This chapter demonstrates the applications of SCOs in project valuation, risk management and financial derivatives pricing.
§7.1 Milestone Project Valuation
This section proposes the Milestone Projection Valuation (MPV) method for multi-stage projects. The projects setting some critical milestones which should be achieved sequentially are called milestone projects (see Figure 7.1 for example). The milestone projects fail if any one of the serial milestones is not completed. The milestone projects are very common in real situations, including R&D management, manufactures, technology development, etc.. Originally, the milestone projects are valuated by methods including the net present values (NPV) and decision trees. The NPV method valuates a project under a rigorous assumption that all future cash flows are certain. Obviously, the uncertainty is ignored in the NPV method and results in symmetric underestimates. Recently, the popular real option approach is applied for flexible consideration and reasonable explanation. Under the framework of financial option theory, the real option approach decomposes the project valuation as several parameters, including the present value, costs, time to maturity, value uncertainty (volatility) and interest rate. Most of the existing real option studies for multi-stage milestone project valuations use one-fold options, while others apply multi-fold options under the assumption of constant parameters through the whole process (Casimon et al., 2004). However, the parameters often change due to the milestone completion and the project values will be misestimated if parameters are assumed constant through all the time. The one-fold real option approach is even inadequate for a multi-stage project.
New Drug FDA Approval
Phase 3 Phase 2
Clinical Phase 1 Testing
Pre-clinical Discovery
Failure
Failure
Failure
Failure
Failure Failure
Figure 7-1 : A Milestone Project Example: the New Drug Development (NDA)
Based on Theorem 3.2 & 3.3, this paper proposes a method called Milestone Projection Valuation (MPV) for multi-stage project valuation. Each milestone completion has the choice to enter the next stage or not, and the sequential project milestone can be viewed by the sequential compound CALL options. The MPV method adopts the results of SCOs and the project is regarded as the corresponding asset in SCOs. Under the same denotations as Theorem 3.2, the MPV valuation formula is expressed as
[ ] [ ]
{ }
j j{ [ ] [ ]
ig j gh j j}
i
j r i
h i i g g i i q
i T e S a e K b
MPV
j u u i
u u
×
= ×
−
×
×
−∑ −
∑
∑= ,
, 1 1
1 , , 0
0) ~ ~
( 1 τ Ν ; ρ 1 τ Ν ; ρ
,
...(7.1.1) where the strikes represent the cost at different stages; the volatilities come from the project value fluctuation and the dividend rates are replaced by the depression rates.
The option features ( ) equal one (for any i, g) due to the underlying compound calls, hence disappear in the MPV pricing formula.
g
Λi,
Compared with the literature, the MPV not only applies multi-fold compound option theory, but also allows the piece-constant parameters to vary with the distinct stages. The different parameters of different stages can adapt to the change of project nature after the milestone completion. More phenomena can be discovered from the parameter comparisons. Under the MPV model, the implicit "project valuation experience" is decomposed as the parameter estimation.
The new drug applications (NDAs) may be the most famous and significant milestone projects. Under the consideration of human health, the NDAs are well-regulated including the stages of pre-clinical trial, phase 1, phase 2, phase 3 and approval phase. Each stage has a definitive milestone. The time- and cost-consuming NDAs are the cores of the pharmaceutical companies because the R&D results from NDAs dominate their future! The MPV model can enhance the NDA valuation under a more reasonable framework and improve the R&D management of these companies.
§7.2 Complex Project Valuation
Projects with tremendous amounts value often have great contribution and impact to the society and catch a lot of public attention. In order to make sure of being executed smoothly, these projects tend to enhance project flexibility by insetting many options, such as growth, switch, abandon, shutting down, contraction or expansion (Trigeorgis 1993, 1996). However, these embedded options will also make the project structure complicated. For this kind of projects, the valuation by real option approach regarding the project still as only a one-fold option is not reasonable. Realistic and
rational valuation should take the project structure into consideration.
The different options and their interactions can be evaluated separately by different SCOs. The MPV case discussed in the previous subsection is just a special case of complex project valuation using SCOs. The effect of revenue guarantee, for example, in a build-operate-transfer (BOT) project of utility construction can be evaluated by SCOs. A company signs the BOT contract with the government to build and operate the construction while related revenue belongs to the company during operating period. The guarantee promised by government ensures the company's minimum revenue. If the actual revenue is less than the minimum, the deficit is subsidized by the government. The company hence owns the operating revenue and the put option written by the government. The put option, with the guarantee amount as its strike price, can enhance the incentives for the BOT project. At the preparation period time prior to construction, the put option can be considered as a 2-fold compound option, call on put. The add-in call option, with the construction cost as its strike price, represents the right to participate in the construction and share the potential revenue.
Similarly, the revenue guarantee of the expansion can be regarded as a 3-fold SCO, call on call on put, at the preparation period. Assume the government will offer corresponding revenue guarantee for the expansion if there is an expansion right embedded in the BOT project. The revenue guarantee of the expansion can be viewed as another put option with its own guarantee amount as the strike price. At the main construction time, the put option can be considered as a 2-fold compound option, call on put. This add-in call option, with the expansion cost as it strike price, stands for the expansion right. At the preparation time, the right can be evaluated as a 3-fold SCO:
call on call on put. The last add-in call option, with the proportional main construction cost as it strike price, represents the right to participate in the main construction. Note that the main construction cost is divided proportionally as the strike prices of both call options for the guarantee of main and expansion construction. The call on call, stacked on the put option, represents the sequential feature that the expansion right exists only when the main construction is executed. The SCOs discussed in this study make the evaluation of complex options possible.
The project valuation considering the intrinsic structure is more logical and acceptable, and is applied gradually (eg: Huang and Chou, 2006). The adoption of SCOs for project valuation can broaden and expand the real option application.
Besides, the sensitivity analysis is more visible under this situation. The change of risk source (such as the asset price, its variance, interest rate) will have different impacts on different parts of the project. The impacts can be quantified by the Greeks of SCO's (Thomassen and Van Wouwe, 2002; Lee et al., 2007) and can be applied for
risk management.
§7.3 American Options
The American options and their exotic styles can be also valuated analytically with SCOs. As is widely known, the optimal time to exercise the American call is only at the time immediately before ex-dividend of the underlying stock. Hence the valuations of American calls in the absence dividends are similar to that of European ones. The Roll-Geske-Whaley model (Roll, 1977; Geske, 1979b, 1981; Whaley, 1981) gives the explicit form for American calls with single dividend by replication of European calls and a 2-fold conventional compound option, while Cassimon et al.
(2007) extend their results to the cases of multiple dividends. Geske and Johnson (1984) propose the American put's closed form formula, which is actually an exotic SCO form.
The SCOs can be adopted for the derivation of pricing formulas for American puts on stocks paying multiple dividends. Besides, the closed form of exotic 2-fold compound options, such as European option on American options, can be proposed explicitly with SCOs.
§7.4 Risk Management
SCOs applied as the instruments for risk control are discussed in this subsection, such as for volatility risk, mortgage pipeline risk and weather risk.
(1) Volatility Risk
Volatility risk, the unobservable but crucial variable, determines the option premiums and the order of the financial system. The notorious Long Term Capital Management (LTCM) crash is just one of the evidences (Lowenstein, 2000).
Originally, the volatility index is designed for hedging volatility fluctuation, e.g. the CBOE Volatility index (VIX). Brenner et al. (2006) and Zsembery (2004) propose exotic 2-fold compound options, the option on a forward-start straddle, in order to improve the efficiency and tradability of volatility hedging.
Under this framework, SCOs can enhance the effectiveness of volatility hedging.
The plain straddle could be replaced with exotic straddles (different strikes and maturities) or complex chooser option (Rubinstein, 1992). Under identical conditions, the complex chooser option is cheaper than the straddle and thus more attractive. The compound option written on these exotic straddles or chooser options can be valuated precisely through SCOs.
(2) Mortgage Pipeline Risk
Mortgage pipeline risk, the unexpected irregular payment caused mainly by
interest rate fluctuation, has been widely hedged by compound option (Bhattacharya, 2005). If the interest rates rise, the mortgage loans fall out of the pipeline and resulting in the lenders' loss. The pipeline risk will be amplified when the loans have been sold. The 2-fold calls on put options provide mortgage corporate the rights to buy put options with cheaper cost. The put options allow the lender to sell the mortgage with higher strike prices to cover loss. There is no need to exercise the put options while the interest rate decreases and this save the cost accordingly.
This kind of pipeline risk hedge can be enhanced through the long position of a pool of SCOs. The central banks (e.g., the FED in the U.S.) often take sequential actions of interest rate hiking to overcome inflation, so the pools of SCOs (combination with 2-fold, 3-fold, etc.) let the lenders make decisions depending on the up-to-date situations with cheaper expenses. The SCOs can also be used for valuating the mortgage with prepayment under the option-adjusted-spread (OAS) framework.
(3) Weather Risk
Insurers and reinsurers pay more attention to the ecosystem evolution than single accidents. They also are more willing to provide long-term management of weather risk than most trading houses. Thus compound options are introduced to give reinsurers (e.g., Swiss Re) the rights to buy an option on the weather risk at a later time (Gakos, 1999). The split-fee feature of compound options can reduce the cost the long-maturity and high-amount hedge of weather risk.
SCOs can offer better hedge effects than 2-fold compound options. Through the combination of different fold/ different maturities SCOs, insurers with a slight up-front premium can lock in coverage at different future exercise dates with additional lower premiums.