The generalized sequential compound options pricing
and sensitivity analysis
Meng-Yu Lee
a,⁎
, Fang-Bo Yeh
b, An-Pin Chen
ca
Institute of Information Management, National Chiao-Tung University, Taiwan b
Department of Mathematics, Tunghai University, Taiwan c
Institute of Information Management, National Chiao-Tung University, Taiwan Received 21 October 2006; received in revised form 30 June 2007; accepted 3 July 2007
Available online 10 July 2007
Abstract
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.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Sequential compound option; Project valuation; Real option; Leibnitz's rule; Option pricing; Risk-neutral JEL classification: G12; G13; G30; C69
Mathematical Social Sciences 55 (2008) 38–54
www.elsevier.com/locate/econbase
⁎Corresponding author. 53, Feng-She Rd., Feng Yuan, Taichung County, 42037 Taiwan. Tel.: +886 952 705142; fax: +886 946 751544.
E-mail addresses:mongyu@iim.nctu.edu.tw,mogyulee@yahoo.com.tw(M.-Y. Lee),f.b.yeh@math.thu.edu.tw
(F.-B. Yeh),apc@iim.nctu.edu.tw(A.-P. Chen).
0165-4896/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2007.07.001
1. Introduction
Compound options, initiating by Geske (1977, 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 byGeske (1977, 1979)and constitutes as precedents with respect to later works. Specific
multi-fold compound option pricing formulas are proposed byGeske and Johnson (1984a)andCarr
(1988)while the pricing formula of sequential compound call (SCC) is proved byThomassen and Van Wouwe (2001) and Chen (2003).Chen (2002) andLajeri-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,
whileAgliardi 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, whileCarr (1988)presents the pricing formula for sequential exchange options. Corporate debt (Chen, 2003; 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 fromMyers (1977)and is followed byBrennan 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 (seeTable 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.
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, 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.
Table 1
Evolutions of compound option theory
Reference Fold Approach Generalization
Number Put-call alternating Time-dependent parameters
Geske (1977, 1979)a 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
2. The pricing formula for generalized sequential compound options
This section defines the notation and derives the pricing formula for multi-fold SCOs using the risk-neutral method. An SCO, composed of European options as building blocks, is the (compound) option on another compound option, where the feature of each folds can be assigned arbitrarily. Each fold option may be either call or put. This section begin by providing notation explanation and a
fundamental theorem that express a k-variate normal integration in terms of (k−1)-variate integrations.
Denote the correlation matrix Qk:= [Q{k},g,h]k×k, where Q{k},g,h is the symmetric (g, h)
entry of the matrix Qk,∀1≤g≤h≤k. Similarly, d{k},g is the gth entry of the vector [d{k},g]k×1.
([Q{k},g,h]k×k)(−i,−j) is the (k−1) by (k−1) matrix which excludes the ith row and the jth column
of [Q{k},g,h]k×k. Define the function fðzÞ ¼
1ffiffiffiffi 2p
p e1
2z2. The k-variate normal integral with upper
bound limit vector [d{k},g]k×1and correlation matrix Qk is characterized as
Nk dfkg;g k1; Qk n o ¼ Z dfkg;1 l Z dfkg;2 l : : :Z dfkg;k l 1 ð2pÞk 2 ffiffiffiffiffiffiffiffiffi jQkj p e21ZVQ 1 k Zdz kdzk1: : :dz1;
where Z′=[z1, z2,…, zk] and N0≡1. The following theorem is the statement about the construction
of multivariate normal integrals. Theorem 1.
(a) The relationship between the (k−1) and k-variate normal integrals (Curnow and Dunnett, 1962)
81 V v V k; Nk dfkg;g k1; Qk n o ¼Z dfkg;v l fðzvÞNk1
f
ð½
dfkg;g Qfkg;v;gzv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q fkg;v;g2 q k1Þ
ðv;Þ ;ð½
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkg;g;h Qfkg;v;gQfkg;v;h 1 Q fkg;v;g2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q fkg;v;h2 q kkÞ
ðv;vÞg
dzv(b) The decomposition of a multivariate normal integral (Schroder, 1989)
Nk dfkg;gk1; Qk n o ¼ Z dfkg;v l Nv1
f½
dfkg;gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkg;g;vzv 1 Q2 fkg;g;v q ðv1Þ1 ;½
Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkg;g;h Qfkg;g;vQfkg;h;v 1 Q2 fkg;g;v q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkg;h;v q ðv1Þðv1Þg
Nkvf½
dfkg;vþgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkg;v;vþgzv 1 Q2 fkg;v;vþg q ðkvÞ1 ;½
Qfkg;vþg;vþhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkg;v;vþgQfkg;v;vþh 1 Q2 fkg;v;vþg q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkg;v;vþh q ðkvÞðkvÞg
fðzvÞdzvwhere Qkis the correlation matrix,∀1≤v≤k.
In Theorem 1, (a) reveals that the k-variate normal integral can be constructed from the (k
−1)-variate by adding another dimension to the upper limit vector and correlation matrix. (b) states further that the specific multivariate normal integral can be partitioned into two integrals of lesser variates. This result can extend the current compound option methodology from 2-fold to
multi-fold by induction, whileChen (2003) just “observes a pattern” to generalize the SCC. Before
applying this theorem to sequential compound option pricing, more pieces of notation are introduced as follows.
Assume Tu−1bTu, the time interval from Tu−1to Tuisτu,∀u≥1. Denote the asset price at time
at time t are given as deterministicσ2(t), r(t) and q(t), respectively. The dividend rate q(u) can
also been regarded as the depreciate rate (Remer et al., 2001).
DenoteΨi(T0) as the i-fold SCO price starting at time T0and expiring at time T1, with strike K1.
Its underlying asset is the (i−1)-fold SCO Ψi−1(T1), which is active from T1to T2. Under the
assumption that the last fold SCO starts from T0, the underlying SCO with fold number (i−u+1),
Ψi−u+1(Tu−1), is valid from Tu−1 to Tuwith strike price Ku. The first fold option,Ψ1(Ti−1), is a
vanilla option with the asset as its underlying asset. It should be noted that fold numbers come in the reverse order.
The option featureΛu,urepresents the call or put attribute of the (underlying) SCO with fold
number (i−u+1) ranging from Tu−1to Tu,∀u≥1. If the SCO of this fold is a call, Λu,u= 1; the
featureΛu,u=−1 is for a put. For example, a call on a put (a 2-fold compound option) starting at
T0 has the option features Λ1,1= 1 and Λ2,2=−1. Denote Kh;g¼ jhu¼g Ku;u; 81 V g V h, and
Λ1,0≡1.Fig. 1shows the notation for an arbitrary i-fold SCO starting from T0.
Under the no arbitrage condition and the assumption that the asset price follows a geometric Brownian Motion in the perfect market, the succeeding theorem derives the pricing formula of an i-fold SCO with alternating arbitrarily calls and puts by the risk-neutral method. Although the SCOs
presented in later sections can start at any time Tu, the SCO in this theorem is starting from T0without
loss of generality. The symbol“ ⁎v ”, meaning “start from time Tv”, is used to indicate time shift in the
sensitivity derivation.
Theorem 2. Generalized sequential compound option pricing formula Denote ai;g;⁎vuai;g;⁎vðSvÞ ¼ ln Sv S#vþg;vþi þ Z Tvþg Tv rðuÞ qðuÞ þ1 2r 2ðuÞ du ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Tvþg Tv r2ðuÞdu s ; 8g z 1 ðaÞ
bi;g;⁎vubi;g;⁎vðSvÞ ¼ ai;g;⁎vðSvÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Tvþg Tv r2ðuÞdu s ; 8gz1 ðbÞ ˜qg;h;⁎v¼ Lvþh1;vþgqg;h⁎v; 8h N g z 1; qg;g;⁎v¼ 1; 8g; qg;h;⁎v¼ qh;g;⁎v; 8h; g; qg;h;⁎v¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Tvþg Tv r2ðuÞdu Z Tvþh Tv r2ðuÞdu v u u u u u u t ; 8l V g b h: ðcÞ ai;g;#vuai;g;⁎vðS#v;iÞ; bi;g;#vubi;g;⁎vðS#v;iÞ ðdÞ
ai;guai;g;⁎0; bi;gubi;g;⁎0;qi;guqi;g;⁎0;˜qg;hu ˜qg;h;⁎0 ðeÞ
(f) Equivalent asset price (EAP) of the underlying
S#g;i¼ KThe asset price which makesi; for g ¼ i W
igðTgÞ ¼ Kg; 81 V g b i (a) (b) (c) (d) (e)
then WiðT0Þ ¼ Ki;1e RTi T0qðuÞduS0Ni Ki;gai;g i1; ˜qg;h ii n o X i j¼1 Kj;1e RTj T0rðuÞduK jNj Ki;gbi;g j1; ˜qg;h jj n o ð1Þ
under the assumption that the EAP (Sg,i) exists,∀1≤g≤i
Proof. see Appendix A. □
Through the induction, the SCO price in Eq. (1) is derived according to the risk-neutral method, by which the physical probability is changed as the risk-neutral measure. Eq. (A.2) is the key of derivation. It means that the current asset price is the expectation of the future price with interest rate discount under the risk neutral probability measure. The above statement accords with public general intuition. The interest rate is a deterministic function, thus the discount factor e
RT1
T0rðuÞducan be dealt ignoring the expectation operator. The other part, max[Λ1,1Ψi(T1)−Λ1,1K1],
within the expectation is derived similarly as the cases of 1-fold and 2-fold options.
According to Eq. (1), the price of an i-fold SCO can be expressed as the weighted asset price minus the sum of weighted strike prices. The weights consist of three factors: the cumulative option features, the discount factor and the in-the-money probabilities. The cumulative option feature is obtained by synthesizing the option features from the current fold to the last fold. The discount factor is a deduction made due to interest rate or depreciation rate compounding. The in the money probabilities are assessed by multivariate normal integrals under different probability measures. The
factors ai,gand bi,gin the integration are similar to the“d1” and “d2” appearing in conventional option
pricing formulas. The correlation matrices of SCOs are similar to those of the sequential compound calls, except for a sign change due to the cumulative option features. Within these 3 weighting factors, the parameters of the last fold have the widely impact on the pricing formula.
The pricing formula of SCOs is more general than those of vanilla options, 2-fold compound options, and sequential compound calls, all of which can be regarded as special cases of SCOs. The main difference between SCOs and sequential compound calls lies in the freedom to alternate
calls and puts, which is represented by a sign changes in the cumulative option features Λh,g,
∀1≤g≤h. In other words, the option prices will depend on the fold features Λh,g. Moreover,
allowing the parameters to vary over time makes the integrated variance and discounting factors
of an SCO quite different from the constant parameters inThomassen and Van Wouwe (2001).
Setting allΛh,gto + 1 in an SCO results in a SCC.
The arbitrary put/call alternation of SCOs causes the EAP existence issue. The existence of
EAPs is crucial (Frey and Sommer, 1998) because the decision whether exercises the SCO or not
is transformed as whether the asset price is greater (or lesser) than the EAP. Similar to the concept
of implied volatility, the EAP can be regarded as the“implied asset price”, solving by the known
(compound) option price (given as the strike price) and other conventional option parameters except the asset price itself. Thus there is no EAP concern in the 1-fold option computation and it
is calculating only for the 2 or more fold compound options. The SCO price (Ψi) is monotone with
respect to the asset price and hence the equivalent asset price (EAP, S#g,i) is unique if it exists.
According to the sensitivity analysis in Theorem 4 (b), the Delta (∂WiðT0Þ
∂SðT0Þ) is a strictly monotone
function. Its increasing or decreasing nature depends on the cumulative option feature (Λi,1).
Therefore the EAP, defined as the asset price making the SCOs price equal to a specific strike price, is unique if it exists. For the SCC, its option price increases with respect to the asset price, thus there is no EAP existence concern. However, the EAP may not exist due to the range limitation of a decreasing SCO price. This is another difference between SCO and SCC. The following Lemma describes explicitly the sufficient conditions of the EAP existence.
Denote ˜Wi;2&3ðT0Þ ¼ Pi j¼1Kj;1e RTj T0rðuÞduK jNjf Ki;gbi;g
j1;½ ˜qg;hjjg, which is the second component of
the SCO pricing formula in Eq. (1). Note that ˜Wi;2&3ðT0Þ may be negative or positive but all SCO
pricesΨi(T0) are always nonnegative.
Lemma 1. The sufficient condition for the existence of existence of equivalent asset price (EAP)
Given g (1≤g≤i−1), the S#g,iexists if
(a) S#ℓ,iexists for all g−1≤ℓ≤i−1,and either the following condition stand.
(b) Λi-g,1= + 1;
(c) Λi-g,1=−1 and Kg≤−Ψ˜i−g,2&3(Tg).
Proof. see Appendix B. □
The condition (a) of Lemma 1 reveals that the existence conditions is also derived based essentially on the induction, by which the multi-fold SCO price is available in Theorem 2. If the EAPs of previous folds exist, the EAP existence of the current fold is discussed according to the
different sign of the cumulative option futureΛi−g,1. The condition (c) states that the strike price
of the current fold Kgis limited by a maximum because the asset price has opposite direction
against the current fold SCO price. The opposite direction is represented by the negative cumulative option feature. For the case of positive cumulative option feature (condition (b)), there is no restriction for the strike price. The non-existing EAP will incur the zero SCO price. 3. The sensitivities of sequential compound options
This section derives some features of multivariate normal integrations and investigates the sensitivities of SCOs based on the derivation.
3.1. Some features of multivariate normal integration
This subsection presents the partial derivatives and separations of multivariate normal
integrations. Thomassen and Van Wouwe (2002) use the partial derivatives of multivariate
generalizes their work, and is applicable to multivariate normal integrations with any kind of correlation matrix. This result will pave the way to the sensitivity analysis of SCOs.
Theorem 3. Partial derivative of the multivariate normal integral Let d{k},g(G1, G2,…, Gp)` d{k},g,, representing a function of G1, G2,…, Gp.
81 V k; 1 V S V p;∂Nkð½dfkg;gk1;½Qfkg;g;hkkÞ ∂GS ¼X k j¼1 f dfkg; j ∂d∂Gfkg; j S Nk1 dfkg;g dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkg; jQfkg; j;g 1 Q2 fkg; j;g q 2 6 4 3 7 5 k1 0 B @ 1 C A ðj;Þ ; Qfkg;g;h Qfkg;j;gQfkg;j;h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkg;j;g 1 Q2 fkg;j;h r 2 6 6 4 3 7 7 5 kk 0 B B B @ 1 C C C A ðj;jÞ 8 > < > : 9 > = > ; ð2Þ
where [Q{k},g,h]k×kis a correlation matrix that is not a function of Gℓ.
Proof. see Appendix C. □
Theorem 3 shows that the partial derivatives of a (k + 1)-variate normal integration can be represented as the k + 1 weighted sum of k-variate normal integrations. As Eq. (C.1) shows, the Leibnitz's rule can be used to decompose the partial derivative into two parts. The first term is a k-variate normal integration with a weighting factor. The second part is an integration of a partial derivative of the (k−1)-variate normal. Theorem 3 proves that this second part turns out has the same form as the first term. This means that Theorem 3 extends the Leibnitz's rule to multivariate normal cases.
The specific partial derivatives presented inThomassen and Van Wouwe (2002)can be viewed
as a special case of Theorem 3. If the elements of the correlation matrix in Eq. (2) as specified as Q{k},g,h= Q{k},h,g= ffiffiffiffi sg sh q , for 1≤g≤h, then Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkg;g;hQfkg; j;gQfkg; j;h ð1Q2 fkg; j;gÞð1Q2fkg; j;hÞ p ¼ 0, for gbjbh or hbjbg.
Another feature of multivariate normal integrations will be presented after the following notation has been defined. Let
tv1;buNv1
f½
Ki;gbi;gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ki;vbi;v˜qg;v1 ð˜qg;vÞ2 q
ðv1Þ1 ;½
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi˜qg;h˜qv;g˜qv;h ½1 ˜qv;g2½1 ˜qv;h2 q ðv1Þðv1Þg
; tj1;b;vutj1;b;vðbi;gÞ ¼ Ni1fð½
Ki;gbi;gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ki;vbi;v˜qv;g 1 ð˜qv;gÞ2 q i1Þ
ðv;Þ ;ð½
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi˜qg;h˜qv;g˜qv;h ½1 ˜qv;g2½1 ˜qv;h2 q i1Þ
ðv;vÞg
; ti1;a;vuti1;a;vðai;gÞLemma 2 shows that the multivariate integrals for SCO sensitivities can be factored into two separated normal integrals.
Lemma 2. (a) ti1;a;v¼ tv1;b Niv Ki;vþgai;g;#v
ðivÞ1; ˜qg;h;⁎v h i ðivÞðivÞ (b) tj1;b;v¼ tv1;b Njv Ki;vþgbi;g;#v ð jvÞ1; ˜qg;h;⁎v h i ð jvÞð jvÞ
Proof. See Appendix D. □
3.2. The sensitivity analysis of SCOs
The sensitivity analysis of SCOs is now possible thanks to the two results demonstrated in the
preceding subsection.Thomassen and Van Wouwe (2002)derived the sensitivities of SCCs and
Theorem 4 extends their analysis to SCOs with the possibility of alternating calls and puts arbitrarily based on Theorem 3. Theorem 4 also shows the interest rate sensitivity under the special case of interest rate fold-wise.
Theorem 4. Sensitivities of SCOs
(a) Delta: ∂WiðT0Þ ∂S0 ¼ Ki;1 e Z Ti T0 qðuÞdu Ni Ki;gai;g i1; ˜qg;h ii n o (b) Gamma: ∂2W iðT0Þ ∂S2 0 ¼Xi v¼1 Kv1;1e 1 2a2i;v RTi T0qðuÞdu S0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p Z Tv T0 r2ðuÞdu s ti1;a;v
(c) Let the interest rate and the variance of asset price be fold-wise constant. In other words, r(t) =
ru,σ(t)=σu,∀Tu-1≤t bTu,1≤u≤i. Under this simplification, the “underscore” labels are
added to the corresponding pieces of notation. The SCO price, the correlation matrix and the
two upper limit vectors are denoted as PWi,
P˜qg;h, Pai;g and Pbi;g, respectively. Thus, the
interest rate sensitivity Rho is:∀1 V ℓ V i,
∂PWiðT0Þ ∂rS ¼ rS Xi j¼S Kj;1Kje Pj u¼1rusuN jf½Ki;g Pbi;gj1;½P˜qg;hjjg:
Proof. See Appendix E. □
As SCOs pricing formulas (Theorem 2) generalize previous results such as vanilla options , 2-fold compound options and SCCs, the SCOs sensitivities given in Theorem 4 are also extension of these previous works intuitively. Again, the sequence of option features will affect the signs of the sensitivities. According to Theorem 4 (a), the value of a SCO is monotonic with respect to the
current asset price S(T0), hence the EAP is unique if it exists.
4. 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 com-pound 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, 1979) and sequential compound calls (Thomassen and Van
Wouwe, 2001; 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 inCasimon 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.
Acknowledgements
The authors give thanks to Dr. Chester Ho for initiating the research idea and are grateful for valuable comments from Son-Nan Chen, Szu-Lang Liao, Huang-Nan Huang, Cheng-Der Fuh, Min-Shann Tsai, Ming-Chi Chang, Hong-Ming Chen and anonymous referees. The first author pay special acknowledgement to S. N. Chen and Liao for their long-term teaching and wholehearted assistance.
Appendix A. Proof of Theorem 2
This theorem is proved by induction. When i = 1,Ψ1(T0) withΛ1,1= 1 andΛ1,1=−1 are the
vanilla call and put formulas respectively. When i = 2,Ψ2(T0) is the 2-fold compound option, such
as call on call (Λ1,1= 1,Λ2,2= 1), put on call (Λ1,1=−1, Λ2,2= 1), call on put (Λ1,1= 1,Λ2,2=−1),
and put on put (Λ1,1=−1, Λ2,2=−1). These generalized 2-fold cases can be extended easily from
Chen (2002)andLajeri-Chaherli (2002).
Assuming that Eq. (1) is true for the i-fold compound option Ψi(T0), it will be shown that
Eq. (1) is also true for the (i + 1)-fold compound option, for anyΛg,g, 1≤g≤i+1.
Because the underlying asset ofΨi+1(T0) isΨi(T1) , instead ofΨi(T0) , the start time of the
i-fold compound option is shifted from T0to T1. All pieces of notation for the i-fold compound
option are changed simultaneously according to this time shift. (In other words, v = 1).
HenceWiðT1Þ ¼ Kiþ1;2e RTiþ1 T1 qðuÞduS1Ni Kiþ1;gþ1a i;g;⁎1 h i i1; ˜qg;h;⁎1 h i ii n o Xi j¼1 Kjþ1;2e RTjþ1 T1 rðuÞduKjþ1Nj Kiþ1;gþ1b i;g;⁎1 h i j1; ˜qg;h;⁎1 h i jj ðA:1Þ
At T1, the maturity time of the i + 1-fold compound option, the option price can be expressed
asΨi+1(T1) = max[Λ1,1Ψi(T1)−Λ1,1K1]. At its starting time T0, the option price is given by
Wiþ1ðT0Þ ¼ ˜Efe
RT1
T0rðuÞdumax½K1;1WiðT1Þ K1;1K1jF0g; ðA:2Þ
according to the fundamental theory of asset pricing (Baxter and Runie, 1996). E˜ is the
expectation operator under the risk-neutral measure, andF0denotes the information available at
time T0from the asset price.
Under the assumption that the asset price follows a geometric Brownian motion, it can be expressed as S1¼ S0e RT1 T0 rðuÞqðuÞ 1 2r 2ð Þu ½ duþz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT1 T0r 2ðuÞdu q ðA:3Þ
where z is a standard normal random number z∼N(0,1), with density function f. Ψi+1(T0) is a
function of S0and hence a function of z. Thus the SCO price can be represented as
Wiþ1ðT0Þ ¼ e RT1
T0rðuÞdu
Z l
lmax½K1;1WiðT1Þ K1;1K1 f ðzÞdz:
Assume that S#1,i+1is the equivalent asset price which makesΨi(T1)−K1= 0. The condition
“S1= S#1,i+1” is then equivalent to “z=−bi+1,1”, where biþ1;1 ¼
ln S0 S#1;iþ1 þRT1 T0 rðuÞqðuÞ 1 2r2ðuÞ ð Þdu ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT1 T0r 2ðuÞdu q
Because the integration range is either [−∞, −bi+1,1] or [−bi+1,1,∞], depending on Λi+1,1(the
sign of S1), the compound option can be expressed in the unified form
Wiþ1ðT0Þ ¼ e RT1 T0rðuÞduKiþ1;1 Z Kiþ1;1l biþ1;1 fK1;1WiðT1Þ K1;1K1gf ðzÞdz: Substituting Eq. (A.1) into the previous equation, it can be obtained that
Wiþ1ðT0Þ ¼ e RT1
T0rðuÞduKiþ1;1Kiþ1;1
ZKiþ1;1l biþ1;1 e RTiþ1 T1 qðuÞduS1Ni Kiþ1;gþ1a i;g;⁎1 h i i1; ˜qg;h;⁎1 h i ii n o fðzÞdz e RT1 T0rðuÞduKiþ1;1X i j¼1 Kjþ1;1 Z Kiþ1;1l biþ1;1 e RTjþ1
T1 rðuÞduKjþ1Nj Kiþ1;gþ1bi;g;⁎1
h i j1; ˜qg;h;⁎1 h i jj fðzÞdz e RT1 T0rðuÞduKiþ1;1K1;1 Z Kiþ1;1l biþ1;1 K1fðzÞdz
≡ ˜Wiþ1;1 ˜Wiþ1;2 ˜Wiþ1;3:
The following paragraphs derivates Ψ˜i+1,1, Ψ˜i+1,2 and Ψ˜i+1,3 explicitly. By Eq. (A.3), S1
can be substituted by the representation of S0 and thus ˜Wiþ1;1¼ Kiþ1;1e
RTiþ1 T0 qðuÞdu S0Kiþ1;1 RKiþ1;1l biþ1;1 p1ffiffiffiffiffiffi2pe 1 2 z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT 1 T0 r 2ðuÞdu q 2
Ni Kiþ1;gþ1˜ai;g;⁎1
h i
i1; ˜qg;h;⁎1
h i
ii
n o
dz; where ˜ai;g;⁎1¼
ln S0 S#gþ1;iþ1 þRTgþ1 T0 rðuÞ qðuÞ þ 1 2r 2ðuÞ ½ du þ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRT1 T0r 2ðuÞdu q RT1 T0r 2ðuÞdu ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RTgþ1 T1 r 2ðuÞdu q ; 81 V g V i:Let z2¼ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT1 T0r 2ðuÞdu q , so that
the above equation can be written as ˜Wiþ1;1 ¼ Kiþ1;1 e
RTiþ1
T0 qðuÞduS0Kiþ1;1RKiþ1;1l
aiþ1;1 p1ffiffiffiffiffiffi2p e1 2z22Ni Kiþ1;gþ1a¯ i;g;⁎1 h i i1; ˜qg;h;⁎1 h i ii n o
dz2; where ¯ai;g;⁎1¼
aiþ1;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiþz2q1;gþ1
1q2
1;gþ1
p ; 81 V g V i:
Then denote z3=−Λi+1,1z2, hence
˜ Wiþ1;1¼ Kiþ1;1e RTiþ1 T0 qðuÞduS0 Z Kiþ1;1aiþ1;1 l e12z23 ffiffiffiffiffiffi 2p p Ni Kiþ1;gþ1aiþ1;gþ1 Kg;1˜q1;gþ1z3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K g;1˜q1;gþ12 q 2 6 4 3 7 5 i1 ; ˜qg;h;⁎1 ii 8 > < > : 9 > = > ;dz3 ¼ Kiþ1;1e RTiþ1
T0 qðuÞduS0Niþ1 Kiþ1;gaiþ1;gðiþ1Þ1; H0;g;h
ðiþ1Þðiþ1Þ
n o
The last equation is obtained by Theorem 1 (a). The following derivation will demonstrate that H0;g;h
According to Theorem 1 (a), H0,1,1= 1; H0,1,g=Λh−1,1ρ1,h,∀2≤g≤i+1; H0,g,h= H0,h,g; and
H0,g,g= 1,∀2≤g≤i+1. Thus ∀2≤gbh≤i+1,
H0;g;h¼ Kg1;1q1;gKh1;1q1;hþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðKg1;1q1;gÞ2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðKh1;1q1;hÞ2 q qg1;h1;⁎1 ¼ Kh1;g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Tg T0 r 2ðuÞdu q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RTh T0 r 2ðuÞdu q ¼ Kh1;gqg;h¼ ˜qg;h:
According to the above statements, [H0,g,h](i+1)×(i+1)= [q˜g,h](i+1)×(i+1)and hence
˜Wiþ1;1¼ Kiþ1;1e
RTiþ1
T0 rðuÞduS
0Niþ1f½Kiþ1;gaiþ1;gðiþ1Þ1;½˜qg;hðiþ1Þðiþ1Þg:
By a similar method,Ψ˜i+1,2andΨ˜i+1,3can be derived:
˜Wiþ1;2¼
Xiþ1 j¼2
Kj;1e RTj
T0rðuÞduKjNj Kiþ1;gbiþ1;g
j1; ˜qg;h jj n o : ˜Wiþ1;3¼ K1;1e RT1 T0rðuÞduK1N1Kiþ1;1biþ1;1:
Eq. (1) is true for any i + 1-fold compound option, provided it is true for the i-fold compound
option. Consequently, Theorem 2 is proved. □
Appendix B. Proof of Lemma 1
According to Theorem 2 (f), the S#g,i will exist only when the EAPs of the previous folds
(S#ℓ,i, g−1≤ℓ≤i−1) exist. Thus the condition (a) holds. According to Theorem 4 (a), the
option priceΨ˜i−g(Tg) is strict monotone and its sign is decided byΛi−g,1. Hence it is discussed as
the cases ofΛi−g,1= + 1 (condition (b)) andΛi-g,1=−1 (condition (c)), respectively. For condition
(b),Ψi−g(Tg) has the same sign with the asset price and thus can ranges from zero to infinity to fit
any nonnegative Kg. For condition (c),Ψi−g(Tg) has the opposite sign with the asset price, then
Ψi-g(Tg) will reach the maximum−Ψ˜i−g,2&3(Tg) while the asset price is zero. Therefore the strike
price Kgcan NOT exceed the maximum in order to keep S#g,iexist. □
Appendix C. Proof of Theorem 3
The theorem is proved by induction. For k = 1,∂N1fdf1g;1g
∂GS ¼ f ðdf1g;1ÞN0. The theorem thus
stands for k = 1.
By the result of k = 1 and Leibnitz's rule, it is obtained that∂N2fdf2g;1;df2g;2;½Qf2g;g;h22g
∂GS ¼ f df2g;1 ∂ðdf2g;1Þ ∂GS N1 df2g;2Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2g;1;2df2g;1 1Q2 f2g;1;2 p ( ) þ ˜N2;1, where ˜N2;1uZl df2g;1 fðzÞ 1ffiffiffiffiffiffi 2p p exp1 2 df2g;2þ Qf2g;1;2z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q f2g;1;22 q 0 B @ 1 C A 2 2 64 3 75 8 > < > : 9 > = > ; ∂ df2g;2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþQf2g;1;2z 1Q2 f2g;1;2 p ! ∂G dz: Denote z4:¼ zþdf2g;2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQf2g;1;2 1Q2 f2g;1;2
p . Thus Ñ2,1can be rewritten as
˜N2;1¼ f d f2g;2 ∂d∂Gf2g;2 S Z df2g;1þQf2g;1;2df2g;2ffiffiffiffiffiffiffiffiffiffiffiffiffi 1Q2 f2g;1;2 p l fðz4Þdz4¼ f d f2g;2 ∂d∂Gf2g;2 S N1 df2g;1 Qf2g;1;2df2g;2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 f2g;1;2 q 8 > < > : 9 > = > ;
∂N2ðdf2g;1; df2g;2; Qf2g;g;h 22Þ ∂GS ¼ X2 j¼1 f d f2g; j ∂df2g; j ∂GS N1 df2g;g df2g; jQf2g; j;g ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 f2g; j;g q 2 6 4 3 7 5 21 0 B @ 1 C A ðj;jÞ 8 > < > : 9 > = > ;
Hence Eq. (2) stands for k = 2.
Assuming that Eq. (2) is true for k, the following proves that it is also true for k + 1. By Leibnitz's rule, ∂Nkþ1f½dfkþ1g;gkþ11;½Qfkþ1g;g;hðkþ1Þðkþ1Þg ∂GS ¼ f ðdfkþ1g;1Þ∂dfkþ1g;1 ∂GS Nk
f
dfkþ1g;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkþ1g;1;gþ1dfkþ1g;1 1 Q2 fkþ1g;1;gþ1 q 2 64 3 75 k1 ; Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkþ1g;g;h Qfkþ1g;1;gQfkþ1g;1;h ð1 Q2 fkþ1g;1;gÞð1 Qfkþ1g;1;h2 Þ q 2 64 3 75 ðkþ1Þðkþ1Þ 0 B B @ 1 C C A ð1;1Þg
þ ˜Nkþ1;1 ðC:1Þ ˜Nkþ1;1u Zl dfkþ1g;1 f zð Þ ∂ ∂GS Nk dfkþ1g;gþ1þ Qfkþ1g;1;gþ1z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkþ1g;1;gþ1 q 2 6 4 3 7 5 k1 ; Qfkþ1g;gþ1;hþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkþ1g;1;gþ1Qfkþ1g;1;hþ1 ð1 Q2 fkþ1g;1;gþ1Þð1 Qfkþ1g;1;hþ12 Þ q 2 6 4 3 7 5 kk 8 > < > : 9 > = > ; 0 B @ 1 C AdzUsing the corresponding result for ∂Nkð½dfkg;gk1;½Qfkg;g;hkkÞ
∂GS , by substituting dfkþ1g;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþQfkþ1g;1;gþ1z 1Q2 fkþ1g;1;gþ1 p and Qfkþ1g;gþ1;hþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1;gþ1Qfkþ1g;1;hþ1 ð1Q2 fkþ1g;1;gþ1Þð1Qfkþ1g;1;hþ12 Þ
p as dk,g, Qk,g,h in Eq. (2) respectively and setting Zkþ1; jþ1¼
zþ dfkþ1g; jþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1 1 Q2 fkþ1g;1; jþ1 q , Ñk +1,1can derived as ˜Nkþ1;1¼ Xk j¼1 f d fkþ1g; jþ1 ∂dfkþ1g; jþ1 ∂GS Z dfkþ1g;1þdfkþ1g; jþ1Qfkþ1g;1; jþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1Q 2 fkþ1g;1; jþ1 p l f Z kþ1;jþ1Nk1 ˜H1; ˜H2 n o dZkþ1; jþ1 ðC:2Þ
The numerator and the denominator of H˜1are multiplied by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkþ1g;1;gþ1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q2 fkþ1g; jþ1;gþ1 q in order to
match the format of Theorem 1. Therefore
˜H1¼ dfkþ1g;gþ1d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fkþ1g; jþ1Qfkþ1g; jþ1;gþ1 1Q2 fkþ1g; jþ1;gþ1p
þ Z
kþ1; jþ1H
1;gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2 1;gq
2 66 4 3 77 5 k1 0 B B @ 1 C C A ðj;Þ and ˜H2¼ H2;g;hkk ðj;jÞ ;where H1;guQfkþ1g;1;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qfkþ1g;1; jþ1Qfkþ1g; jþ1;gþ1 ð1 Q2 fkþ1g;1; jþ1Þð1 Qfkþ1g; jþ1;gþ12 Þ q ; 81 V g V k; H2;g;h¼ Qfkþ1g;gþ1;hþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1;gþ1Qfkþ1g;1;hþ1 ð1Q2 fkþ1g;1;gþ1Þð1Qfkþ1g;1;hþ12 Þ p Qfkþ1g; jþ1;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1Qfkþ1g;1;gþ1 ð1Q2 fkþ1g;1; jþ1Þð1Qfkþ1g;1;gþ12 Þ p Qfkþ1g; jþ1;hþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1Qfkþ1g;1;hþ1 ð1Q2 fkþ1g;1; jþ1Þð1Qfkþ1g;1;hþ12 Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Qfkþ1g; jþ1;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1Qfkþ1g;1;gþ1 ð1Q2 fkþ1g;1; jþ1Þð1Qfkþ1g;1;gþ12 Þ p !2 2 4 3 5 1 Qfkþ1g; jþ1;hþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1Qfkþ1g;1;hþ1 ð1Q2 fkþ1g;1; jþ1Þð1Qfkþ1g;1;hþ12 Þ p !2 2 4 3 5 v u u u t ; ∀1≤g, h≤k.
The integration of Ñk +1,1can be performed by applying Theorem 1. Hence,
˜Nkþ1; 1 ¼X k j¼1 f d fkþ1g; jþ1 ∂dfkþ1g; jþ1 ∂GS Nk dfkþ1g;g dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkþ1g; jþ1Qfkþ1g; jþ1;g 1 Q2 fkþ1g; jþ1;g q 2 6 4 3 7 5 ðkþ1Þ1 0 B B @ 1 C C A ðj1;1Þ ; ˜H3 8 > > < > > : 9 > > = > > ; ðC:3Þ where ˜H3¼ 1 ˜HT4 ˜H4 ˜H5 2 4 3 5; ˜H4¼ Qfkþ1g;1;gþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfkþ1g;1; jþ1Qfkþ1g; jþ1;gþ1 ð1Q2 fkþ1g;1; jþ1Þð1Q2fkþ1g; jþ1;gþ1Þ p " # k1 !ðj;Þ ,
By Theorem 1, H˜3 and H˜5 are symmetric with diagonal elements equal to 1. For 1V
gb h; H5;g;h¼ H1;gH1;hþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 H2 1;gÞð1 H1;h2 Þ q H2;g;h. Thus ˜H5 ¼ Qfkþ1g;gþ1;hþ1 Qfkþ1g; jþ1;gþ1Qfkþ1g; jþ1;hþ1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 Q2 fkþ1g; jþ1;gþ1Þð1 Q2fkþ1g; jþ1;hþ1Þ q 2 6 4 3 7 5 kk 0 B @ 1 C A ðj;jÞ and ˜H3 ¼ Qfkþ1g;g;h Qfkþ1g; jþ1;gQfkþ1g; jþ1;h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 Q2 fkþ1g; jþ1;gÞð1 Q2fkþ1g; jþ1;hÞ q 2 6 4 3 7 5 ðkþ1Þðkþ1Þ 0 B B @ 1 C C A ðj1;j1Þ :
Substitute H˜3into Eq. (C.3) and change the index j to obtain
˜Nkþ1;1¼X kþ1 j¼2 1ffiffiffiffiffiffi 2p p f dfkþ1g; j ∂dfkþ1g; j ∂GS Nk dfkþ1g;g dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkþ1g; jQfkþ1g; j;g 1 Q2 fkþ1g; j;g q 2 64 3 75 ðkþ1Þ1 0 B B @ 1 C C A ðj;Þ ; Qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifkþ1g;g;h Qfkþ1g; j;gQfkþ1g; j;h ð1 Q2 fkþ1g; j;gÞð1 Q2fkþ1g; j;hÞ q 2 64 3 75 ðkþ1Þðkþ1Þ 0 B B @ 1 C C A ðj;jÞ 8 > > < > > : 9 > > = > > ; ðC:4Þ Substituting the above result into Eq. (C.1), the consequence is obtained: ∂Nkþ1f½dfkþ1g;gðkþ1Þ1;½Qfkþ1g;g;hðkþ1Þðkþ1Þg
∂GS
. □
Appendix D. Sketch Proof of Lemma 2
The left-hand sides of Theorem 2 (a) and (b) are identical, hence the on right-hand sides are also the same. Lemma 2 can be proved according to the above result. Lemma 2 can also be proved
directly through a multivariate normal integration whose correlation matrix can be partitioned into “four quadrants”. The top-right and the bottom-left quadrants are zero matrices, so the integrals
can be represented as the product of two uncorrelated normal integrals (Bickel and Doksum,
2001, Theorem B.6.4). □
Appendix E. Proof of Theorem 4
For part (a), ∂WiðT0Þ
∂S0 ¼ Ki;1e RTi T0qðuÞduNi Ki;gai;g ii; ˜qg;h ii n o þ ˜W∂S;1 ˜W∂S;2; where ˜W∂S;1uKi;1e RTi T0qðuÞduS0Pi v¼1 p1ffiffiffiffi2pe 1
2a2i;v Ki;v∂ai;v ∂S0 ti1;a;v; ˜W∂S;2u Xi j¼1 Kj;1e RTj T0rðuÞduKj Xj v¼1 1ffiffiffiffiffiffi 2p p e12b 2
i;v Ki;v∂bi;v ∂S0
tj1;b;v
The sequential paragraphs demonstrateΨ˜∂S,1−Ψ˜∂S,2= 0.
By definition, e12a2i;v ¼S#v;i S0 e 1 2b 2 i;v RTv T0rðuÞdu; 81 V v V i: ðE:1Þ
The Ψ˜∂S,3 is denoted as W˜∂S;3¼ p1ffiffiffiffi2pe12b2i;v Ki;v∂bi;v ∂S0
tv1;be
RTv
T0rðuÞdu for convenience.
According to Lemma 2, Eq. (E.1) and the fact that∂ai;v
∂S0 ¼ ∂bi;v ∂S0 ; ˜W∂S;1 can be reformulated as ˜W∂S;1¼X i v¼1 ˜ W∂S;3 1fv b ige RTi TvqðuÞduþ 1fv¼ig
Ki;1S#v;iNiv Ki;vþgai;g;#vðivÞ1; ˜qg;h;⁎v
h i ðivÞðivÞ : ˜W∂S;2¼ Xi j¼1 Kj;1e RTj T0rðuÞduK j Xj v¼1 1ffiffiffiffiffiffi 2p p e12b 2 i;v Ki;v∂bi;v ∂S0 tv1;b Njv Ki;vþgbi;g; #vð jvÞ1; ˜qg;h;⁎vð jvÞð jvÞ n o ¼Xi v¼1 ˜W∂S;3
ð
Kv;1Kvþ 1fv b ig Xiv j¼1 e RTvþj Tv rðuÞdu Kvþj;1KvþjNjf½
Ki;vþgbi;g;#vð jvÞ1;½
˜qg;h;⁎vð jvÞð jvÞgÞ
The last equality is obtained by interchange of the two summations.˜W∂S;1 ˜W∂S;2¼ ˜W∂S;3ðKi;1S#i;iN0 Ki;1KiÞ þ Xi1 v¼1˜W∂S;3 ðKv;1˜W∂S;4 Kv;1KvÞ; where ˜W∂S;4 ¼ e RTi
TvqðuÞduKi;vþ1S#v;iNiv f K i;vþgai;g;#v
ðivÞ1;½ ˜qg;h;⁎v ðivÞðivÞg Piv
j¼1e
RTvþj
Tv rðuÞduKvþj;vþ1KvþjNjf K i;vþgbi;g;#v
ð jvÞ1;½ ˜qg;h;⁎vð jvÞð jvÞg:
By definitions, S#i,i= Ki, hence Λi,1S#i,i N0−Λi,1KI= 0. Ψ˜∂S,4 is the (i−v)-fold compound
option price with start time Tv(instead of T0). In other words,Ψ˜∂S,4=Ψi−v(Tv) with initial asset
price S#v,1. Thus, by definitions,Ψ˜∂S,4= Kv, andΨ˜∂S,1−Ψ˜∂S,2= 0. Part (b) and (c) can be proved by
References
Agliardi, E., Agliardi, R., 2003. A generalization of Geske formula for compound options. Mathematical Social Sciences 45, 75–82.
Agliardi, E., Agliardi, R., 2005. A closed-form solution for multicompound options. Risk Letters 1, 1–2.
Bhattcaharya, A.K., 2005. Chpater 26: interestrate caps and floors and compound options, In: Fabozzi, F.J. (Ed.), The Handbook of Fixed Income Securities, 7th Ed. McGraw-Hill, NY, pp. 1283–1300.
Baxter, M., Runie, A., 1996. Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press, UK. Bickel, P.J., Doksum, K.A., 2001. 2nd ed. Mathematical Statistics: Basic Ideas and Selected Topics, vol. 1. Prentice Hall,
New Jersey, US.
Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–657. Bowman, E.H., Moskowitz, G.T., 2001. Real options analysis and strategic decision making. Organization Science 12, 772–777. Brennan, M.J., Schwartz, E.S., 1985. Evaluating natural resource investment. Journal of Business 58, 135–157. Carr, P., 1988. The valuation of sequential exchange opportunities. Journal of Finance 43, 1235–1256.
Casimon, D., Engelen, P.J., Thomassen, L., Van Wouwe, M., 2004. The valuation of a NDA using a 6-fold compound option. Research Policy 33, 41–51.
Chen, S.N., 2002. Financial Engineering (Chinese Edition). Hua-Tai, Taipei.
Chen, R.R., 2003. The extended Geske–Johnson model and its consistency with reduced form models. Working Paper. Rutgers Business School, Rutgers University.
Copeland, T., Antikarov, V., 2003. Real Options: A Practitioner's Guide. Thomson, UK.
Cortazar, G., Schwartz, E.S., 1993. A compound option model of production and intermediate inventories. Journal of Business 66, 517–540.
Curnow, R.N., Dunnett, C.W., 1962. The numerical evaluation of certain multivariate normal integrals. Annals of Mathematical Statistics 33, 571–579.
Duan, C.W., Lin, W.T., Lee, C.F., 2003. Sequential capital budgeting as real options: the case of a new DRAM chipmaker in Taiwan. Review of Pacific Basin Financial Markets and Polices 6, 87–112.
Frey, F., Sommer, D., 1998. The generalization of the Geske-formula for compound options to stochastic interest rates is not trivial— a note. Journal of Applied Probability 35, 501–509.
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.
Geske, R., Johnson, H.E., 1984a. The American put option valued analytically. Journal of Finance 39, 1511–1524. Geske, R., Johnson, H.E., 1984b. The valuation of corporate liabilities as compound options: a correction. Journal of
Financial and Quantitative Analysis 19, 231–232.
Kauffman, R.J., Li, X., 2005. Technology competition and optimal investment timing: a real options perspective. IEEE Transactions on Engineering Management 52, 15–29.
Kogut, B., 1991. Joint ventures and the option to expand and acquire. Management Science 37, 19–33.
Lajeri-Chaherli, F., 2002. A note on the valuation of compound options. Journal of Futures Markets 22, 1103–1115. Musiela, M., Rutkowski, M., 1998. Martingale Methods in Financial Modelling. Springer, Berlin, Germany. Myers, S.C., 1977. Determinants of corporate borrowing. Journal of Financial Economics 5, 147–175.
Pindyck, R.S., 1988. Irreversible investment, capacity choice and the value of the firm. American Economic Review 79, 969–985.
Remer, S., Ang, S.H., Baden-Fuller, C., 2001. Dealing with uncertainties in the biotechnology industry: the use of real options reasoning. Journal of Commercial Biotechnology 8, 95–105.
Rubinstein, M., 1992. Options for the undecided. In: Tompkins, R. (Ed.), From Black Scholes to Black Holes— New Frontiers in Options. Risk Publication, UK, pp. 187–189.
Schroder, M., 1989. A reduction method applicable to compound option formulas. Management Science 35, 823–827. Thomassen, L., Van Wouwe, M., 2001. The n-fold compound option. Research Paper 2001–041. University of Antwerp,
Department of Mathematics and Statistics.
Thomassen, L., Van Wouwe, M., 2002. A Sensitivity Analysis for the n-fold Compound Option. Research Paper 2002– 014. University of Antwerp, Department of Mathematics and Statistics.
Trigeorgis, L., 1993. The Nature of Option Interactions and the Valuation of Investments with Multiple Real Options. Journal of Financial and Quantitative Analysis 28, 1–20.
