoyanov***
* Fraunhofer Institute for Systems alld Illnovatioll Research,
Karlsruhe, Germany ** *** BRA VD Consulting and University of Sofia, Sofia. Bulgaria ** Departmellt of Economics and Business Administratioll *** Department of Mathematics and Computer Science
Abstract: The theoretical rationale is presented for a simple Black-Scholes analysis to obtain the main characterising features of a price process which determines the value of bio-pharmaceutical firms . This information is used to calibrate a compound option model. Theoretical considerations and an analysis of simulations of the compound option framework, assuming different degrees of information, indicate that the compound option analogy portrays the actual situation of sequential research phases in biopharmaceutical drug development more appropriately than conventional Black-Scholes applications. Copyright © 2001 IFAC Keywords : Real options, option theory, compound options, valuation, biotechnology
(1973) in their seminal paper on option pricing. They found a solution for a partial differential equation characterising the stochastic process of stock returns. Changes in the value of the underlying of a real option are stochastic over time as well . It is fair to say that there is no technique to predict the movements of the underlying process exactly. The changes can take place at any points in time. They may take virtually any value. The stochastic process is thus continuous in time and continuous in the underlying variable. Theoretically, the estimate of continuous time, continuous variable processes to model the stochastically changing underlying should therefore be a reasonable proxy in real options applications.
1. INTRODUCTION The valuation of biotechnology companies is a perplexing issue. Due to the high degree of uncertainty, long time horizons and negative cash-flows, the net .present value technique to discount future cash-flows alone may not satisfy the investor. A biotech firm may not be profitable for ten years or more. During this time, more cash may be spent than comes in through earnings. However, biotechnology companies are traded, on various stock exchanges, with a considerable positive value. Clearly the value of a company that loses money, as is the case for most firms in the biotechnology sector, is based on its potential for future earnings. Based on theoretical considerations, an options tool to analyse the underlying assets behind the observed market value of venture firms is delineated. This tool is evaluated in the empirical part of the study.
Five variables determine the value of a simple financial option according to Black and Scholes (1973): an underlying S, an exercise price X, the time to maturity T. the volatility (J of S, and the risk free interest rate r.
2. SIMPLE OPTION VARIANT
The basic principle of their approach is that it is possible at any time to duplicate the payoff stream of an option by purchasing a combination of riskless bonds and a certain number of the asset underlying the
Considering the shares of a levered firm as an option on its value was first described by Black and Scholes
231
option. The seller of an option can buy this replicating portfolio of bond and commodity and hedge his or her risk from selling the option. A risk free position can be maintained by adjusting the portfolio to changes in the option value. Such a portfolio will then yield the risk free interest rate. As all investors agree on the return of the portfolio, it must have a unique value for any investor in the risk neutral world.
process is known as duplication in finance theory. The option on the traded asset, or twin asset spanned by investment opportunities in capital markets, is then valued using ordinary option pricing theory, thus providing an estimation of the price the real option would have in capital markets. Duplicating risk profiles of non-traded assets is only possible in complete markets, where the spanning condition holds. The spanning condition is not a specific requirement of option valuation but any technique that aims to calculate a (capital) market value (including the discounted cash-flow method). However, the results of Sick (1995) imply that not only options on nontraded assets but also options on nonexisting assets can be priced using the principles of ordinary option pricing theory. Identifying a traded asset or portfolio of assets with the same risk profile as the underlying of the real option is theoretically possible in most of the cases. It becomes trivial if the underlying is traded itself, e.g. raw material, oil or metals. In the case of applications in capital budgeting to value the flexibility associated with investment decisions, it seems reasonable to believe that once the risk profiles of the investment alternatives have been described, finding or creating identical profiles should be feasible by applying modern techniques of financial engineering.
In the case of real assets, the underlying is equivalent to the market value of the firm. If at maturity the market value of the firm exceeds the face value of the debt, the owners of the firm will exercise their (call) option by paying off the debt. If the value of the bonds is higher than the market value of the company, the shareholders will default and let their option expire worthless. In the real world, stock prices of bio-pharmaceutical firms incorporate preferences towards risk as well as beliefs about the outcome of research efforts on novel drugs. These preferences are immanent in observed financial share prices which are understood as prices of real options as explained above. The estimated implicit distributions and underlying parameters of the stochastic process solely represent the risk-neutral (martingale equivalent) world . Empirical evidence on oil futures suggests that the recovered risk-neutral parameters should be quite close to actual parameters (Kumar, 1992; Deaves and Krinsky, 1992).
This endeavour should be much more difficult, if not impossible, when the underlying assets are inventions, like new drugs. The reason is that innovations (inventions brought onto the market) expand the space of possible investment alternatives, potentially introducing novel sources of risk. The sources of risk could be the internal development risk of unproven technologies or the exogenous risk associated with the impact of the product under development after its release. In genetic engineering, high exogenous risk is apparent as the long term effects of modifications in the DNA structure of living organisms have not been thoroughly studied. If the source of risk is unparalleled, it may be impossible to hedge this risk by selling a portfolio with the same risk profile because such a hedging portfolio is just not available. This means that the risk neutral approach of pricing a known underlying in incomplete markets falls short of individual risk preferences. The actual value of the underlying should thus be lower than the values calculated empirically (using the procedure described below). The difference points to a general problem caused by incomplete markets for investment opportunities.
Most bio-pharmaceutical ventures are mainly financed with equity rather than debt. The reason is the high risk profile of these companies. The risk premium to be paid by the firm would be too high to make bonds attractive to the company. In the case of bio-pharmaceuticals, the option of the owners of the firm therefore concerns the sequential payments necessary to carry on with the next phase in R&D rather than the compensation of the creditors. If the board of the corporation comes to the conclusion that it is worthwhile proceeding with the next phase in pharmaceutical R&D, they will bear the cost of the subsequent steps to bring the drug onto the market. They may even increase the speed of R&D (i.e. spend more money) if the outcome of their research is very favourable. If they find that the required expenses are not justified, that the technology is not mature enough or that the past results are less promiSing than expected, they may abandon, defer or otherwise alter their research project accordingly. In the case of mUltiple projects under way, interdependencies among these projects become relevant as well. This is where real options usually come into play in capital budgeting.
Completeness of markets can be formally defined by the dimension of the market subspace M (Magill and Quinzii, 1996). M spans a linear subspace of net present values of investment opportunities. If M comprises all possible P portfolios of securities, the market is called complete. If dim{M} < P, the market is said to be incomplete. In this case, investment returns will not necessarily coincide in a single equilibrium allocation p' (cf. Karatzas and Shreve, 1998;
The basic idea behind real options valuation is to find or create an underlying asset that is traded on financial markets. This commodity should show the same risk profile as the underlying of the real option. This
232
p. 24 for the proof) . In incomplete markets it is rarely possible to construct a portfolio of assets that shows the same payoff structure as the option's payoff. Hence, valuation via hedging fails. Two strands in the literature have emerged to tackle this problem (van Hulle, 1988). The first captures techniques to introduce upper and lower boundaries of option prices, thus releasing the claim of uniqueness of the so derived option price. Stochastic dominance arguments are used to determine a range in which the real price of a contingent claim will lie. The range depends on the cost of capital and on the probability of an up- or down movement in discrete time. If risk preferences of investors and/or the market price of risk are known, the range can be narrowed down to a unique solution (Garman, 1976; Levy, 1985; Gerber and Shius, 1988; Merton, 1973; Perrakis and Ryan, 1984; Ritchken, 1985; Lo, 1987). The second cohort of authors applies utility theory and capital market equilibrium models to delineate option prices, which are unique under the constraints imposed by the assumptions of the given models. By defining the type of the density function of the terminal asset value relative to its initial price, a risk neutral valuation equation can be derived under the condition of a market equilibrium by introducing a characteristic density function for the state prices and assuming that the equilibrium is independent of individual attitudes towards risk or the wealth positions of individual investors (Brennan and Kraus, 1978; Brennan, 1979). The problem has been solved in a one period trading model for normal, lognormal and multinomial density functions (Rubinstein, 1976; Brennan, 1979; Stapleton and Subrahmaniam, 1984).
After a review process it is decided whether the firm may go on with the trials in the subsequent stage. Proceeding is subject to approval by the FDA. It thus seems reasonable to divide the R&D process (which may take eight years or more) into several periods according to the regulatory principles and to model each phase as an option on the subsequent phase. The corresponding duration of each phase then defines the time to maturity of the option. It is only known ex-post, although reliable estimations may be feasible ex-ante.
2.2 Exercise Price The exercise price of the option during a particular phase is considered to be the investment costs that have to be incurred by the company if it decides to pursue the subsequent phase of pharmaceutical R&D. At the end of the lifetime of the real option, the company is supposed to decide whether to spend the investment costs to enter the next phase, i.e. exercise the real option and acquire another real option, or to cancel the development. The best guess for the exercise price of the option is thus the net present value (NPV) of the cost of the next phase. The company tries in stage j to gain sufficient information to be allowed to start stage j+ 1, and in effect having to pay the NPV of the cost of the following phase as an exercise price if successful. If the firm is not successful, the exercise price stays the same but it is now higher than the value of the underlying because the probability that the expected cash-flows will ever occur goes to zero. Virtually all of the sample firms should calculate the expected cost over the next years in their financial planning. Unfortunately, such information is not regularly reported and the management of the sample firms was usually not inclined to provide it. As an estimate for the actual number, the burn rate of the firm during the whole time of the corresponding research phase is used, which can be extracted from publicly available data.
The aim of this research is to use market information as efficiently as possible when pricing the shares of a bio-pharmaceutical company instead of a financial option but using similar techniques. The need to use expected future commodity prices or risk-adjusted discount rates is thereby mitigated. Traditional methods for calibrating the model did not appear fruitful. Accounting measures (e.g. to estimate a fair value) are rather useless for science based high-tech firms because they fail to measure the intangible assets of a science based firm which determine its market value. Discounted cash-flow techniques face the same difficulties, providing less accurate information (Mason and Merton, 1985). The analogies between the input parameters of the option pricing theory and the (real) option theory of this study are discussed in the remaining paragraphs of this chapter.
2.3 Risk Free Interest Rate The Black-Scholes formula is independent of risk preferences. None of its factors imply risk preferences of individual investors. They can thus be considered risk neutral. When moving to the risk neutral world, the volatility of the variables (and correlation between variables) is not changed. Risk neutral valuation produces consistent results not only in the risk neutral world but in all other worlds, if the growth rates of asset values are adjusted appropriately. This does not mean that all investors in the real world are actually risk neutral. It just makes no difference to the solution if risk preferences are taken into account. While adjusting to a risk averse or risk loving world, the growth rate of the return process and the interest rate to discount future pay-offs would change. These opposite effects would exactly offset
2.1 Time To Maturity Pharmaceutical R&D takes place in separate steps, or phases, following strict rules imposed by regulatory offices. For the case of the USA, the regulatory office is the Food and Drug Administration (FDA). To move on to the next phase, a firm (the sponsor) has to deliver certain documents to the regulation office.
233
each other. Assuming risk neutrality, which means that no premium is required for investors to take a risk, simplifies the application of the Black-Scholes formula because the risk free interest rate can be used to discount any pay-off. The risk free interest rate is the return to be earned from holding an asset without any risk.
derived (FIeming, 1998), a computational procedure was implemented in Matlab 6.0. The idea behind the chosen approach is to numerically solve a system of non-linear equations. In these calculations no assumptions are made about the price of the underlying or about its volatility.
2.5 Volatility 2.4 Implicit Underlying The volatility captures the stochastic moment of the underlying price process. The volatility of the underlying commodity O"c is derived from the standard deviation of the return on the commodity prices in a certain period of time, normally a year. The volatility is the standard deviation of the natural logarithm of the price of the commodity (Hull, 1999).
The underlying S is considered to be unknown. (Future) profitability is crucial for the value of a biopharmaceutical firm. This may be measured by discounting expected cash-flows generated by the new product under development. Other factors influencing profitability regard protection of the firm's intellectual property, e.g. by patents, competence of its management, liquidity, and general investment climate.
The fact that 0;. is used to calculate S introduces cross-references. For the initial calculation of the volatility of the financial stock to estimate S, the notation presented in Hull (1999) is used.
These factors alter the likelihood of success in R&D and its duration and define the share of the potential market the company may be able to explore. It should be reasonable to believe that this argumentation can be reduced by assuming a virtual project value, determined by the size of the potential market for the drug and the probability that the firm will gain a certain share of this market. The influence of the investment climate should average out over time. The observed market value of a traded enterprise is thus the consensus of many investors on the value of the underlying. Their guesses will change over time, e.g. whenever new information arrives. New information may concern advances in R&D, recruitment of new employees, fund raising, etc. It is because of this research that success and failure in R&D is most important for a bio-pharmaceutical firm and that the phases in pharmaceutical R&D regulated by the FDA create a framework for a (real) options analysis.
2.6 Empirical findings In order to empirically assess the Black-Scholes approach for the purpose of company valuation, a Matlab 6.0 program was developed based on the formal rationale discussed in the preceding paragraphs of this chapter. The program is applied in a case study on a small bio-pharmaceutical company. The aim of this case study is to illustrate how publicly available information about a particular company can be used to estimate input parameters for real options valuation and to investigate the accuracy of the simple option variant. The Black-Scholes approach is the most frequently used technique in real option pricing. In its original form, it was designed to price a European call option on a non-dividend paying asset. Instead of taking a financial commodity as the underlying asset of the option, an R&D project of a bio-pharmaceutical company is used.
The project value(s) as described above should be the major driving force behind the share price of the firm, serving the role of a state variable in the options analysis. The first stage of the computing should calculate the value of the underlying using the above parameters and the fact that the value of the real option price is known because the daily prices of the financial stock can be observed, thus working as the equivalent for the real option's price in the model. The Black-Scholes formula is thus used to measure the market opinion about the underlying assets of the company. Mainly, these assets should refer to its R&D projects. It is taken advantage of the fact that the (call) option price is an increasing function of the price of the underlying. In analogy to Cox' and Rubinstein's (1985) implicit volatility, the underlying S will be called the implicit underlying.
The case study company NeoTherapeutics (Irvine, California) is a development-stage biopharmaceutical company engaged in the discovery and development of novel therapeutic drugs intended to treat neurological diseases and conditions. It was incorporated in 1987. In 2000, it employed 33 people. It is listed on the NASDAQ under the symbol NEOT. Solving the Black-Scholes formula numerically for the underlying S using the input data as described above and plotting the values of the implied underlying yields a stochastically moving process (see below). The price of the implied underlying can be seen as the project value that explains the observed share price within a Black-Scholes real option analysis. As expected, the underlying and the share price are highly correlated. Hence, if the stochastic process could be estimated based on historical data, expected
In an attempt to simultaneously find the implied price of the underlying and its implied volatility, thus extending the way how implied volatilities are usually
234
prices for the underlying and therefore for the share price could be simulated.
of financial shares. Compared to the value of the implied underlying for the case of the single option analysed in section 2 using the Black-Scholes analogy, the implied underlying option value in the compound option model should be lower. This becomes clear after recalling that the impact of the volatility of the second underlying asset is reduced by the correlation coefficient in Geske's formula which should be below one in most cases. The value of the underlying option is just the Black-Scholes price of a European call option. This is analogous to the case of a compound option model where the exercise price of the first option is zero. Figure 1 depicts the comparison between the value of the underlying option in the compound option model (dotted line) and the underlying project value derived by inverting the BlackScholes formula (solid line).
A sensitivity analysis reveals that the estimation errors of a too optimistic value for the underlying project translate most significantly into errors in the company value. Deviations from the base case in the exercise price, the duration of the respective phase of R&D, or the risk free interest rate affect the outcome of the valuation process much less. It is surprising that the same applies to errors in guessing the volatility of the underlying price process. Sensitivity analyses on the interaction between changes in the input parameters and corresponding values of financial options mostly lead to the conclusion that their prices usually react quite considerably to changes in the volatility measure (Cox and Ross 1976, Conner 1981, Chen and Sears 1990). After calculating average values for the input parameters of the simple option variant, they can be used in a Black-Scholes framework to predict the share prices of the firm under consideration. The aggregate error is calculated by the mean absolute deviation between actual and predicted share prices. The numbers indicate that the error obtained within the simple option framework is in the same order of magnitude as the error caused by a much simpler forecasting method of linear trend extrapolation of share prices.
Fig. l. Comparison between the value of the underlying option in the compound option model and the underlying project value derived by inverting the Black-Scholes formula for the case company Neotherapeutics.
3. COMPOUND OPTION EXTENSION A compound option is an option whose underlying asset is an option. The analogy for the case of a biopharmaceutical firm is that the firm value, the "visible" asset, is an option on a certain project phase, the "hidden" asset, which itself is an option on all following project phases. It is evident that this shows many characteristics of growth options discussed in real options literature. In fact, growth options in capital budgeting are often valued using the compound option technique. The rationale behind compound option thinking is much like the Black-Scholes approach of valuing a single option, where the hidden asset (the underlying) is just a commodity rather than a derivative. It is thus not surprising that a prominent compound option model created by Geske (1979) is a generalisation of the Black-Scholes formula. Geske prices an option on the value of the firm, which is considered to be an option on the firm's assets itself. The idea is that both the option value and the company value follow a Brownian motion with drift.
Further empirical evidence for a sample of 13 neurological companies confirms that the implied underlying option value according to the compound option model is systematically lower or equal to the implied underlying S of the option derived by the BlackScholes logic.
4. CONCLUSIONS The limitations of the Black-Scholes analysis for the valuation of bio-pharmaceutical companies have become evident. It is not only that some of the assumptions of Black and Scholes (1973) are violated due to the fact that we are concerned with incomplete markets in that particular application, but that the analogy of a single option should only loosely resemble reality. The compound option analogy portrays the actual situation of sequential research phases in bio-pharmaceutical drug development more appropriately because the value of the growth option associated with the current phase in research is better captured than in the Black-Scholes framework, where
The compound option value depends on the value of the underlying option, which is described by the Black-Scholes formula, the exercise price and the time to maturity of the two options. As the compound price is monotonous in the price of the underlying option, the price of the latter can be inferred from the observed compound option values trading in the form
235
Geske, R (1979). Valuation of compound options. Journal of Financial Economics, 7, 63-81. Hull, J.c. (1999). Options. futures, and other derivatives. Prentice Hall. Fourth edition. Karatzas, I. and S.E. Shreve (1998). Methods of mathematical finance . Springer-Verlag, New York, Berlin, Heidelberg. Kumar, M. (1992). The forecasting accuracy of crude oil futures prices. IlIternational Monetary Fund Staff Papers, 39, 432-461. Levy, H. (1985). Upper and lower bounds of put and call option value: stochastic dominance approach. Journal of Finance, 40, 1197-1217. Lo, AW. (1987). Semi-parametric upper bounds for option prices and expected payoffs. Journal of Financial Economics, 19, 373-387. Magill, M. and M. Quinzii (1996). Theory of incomplete markets. MIT Press, Cambridge, Massachusetts. Mason, S.P. and RC. Merton (1985). The role of contingent claims analysis in corporate finance. In: Recents advances in corporate finance (E. Altman, M. Subrahmanyam, eds.). Homewood, Illinois. Richard D. Irwin, pp.7-54. Merton, R.C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Sciences, 4,141-183. Perrakis, S. and PJ. Ryan (1984). Option pricing bounds in discrete time. The Journal of Finance, 39,519-527. Ritchken, P. (1985). On option pricing bounds. Journal of Finance, 40,1219-1233. Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics, 7, 407. Sick, J. (1995). Real options. In: Finance. Volume 9 of Handbooks in OR and Management Science (R Jarrow, V. Maksimovic, W.T . Ziemba, eds.): Elsevier North-Holland. Stapleton, RC., and M.G. Subrahmanyam (1984). The valuation of multivariate contingent claims in discrete time models. Journal of Finance, 39, 207-228. van Hulle, C. (1988). Option pricing methods: an overview. Insurance, Mathematics & Economics, 7,139-152.
it is rather implicit in the general value for the underlying asset. Up to now, the case of a firm that pursues a single project exclusively has been investigated. It offers many advantages to find realistic measures for the required input variables. However, the case of a portfolio with numerous R&D projects is more frequently observed. Modelling this situation is much more challenging and is discussed in a forthcoming article.
5. REFERENCES Black, F. and M. Scholes (1973). The pncmg of options on corporate liabilities. The Journal of Political Economy, 81, 637-659. Brennan, MJ. (1979). The pricing of contingent claims in discrete time models. Journal of Finance,34. Brennan, MJ. and A Kraus (1978). Necessary conditions for aggregation in securities markets. Journal of Financial and Quantitative Analysis, 13,407-418. Chen, K.c. and RS. Sears (1990). Pricing the SPIN. Financial Management, 19, 36-47. Conner, R.E. (1981). Determining the fair value of stock options. Pensions & Investment Age, 9, 2526. Cox, J.c. and S.A Ross (1976). A survey of some new results in financial option pricing theory. The Journal of Finance, 31, 383-402. Cox, J.c. and M. Rubinstein (1985). Options Markets. Prentice Hall, Englewood Cliffs, New Jersey. Deaves, R. and I. Krinsky (1992). Risk premiums and efficiency in the market for crude oil futures . Energy Journal, 13,93-118. Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option prices. Journal of Empirical Finance,S, 317-345. Garman, M. (1976). An algebra for evaluating hedge portfolios. Journal of Financial Economics, 3, 403-428. Gerber, H.U. and E.S.W. Shiu (1988). Nonuniqueness of option prices. Insurance. Mathematics & Economics, 7, 67-69.
236
Copyright@IFAC Modeling and Control of Economic Systems. Klagenfurt. Austria. 2001
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
MODELLING A BIO-PHARMACEUTICAL COMPANY AS A COMPOUND OPTION Stefan Worner*. Boryana Racheva-Iotova**, Stoyan Stoyanov***
* Fraunhofer Institute for Systems alld Illnovatioll Research,
Karlsruhe, Germany ** *** BRA VD Consulting and University of Sofia, Sofia. Bulgaria ** Departmellt of Economics and Business Administratioll *** Department of Mathematics and Computer Science
Abstract: The theoretical rationale is presented for a simple Black-Scholes analysis to obtain the main characterising features of a price process which determines the value of bio-pharmaceutical firms . This information is used to calibrate a compound option model. Theoretical considerations and an analysis of simulations of the compound option framework, assuming different degrees of information, indicate that the compound option analogy portrays the actual situation of sequential research phases in biopharmaceutical drug development more appropriately than conventional Black-Scholes applications. Copyright © 2001 IFAC Keywords : Real options, option theory, compound options, valuation, biotechnology
(1973) in their seminal paper on option pricing. They found a solution for a partial differential equation characterising the stochastic process of stock returns. Changes in the value of the underlying of a real option are stochastic over time as well . It is fair to say that there is no technique to predict the movements of the underlying process exactly. The changes can take place at any points in time. They may take virtually any value. The stochastic process is thus continuous in time and continuous in the underlying variable. Theoretically, the estimate of continuous time, continuous variable processes to model the stochastically changing underlying should therefore be a reasonable proxy in real options applications.
1. INTRODUCTION The valuation of biotechnology companies is a perplexing issue. Due to the high degree of uncertainty, long time horizons and negative cash-flows, the net .present value technique to discount future cash-flows alone may not satisfy the investor. A biotech firm may not be profitable for ten years or more. During this time, more cash may be spent than comes in through earnings. However, biotechnology companies are traded, on various stock exchanges, with a considerable positive value. Clearly the value of a company that loses money, as is the case for most firms in the biotechnology sector, is based on its potential for future earnings. Based on theoretical considerations, an options tool to analyse the underlying assets behind the observed market value of venture firms is delineated. This tool is evaluated in the empirical part of the study.
Five variables determine the value of a simple financial option according to Black and Scholes (1973): an underlying S, an exercise price X, the time to maturity T. the volatility (J of S, and the risk free interest rate r.
2. SIMPLE OPTION VARIANT
The basic principle of their approach is that it is possible at any time to duplicate the payoff stream of an option by purchasing a combination of riskless bonds and a certain number of the asset underlying the
Considering the shares of a levered firm as an option on its value was first described by Black and Scholes
231
option. The seller of an option can buy this replicating portfolio of bond and commodity and hedge his or her risk from selling the option. A risk free position can be maintained by adjusting the portfolio to changes in the option value. Such a portfolio will then yield the risk free interest rate. As all investors agree on the return of the portfolio, it must have a unique value for any investor in the risk neutral world.
process is known as duplication in finance theory. The option on the traded asset, or twin asset spanned by investment opportunities in capital markets, is then valued using ordinary option pricing theory, thus providing an estimation of the price the real option would have in capital markets. Duplicating risk profiles of non-traded assets is only possible in complete markets, where the spanning condition holds. The spanning condition is not a specific requirement of option valuation but any technique that aims to calculate a (capital) market value (including the discounted cash-flow method). However, the results of Sick (1995) imply that not only options on nontraded assets but also options on nonexisting assets can be priced using the principles of ordinary option pricing theory. Identifying a traded asset or portfolio of assets with the same risk profile as the underlying of the real option is theoretically possible in most of the cases. It becomes trivial if the underlying is traded itself, e.g. raw material, oil or metals. In the case of applications in capital budgeting to value the flexibility associated with investment decisions, it seems reasonable to believe that once the risk profiles of the investment alternatives have been described, finding or creating identical profiles should be feasible by applying modern techniques of financial engineering.
In the case of real assets, the underlying is equivalent to the market value of the firm. If at maturity the market value of the firm exceeds the face value of the debt, the owners of the firm will exercise their (call) option by paying off the debt. If the value of the bonds is higher than the market value of the company, the shareholders will default and let their option expire worthless. In the real world, stock prices of bio-pharmaceutical firms incorporate preferences towards risk as well as beliefs about the outcome of research efforts on novel drugs. These preferences are immanent in observed financial share prices which are understood as prices of real options as explained above. The estimated implicit distributions and underlying parameters of the stochastic process solely represent the risk-neutral (martingale equivalent) world . Empirical evidence on oil futures suggests that the recovered risk-neutral parameters should be quite close to actual parameters (Kumar, 1992; Deaves and Krinsky, 1992).
This endeavour should be much more difficult, if not impossible, when the underlying assets are inventions, like new drugs. The reason is that innovations (inventions brought onto the market) expand the space of possible investment alternatives, potentially introducing novel sources of risk. The sources of risk could be the internal development risk of unproven technologies or the exogenous risk associated with the impact of the product under development after its release. In genetic engineering, high exogenous risk is apparent as the long term effects of modifications in the DNA structure of living organisms have not been thoroughly studied. If the source of risk is unparalleled, it may be impossible to hedge this risk by selling a portfolio with the same risk profile because such a hedging portfolio is just not available. This means that the risk neutral approach of pricing a known underlying in incomplete markets falls short of individual risk preferences. The actual value of the underlying should thus be lower than the values calculated empirically (using the procedure described below). The difference points to a general problem caused by incomplete markets for investment opportunities.
Most bio-pharmaceutical ventures are mainly financed with equity rather than debt. The reason is the high risk profile of these companies. The risk premium to be paid by the firm would be too high to make bonds attractive to the company. In the case of bio-pharmaceuticals, the option of the owners of the firm therefore concerns the sequential payments necessary to carry on with the next phase in R&D rather than the compensation of the creditors. If the board of the corporation comes to the conclusion that it is worthwhile proceeding with the next phase in pharmaceutical R&D, they will bear the cost of the subsequent steps to bring the drug onto the market. They may even increase the speed of R&D (i.e. spend more money) if the outcome of their research is very favourable. If they find that the required expenses are not justified, that the technology is not mature enough or that the past results are less promiSing than expected, they may abandon, defer or otherwise alter their research project accordingly. In the case of mUltiple projects under way, interdependencies among these projects become relevant as well. This is where real options usually come into play in capital budgeting.
Completeness of markets can be formally defined by the dimension of the market subspace M (Magill and Quinzii, 1996). M spans a linear subspace of net present values of investment opportunities. If M comprises all possible P portfolios of securities, the market is called complete. If dim{M} < P, the market is said to be incomplete. In this case, investment returns will not necessarily coincide in a single equilibrium allocation p' (cf. Karatzas and Shreve, 1998;
The basic idea behind real options valuation is to find or create an underlying asset that is traded on financial markets. This commodity should show the same risk profile as the underlying of the real option. This
232
p. 24 for the proof) . In incomplete markets it is rarely possible to construct a portfolio of assets that shows the same payoff structure as the option's payoff. Hence, valuation via hedging fails. Two strands in the literature have emerged to tackle this problem (van Hulle, 1988). The first captures techniques to introduce upper and lower boundaries of option prices, thus releasing the claim of uniqueness of the so derived option price. Stochastic dominance arguments are used to determine a range in which the real price of a contingent claim will lie. The range depends on the cost of capital and on the probability of an up- or down movement in discrete time. If risk preferences of investors and/or the market price of risk are known, the range can be narrowed down to a unique solution (Garman, 1976; Levy, 1985; Gerber and Shius, 1988; Merton, 1973; Perrakis and Ryan, 1984; Ritchken, 1985; Lo, 1987). The second cohort of authors applies utility theory and capital market equilibrium models to delineate option prices, which are unique under the constraints imposed by the assumptions of the given models. By defining the type of the density function of the terminal asset value relative to its initial price, a risk neutral valuation equation can be derived under the condition of a market equilibrium by introducing a characteristic density function for the state prices and assuming that the equilibrium is independent of individual attitudes towards risk or the wealth positions of individual investors (Brennan and Kraus, 1978; Brennan, 1979). The problem has been solved in a one period trading model for normal, lognormal and multinomial density functions (Rubinstein, 1976; Brennan, 1979; Stapleton and Subrahmaniam, 1984).
After a review process it is decided whether the firm may go on with the trials in the subsequent stage. Proceeding is subject to approval by the FDA. It thus seems reasonable to divide the R&D process (which may take eight years or more) into several periods according to the regulatory principles and to model each phase as an option on the subsequent phase. The corresponding duration of each phase then defines the time to maturity of the option. It is only known ex-post, although reliable estimations may be feasible ex-ante.
2.2 Exercise Price The exercise price of the option during a particular phase is considered to be the investment costs that have to be incurred by the company if it decides to pursue the subsequent phase of pharmaceutical R&D. At the end of the lifetime of the real option, the company is supposed to decide whether to spend the investment costs to enter the next phase, i.e. exercise the real option and acquire another real option, or to cancel the development. The best guess for the exercise price of the option is thus the net present value (NPV) of the cost of the next phase. The company tries in stage j to gain sufficient information to be allowed to start stage j+ 1, and in effect having to pay the NPV of the cost of the following phase as an exercise price if successful. If the firm is not successful, the exercise price stays the same but it is now higher than the value of the underlying because the probability that the expected cash-flows will ever occur goes to zero. Virtually all of the sample firms should calculate the expected cost over the next years in their financial planning. Unfortunately, such information is not regularly reported and the management of the sample firms was usually not inclined to provide it. As an estimate for the actual number, the burn rate of the firm during the whole time of the corresponding research phase is used, which can be extracted from publicly available data.
The aim of this research is to use market information as efficiently as possible when pricing the shares of a bio-pharmaceutical company instead of a financial option but using similar techniques. The need to use expected future commodity prices or risk-adjusted discount rates is thereby mitigated. Traditional methods for calibrating the model did not appear fruitful. Accounting measures (e.g. to estimate a fair value) are rather useless for science based high-tech firms because they fail to measure the intangible assets of a science based firm which determine its market value. Discounted cash-flow techniques face the same difficulties, providing less accurate information (Mason and Merton, 1985). The analogies between the input parameters of the option pricing theory and the (real) option theory of this study are discussed in the remaining paragraphs of this chapter.
2.3 Risk Free Interest Rate The Black-Scholes formula is independent of risk preferences. None of its factors imply risk preferences of individual investors. They can thus be considered risk neutral. When moving to the risk neutral world, the volatility of the variables (and correlation between variables) is not changed. Risk neutral valuation produces consistent results not only in the risk neutral world but in all other worlds, if the growth rates of asset values are adjusted appropriately. This does not mean that all investors in the real world are actually risk neutral. It just makes no difference to the solution if risk preferences are taken into account. While adjusting to a risk averse or risk loving world, the growth rate of the return process and the interest rate to discount future pay-offs would change. These opposite effects would exactly offset
2.1 Time To Maturity Pharmaceutical R&D takes place in separate steps, or phases, following strict rules imposed by regulatory offices. For the case of the USA, the regulatory office is the Food and Drug Administration (FDA). To move on to the next phase, a firm (the sponsor) has to deliver certain documents to the regulation office.
233
each other. Assuming risk neutrality, which means that no premium is required for investors to take a risk, simplifies the application of the Black-Scholes formula because the risk free interest rate can be used to discount any pay-off. The risk free interest rate is the return to be earned from holding an asset without any risk.
derived (FIeming, 1998), a computational procedure was implemented in Matlab 6.0. The idea behind the chosen approach is to numerically solve a system of non-linear equations. In these calculations no assumptions are made about the price of the underlying or about its volatility.
2.5 Volatility 2.4 Implicit Underlying The volatility captures the stochastic moment of the underlying price process. The volatility of the underlying commodity O"c is derived from the standard deviation of the return on the commodity prices in a certain period of time, normally a year. The volatility is the standard deviation of the natural logarithm of the price of the commodity (Hull, 1999).
The underlying S is considered to be unknown. (Future) profitability is crucial for the value of a biopharmaceutical firm. This may be measured by discounting expected cash-flows generated by the new product under development. Other factors influencing profitability regard protection of the firm's intellectual property, e.g. by patents, competence of its management, liquidity, and general investment climate.
The fact that 0;. is used to calculate S introduces cross-references. For the initial calculation of the volatility of the financial stock to estimate S, the notation presented in Hull (1999) is used.
These factors alter the likelihood of success in R&D and its duration and define the share of the potential market the company may be able to explore. It should be reasonable to believe that this argumentation can be reduced by assuming a virtual project value, determined by the size of the potential market for the drug and the probability that the firm will gain a certain share of this market. The influence of the investment climate should average out over time. The observed market value of a traded enterprise is thus the consensus of many investors on the value of the underlying. Their guesses will change over time, e.g. whenever new information arrives. New information may concern advances in R&D, recruitment of new employees, fund raising, etc. It is because of this research that success and failure in R&D is most important for a bio-pharmaceutical firm and that the phases in pharmaceutical R&D regulated by the FDA create a framework for a (real) options analysis.
2.6 Empirical findings In order to empirically assess the Black-Scholes approach for the purpose of company valuation, a Matlab 6.0 program was developed based on the formal rationale discussed in the preceding paragraphs of this chapter. The program is applied in a case study on a small bio-pharmaceutical company. The aim of this case study is to illustrate how publicly available information about a particular company can be used to estimate input parameters for real options valuation and to investigate the accuracy of the simple option variant. The Black-Scholes approach is the most frequently used technique in real option pricing. In its original form, it was designed to price a European call option on a non-dividend paying asset. Instead of taking a financial commodity as the underlying asset of the option, an R&D project of a bio-pharmaceutical company is used.
The project value(s) as described above should be the major driving force behind the share price of the firm, serving the role of a state variable in the options analysis. The first stage of the computing should calculate the value of the underlying using the above parameters and the fact that the value of the real option price is known because the daily prices of the financial stock can be observed, thus working as the equivalent for the real option's price in the model. The Black-Scholes formula is thus used to measure the market opinion about the underlying assets of the company. Mainly, these assets should refer to its R&D projects. It is taken advantage of the fact that the (call) option price is an increasing function of the price of the underlying. In analogy to Cox' and Rubinstein's (1985) implicit volatility, the underlying S will be called the implicit underlying.
The case study company NeoTherapeutics (Irvine, California) is a development-stage biopharmaceutical company engaged in the discovery and development of novel therapeutic drugs intended to treat neurological diseases and conditions. It was incorporated in 1987. In 2000, it employed 33 people. It is listed on the NASDAQ under the symbol NEOT. Solving the Black-Scholes formula numerically for the underlying S using the input data as described above and plotting the values of the implied underlying yields a stochastically moving process (see below). The price of the implied underlying can be seen as the project value that explains the observed share price within a Black-Scholes real option analysis. As expected, the underlying and the share price are highly correlated. Hence, if the stochastic process could be estimated based on historical data, expected
In an attempt to simultaneously find the implied price of the underlying and its implied volatility, thus extending the way how implied volatilities are usually
234
prices for the underlying and therefore for the share price could be simulated.
of financial shares. Compared to the value of the implied underlying for the case of the single option analysed in section 2 using the Black-Scholes analogy, the implied underlying option value in the compound option model should be lower. This becomes clear after recalling that the impact of the volatility of the second underlying asset is reduced by the correlation coefficient in Geske's formula which should be below one in most cases. The value of the underlying option is just the Black-Scholes price of a European call option. This is analogous to the case of a compound option model where the exercise price of the first option is zero. Figure 1 depicts the comparison between the value of the underlying option in the compound option model (dotted line) and the underlying project value derived by inverting the BlackScholes formula (solid line).
A sensitivity analysis reveals that the estimation errors of a too optimistic value for the underlying project translate most significantly into errors in the company value. Deviations from the base case in the exercise price, the duration of the respective phase of R&D, or the risk free interest rate affect the outcome of the valuation process much less. It is surprising that the same applies to errors in guessing the volatility of the underlying price process. Sensitivity analyses on the interaction between changes in the input parameters and corresponding values of financial options mostly lead to the conclusion that their prices usually react quite considerably to changes in the volatility measure (Cox and Ross 1976, Conner 1981, Chen and Sears 1990). After calculating average values for the input parameters of the simple option variant, they can be used in a Black-Scholes framework to predict the share prices of the firm under consideration. The aggregate error is calculated by the mean absolute deviation between actual and predicted share prices. The numbers indicate that the error obtained within the simple option framework is in the same order of magnitude as the error caused by a much simpler forecasting method of linear trend extrapolation of share prices.
Fig. l. Comparison between the value of the underlying option in the compound option model and the underlying project value derived by inverting the Black-Scholes formula for the case company Neotherapeutics.
3. COMPOUND OPTION EXTENSION A compound option is an option whose underlying asset is an option. The analogy for the case of a biopharmaceutical firm is that the firm value, the "visible" asset, is an option on a certain project phase, the "hidden" asset, which itself is an option on all following project phases. It is evident that this shows many characteristics of growth options discussed in real options literature. In fact, growth options in capital budgeting are often valued using the compound option technique. The rationale behind compound option thinking is much like the Black-Scholes approach of valuing a single option, where the hidden asset (the underlying) is just a commodity rather than a derivative. It is thus not surprising that a prominent compound option model created by Geske (1979) is a generalisation of the Black-Scholes formula. Geske prices an option on the value of the firm, which is considered to be an option on the firm's assets itself. The idea is that both the option value and the company value follow a Brownian motion with drift.
Further empirical evidence for a sample of 13 neurological companies confirms that the implied underlying option value according to the compound option model is systematically lower or equal to the implied underlying S of the option derived by the BlackScholes logic.
4. CONCLUSIONS The limitations of the Black-Scholes analysis for the valuation of bio-pharmaceutical companies have become evident. It is not only that some of the assumptions of Black and Scholes (1973) are violated due to the fact that we are concerned with incomplete markets in that particular application, but that the analogy of a single option should only loosely resemble reality. The compound option analogy portrays the actual situation of sequential research phases in bio-pharmaceutical drug development more appropriately because the value of the growth option associated with the current phase in research is better captured than in the Black-Scholes framework, where
The compound option value depends on the value of the underlying option, which is described by the Black-Scholes formula, the exercise price and the time to maturity of the two options. As the compound price is monotonous in the price of the underlying option, the price of the latter can be inferred from the observed compound option values trading in the form
235
Geske, R (1979). Valuation of compound options. Journal of Financial Economics, 7, 63-81. Hull, J.c. (1999). Options. futures, and other derivatives. Prentice Hall. Fourth edition. Karatzas, I. and S.E. Shreve (1998). Methods of mathematical finance . Springer-Verlag, New York, Berlin, Heidelberg. Kumar, M. (1992). The forecasting accuracy of crude oil futures prices. IlIternational Monetary Fund Staff Papers, 39, 432-461. Levy, H. (1985). Upper and lower bounds of put and call option value: stochastic dominance approach. Journal of Finance, 40, 1197-1217. Lo, AW. (1987). Semi-parametric upper bounds for option prices and expected payoffs. Journal of Financial Economics, 19, 373-387. Magill, M. and M. Quinzii (1996). Theory of incomplete markets. MIT Press, Cambridge, Massachusetts. Mason, S.P. and RC. Merton (1985). The role of contingent claims analysis in corporate finance. In: Recents advances in corporate finance (E. Altman, M. Subrahmanyam, eds.). Homewood, Illinois. Richard D. Irwin, pp.7-54. Merton, R.C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Sciences, 4,141-183. Perrakis, S. and PJ. Ryan (1984). Option pricing bounds in discrete time. The Journal of Finance, 39,519-527. Ritchken, P. (1985). On option pricing bounds. Journal of Finance, 40,1219-1233. Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics, 7, 407. Sick, J. (1995). Real options. In: Finance. Volume 9 of Handbooks in OR and Management Science (R Jarrow, V. Maksimovic, W.T . Ziemba, eds.): Elsevier North-Holland. Stapleton, RC., and M.G. Subrahmanyam (1984). The valuation of multivariate contingent claims in discrete time models. Journal of Finance, 39, 207-228. van Hulle, C. (1988). Option pricing methods: an overview. Insurance, Mathematics & Economics, 7,139-152.
it is rather implicit in the general value for the underlying asset. Up to now, the case of a firm that pursues a single project exclusively has been investigated. It offers many advantages to find realistic measures for the required input variables. However, the case of a portfolio with numerous R&D projects is more frequently observed. Modelling this situation is much more challenging and is discussed in a forthcoming article.
5. REFERENCES Black, F. and M. Scholes (1973). The pncmg of options on corporate liabilities. The Journal of Political Economy, 81, 637-659. Brennan, MJ. (1979). The pricing of contingent claims in discrete time models. Journal of Finance,34. Brennan, MJ. and A Kraus (1978). Necessary conditions for aggregation in securities markets. Journal of Financial and Quantitative Analysis, 13,407-418. Chen, K.c. and RS. Sears (1990). Pricing the SPIN. Financial Management, 19, 36-47. Conner, R.E. (1981). Determining the fair value of stock options. Pensions & Investment Age, 9, 2526. Cox, J.c. and S.A Ross (1976). A survey of some new results in financial option pricing theory. The Journal of Finance, 31, 383-402. Cox, J.c. and M. Rubinstein (1985). Options Markets. Prentice Hall, Englewood Cliffs, New Jersey. Deaves, R. and I. Krinsky (1992). Risk premiums and efficiency in the market for crude oil futures . Energy Journal, 13,93-118. Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option prices. Journal of Empirical Finance,S, 317-345. Garman, M. (1976). An algebra for evaluating hedge portfolios. Journal of Financial Economics, 3, 403-428. Gerber, H.U. and E.S.W. Shiu (1988). Nonuniqueness of option prices. Insurance. Mathematics & Economics, 7, 67-69.
236