قیمت گذاری گزینه های ترکیب متوالی تعمیم یافته و تجزیه و تحلیل حساسیت

This paper proposes a generalized pricing formula and sensitivity analysis for sequential compound options (SCOs). Most compound options described in literatures, initiating by Geske [Geske, R., 1977. The Valuation of Corporate Liabilities as Compound Options. Journal of Finance and Quantitative Analysis, 12, 541–552; Geske, R., 1979. The Valuation of Compound Options. Journal of Financial Economics 7, 63–81.], are simple 2-fold options. Existing research on multi-fold compound options has been limited to sequential compound CALL options whose parameters are constant. The multi-fold sequential compound options proposed in this study are defined as compound options on (compound) options where the call/put property of each fold can be arbitrarily assigned. In addition, the deterministic time-dependent parameters, including interest rate, depression rate and variance of asset price, make the SCOs more flexible. The pricing formula is derived by the risk-neutral method. The partial derivative of a multivariate normal integration, which is an extension of Leibnitz's Rule, is derived in this study and used to derive the SCOs sensitivities. The general results for SCOs presents in this paper can enhance and broaden the use of compound option theory in the study of real options and financial derivatives.

Compound options, initiating by Geske, 1977 and Geske, 1979, are options with other options as underlying assets. The fold number of a compound option counts the number of option layers tacked directly onto underlying options. The original closed form of 2-fold compound option is proposed by Geske, 1977 and Geske, 1979 and constitutes as precedents with respect to later works. Specific multi-fold compound option pricing formulas are proposed by Geske and Johnson (1984a) and Carr (1988) while the pricing formula of sequential compound call (SCC) is proved by Thomassen and Van Wouwe (2001) and Chen (2003). Chen (2002) and Lajeri-Chaherli (2002) simultaneously derive the price formula for 2-fold compound options through the risk-neutral method. Agliardi and Agliardi (2003) generalize the results to 2 fold compound calls with time-dependent parameters, while Agliardi and Agliardi (2005) extend the multi-fold compound calls to parameters varying with time. Financial applications based on compound option theory are widely employed. Geske and Johnson (1984a) use exotic multi-fold compound options for the American put option, while Carr (1988) presents the pricing formula for sequential exchange options. Corporate debt (Chen, 2003 and Geske and Johnson, 1984b) and chooser options (Rubinstein, 1992), as well as capletions and floortions (options on interest rate options) ( Musiela and Rutkowski, 1998) are also priced by compound options. In addition to the pricing of financial derivatives, compound option theory is widely used in the real option study. This approach originates from Myers (1977) and is followed by Brennan and Schwartz (1985), Pindyck (1988), Trigeorgis (1993, 1996) and so forth. Examples include project valuation of new drugs (Casimon et al., 2004), production and inventory (Cortazar and Schwartz, 1993) and capital budget decision (Duan et al., 2003). Compound option methodology turns out to be very common, and the theory is versatile enough to treat many real-world cases (Copeland and Antikarov, 2003). However, the sophisticated structure of financial derivatives and their wide deployment in the real options field have revealed the limitations of the current compound option methodology. 2-fold compound options cannot be used as further building blocks to model other financial innovations, but results concerning multi-fold compound options so far have focused only on sequential compound calls. Although Remer et al. (2001, p.97) mention that “… in practice, different project phases often have different risks that warrant different discount rates,” the important feature of time-dependent (or fold-dependent) parameters is rarely taken into account by current methodologies. This paper, using vanilla European options as building blocks, extends the compound option theory to multi-fold sequential compound options (SCOs) with time-dependent parameters as well as alternating puts and calls arbitrarily (see Table 1). An SCO is defined as a (compound) option written on another compound option, where the call/put feature of each fold can be assigned arbitrarily. The SCOs presented in this study also allow deterministic parameters (such as interest rate, depression rate and variance of asset price) to vary over time, hence entitle this paper as a “generalized” SCOs and regard the situation of fold-wise parameters as its special case. This study derives an explicit valuation formula for SCOs by the risk-neutral method, and performs the sensitivity analysis on the result. Compared with the P.D.E. method, more financial intuition is gained by the risk-neutral derivation. Moreover, the partial derivative of a multivariate normal integration (an extension of Leibnitz's rule), is also derived here for the sensitivity analysis. Table 1. Evolutions of compound option theory Reference Fold Approach Generalization Number Put-call alternating Time-dependent parameters Geske, 1977 and Geske, 1979a 2 PDE Put/Call No Agliardi and Agliardi (2003) 2 PDE Call Yes Chen (2002), Lajeri-Chaherli (2002) 2 Risk-neutral Put/call No Carr (1988), Chen (2003) Multiple Risk-neutral Call No Thomassen and Van Wouwe (2001) Multiple PDE Call No Agliardi and Agliardi (2005) Multiple Risk-neutral Call Yes This Paper Multiple Risk-neutral Put/call Yes a The seminal compound option paper series. Table options Multi-fold SCOs with alternating puts and calls and time-dependent parameters can greatly enhance the number of practical applications for compound options, especially in the real option field. Real world cases can often be expressed in terms of options, such as expansion, contraction, shutting down, abandon, switch, and/or growth (Trigeorgis, 1993 and Trigeorgis, 1996). These options with different types can be evaluated by the 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 its 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 its 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 SCOs can also be applied to the existing real option applications, such as the competing technology adoption (Kauffman and Li, 2005), joint ventures behavior analysis (Kogut, 1991) and strategic project examination (Bowman and Moskowitz, 2001). Furthermore, the pricing of exotic financial derivatives, such as exotic chooser options and capletions, can also be accomplished using SCO methodology. This paper is arranged as the follows. Section 2 presents the SCOs pricing formula. Section 3 presents some features of multivariate normal distributions, and derives some comparative statistics as its application. The paper ends with the conclusion

The present study defines and derives the pricing formula of sequential compound options (SCOs), where the parameters vary over time and each fold option may have different put/call attribute. The SCO price can be evaluated by a linear combination of the asset and strike prices weighted by different variate normal integrations. The risk-neutral method enriches the SCOs pricing formula derivation with more financial implications than P.D.E. method. The partial derivative of a multivariate normal integration is derived in this paper as an extension of Leibnitz's Rule, and is used to derive the sensitivities of SCOs. Previous results have analyzed 2-fold puts/calls-alternating compound options or multi-fold “sequential compound calls” where all options are of call-type. Fold-wise differences are rarely taken into consideration. The SCOs presented in this paper have the following qualities. First of all, multi-fold SCOs enable arbitrary option feature (call/put) assignments, greatly enhancing the range of practical applications that can be treated by compound option theory. Second, in real-world problems option parameters often vary over time; SCOs enabling time-dependent parameters (interest rate, depression rate and variance of the asset price) can capture the "sequential" features. Third, SCOs can accommodate an arbitrary number of folds. Furthermore, SCOs can be used to demonstrate some features of multivariate normal integrals, such as their partial derivatives. The Leibnitz's rule can be used to decompose the partial differential of (k + 1)-variate integration into two parts: a k-variate normal integration and an integration with the integrand of a partial derivative. This paper proves that, under the multivariate normal cases, these two parts can be presented in a unified form. Based on the result, sensitivities of SCOs to asset price (and its change) and interest rate (under the case of interest rate fold-wise) are derived. SCOs generalize the methodology of European Options (Black and Scholes, 1973), 2-fold compound options (Geske, 1977 and Geske, 1979) and sequential compound calls (Thomassen and Van Wouwe, 2001 and Agliardi and Agliardi, 2005), and can be regarded intuitively as multi-dimensional options extending from their work. Moreover, the sensitivities of SCOs can also be expressed explicitly as generalized versions of those of their works. The generalized parameters presented in this study regard the parameters as deterministic time-dependent functions. This kind of parameter setting considers the constant or fold-wise constant situations as their special cases and allows the SCOs more flexible. However, the case of stochastic interest rate for compound options should under an unreasonable and unacceptable condition (Frey and Sommer, 1998). Thus the SCOs are not extended to stochastic cases for realistic consideration. For the advantages of using 2fold compound options as financial instruments (Bhattcaharya, 2005), such as split-fee, decision postponement and risk management, SCOs can do better. SCOs buyers pay a few premiums at the initial time and own the privilege to pay again while they exercise to gain the next fold SCOs. The SCOs will be discarded while they are not worth holding in sacrificing previous payment. This split-fee property let the SCOs owners to pay proportionally according to available information at that time, instead of sinking option premium at the beginning. Thus the decision-making can be postponed under indefinite environments and more flexibility is offered to SCOs holders. The feature of SCOs with high profit potential under constrained cost can provide greater leverage and yield enhancement for SCOs owners. SCOs can also be tailored for financial institutions as risk management instruments, such as hedging or mortgage pipeline risk. SCOs can enhance and broaden the use of compound option theory in real option and financial derivative fields. Real options often incorporate multiple options of different types with sophisticated interactions, but such situations can be evaluated by aggregating various SCOs. Some complex options can be regarded as exotic SCOs and can applied the similar derivation in this study to get explicit pricing formulas. Even milestone projects, which must decide whether or not a project has terminated according to the milestone achievement, can be evaluated through the use of SCOs. Compared with the constant variance and interest rate of the SCC assumed in Casimon et al. (2004), allowing parameters to vary with different periods makes this method of project valuation more precise and flexible. Finally, a number of complex financial derivatives can be developed or evaluated using SCOs in the same way that chooser options and capletions can be priced by 2-fold compound options. These applications of SCOs with real-world cases will be the subject of probable future study.