行政院國家科學委員會專題研究計畫 成果報告
序列複合買權法應用於履約保證下多期 BOT 基礎建設專案
評價之研究
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 100-2221-E-009-137- 執 行 期 間 ： 100 年 08 月 01 日至 101 年 07 月 31 日 執 行 單 位 ： 國立交通大學土木工程學系（所） 計 畫 主 持 人 ： 黃玉霖 計畫參與人員： 博士班研究生-兼任助理人員：畢佳琪 博士班研究生-兼任助理人員：林岑縉 公 開 資 訊 ： 本計畫可公開查詢中 華 民 國 101 年 10 月 31 日
中 文 摘 要 ： 利用 BOT 模式來推動基礎建設民營化，尤其是大規模基礎建 設投資專案， 已成為全球營建市場一種成熟的趨勢。然而， 在不確定的市場環境與無可預期的各種變化之下，這些大規 模的 BOT 投資專案必須面對極大的投資風險，特別是針對特 定市場量身制裁的長期專屬資產(dedicated assets)投資所 形成沉默成本投資(sunk cost)。為了吸引私部門進行這些投 資，各國政府提供各種特許權利誘因措施，以協助投資者進 行 BOT 專案的風險控管，如：還款保證、最低營收保證、分 期投資擴張規模的權利，以及特許期間屆滿前的放棄權利 等。這些特許誘因的出現，意味著傳統的專案評價方法，如 折現現金流模型(discounted cash flow model, DCF )，已 不再適用於 BOT 專案評價。為了解決較複雜的 BOT 評價的問 題，各國的研究人員已經發展出許多不同的實質選擇權評價 模型(real-option valuation models)。
雖然實質選擇權的評價方法很盛行，但現有的 BOT 實質選擇 權評價模型，尚未針對履約保證金對專案價值的影響，提出 任何評估。特別是，BOT 專案價值在特許契約簽訂時是不確 定的，必須依據契約執行期間所獲取的專案新訊息，不斷地 進行重新評價。這將產生＇套牢＇ (＇hold-ups＇) 的可能 性。亦即，即使某項建設經重新評價之後認定是不再值得投 資的，政府仍然可能會利用已簽屬的合約條款向法院提出強 制執行的請求，要求特許公司繼續進行該項不可行的投資。 因此，為了迴避風險， BOT 特許契約通常會賦予特許公司， 在契約執行期間提前放棄投資的權利。然而，政府也會藉由 履約保證金的設定，確保投資者不會任意行使該項放棄權 利。 在實質選擇權評價理論下，提前放棄投資的權利是具有潛在 價值的；但是，履約保證金的設置將會降低此潛在價值。即 使履約保證金可以避免任意行提前放棄的權利，這還是會帶 來問題。更具體的說：如果 BOT 專案實質選擇權的評價不考 量履約保證金的負面影響，評價結果將會高估專案投資價 值，甚至誤導投資。 有鑑於此，本研究將建立序列複合買權評價模型(sequential compound call option valuation model)，推導多期 BOT 投 資專案在履約保證之下的評價封閉解。本研究也將提供敏感 性分析來探討履約保證金對於 BOT 專案價值的影響。同時， 本研究將利用 MATLAB 程式來撰寫 SCCO 的專案評價軟體，選 定實際的投資案例，進行專案評價的應用與分析。 中文關鍵詞： BOT、民營化、多期基礎建設投資、履約保證、序列複合買 權、評價、敏感度分析
英 文 摘 要 ： 英文關鍵詞：
行政院國家科學委員會專題研究計畫成果報告
序列複合買權法應用於履約保證下多期BOT 基礎建設專案評價之研究
Sequential compound call option valuation of multistage BOT
infrastructure projects under performance bonding
計畫編號：NSC 100－2221－E－009－137 執行期限：100年8月01日 至 101年7月31日 主 持 人：黃玉霖 交通大學土木系 教授 計畫參與人員：畢佳琪、林岑縉 中文摘要 利用 BOT 模式來推動基礎建設民營化，尤其 是大規模基礎建設投資專案， 已成為全球營建市 場一種成熟的趨勢。然而，在不確定的市場環境 與無可預期的各種變化之下，這些大規模的 BOT 投資專案必須面對極大的投資風險，特別是針對 特 定 市 場 量 身 制 裁 的 長 期 專 屬 資 產 (dedicated assets)投資所形成沉默成本投資(sunk cost)。為了 吸引私部門進行這些投資，各國政府提供各種特 許權利誘因措施，以協助投資者進行 BOT 專案的 風險控管，如：還款保證、最低營收保證、分期 投資擴張規模的權利，以及特許期間屆滿前的放 棄權利等。這些特許誘因的出現，意味著傳統的 專案評價方法，如折現現金流模型(discounted cash flow model, DCF )，已不再適用於 BOT 專案評價。 為了解決較複雜的 BOT 評價的問題，各國的研究 人員已經發展出許多不同的實質選擇權評價模型 (real-option valuation models)。
雖然實質選擇權的評價方法很盛行，但現有 的 BOT 實質選擇權評價模型，尚未針對履約保證 金對專案價值的影響，提出任何評估。特別是， BOT 專案價值在特許契約簽訂時是不確定的，必 須依據契約執行期間所獲取的專案新訊息，不斷 地進行重新評價。這將產生“套牢” (“hold-ups”) 的 可能性。亦即，即使某項建設經重新評價之後認 定是不再值得投資的，政府仍然可能會利用已簽 屬的合約條款向法院提出強制執行的請求，要求 特許公司繼續進行該項不可行的投資。因此，為 了迴避風險， BOT 特許契約通常會賦予特許公司， 在契約執行期間提前放棄投資的權利。然而，政 府也會藉由履約保證金的設定，確保投資者不會 任意行使該項放棄權利。 權利是具有潛在價值的；但是，履約保證金的設 置將會降低此潛在價值。即使履約保證金可以避 免任意行提前放棄的權利，這還是會帶來問題。 更具體的說：如果 BOT 專案實質選擇權的評價不 考量履約保證金的負面影響，評價結果將會高估 專案投資價值，甚至誤導投資。 有鑑於此，本研究將建立序列複合買權評價 模 型 (sequential compound call option valuation model)，推導多期 BOT 投資專案在履約保證之下 的評價封閉解。本研究也將提供敏感性分析來探 討履約保證金對於 BOT 專案價值的影響。同時， 本研究將利用 MATLAB 程式來撰寫 SCCO 的專案 評價軟體，選定實際的投資案例，進行專案評價 的應用與分析。 關鍵詞：BOT、民營化、多期基礎建設投資、履 約保證、序列複合買權、評價、敏感度分析 Abstract
The build-operate-transfer (BOT) model is a popular approach to infrastructure privatization, especially for large-scale infrastructure investments. However, large-scale BOT projects usually require long-term sunk investments in infrastructure facilities that are exposed to uncertain market conditions and unforeseen contingencies. To attract private-sector investments in BOT projects, host governments have offered a variety of concession arrangements for BOT risk management, such as loan repayment guarantee, minimum-revenue guarantee, the rights to expand incrementally, and the rights to abandon prematurely. The presence of these arrangements means that traditional valuation methods, such as the discounted cash flow model,
Researchers have developed various real-option models to treat the more complex BOT valuation issues.
Although the real-option approach is popular, existing BOT real-option valuation models have not incorporated the impact of performance bonds on project value. In particular, the value of BOT projects is uncertain at contract signing, and must be re-valuated when new project information is available during contract execution. This produces the possibility of a “hold-up:” the government may pursue court enforcement of literal contract terms, asking the concessionaire to invest in the underlying project even when the project is deemed infeasible by a re-valuation. To avoid the “hold-up” risk, BOT concession contracts often grant concessionaires voluntary abandonment rights during contract execution. However, host governments also require performance bonding by concessionaires as a security in order to ensure that voluntary termination is not exercised arbitrarily.
In real option theories, voluntary abandonment is potentially valuable, but imposing a bonding requirement creates a penalty term upon voluntary abandonment. Performance bonding reduces the value of voluntary abandonment rights even though the arbitrary exercise of the rights can be avoided. If the impact of performance bonds on BOT project value is not assessed in BOT real option valuation, the resulting project value would tend to be overstated.
Accordingly, this research will apply the theory of sequential compound call option (SCCO) to derive a closed form solution to the valuation of multistage BOT infrastructure projects under performance bonding. This research will also provide sensitivity analysis to examine the impact of performance bonding on BOT project value. A
computer program will be written to support the numerical implementation of the closed solution using MATLAB, and a real-world BOT case will be chosen to demonstrate numerically the applicability of the proposed valuation model.
Keywords: Build-operate-transfer; multistage infrastructure investment, privatization, performance bond, sequential compound call option, valuation, sensitivity analysis
INTRODUCTION
For nearly three decades infrastructure privatization has gradually become a popular, well-established approach to the delivery of infrastructure services. For the early developments of this trend, Huang (1995) documented over eighty privatized infrastructure projects in the world. Huang (1995) focused on the institutional and regulatory designs of these projects. For more recent developments, Tam (1999) and Kumaraswamy and Morris (2002) investigated build-operate-transfer (BOT) infrastructure projects in Asia. Chen and Messner (2005) investigated BOT water supply projects in China. Kleiss and Imura (2006) investigated private finance initiative (PFI) in Japan. Winch (2000) investigated PFI public works projects in the United Kingdom. Koch and Buser (2006) investigated public-private partnership (PPP) governance in Denmark. Fischer, Jungbecker, and Alfen (2006) investigated PPP infrastructure developments in Germany. Vazquez and Allen (2004) investigated BOT highway projects in Central America and Mexico. Algarni, Arditi, and Polat (2007) investigated BOT infrastructure projects in the United States.
Between PFI and other types of infrastructure privatization approaches, the BOT model is popular for large-scale infrastructure developments.
Large-scale BOT projects usually require long-term sunk investments in infrastructure facilities that are exposed to uncertain market conditions and unforeseen contingencies (for example Grimsey and Lewis [2002]). To attract private-sector investments in BOT projects, host governments have offered a variety of concession arrangements for BOT risk management, such as loan repayment guarantee, minimum-revenue guarantee, the rights to expand incrementally, and the rights to abandon prematurely (for example Huang [1995] and Wibowo [2004]). The presence of these arrangements means that traditional valuation methods, such as the discounted cash flow model, are no longer satisfactory for BOT project valuation. Researchers have developed various real-option models to treat the more complex BOT valuation issues. For example, Rose (1998) evaluated interacting toll road investment options. Smit (2003) provided a real-option-based game theory model to evaluate airport expansions. Garvin and Cheah (2004) proposed a real-option pricing model for analyzing toll road investments. Wand and Min (2006) evaluated the interrelationships among power generation projects. Huang and Chou (2006) evaluated minimum-revenue guarantees. Damnjanovic, Duthie, and Waller (2008) evaluated the interconnectivity and flexibility of toll road expansions. Huang and Pi (2009) developed a European-style sequential compound call option (SCCO) model to evaluate multi-stage BOT projects involving dedicated asset investments. Huang and Pi (2011) extended the SCCO model to assess the impacts of competition and technological obsolescence on project value in privatized infrastructure markets.
In fact real-option models are powerful valuation tools not only for complex BOT concession arrangements. The real-option valuation
approach has also been applied for information technology and other types of investment projects. For example, Panayi and Trigeorgis (1998) developed a real-option model for the valuation of multistage information technology projects. Yeo and Qiu (2002) discussed the valuation of investment flexibility of technology investment projects by the real-option approach. Chen, Zhang, and Lai (2009) developed an integrated real-option approach for the valuation of information technology projects. Eckhause, Hughes, and Gabriel (2009) developed a real-option approach for vendor selection in multi-stage R&D acquisitions.
Although the real-option approach is popular and powerful, existing BOT real-option valuation models have not incorporated the impact of performance bonds on project value. In particular, the value of BOT projects is uncertain at contract signing, and must be revaluated when new project information is available during contract execution. This produces the possibility of a “hold-up:” the government may pursue court enforcement of literal contract terms, asking the concessionaire to invest in the underlying project even when the project is deemed infeasible by a re-valuation. To avoid the “hold-up” risk, BOT concession contracts often grant concessionaires voluntary abandonment rights during contract execution. According to Klein (1996), voluntary termination can avoid “hold-ups.” However, host governments also require performance bonding by concessionaires as a security. As Vandegrift (1999) suggested, performance bonding can ensure that voluntary termination is not exercised arbitrarily.
In real option theories, voluntary abandonment is a type of flexibility to avoid irreversible sunk investments under adverse market conditions (for example Dixit and Pinkyck [1994]). This type of
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hen, the fin ven as
1,1 1 1 , 1 max m ) , ( S t S Therefor itial payoff ith performa solution t all option ( of perform ution is ba he assumpti wed the w y Black and syncratic as umed to fol Q t dz enotes the he standard nvironment variance expected ri inistic divid sume that markets. Als would inves option C 1 ,1 1,1 1 B . e of the on he amount nal payoff o
1,1 1,1
1 , 1 , max B K K S re, then risk f of a one-ance bond a to a Euro (European mance bond ased on ris ions of the well known d Scholes (1 ssets of a B llow a stoch stochastic rd Brownia t Q. The pa of the re isk-free rate dend payout there is so, for a o st only if the ,1(S,t1) is By definit ne-fold call of the pe of a one-fo
1,1 1 , 1 1 , 1 0 , , B B k-neutral pr -fold Europ as: opean Sequ SCCO) wit d. In additio k-neutral p e option p n B-S-M m 1973) and M BOT project hastic proc asset valu an motion arameter 2 eturn, r i e of return, t rate. To ob no-arbitrag one-stage pr e time-t1 va “in the mo tion, K 1,1i option at t erformanceold call opt
ricing give pean call op uential th the on, the pricing pricing model Merton are at ess of (1) e and under is the is the and q btain a ge in roject, alue of oney”, is the time-t1 bond. tion is (2) s the ption
1,1 1,1 1,1
) ( 0 1 , 1 , max ) , ( 1 0 E S K B e t S C Q du u r t t (3)where EQ denotes a conditional expectation
operator.
In addition, for a 2-stage project, the investor
would invest in the second stage at time t1 only if the
balance between C 2,1(S,t1) and K 2,1 is higher than
the amount of the performance bondB 2,1. Here the
boundary condition can be known as
2,1(S,t1) K 2,1 B 2,1
C , and the time-t1 value of
2,2(S,t1) C is given by
2,1 2,1 2,1
2,1 1 , 2 1 , 2 1 , 2 1 2 , 2 0 , max , max ) , ( B B K S B K S t S C (4)By induction, for the n-stage project, denote
n,n i1(S,ti)
C and C n,ni(S,ti) respectively as the
time-ti value of the n-(i-1)th SCCO and the (n-i)th
fold SCCO. Then the boundary condition for exercising the n-(i-1)th SCCO is given by
n ni i n i n i
ni i n i n i i n n i i n n B B K t S C B K t S C t S C , , , , , , , 1 , 0 , ) , ( max , ) , ( max ) , ( (5)Applying the idiosyncratic asset value process in (1) and the boundary conditions in (5) gives the following theorem.
Theorem. (The European SCCO Pricing Formula)
{[ ] ;[ ] } } ] [ ; ] {[ } ] [ ; ] {[ ) , ( 1 1 , 1 1 , 1 ) ( , 1 , 1 , ) ( , , 1 , 1 , ) ( 0 , 0 , 0 0 1 0
m m j i m i n m du u r m n n m m m j i m i n m du u r m n m n n m n n j i n i n n du u q n n n h N e B h N e B K g N e S t S C m t t m t t n t t (6) where i n du u du u u q u r S S g i i t t t t n i i n
1 ; ) ( )] ( 2 1 ) ( ) ( [ ) ln( 0 0 2 2 , 0 , i n du u du u u q u r S S h i i t t t t n i i n
1 ; ) ( )] ( 2 1 ) ( ) ( [ ) ln( 0 0 2 2 , 0 , Proof.1-fold European call option
The final payoff of the option is written as equation (3). Base on the risk-neutral pricing method, provided by Cox and Ross (1976), Harrison and Kreps (1979) and Harrison and Pliska (1981); the fair price of a vanilla European call option with the consideration of the performance bond is equal to
1,1 1,1 1,1
) ( , max 1 0 E S K B e Q du u r t t under the measure Q. Then
1 0 1 , 1 1 , 1 1 , 1 1 0 1 , 1 1 , 1 1 , 1 1 0 1 0 1 , 1 1 , 1 1 , 1 1 0 1 0 1 0 ) ( 1 , 1 )) ( ) ( 1 , 1 1 , 1 ) ( 1 , 1 ) ( ) ( 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 ) ( 0 0 , 1 1 ) ( 1 1 )) ( ( ] 0 ), ( max[ , max ) , ( t t t t t t t t t t t t t t du u r B K S Q du u r B K S Q du u r du u r B K S Q du u r Q du u r Q du u r e B E e B K S E e e B B K S E e B B K S E e B K S E e t S C (7)According to equation (1), the solution of the stochastic differential equation (S.D.E.) is given as
10 2 1 0 2( )] ( ) 2 1 ) ( ) ( [ 0 , 1 1 , 1 t t Q t t du u z du u u q u r e S S (8) Then we can use equation (8) to substitute the
1,1
S in equation (3). The pricing formula becomes
1 0 1 , 1 1 , 1 1 , 1 1 0 1 , 1 1 , 1 1 , 1 1 0 2 1 0 2 1 0 ) ( 1 , 1 )) ( ) ( 1 , 1 1 , 1 ) ( ) ( ) ( 2 1 ) ( 0 , 1 0 0 , 1 1 ) ( 1 ) , ( t t t t t t Q t t t t du u r B K S Q du u r B K S du u z du u Q du u q e B E e B K e E e S t S C (9)Before deriving the pricing formula of a European call option with the consideration of the performance bond, we have to eliminate the
uncertain term in the expectation operator EQ.
Therefore, we use Girsanov’s Theory to change the probability measure, and the Radon-Nikodym derivative is defined as 1 0 1 02 ( ) ( ) 1 2 t 2 t Q t t u du z u du dQ dR (10)
Base on the equation (10), the Brownian motion term before and after change of measure can be defined as 1 0 ( ) 2 t t R t Q t dz u du dz (11) According to Girsanov’s Theorem and equation (11), we can rewrite equation (1) as
R t dz dt q r S dS 2 (12) where R tdz represent the standard Brownian motion
under the measure R, underlying asset as numeraire.
Since the underlying asset is log-normally distributed, we use the Ito’s lemma to find the solution of equation (12). After some calculation, the dynamic price of the asset S under measure R is finally found and shown as follows
10 2 1 0 2 ) ( )] ( 2 1 ) ( ) ( [ 0 , 1 1 , 1 t t R t t ru qu u du z udu e S S (13) Next, we can put (13) back into to the pricing formula (9) and change the measure from Q to R, and recalculate the probability,
1 0 1 0 1 0 1 0 1 , 1 1 , 1 1 , 1 1 0 1 , 1 1 , 1 1 , 1 1 0 1 0 1 , 1 1 , 1 1 , 1 1 0 1 , 1 1 , 1 1 , 1 1 0 ) ( 1 , 1 1 , 1 1 , 1 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 ) ( 0 , 1 ) ( 1 ) ( ) ( 1 1 ) ( ) ( 0 , 1 ) ( 1 , 1 )) ( ) ( 1 , 1 1 , 1 ) ( ) ( 0 , 1 0 0 , 1 ) ln( ln ) ln( ln 1 1 1 1 ) , ( t t t t t t t t t t t t t t t t t t du u r Q du u r R du u q du u r B K S Q du u r B K S R du u q du u r B K S Q du u r B K S Q du u q e B B K P e B K B K S P e S e B E e B K E e S e B E e B K dQ dR E e S t S C where 1,1 2 2 1 , 1 1 , 1 0 , 1 1 , 1 1 , 1 2 2 0 , 1 1 , 1 1 , 1 1 , 1 1 0 1 0 1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ln ) ln( ) ( )] ( 2 1 ) ( ) ( [ ln ) ln( ln g z P du u du u u q u r B K S z P B K du u z du u u q u r S P B K S P R R t t t t R R t t R t t R R With the same pattern, we can have
lnS1,1 ln(K1,1 B1,1)
P
z h 1,1
PQ R Q where 1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ln 2 2 1 , 1 1 , 1 0 , 1 1 , 1 t t t t du u du u u q u r B K S h Therefore, the pricing model becomes to
1 0 1 0 1 0 1 0 1 0 1 0 ) ( 1 , 1 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 ) ( 0 , 1 ) ( 1 , 1 2 ) ( 1 , 1 1 , 1 1 ) ( 0 , 1 0 0 , 1 ) ( ) ( ) , ( t t t t t t t t t t t t du u r du u r du u q du u r Q Q du u r R R du u q e B h N e B K g N e S e B d z P e B K d z P e S t S C (14)Both zR and zQ are the standard Brownian Motion,
given the fact thatz~ N
0,1 , and 1N(h 1,1)N(h 1,1)；1 ) 0 (
N .
2-fold compound call option
For the 2-stage project, the investor would
invest in the second stage at time t1 only if
2,1(S,t1)
K 2,1 B 2,1
2,2(S,t1)
C given as equation (4).
According to the boundary condition given by
expression (4), the time-t0 price of the two-fold
SCCO can be found by the risk-neutral pricing approach elaborated by Lajeri-Chaherli (2002):
01 1 0 1 0 ) ( 1 , 2 1 , 2 1 , 2 1 1 , 2 ) ( 1 , 2 1 , 2 1 , 2 1 1 , 2 ) ( 0 2 , 2 ) ( ) , ( , 0 max ) , ( , 0 max ) , ( t t t t t t du u r du u r Q du u r e B dz z f B K t S C e B B K t S C E e t S C (15)
Since C 2,1(S,t1)is a one-fold call option, we can use
the result of equation (14) and shift the initial time
from t0 to t1, which is
2 1 2 1 2 1 ) ( 2 , 2 1 ,* 1 , 1 ) ( 2 , 2 2 , 2 1 ,* 1 , 1 ) ( 1 , 2 1 1 , 2 ) ( ) ( ) , ( t t t t t t du u r du u r du u q e B h N e B K g N e S t S C .Also, set S1, 2be the equivalent asset value found at
the point where the underlying option finishes “at the money”, i.e., C 2,1(S,t1)
K 2,1B 2,1
, and it canbe found at the point where the underlying option finishes ‘at the money’. Set
1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ) ln( 2 2 2 , 1 0 , 2 1 , 2 t t t t du u du u u q u r S S h , and it meansS 2,1S1, 2if
and only if zh 2,1. With the consideration of
boundary condition and
10 2 1 0 2()] () 2 1 ) ( ) ( [ 0 , 2 1 , 2 t t t t ru qu u du z udu e S
S , then the equation (15)
can be rewritten as follows,
1 0 2 1 , 2 2 0 1 , 2 2 1 0 2 0 1 , 2 2 2 0 2 1 0 2 1 , 2 2 0 1 0 1 , 2 1 0 1 , 2 2 0 1 , 2 1 0 1 0 2 1 0 2 1 , 2 2 1 1 0 1 0 1 , 2 2 1 2 1 2 1 1 0 ) ( 1 , 2 2 1 ) ( 2 , 2 2 1 ) ( 1 , 2 1 , 2 ) ( 1 ,* 1 , 1 2 1 ) ( 2 , 2 2 , 2 1 ,* 1 , 1 ] ) ( [ 2 1 ) ( 0 , 2 ) ( 1 , 2 ) ( 1 , 2 1 , 2 ) ( 2 , 2 1 ,* 1 , 1 ) ( 2 , 2 2 , 2 1 ,* 1 , 1 ) ( )] ( 2 1 ) ( ) ( [ 0 , 2 ) ( ) ( ) ( 1 , 2 1 , 2 1 , 2 ) ( 2 , 2 1 ,* 1 , 1 ) ( 2 , 2 2 , 2 1 ,* 1 , 1 ) ( 1 , 2 ) ( 0 2 , 2 2 1 2 1 ) ( 2 1 ) ( 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( max ) , ( t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t du u r z h du u r h z du u r du u r h z du u r du u z h du u q du u r h du u r h du u r h du u r du u z du u u q u r h du u q du u r du u r h du u r du u r du u q du u r e B dz e e B dz e e B K dze h N e e B K dz g N e e S e B dz z f e B K dz z f e B dz z f h N e B K dz z f g N e S e e e B dz z f B K e B h N e B K g N e S e t S C Set e e N g dz S C t t t t z udu h du u q ) ( 2 1 1 ,* 1 , 1 ] ) ( [ 2 1 ) ( 0 , 2 1 2 1 0 2 1 , 2 2 0
; e dz e B K dze h N e e B K C h z du u r du u r h z du u r t t t t t t 1 , 2 2 1 0 2 0 1 , 2 2 2 0 2 1 ) ( 1 , 2 1 , 2 ) ( 1 ,* 1 , 1 2 1 ) ( 2 , 2 2 , 2 2 2 1 ) ( 2 1 ;
1 0 2 1 , 2 2 0 () 1 , 2 2 1 ) ( 2 , 2 3 2 1 t t t t z rudu h du u r e B dz e e B C ; where i i C t S C 3 1 0 2 , 2 ( , ) .Use bivariate normal distribution function to
reform C1 and C2 where the factor g 1,1,*1 and
1,1,*1
2 0 2 1 2 0 1 0 2 0 2 0 2 1 2 0 1 0 2 1 1 0 1 0 2 1 2 1 2 1 2 1 ) ( ) ( ) ( ) ( ) ( )] ( 2 1 ) ( ) ( [ ) ln( ) ( ) ( )] ( 2 1 ) ( ) ( [ ) ln( ) ( ) ( )] ( 2 1 ) ( ) ( [ ) ( )] ( 2 1 ) ( ) ( [ ) ln( ) ( )] ( 2 1 ) ( ) ( [ ) ln( 2 2 2 2 2 2 2 2 , 2 2 , 2 0 , 2 2 2 2 2 2 , 2 2 , 2 0 , 2 2 2 2 2 2 2 , 2 2 , 2 0 , 2 2 2 2 , 2 2 , 2 1 , 2 1 ,* 1 , 1 t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t du u du u du u du u z du u du u u q u r B K S du u du u z du u u q u r B K S du u du u z du u u q u r du u du u u q u r B K S du u du u u q u r B K S g where 1 0 ( ) 2 2 t t u du z z and 2 2 , 1 2 2 1 ) ( ) ( 2 0 2 1 t t t t du u du u .Finally, g 1,1,*1 can be reform as
2 2 , 1 2 2 , 1 2 , 2 1 z g . with
the same deriving pattern, h 1,1,*1 can be reform as
2 2 , 1 2 , 1 2 , 2 1 z h . ( , ; ) ) 1 ( 2 1 2 , 1 2 , 2 1 , 2 2 ) ( 0 , 2 2 2 2 , 1 2 2 , 1 2 , 2 2 1 ) ( 0 , 2 1 2 0 2 2 1 , 2 2 0 g g N e S dz z g N e e S C t t t t du u q z g du u q
2,1 ) ( 1 , 2 1 , 2 2 , 1 2 , 2 1 , 2 2 ) ( 2 , 2 2 , 2 2 1 ) ( 1 , 2 1 , 2 2 2 , 1 2 , 1 2 , 2 2 1 ) ( 2 , 2 2 , 2 2 1 0 2 0 1 , 2 2 1 0 1 , 2 2 2 0 ; , 2 1 1 2 1 h N e B K h h N e B K dz e e B K dz z h N e e B K C t t t t t t t t du u r du u r h z du u r h z du u r and C3 is reformed by using traditional normal
distribution function shown as follows
1 0 2 0 1 0 2 1 , 2 2 0 ) ( 1 , 2 1 , 2 ) ( 2 , 2 ) ( 1 , 2 2 1 ) ( 2 , 2 3 2 1 t t t t t t t t du u r du u r du u r z h du u r e B h N e B e B dz e e B C Finally, the two-fold SCCO with consideration of performance bond is
1 0 2 0 1 0 2 0 2 0 ) ( 1 , 2 1 , 2 ) ( 2 , 2 1 , 2 ) ( 1 , 2 1 , 2 2 , 1 2 , 2 1 , 2 2 ) ( 2 , 2 2 , 2 2 , 1 2 , 2 1 , 2 2 ) ( 0 , 2 0 2 , 2 ; , ) ; , ( ) , ( t t t t t t t t t t du u r du u r du u r du u r du u q e B h N e B h N e B K h h N e B K g g N e S t S C (16) where i i t t t t i i du u du u u q u r S S g 0 0 ) ( )] ( 2 1 ) ( ) ( [ ln 2 2 2 , 0 , 2 , 2 ; i i t t t t i i du u du u u q u r S S h 0 0 ) ( )] ( 2 1 ) ( ) ( [ ln 2 2 2 , 0 , 2 , 2 Equation (16) can be rewritten as
-1
2, -11
, -1 -1
) ( 2 1 , 2 , 1 , 2 ) ( , 2 , 2 2 1 2 2 , 1 2 , 2 2 ) ( 0 , 2 0 2 , 2 ; ; ; ) , ( 0 0 2 0 m m j i m i m du u r m m m m j i m i m du u r m m m j i i du u q h N e B h N e B K g N e S t S C m t t m t t t t
(17)n-fold compound call option
By induction, the n-fold SCCO pricing model shows as follows
{[ ] ;[ ] } } ] [ ; ] {[ } ] [ ; ] {[ ) , ( 1 1 , 1 1 , 1 ) ( , 1 , 1 , ) ( , , 1 , 1 , ) ( 0 , 0 , 0 0 1 0
m m j i m i n m du u r m n n m m m j i m i n m du u r m n m n n m n n j i n i n n du u q n n n h N e B h N e B K g N e S t S C m t t m t t n t t (18) where i i t t t t n i n i n du u du u u q u r S S g 0 0 ) ( )] ( 2 1 ) ( ) ( [ ln 2 2 , 0 , , ; i i t t t t n i n i n du u du u u q u r S S h 0 0 ) ( )] ( 2 1 ) ( ) ( [ ln 2 2 , 0 , ,
The correlation matrix is symmetric, i.e.,
i j j i, ,
, and given by:
n j i du u du u j i j i t t t t j i 1 ; ) ( ) ( ; 1 0 0 2 2 ,
If the above solution for the n-fold SCCO, then it is also true for an (n+1)-fold SCCO. To prove that, present value of the (n+1)-fold SCCO can be found by the sane risk-neutral approach as:
-1
1,
-1 1
, -1 -1
) ( 1 1 , 1 , 1 , 1 ) ( , 1 , 1 1 1 1 1 , 1 1 , 1 1 ) ( 0 , 1 0 1 , 1 ; ; ; ) , ( 0 0 1 0 m m j i m i n m du u r n m m n m m j i m i n m du u r m n m n n m n n j i n i n n du u q n n n h N e B h N e B K g N e S t S C m t t m t t n t t
where g n1,iand h n1,iare the i-th g, h values of
the (n+1)-fold SCCO. The solution can be obtained directly from Equation (17) by adding one more fold layer. The following provides a more complete outline of this proof. First, first denote
1, 1(S,t1)
Cn n as the time-t1 value of the (n+1)-fold
SCCO. The value is given by the boundary condition 1, 1
1,1 1,1
1,1 1 , 1 1 , 1 1 , 1 1 1 , 1 ] ) , ( , 0 max[ ] , ) , ( max[ ) , ( n n n n n n n n n n n B B K t S C B K t S C t S C (19)which states that the underlying n-fold SCCO
1, (S,t1)
Cn n will be exercised at time-t1 if its time-t1
value is greater than or equal to the difference of exercise price, K n1,1, and performance bond,B n1,1.
Under risk-neutral pricing, the time-t0 value of the
(n+1)-fold SCCO can be given by
1, 1 1,1 1,1 1,1
) ( 0 1 , 1 ) , ( , 0 max ) , ( 1 0 n n n n n Q du u r n n B B K t S C E e t S C t t (20)where EQ is the conditional expectation operator
with respect to a risk-neutral probability measure. Assume the underlying asset value process is a standard geometric Brownian motion in Equation
(1). Then the time-t1 value of the asset can be given
by: 01 2 1 0 2 ) ( )] ( 2 1 ) ( ) ( [ 0 , 1 1 , 1 t t Q t t ru qu u du z udu n n S e S (21)
where z is a standard normal random number with density function 2 2 1 2 1 ) (z e z f ,z~ N(0,1).
In addition, the time-t1 value of the underlying
n-fold SCCO can be calculated by the value of the original n-fold SCCO with a shift of start time from
t0 to t1. That is, on the basis of Equation (18),
1 1 1 ,* , 1 1 1 ,* , 1 1 ) ( 1 , 1 1 1 ,* , 1 1 ,* , 1 ) ( 1 , 1 1 , 1 1 1 ,* , 1 1 ,* , ) ( 1 , 1 1 , 1 ] [ ; ] [ ] [ ; ] [ ] [ ; ] [ ) , ( 1 1 1 1 1 1 m m j i m i n m du u r m n n m m m j i m i n m du u r m n m n n m n n j i n i n n du u q n n n h N e B h N e B K g N e S t S C m t t m t t n t t (22)where *1 denotes the time shift. Then substituting equation (21) and equation (22) into equation (20) gets
1 0 1 1 1 1 1 1 1 0 1 0 ) ( 1 , 1 1 , 1 1 , 1 1 1 1 ,* , 1 1 1 ,* , 1 1 ) ( 1 , 1 1 1 ,* , 1 1 ,* , 1 ) ( 1 , 1 1 , 1 1 1 ,* , 1 1 ,* , ) ( 1 , 1 ) ( 1 , 1 1 , 1 1 , 1 1 , 1 ) ( 0 1 , 1 ) ( ] [ ; ] [ ] [ ; ] [ ] [ ; ] [ , 0 max ] ) , ( , 0 max[ ) , ( t t m t t m t t n t t t t t t du u r n n n m m j i m i n m du u r m n n m m m j i m i n m du u r m n m n n m n n j i n i n n du u q n du u r n n n n n Q du u r n n e B dz z f B K h N e B h N e B K g N e S e B B K t S C E e t S C
Now, let S1,n1 be the time t1 equivalent value of
the underlying asset such that at time-t1, the value
of the underlying n-fold SCCO is ‘at the money’, i.e. C n1,n(S,t1)
K n1,1B n1,1
0. Accordingly, 1 0 1 0 2 ( )] ( ) 1 ) ( ) ( [ ) ln( 2 2 0 , 1 1 , 1 t t t t n n du u z du u u q u r S S and therefore 1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ) ln( 2 2 1 , 1 0 , 1 t t t t n n du u du u u q u r S S z Further let
1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ) ln( 2 2 1 , 1 0 , 1 1 , 1 t t t t n n n du u du u u q u r S S h BecauseS n1,0S1, n1 , zh n1,1Given above, the
value of the (n+1)-fold can be solved by:
1 0 2 1 , 1 1 0 2 1 , 1 1 0 2 1 , 1 1 0 2 1 0 2 1 0 2 1 1 1 , 1 1 0 ) ( 1 , 1 2 1 ) ( 1 , 1 1 , 1 1 1 1 ,* , 1 1 1 ,* , 1 1 2 1 ) ( 1 , 1 1 1 ,* , 1 1 ,* , 1 2 1 ) ( 1 , 1 1 , 1 1 1 ,* , 1 1 ,* , 2 1 ) ( )] ( 2 1 ) ( ) ( [ ) ( 0 , 1 ) ( 0 1 , 1 2 1 ] [ ; ] [ 2 1 ] [ ; ] [ 2 1 ] [ ; ] [ 2 1 ) , ( t t n t t n m t t n m t t t t t t n t t n t t du u r n z h du u r n n m m j i m i n m z h du u r m n n m m m j i m i n m z h du u r m n m n n m n n j i n i n n z du u z du u u q u r du u q n h du u r n n e B dz e e B K dz h N e e B dz h N e e B K dz g N e e S e t S C The solution can be found by deriving the pricing components separately. Denote the components as
1
C、C2、and C3 respectively, and start the derivation
with the first component. Recall the asset price process in equation (21), and rewrite the value of
1 C as: e e N
g
dz S C n ni n ij nn h du u z du u q n n t t n t t
[ ,,*1] 1;[ ,,*1] 2 ) ( 2 1 ) ( 0 , 1 1 1 , 1 1 0 2 1 0 To specify the parameters of the n-variate normal
integration, let 1 0 ( ) 2 2 t t u du z z . Then 1,1 2 1 , 1 2 1 , 1 ( ) ( ( ) ) 1 0 1 0 n t t n t t n u du h u du g h
where the factor g n1,1 is given by
1 0 1 0 ) ( )] ( 2 1 ) ( ) ( [ ) ln( 2 2 1 , 1 0 , 1 1 , 1 t t t t n n n du u du u u q u r S S g Accordingly, the value of C1 is further rewritten
as:
,,*1 1 ,,*1
2 2 1 ) ( 0 , 1 1 [ ] ;[ ] 2 2 1 , 1 1 0 e N g dz e S C z n n i n ij nn g du u q n n n t t
The value of the factor gn1,i with the time
shift can further be found by algebraic manipulations as
2 1 , 1 1 , 1 2 1 , 1 2 2 2 2 1 , 1 0 , 1 2 1 2 1 , 1 1 , 1 1 ,* , 1 ) ( ) ( )] ( 2 1 ) ( ) ( [ )] ( 2 1 ) ( ) ( [ ) ln( 1 ; ) ( )] ( 2 1 ) ( ) ( [ ) ln( 1 1 1 0 1 0 1 1 1 1 1 i i i n t t t t t t t t n i n t t t n i n i n z g du u du u z du u u q u r du u u q u r S S n i du u du u u q u r S S g i i i i
where the correlation coefficient is given by
1 0 1 0 ) ( ) ( 2 2 1 , 1 i t t t t i du u du u
According to Theorem 1(a) of
Lee et al. (2008), the value of the correlation coefficient with the time shift can be given by:
1,1, 1
2 2 1 , 1 , 1 1 , 1 , 1 1 , 1 , 1 1 , 1 , 1 , , 1 ,* , 1 1 j n i n j n i n j i n j i n j i Q Q Q Q Q Q where Q n,i,j is a symmetric entry (i, j) of the n by
n correlation matrix of the n-variate normal integral. Further algebraic manipulations get
n j i du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u du u Q Q Q Q Q Q j i t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t j n i n j n i n j i n j i n j i j i i j i j i i j i j i j i 1 ; ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( 1 1 1 , 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 , 1 , 1 1 , 1 , 1 2 1 , 1 , 1 2 1 , 1 , 1 , , 1 , 1 , 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 Therefore, [i,j,*1]nn [i,j] n1n1 . Given above,
the value of C1 is found as:
1
, 1 1 , 1 1
) ( 0 , 1 1 [ ] ;[ ] 1 0 n n j i n i n n du u q n e N g S C n t t By a similar approach, the values of C2and C3 can
be found, respectively, as:
m
n i m ij mm
du u r m n m n n m z h du u r n n m m j i m i n m z h du u r m n m n n m h N e B K dz e e B K dz h N e e B K C m t t n t t n m t t
] [ ; ] [ 2 1 ] [ ; ] [ 2 1 , 1 , 1 ) ( , 1 , 1 1 1 2 1 ) ( 1 , 1 1 , 1 1 ,* , 1 1 ,* , 1 2 1 ) ( 1 , 1 1 , 1 1 2 0 2 1 , 1 1 0 2 1 , 1 1 0
1
1,,*1 11 ,,*1 1 1
) ( , 1 1 1 ) ( 1 , 1 1 1 1 ,* , 1 1 1 ,* , 1 1 2 1 ) ( 1 , 1 1 3 ] [ ; ] [ ] [ ; ] [ 2 1 1 1 0 2 1 , 1 1 0
m m j i m i n m du u r m n n m du u r n m m j i m i n m z h du u r m n n m h N e B e B dz h N e e B C m t t t t n m t t The solution can be obtained directly from equation (18) by adding one more fold layer. Equation (18) is proven by induction.
INFLUENCE OF PERFORMANCE BOND ON PROJECT VALUE
To see the influence of the performance bond to the project value, the following derives the partial
derivatives of C0 with respect to the parameter B.
Proposition. From the pricing formula (18),
0 ; ; , 1 1 , , 1 1 , 1 , , 1 , 1 , 0 , 1 0
l l j i n l i n l l l j i n l i n l du u r n l l n n n n l h N h N e B t V C l t t (23) Proof.For n = 1, (9) reduces to a vanilla European call option, and therefore
