Common Decomposition of Correlated Brownian Motions and its Financial Applications
CCommon Decomposition of Correlated Brownian Motions and itsFinancial Applications
Tianyao Chen, Xue Cheng, Jingping YangJuly 9, 2019
Abstract
In this paper, we develop a theory of common decomposition for two correlated Brownian motions, in which, byusing change of time method, the correlated Brownian motions are represented by a triple of processes, ( X , Y , T ) ,where X and Y are independent Brownian motions. We show the equivalent conditions for the triple being inde-pendent. We discuss the connection and difference of the common decomposition with the local correlation model.Indicated by the discussion, we propose a new method for constructing correlated Brownian motions which per-forms very well in simulation. For applications, we use these very general results for pricing of two-factor financialderivatives whose payoffs rely very much on the correlations of underlyings. And in addition, with the help ofnumerical method, we also make a discussion of the pricing deviation when substituting a constant correlationmodel for a general one. The correlation between assets plays an important role in finance. Whenever we meet a problem involving twostochastic factors, the correlation risk is unavoidable. The problem may be from areas of asset allocation, pairstrading, risk management and typically, multi-assets derivative’s pricing. Generally speaking, there are two meth-ods in financial modelling to induce dependence between assets, one is by copula , the other is in SDE models byassuming a correlation structure for processes driving the model. Since Brownian motion is the most commonlyused driving process stemming from Bachelier, correlation between Brownian motions is crucially important inthe latter.To formulate correlated Brownian motions, many models adopt the constant local correlation assumption,i.e., d [ B , W ] t = ρ dt or conventionally, dB t dW t = ρ dt , for Brownian motions B and W and a constant ρ ∈ R .However, more and more empirical works proved that the dependence between financial factors varies over timeand depending on the economic status. There is a short review of literatures rejecting constant correlation beforethe 2008 crisis in Buraschi et al. 2010, and the authors themselves studied the joint correlation of stock index aroundthe crisis. Other empirical evidences include Chiang et al. 2007 finding a significant increasing for correlationsbetween Asian market after the crisis, Syllignakis and Kouretas 2011 and Junior and Franca 2012 getting similarresults for the European and global markets, Xiong et al. 2018 for time-varying correlation between policy indexand stock return in China and Balcilar et al. 2018 for dynamic correlation between oil price and inflation in SouthAfrica.Probably for this reason, there is a growing literature in recent years applying dynamic local correlation forfinancial problems. Since the value of local correlation, i.e., ρ introduced above, must be in [ −
1, 1 ] , these literatures1 a r X i v : . [ q -f i n . M F ] J u l dopted various techniques to assure this. Osajima 2007 and Fern´andez et al. 2013 modeled ρ as a bounded deter-ministic function of time t for SABR model while Teng et al. 2015 adopted the same idea in geometric-Brownian-motion model and applied it to pricing Quanto. Note that in these models, ρ is dynamic but nonstochastic. Forstochastic ρ , Van Emmerich 2006, Langnau 2010, Teng et al. 2016 and Carr 2017 expressed ρ as a bounded functionof some stochastic state processes and applied it in derivatives’ pricing problems. And some literatures modeled ρ directly by a bounded stochastic process. For example, bounded Jacobi process is a kind of bounded diffusionprocess driven by Brownian motion and was introduced to model ρ with applications in option pricing, includingvanilla option (Teng et al. 2016), correlation swap (Meissner 2016), Quanto (Ma 2009) and multi-asset option (Ma2009), and in portfolio selection and risk management (Buraschi et al. 2010). Hull et al. 2010 modeled the localcorrelation as a step process where each step is a beta-distributional random variable. M´arkus and Kumar 2019made a comparison of several stochastic local correlation models. Moreover, regime switching model is a wellused model in finance where all the parameters, including ρ , could be driven by a common continuous-time finite-state stationary Markov process, and thus provide another way to model stochastic local correlation, e.g. Zhouand Yin 2003.The main focus of this paper is on proposing a new method for formulation and analysis of the dependencestructure for general correlated Brownian motions.Dynamic correlation is widely considered in discrete time model, Engle 2002 proposed DCC-GARCH model togenerate conditional correlation and Christodoulakis and Satchell 2002 used the correlated ARCH process. Tsay2005 investigated a large number of multivariate time series and multivariate volatility models, and apply theminto financial markets.Copula is a well developed method for characterizing dependency structure between random variables, Ja-worski and Krzywda 2013 and Bosc 2012 apply Copula to correlated Brownian motions. Given the Copula ofcorrelated Brownian motions, they solved local correlation from PDEs and found that the Copula of Brownianmotions can not be Gumbel and Clayton Copula.Wishart process can establish stochastic covariance directly, and the local correlation obtained from covariancematrix is stochastic as well. Da Fonseca et al. 2007 discussed the Wishart process for multi-asset option pricing andfound that there is a correlation leverage affect in call on max style option. Double Heston model also allows aspecial kind of local correlation between asset and stochastic volatility, see Costabile et al. 2012 and Christoffersenet al. 2009 for more details.Except correlated Brownian motions, there are also other ways to construct correlated stochastic processes.Wang 2009 obtained correlated variance gamma processes by Brownian motions with constant correlation com-pound with time changes. Mendoza-Arriaga and Linetsky 2016 and Barndorff-Nielsen et al. 2001 describe corre-lated stochastic processes by independent background stochastic processes with dependent L´evy subordinators.Ballotta and Bonfiglioli 2016 proposed factor model for L´evy process, each asset is governed by a systematic com-ponent and a specific component.In this article, we construct correlated Brownian motions by change-of-time. Change-of-time is a developedtechnique to construct stochastic processes (see Barndorff-Nielsen and Shiryaev 2015), and is widely applied tomathematical finance (e.g. Carr et al. 2003 and Geman et al. 2001). However, as far as we know, there are fewworks apply change-of-time technique into modeling correlated Brownian motions.After constructing correlated Brownian motions, we apply our method into multi-asset option pricing byFourier transform. Fourier transform method in option pricing is developed by Carr and Madan 1999, morerecent papers studied Fourier transform method to price multi-asset options, e.g. Hurd and Zhou 2010 for spreadoption, Wang 2009 for rainbow options and Leentvaar and Oosterlee 2008 gave a numerical method for multi-assetoptions without explicit expression.Recall the main result in Chen et al. 2018, if { B n } n ≥ and { W n } n ≥ are two random walks with filtration2 F n } n ≥ , satisfy P ( B n − B n − = |F n − ) = P ( B n − B n − = − |F n − ) = P ( W n − W n − = |F n − ) = P ( W n − W n − = − |F n − ) =
12 .Then { B n } n ≥ and { W n } n ≥ can be decomposed as ( B n , W n ) = ( X T n + Y S n , X T n − Y S n ) ,where { X n } n ≥ , { Y n } n ≥ are two independent random walks, and T n + S n = n . Moreover, if P ( B n − B n − = W n − W n − = |F n − ) = ϑ is constant, then { X n } n ≥ , { Y n } n ≥ and { T n } n ≥ are mutually independent, and { T n } n ≥ is increment independent with P ( T n − T n − = ) = ϑ , P ( T n − T n − = ) = − ϑ .As we all know, by Donsker’s theorem, ( B [ nt ] √ n , W [ nt ] √ n ) d → ( ˜ B t , ˜ W t ) , ( X [ nt ] √ n , Y [ nt ] √ n ) d → ( ˜ X t , ˜ Y t ) ,where ( ˜ B t , ˜ W t ) are two Brownian motions with constant correlation coefficient 4 ϑ − ( ˜ X t , ˜ Y t ) are two independentBrownian motions. According to strong law of large numbers, T [ nt ] n a . s . → ϑ t , S [ nt ] n a . s . → t − ϑ t . Hence, by Billingsley1968[Section 14, Lemma] ( X T [ nt ] √ n , Y S [ nt ] √ n ) d → ( ˜ X ϑ t , ˜ Y ( − ϑ ) t ) .From lim n → ∞ ( B [ nt ] √ n , W [ nt ] √ n ) = lim n → ∞ ( X T [ nt ] √ n + Y S [ nt ] √ n , X T [ nt ] √ n − Y S [ nt ] √ n ) .we know that ( ˜ B t , ˜ W t ) d = ( ˜ X ϑ t + ˜ Y ( − ϑ ) t , ˜ X ϑ t − ˜ Y ( − ϑ ) t ) .Since we have proved two Brownian motions with constant correlation coefficient have the decompositionsimilar with Chen et al. 2018 in the sense of distribution, in the rest of this article, we consider the general twocorrelated Brownian motions. Furthermore, we decompose Brownian motions in the sense of almost surely.From another view, every step have the same size in random walk, so we can say common or counter move-ments and put the common (counter) movements together. However, size of two Brownian motions movementsmay very different in a very short time period. Thus, we need to develop a new kind of common and countermovements. In this section, we consider the dependency structure of two correlated Brownian motions. In Section 2.1, wepropose the definition of common decomposition of two correlated Brownian motions and give some notations.In Section 2.2, we investigate the independency property of stochastic processes obtained from the common de-composition. In Section 2.3, we study the connection of the common decomposition and local correlation of twocorrelated Brownian motions. 3 .1 Model Setup
Throughout this section we consider, on a complete probability space ( Ω , F , P ) , two correlated Brownian motions, { B t } t ≥ and { W t } t ≥ , with respect to the same filtration F = {F t } t ≥ which is assumed to satisfy the usualconditions.Define T t (cid:44) t + [ B , W ] t S t (cid:44) t − [ B , W ] t s < t then we have − t + s = − [ B ] t − [ W ] t + [ B ] s + [ W ] s ≤ [ B , W ] t − [ B , W ] s ≤ [ B ] t + [ W ] t − [ B ] s − [ W ] s = t − s ,hence 0 ≤ T t − T s ≤ t − s , 0 ≤ S t − S s ≤ t − s .That is to say, T t and S t are increasing processes with T t + S t = t . And they are both absolutely continuous withrespect to t . T and S could be regarded as special “timers” that record the time with special correlation information. Example 2.1.
For example, if the correlation coefficient of B and W is a constant ρ , i.e., [ B , W ] t = ρ t, then T t = + ρ t andS t = − ρ t. Specifically, • when B and W are completely positive correlated, i.e. [ B , W ] t = t, T t = t and S t = ; • when B and W are completely negative correlated, i.e. [ B , W ] t = − t, T t = and S t = t; • when B and W are independent with each other, T t = S t = t . Let τ t = inf { u : T u > t } , ς t = inf { u : S u > t } , (2)by definition, { τ t } t ≥ and { ς t } t ≥ are time changes of filtration F . By contrast, T is a time change of {F τ t } t ≥ and S is a time change of {F ς t } t ≥ .Loosely speaking, the so-called common decomposition in this article could be given through time-changed pro-cesses X t (cid:44) B τ t + W τ t , Y t (cid:44) B ς t + W ς t . By definitions of T , S , τ and ς , we observe that, in most cases, X T t = B t + W t and Y S t = B t − W t .However, given ω ∈ Ω , when the limit of T u ( ω ) exists and is finite, i.e., whenever T ∞ ( ω ) (cid:44) lim u → ∞ T u ( ω ) is finite, X t ( ω ) is not well-defined for t ≥ T ∞ ( ω ) . For example, if B and W are completely negative correlated, then [ B , W ] t = − t , T t = t ≥
0, and τ t = ∞ . The same happens to S and Y . (Here let S ∞ ( ω ) (cid:44) lim u → ∞ S u ( ω ) .)In order to overcome this limitation, we apply the similar method as in Revuz and Yor 2013[Chapter V] tomodify the definition of X and Y . We assume the probability space ( Ω , F , P ) are rich enough to support Brownianmotions that are independent of known Brownian motions and F ∞ . A time change C is a family C s , s ≥
0, of stopping times such that the map s → C s are a.s. increasing and right-continuous (Revuz and Yor2013,Chapter V, Definition 1.2). { ˜ X t , ˜ Y t } t ≥ is a 2-dimensional Brownian motion independent from F ∞ . We modify the definition of { X t } t ≥ and { Y t } t ≥ as follows . X t (cid:44) (cid:40) B τ t + W τ t , if t < T ∞ X ∞ + ˜ X t − T ∞ , if t ≥ T ∞ , Y t (cid:44) (cid:40) B ς t − W ς t , if t < S ∞ Y ∞ + ˜ Y t − S ∞ , if t ≥ S ∞ . (3)By the definitions of X , Y , T and S , B t = X T t + Y S t , W t = X T t − Y S t . (4)Thus we obtained a representation of ( B , W ) through the three new-defined processes X , Y , and T (it always holdsthat S t = t − T t ). We call this the common decomposition of ( B , W ) and denote it by ( X , Y , T ) . Remark 2.2.
The choice of ( ˜ X , ˜ Y ) can only affect definition of ( X , Y ) , but has no influence on the decomposition of B andW. To be more specific, for ∀ t ≥ , if T t < T ∞ , by definition, X T t does not depend on ˜ X. If T t = T ∞ , X T t = X T ∞ = X ∞ ,does not depend on ˜ X, either. The same is true for Y.
For the readers’ convenience, we introduce some notations here: • F Xt : natural filtration of stochastic process { X t } t ≥ . It is remarkable that F Tt = σ ( T u : u ≤ t ) = σ ( S u : u ≤ t ) = F St . • A ⊥ B | C : A and B are conditional independent under C . In the previous section, we introduced the so called common decomposition ( X , Y , T ) of Brownian motions B and W . In this part we give some basic properties of this decomposition. Proofs can be found in the Section 6.Our first result is to illustrate the essence of and the relationship between X and Y . Theorem 2.3.
Given Brownian motions { B t } t ≥ and { W t } t ≥ and their decomposition ( X , Y , T ) as in Section 2.1, wehave that { X t } t ≥ (resp., { Y t } t ≥ ) is a Brownian motion of the filtration {F τ t } t ≥ (resp., {F ς t } t ≥ ), and that X and Y areindependent. From Theorem 2.3, the common decomposition represents B (resp. W ) as the sum (resp. difference) of two time-changed Brownian motions. The dependency structure of B and W is embodied in T as well as in the dependenciesbetween it and the two new-defined Brownian motions. Hence for clarity and convenience, the independency of X , Y and T is worth studying. In the following theorem, a sufficient and necessary condition is given for mutualindependency of them. Theorem 2.4.
Under the conditions and notations as in Theorem 2.3, the processes of decomposition triple, i.e., X, Y and T,are mutually independent if and only if: (C1) F B ∞ ⊥ F T ∞ |F B , Wt and F W ∞ ⊥ F T ∞ |F B , Wt . As an example to understand the condition, when B and W has a constant correlation say, ρ , (C1) is satisfiedsince T t = + ρ t and F T ∞ is a trivial σ -algebra. More general cases will be discussed in Section 2.3 later. Please note that according to Revuz and Yor 2013[Chapter V, Proposition 1.8], X ∞ (cid:44) lim t → ∞ B t + W t and Y ∞ (cid:44) lim t → ∞ B t − W t exist. T t or S t . And their revolutionsare decomposed thereupon according to the new clocks. By Theorem 2.3, under the new clocks, they keep theirBrownian-motion features and these features are independent under the two clocks. Thus dependency structuresand Brownian features are decomposed either. By Theorem 2.4, if they satisfy condition (C1), their dependencyinformation is only contained in T . The decomposition is quite complete and clear. In this case, if one studied thedependency of two correlated Brownian motions, all he need is to consider precess T in common decomposition.The following proposition gives an equivalent condition of (C1) from another aspect. Proposition 2.5.
Suppose the assumptions in Theorem 2.3 hold. Then condition (C1) is equivalent with the followingstatement. (C2)
Given two processes { φ t } t ≥ and { φ t } t ≥ , which are progressively measurable with {F Tt } t ≥ thatE (cid:20) exp (cid:18) (cid:90) t ( φ u ) dT u + (cid:90) t ( φ u ) dS u (cid:19)(cid:21) = E (cid:20) exp (cid:18) (cid:90) t (cid:16) ( φ u ) − ( φ u ) (cid:17) dT u + (cid:90) t ( φ u ) du (cid:19)(cid:21) < ∞ , ∀ t , let D φ t (cid:44) exp (cid:18) (cid:90) t φ u dX T u + (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dT u − (cid:90) t ( φ u ) dS u (cid:19) , then D φ is a martingale and dQdP | F t = D φ t defines a probability measure such that ( X φ T , Y φ S ) Q d = ( X T , Y S ) P , (5) where X φ T t = X T t − (cid:82) t φ u dT u , Y φ S t = Y S t − (cid:82) t φ u dS u .This proposition link the independency of the decomposition triple with conditions similar to Girsanov theo-rem. Undoubtedly it may attract our attention to consider its connection with financial modelling. Example 2.6.
In financial models, the Girsanov transform is typically used to change the drift parts of diffusions thatmodeling the prices. Consider two drifted Brownian motions, (cid:90) t θ u du + B t , (cid:90) t θ u du + W t , where θ i , i =
1, 2 are bounded, progressively measurable with {F Tt } t ≥ . According to Theorem 2.3, these two processes canbe represented as (cid:90) t θ u du + X T t + Y S t , (cid:90) t θ u du + X T t − Y S t . Let λ and µ denote the densities of T and S, λ t (cid:44) dT t dt , µ t (cid:44) dS t dt , and suppose that inf { t ≥ λ t , µ t } > .If ( B , W ) satisfies condition (C1), then from Proposition 2.5, the two drifted Brownian motions can be transformed to (cid:90) t θ u du + B t = X φ T t + Y φ S t : = B φ t , (cid:90) t θ u du + W t = X φ T t − Y φ S t : = W φ t , (6)6 here φ = ( φ , φ ) is defined as φ t = θ t + θ t λ t , φ t = θ t − θ t µ t . Under the probability Q as defined in Proposition 2.5, it is notable that by (5) , ( B , W ) P d = ( B φ , W φ ) Q , (7) thus the drift parts vanish after change of probability measure.Consider the common decomposition of ( B φ , W φ ) , denoted by ( X φ , Y φ , T φ ) . From (6) ,T t = T φ t . (8) Moreover, from (5) we have ( { T t } t ≥ ) P d = ( { T t } t ≥ ) Q . (9) Remark 2.7. (8) and (9) reveal the invariance property of T under change of measure. From the application point of view,this implies that in financial modelling after change of numeraire, the common decomposition method is still valid. And fromempirical view, we can estimate parameters from real probability measure and apply to risk neutral measure directly. Forexample, Ballotta and Bonfiglioli 2016 bring correlation matrix estimated from observed asset prices in option pricing model,and we give a theoretical foundation. This is quite convenient for derivatives pricing which are lack of public data.This also shows that we can simplify two correlated Brownian motions with drifts by changing of measure, and keep thedependency structure of them.
Let ρ t (cid:44) d [ B , W ] t dt , ρ is called the local correlation process of B and W .When ρ is a constant, i.e., d [ B , W ] t = ρ dt , this is the most commonly used model for correlated Brownianmotions, so common that it sometimes looks like a routine operation to add a ρ to include dependency structurein a Brownian-motion-driven model. Without doubt, when dependency is a critical element for the issues understudy, the constant local correlation assumption seems neither realistic (e.g.,Chiang et al. 2007) nor sufficientlyprecise(e.g., Driessen et al. 2013). Some researchers have noticed this problem and brought in more realistic as-sumptions for ρ . For example, among others, Osajima 2007 and Teng et al. 2015 supposed ρ to be deterministic buttime variant, Langnau 2010 modeled ρ as a function of the asset price thus is dynamic and stochastic; and someintroduced a SDE to directly model a dynamic and stochastic ρ , such as Ankirchner and Heyne 2012, Jaworski andKrzywda 2013 Bosc 2012, Langnau 2010, etc.In this section, we take a new look at the common decomposition via the local correlation process. We considerthe differences and connections of the common decomposition method and the local correlation model. Still, theproofs can be found in Section 6. Let us first recall a well used ρ -based decomposition method representing correlated Brownian motions as linearcombinations of independent Brownian motions. Suppose ˜ Z is a Brownian motion independent of F ∞ and ( ˜ X , ˜ Y ) in (3), define 7 t (cid:44) (cid:90) t { ρ u (cid:54) = ± } (cid:112) − ρ u ( dW u − ρ u dB u ) + (cid:90) t { ρ u = ± } d ˜ Z u . (10)Particularly, if ρ t (cid:54) = ±
1, a.s. ∀ t , then Z t = (cid:90) t (cid:112) − ρ u dW u − (cid:90) t ρ u (cid:112) − ρ u dB u .It is not difficult to verify that { Z t } t ≥ is a Brownian motion independent of { B t } t ≥ , and that the local correlationof Z and W is (cid:113) − ρ t .By definition of Z t , we have the local-correlation based decomposition of ( B , W ) , ( B t , W t ) = ( B t , (cid:90) t ρ s dB s + (cid:90) t (cid:113) − ρ s dZ s ) . (11)If we start from the right side of the equation, i.e., starting from independent Brownian motions B , Z and localcorrelation process ρ , we have got a commonly used model for construction correlated ( B , W ) are Brownianmotions.As a comparison, recall that the common decomposition in the current paper of ( B , W ) is ( B t , W t ) = ( X T t + Y S t , X T t − Y S t ) .Similarly, if we can start from the right side, i.e., from independent Brownian motions X , Y and time-changeprocess T , and make the construction, under some conditions, to be correlated Brownian motions, then we havegot a new construction method of ( B , W ) . We will make further discussions of this in the next section. Remark 2.8.
The different ideas behind the two methods look clear from the above comparison: the local-correlation methodmodel dependency of the Brownian motions from a spatial perspective while the common-decomposition method from a tem-poral perspective.
The next proposition give a connection between local-correlation based decomposition and common decompo-sition. The two method would share the same equivalent conditions when considering completely-independentdecomposition.
Proposition 2.9.
Under the conditions stated in Theorem 2.3, X, Y and T are mutually independent if and only if thefollowing condition holds: (C3) ρ , B and Z in local-correlation model (11) are mutually independent. T From the setup, we can see the important role T played in the common decomposition. Since X and Y are inde-pendent, T is the one relevant to the dependency structure of ( B , W ) in the decomposition triple (in the case of The second part of Z is obviously well defined. Consider the first part (cid:82) t { ρ u (cid:54) = ± } √ − ρ u ( dW u − ρ u dB u ) , let M t (cid:44) W t − (cid:82) t ρ u dB u , [ M ] t = t + (cid:82) t ρ u du − (cid:82) t ρ u d [ B , W ] u = (cid:82) t ( − ρ u ) du . For any 0 < t < ∞ , (cid:82) t { ρ u (cid:54) = ± } − ρ u d [ M ] u ≤ (cid:82) t du = t < ∞ , which implies the second part is welldefined, too. Some conditions need to be added to ensure that the construction is well posed and B , W X , Y and T are independent, T contains all the dependency information). On theother hand, if we treat T as a special timer, a ”clock”, it is obvious that this clock’s movings are affected by thecorrelation degree of ( B , W ) . In this section, we make some more specific discussions of T via ρ to get a betterunderstanding of the common decomposition.First, by definitions, there is a connection between T and ρ : T t = (cid:90) t + ρ u du , S t = (cid:90) t − ρ u du ,in which, 1 + ρ t is in fact the distance between local correlation ρ t and −
1, 1 − ρ t is the distance between ρ t and +
1, and the denominator 2 is the distance between − +
1. Thus the integrands could be regarded asnormalizations of the deviation of ( B , W ) ’s correlation from complete correlation. Think of the case when ρ isalways close to 1 and far away from −
1, then the ”clock T ” runs runs faster than S , and it is the clock focusing onpositive correlation.Consider the readings of the two clocks, at any time t , they satisfy T t + S t = t , T t − S t = (cid:90) t ρ u du .That is to say, the sum of the readings represents the calender time, while the difference of them shows the cumu-lated correlation of ( B , W ) till time t .And the average correlation coefficient process which is defined as¯ ρ t = t (cid:90) t ρ u du , (12)could also be represented by T and S , ¯ ρ t = T t − S t t = T t − S t T t + S t .The change-of-time technique did be used in stochastic correlations, Barndorff-Nielsen et al. 2001 and Mendoza-Arriaga and Linetsky 2016 describe dependent stochastic processes by independent stochastic processes with de-pendent L´evy subordinators. However as far as we know, there are few works model correlated Brownian motionsby time-changed process.In two-factor derivative’s pricing, when local correlation of the two factors varies stochastically over time, it isalways difficult to obtain the option prices. The average correlation coefficient process, ¯ ρ , usually plays an impor-tant role under this circumstances. For example, in Ma 2009, the price of foreign equity option was approximatedby the moments of ¯ ρ , in Van Emmerich 2006 and Teng et al. 2016, the price of a Quanto is determined by theLaplace transform of ¯ ρ . Note that ¯ ρ is an integral in local correlation model, but in our method, ¯ ρ t = T t / t − Example 2.10.
Suppose B and W are two Brownian motions with constant correlation ρ ∈ ( −
1, 1 ) . then by local-correlationmethod, ( B t , W t ) = ( B t , ρ B t + (cid:113) − ρ Z t ) .9 n this case, the condition in Proposition 2.9 is satisfied, thus the processes of common-decomposition triple, X , Y and T,are mutually independent. And they can be calculated accurately:T t = + ρ t , S t = − ρ t , X t = B + ρ t + W + ρ t , Y t = B − ρ t − W − ρ t , and the decomposition of ( B , W ) , ( B t , W t ) = ( X + ρ t + Y − ρ t , X + ρ t − Y − ρ t ) , In this example, it shows clearly that, T and S conform a decomposition of the “calender time” in any time period. They are composed by special “time points”picked out according to the correlation structure of ( B , W ) . They can be considered as special clocks that moves only atspecial time. If ρ > , clock T runs faster than clock S, vice versa. Consider the family of convex combinations of B and W, C = { α B + ( − α ) W | ≤ α ≤ } . The local correlation ofevery two processes in C with parameters α and β is ρ α , β = ( − ρ )[( α − ) β − α ] + If α = , ρ α , β = ρ + > ∀ β ∈ [
0, 1 ] . Otherwise, there always exists a β ∈ [
0, 1 ] such that ρ α , β ≤ . That is to say, B + W is the only process in C that is strictly positive correlated with any other process in C . Note that this process is infact X under clock T, thus X represents the common structures in B and W. Remark 2.11.
Actually, if the local correlation of B and W is not constant, the three descriptions for Example 2.10 remainvalid. For and , the results remain the same. For , we can prove t Cov ( α B t + ( − α ) W t , β B t + ( − β ) W t ) = ( − Corr ( B t , W t ))[( α − ) β − α ] + where Cov and Corr denote covariance and correlation respectively. With the similar discussion, B + W is the only process in C that is strictly positive correlated with any convex combination of B , W. T and S The example in previous section demonstrated what the processes in common decomposition look like and howto construct the clock T when ρ t ≡ ρ ∈ (
0, 1 ) . In this section, similar analysis is carried from a distributional aspectfor general cases by discretizing ρ . In this part, we also start with two correlated Brownian motion B and W withlocal correlation process ρ , and all the other notations defined in previous sections are followed.Given t ≥
0, let Π be a partition of [ t ] :0 = t < t < t < · · · < t n = t ,and write || Π || = max { t i − t i − : i =
1, . . . , n } . For ∀ ω ∈ Ω , define A Π ( ω ) = n − (cid:91) i = ( t i , t i + + ρ t i ( ω ) ∆ t i ] .10ote that by the construction of A Π , the stochastic processes { { u ∈ A Π } } ≤ u ≤ t and { [ u / ∈ A Π ] } ≤ u ≤ t are predictable.Set ˜ X Π s (cid:44) (cid:90) s { u ∈ A Π } dB u , ˜ Y Π s (cid:44) (cid:90) s { u / ∈ A Π } dB u ,i.e., ˜ X Π keeps in step with B in A Π and stays still at other time while Y Π does the opposite. Let˜ W Π s (cid:44) ˜ X Π s − ˜ Y Π s = (cid:90) s { u ∈ A Π } dB u − (cid:90) s { u / ∈ A Π } dB u . (13)Then ˜ W Π is a Brownian motion moving commonly with B in A Π and oppositely in ( A Π ) c . And X Π and Y Π represent the common movements and counter movements of B and W Π .At any time s ≤ t , the time period [ s ] is divided into two parts: the commonly-moving period A Π (cid:84) [ s ] and the oppositely-moving period ( A Π ) c (cid:84) [ s ] , whose total lengths could be calculated respectively as (suppose t i < s ≤ t i + ) ˜ T Π s ( ω ) (cid:44) m (cid:16) [ s ] ∩ A Π ( ω ) (cid:17) = t i + ∑ ik = ρ t k ( ω ) ∆ t k + m (cid:16) ( t i , s ] ∩ A Π ( ω ) (cid:17) ,˜ S Π s ( ω ) (cid:44) m (cid:16) [ s ] / E Π ( ω ) (cid:17) = t i − ∑ ik = ρ t k ( ω ) ∆ t k + m (cid:16) ( t i , s ] / A Π ( ω ) (cid:17) ,where m ( · ) denotes the Lebesgue measure on R . Obviously,lim || Π ||→ ˜ T Π s = s + (cid:82) s ρ u du = T s , lim || Π ||→ ˜ S Π s = s − (cid:82) s ρ u du = S s , ∀ s ∈ [ t ] . (14)The following proposition considers the limitation of X Π , Y Π and W Π in distribution. Proposition 2.12.
Suppose the assumptions in model setup and the conditions in Proposition 2.9 hold, under the formernotations, for any given ≤ u < u < · · · < u K < ∞ , 0 ≤ v < v < · · · < v L < ∞ , as || Π || → , we have ( B u , B u , . . . , B u K , ˜ W Π v , ˜ W Π v , . . . , ˜ W Π v L ) d −→ ( B u , B u , . . . , B u K , W v , W v , . . . , W v L ) .Proposition 2.12 guarantees that ( B , ˜ W Π ) converge to ( B , W ) in the sense of distribution as || Π || →
0. Forsimplicity, we still denote this distributional convergence of processes by ” d −→ ”. Thus, ( B , ˜ W Π ) d −→ ( B , W ) .as a consequence ( ˜ X Π , ˜ Y Π ) = ( B + ˜ W Π B − ˜ W Π ) d −−−−→ || Π ||→ ( X T , Y S ) . (15)The convergence properties (15) and (14) reveal the connections of X and Y with common and counter movementsof ( B , W ) in some sense, and give an intuitive explanation for T and S to be considered as clocks recording positivecorrelation and negative correlation of ( B , W ) . 11 A New Method for Construction of Correlated Brownian Motions
In the previous section, we demonstrated the common decomposition representation of two correlated Brownianmotions. For any two Brownian motions B and W , we can find a decomposition triple ( X , Y , T ) to represent themby change of time method. While in practice, a converse problem may be also worthy of concern and research.That is, is it possible to construct two Brownian motions with desired dependency structure from two independentBrownian motions by common decomposition method? In this section we will focus on this problem.In tradition, in order to construct correlated Brownian motions, local correlation model and copula method areboth considered. Ma 2009, Ma 2009 and Teng et al. 2016 modelled local correlation by bounded Jacobi process. Tenget al. 2016 studied stochastic local correlation by an O-U process compound with hyperbolic function tanh sincelocal correlation is bounded. Bosc 2012 and Jaworski and Krzywda 2013 discussed correlated Brownian motionsby copula. In the following, we construct correlated Brownian motions by common decomposition method. Theorem 3.1.
Let ( X , Y ) be a 2-dimensional standard Brownian motion and { T t } t ≥ , { S t } t ≥ be time changes with respectto F . If F Y S t ⊥ F X T ∞ |F X T t and F X T t ⊥ F Y S ∞ |F Y S t , then { X T t } t ≥ and { Y S t } t ≥ are martingales with respect to F X T , Y S . IfT t + S t = t , ∀ t ≥ , then B t (cid:44) X T t + Y S t and W t (cid:44) X T t − Y S t are two correlated Brownian motions with respect to F B , W with [ B , W ] t = T t − S t . Immediately, we have a convenient way to construct correlated Brownian motions from Theorem 3.1.
Corollary 3.2.
Suppose that T , S are increasing processes satisfying T t + S t = t , ∀ t ≥ , and X , Y are independentBrownian motions. If X , Y , T are mutually independent, thenB t (cid:44) X T t + Y S t and W t (cid:44) X T t − Y S t are two correlated Brownian motions with respect to F B , W with [ B , W ] t = T t − S t . Regime switching is a commonly used model in finance, it can capture features in financial data well, for exam-ple see Schaller and Norden 1997. Also, Casarin et al. 2018 and Pelletier 2006 have considered regime switchingmodel for correlations in discrete time. In the next example, we consider regime switching model to constructcorrelated Brownian motions by common decomposition method.
Example 3.3. (Regime switching model) Suppose Q t is a continuous time stationary Markov Chain taking value in a finitestate space { e , e , . . . , e n } , where e i = (
0, . . . , 0, 1, 0, . . . , 0 ) denotes the unit vector. The Markov chain Q t has a stationarytransition probability matrix P ( t ) = ( p ij ( t )) n × n , wherep ij ( t ) = P ( Q t + s = e j | Q s = e i ) . And the homogeneous generator A = ( a ij ) n × n is defined as A (cid:44) lim t ↓ P ( t ) − I t , I denotes the identity matrix. Then we have d P ( t ) dt = AP ( t ) = P ( t ) A ,12 olve the ODE we obtain P ( t ) = e A t . (16) Let ω = [ ω , ω , . . . , ω n ] T , ω i ∈ [
0, 1 ] , ∀ i andT t = (cid:90) t ω T Q s ds , S t = t − T t = (cid:90) t ( − ω ) T Q s ds . Obviously, { T t } t ≥ , { S t } t ≥ are increasing processes. Let { X t } t ≥ , { Y t } t ≥ be 2-dimensional standard Brownian motionindependent with Q t . Then from Corollary 3.2, we have { X T t + Y S t } t ≥ and { X T t − Y S t } t ≥ are two correlated Brownianmotions. The common decomposition method also gives a new way to simulate correlated Brownian motions. One cansimulate T t at first, then simulate X T t and Y S t under the condition of T t . Particulary, if X , Y and T are mutuallyindependent, the distribution of X T t and Y S t under the condition of T t is normal distribution, this may bringadvantages of our construction compared with local correlation model.One of the most common simulation method for local correlation model is Euler-Maruyama scheme (see Kloe-den and Platen 2013). Firstly, given a partition Π of [ t ] , let W Π t = (cid:90) t ρ Π u dB u + (cid:90) t (cid:113) − ( ρ Π u ) dZ u = n − ∑ k = ( ρ t k ∆ B t k + (cid:113) − ρ t k ∆ Z t k ) , (17)where ∆ B t k = B t k + − B t k , ∆ Z t k = Z t k + − Z t k and { ρ Π u } ≤ u ≤ t is defined as ρ Π u = ρ t i , t i ≤ u < t i + .Secondly, we simulate ( B , W ) by (17). Thus, the simulation result is ( B , W Π ) eventually, and there will be a simu-lation error.Under the condition X , Y and T are mutually independent, Table 1 and Table 2 show the specific steps ofsimulation of common decomposition method when we do not have the explicit expression of T ’s distribution andcomparing with the Euler-Maruyama scheme of local correlation model. The common decomposition of ( B , W Π ) isdenoted as ( X Π , Y Π , T Π ) . From Table 1, the advantages of common decomposition method in simulation include: • If we only need B t and W t at time t , and the trajectory is not necessary, common decomposition method canreduce the time of simulations. If ρ t is a stochastic process, we need to simulate 3 n random numbers, i.e. ∆ B t i , ∆ Z t i , ρ t i , i =
0, 1, . . . , n −
1. However, in common decomposition method we only need to simulate n + T Π t , T Π t , . . . , T Π t n , X Π T Π t and Y Π S Π t . Moreover, if we have the explicit expression of T ’sdistribution, we can simulate T t directly, then we only need to simulate X T t and Y S t , hence simulation can bereduced to 3 times. Remark 3.4.
The simulation error can be controlled as long as the simulation error of T Π t can be controlled, sinceE | X T t − X T Π t | = E | T t − T Π t | ≤ ( E | T t − T Π t | ) . Therefore, if the explicit expression of T’s distribution is obtained, one can simulate T directly, and then simulate X T and Y S with the similar steps in Table 1 and Table 2. There is no simulation error for T, hence we can simulate ( B , W ) accuratelywhile this is impossible for local correlation model. If the explicit expression of T’s distribution is unobtained, the simulationerror of two methods is same, because both the simulation result of two methods is ( B , W Π ) . T ’s distribution , there isno much difference in simulation for two methods.Table 1: Simulate ( B t , W t ) (Explicit expression of T ’s distribution is unobtained)Common decomposition method Local correlation model (Euler-Maruyama scheme)Simulate T Π t , T Π t , . . . , T Π t n one by one Simulate ρ t , ρ t , . . . , ρ t n − one by oneSimulate X Π T Π t and Y Π S Π t Simulate ∆ B t , ∆ B t , . . . , ∆ B t n − and ∆ Z t , ∆ Z t , . . . , ∆ Z t n − Calculate ( B t , W Π t ) Calculate ( B t , W Π t ) Table 2: Simulate trajectory of ( B , W ) in [ t ] (Explicit expression of T ’s distribution is unobtained)Common decomposition method Local correlation model (Euler-Maruyama scheme)Simulate T Π t , T Π t , . . . , T Π t n one by one Simulate ρ t , ρ t , . . . , ρ t n − one by oneSimulate ∆ X Π T Π t , ∆ X Π T Π t , . . . , ∆ X Π T Π tn − and ∆ Y Π S Π t , ∆ Y Π S Π t ,. . . , ∆ Y Π S Π tn − Simulate ∆ B t , ∆ B t , . . . , ∆ B t n − and ∆ Z t , ∆ Z t , . . . , ∆ Z t n − Calculate B t , B t , . . . , B t n and W Π t , W Π t , . . . , W Π t n Calculate B t , B t , . . . , B t n and W Π t , W Π t , . . . , W Π t n According to T Π t = t + ∑ n − i = ρ ti ∆ t i , simulate T Π t , T Π t , . . . , T Π t n is equivalent with simulate ρ t , ρ t , . . . , ρ t n − . Under the condition of T Π t , T Π t , . . . , T Π t n − , ∆ X Π T Π t , ∆ X Π T Π t , . . . , ∆ X Π T Π tn − and ∆ Y Π S Π t , ∆ Y Π S Π t , . . . , ∆ Y Π S Π tn − are indepen-dent normal distributions with mean zero and variance ∆ T Π t , ∆ T Π t , . . . , ∆ T Π t n − , ∆ S Π t , ∆ S Π t , . . . , ∆ S Π t n − respec-tively.Figure 1(a), Figure 1(b), Figure 1(c) display how we simulate the trajectory of ( B , W ) in [ t ] through commondecomposition method (explicit expression of T ’s distribution is unobtained) step by step, parameters are taken asfollow, Q = [
1, 0, 0 ] T , ω = [ ] T , A = − − − , t = ∆ t i = ∀ i . (18) (a) Step 1: Simulate T t (b) Step 2: Simulate X T t and Y S t (c) Step 3: Calculate B t and W t Figure 1: Simulate ( B t , W t ) by Common Decomposition Method (Explicit expression of T ’s distribution is unob-tained) 14n order to compare two simulation methods in practical, we consider the regime switching model in Example3.3. Thanks to (16), simulation for regime switching model is feasible. We calculate the mean of B t + W t by simulate ( B t , W t ) with N = B t + W t is 0 in theoretical, Table 3 shows that the simulation error of two methods arevery close, because their standard deviation are truly close. And common decomposition method runs much fasterthan local correlation model with Euler-Maruyama scheme.Table 3: Comparing two simulation methods E ( B t + W t ) Std Dev Running timeCommon decomposition method(explicit expression of T ’s distribution is unobtained) -0.0034 1.8593 × − × − In financial derivatives’ pricing, there are quite a few chances to meet with the situation of handling two stochasticfactors. For example, in stochastic volatility models, the risky price and the stochastic volatility are two factors;in cross-currency derivatives, the evolution of two currencies are driven by different stochastic factors; in two-asset or multi-asset derivatives, the price movements may be modeled by two stochastic processes, etc. In theseproblems, modeling the stochastic factors by two Brownian motions has been a common-used method, see, amongothers, Heston 1993,Dai et al. 2004 and Hurd and Zhou 2010. In most situations, from a practical aspect, the twostochastic factors (hence the two Brownian motions) should be correlated to each other. And empirical researcheshas indicated that their dependence changed over time and depending on the market conditions, e.g., [2] for cross-currency derivatives, Engle and Sheppard 2001 for multi-asset and Benhamou et al. 2010 for stochastic volatilitymodels.In the previous two sections, we considered the common decomposition of two Brownian motions, whosedependence structure could be very general. In Section 2, we showed how to decompose Brownian motions ( B , W ) to a triplet ( X , Y , T ) , and in Section 3 we answered how to construct two correlated Brownian motions froma given triplet ( X , Y , T ) . In this section, we will apply the common decomposition method to study the pricing ofsome typical two-factor derivatives that modeled by two correlated Brownian motions. We first give two examplesshowing direct usage of the decomposing triplet ( X , Y , T ) in deriving pricing formula. And then we will focus onthe pricing of two-color rainbow options, because a wide variety of contingent claims have a payoff function whichincludes two risky assets. There are several examples for two-color rainbow options, one is given by option-bonds,see Stulz 1982 for details; besides, a special kind of two-color rainbow options, spread options, are ubiquitous infinancial markets, including equity, fixed income, foreign exchange, commodities and energy markets, Carmonaand Durrleman 2003 present a overview of examples and common features of spread options.For simplicity, we assume that X , Y and T are mutually independent in this section, i.e., ρ is independent from( B , Z ) in the local correlation model by Proposition 2.9. This assumption is not so rigorous as to go against the Note that we do not need to simulate the trajectory here. ρ , B and Z from an empirical view. Options dependent on exchange rate movements, such as those paying in a currency different from the underlyingcurrency, have an exposure to movements of the correlation between the asset and the exchange rate. This riskmay be eliminated by two ways, a straightforward approach is Quanto option and will be discussed in Section 4.2;the other approach that we focus on this section is
Covariance Options or Correlation Options , see Swishchuk 2016 formore details. By combining variance and covariance options, the realised variance of return on a portfolio can belocked in. Carr and Madan 1999 illustrated that the covariance swaps can be constructed by options and futures, inother words, options can be perfectly hedged by covariance swaps and futures. In the following part, we considerthe so called covariance options which is designed to cope with the covariance risks of two underlying assets.Suppose that the prices of the two assets, ( S , S ) , can be characterized as dS t S t = µ dt + σ dB t , dS t S t = µ dt + σ dW t . (19)where the drifts and volatilities of underlying prices are assumed to be constant. Example 4.1 (Swap and Option on Realized Covariance of Returns) . Consider two risky assets whose prices evolve asin (19). Then according to Example 2.6, ( S , S ) could be transformed to, under proper conditions,dS t S t = rdt + σ d ˜ B t , dS t S t = rdt + σ d ˜ W t , where ˜ B and ˜ W are Brownian motions under the risk neutral measure Q, and r denotes the constant risk free interest rate.Continuously compounded rate returns of two assets are ln S t / S and ln S t / S . Accordingly, define the realized covari-ance of returns of two underlying assets as the quadratic covariation of ln S t / S and ln S t / S Cov R ( S t , S t ) (cid:44) [ ln S S , ln S S ] t , then the payoff of covariance swap and covariance option of the underlying equity S and S at expiration isCov R ( S t , S t ) − K , and max { Cov R ( S t , S t ) − K , 0 } , where K represent the strike price. SoCov R ( S t , S t ) = [ ln S S , ln S S ] t = (cid:90) t σ σ d [ B , W ] t = σ σ ( T t − S t ) = σ σ ( T t − t ) , the price of covariance swap and covariance option only depend on the expectation and distribution of T t . The financialderivatives correlation swap and correlation option is similar. .2 Pricing of Quanto Option Quanto option is a famous cross-currency financial product trading in organized exchanges as well as in OTC.Its payoff is calculated in one currency but is settled in another currency at a fixed exchange rate. It is designedto hedge the risks of delivering foreign investments to domestic currency. Hence the correlation between theunderlying price and the exchange rate plays an ultimate role in pricing. Usually, this correlation structure ismodeled by two correlated Brownian motons. In Section 2, we have showed that part of the dependency of twoBrownian motions could be described by T in common decomposition. In the following example, we will showthe essential role of T in the pricing of an European-style Quanto. Example 4.2.
Let us consider an European-style Quanto. Suppose the price of underlying equity S in foreign currency andthe exchange rate R are modeled, under the risk neutral probability in the domestic currency, as follows:dS t = µ S t dt + σ S t dB t , dR t = µ R t dt + σ R t dW t , the payoff of a Quanto put option is R max ( K − S t ) . Under the arbitrage free assumption in domestic currency world, let r , r represent the risk free interest under domesticcurrency and foreign currency respectively, one can getR = exp ( − r t ) E [ exp ( r t ) R t ] , (20) S R = exp ( − r t ) E [ S t R t ] . (21)(20) represent the bank account in foreign currency. Note thatE [ R t ] = R exp ( µ t ) , E [ S t R t ] = S R exp (cid:18) ( µ + µ − σ − σ ) t (cid:19) E [ exp ( σ B t + σ W t )] , and under the condition (C3), we haveE [ exp ( σ B t + σ W t )] = exp (cid:18) ( σ + σ ) t (cid:19) E (cid:20) exp (cid:18) σ σ (cid:90) t ρ u du (cid:19)(cid:21) . After simple calculations, µ = r − r , µ = r − µ − t ln E (cid:20) exp ( σ σ (cid:90) t ρ u du ) (cid:21) . According to Van Emmerich 2006 and Teng et al. 2016, Quantos’ price is a function of ln E (cid:104) exp ( σ σ (cid:82) t ρ u du ) (cid:105) ,P Quanto = R (cid:18) Ke − r t N ( − d ) − S e − ( r t − r t + ln E (cid:104) exp ( σ σ (cid:82) t ρ u du ) (cid:105) ) N ( − d ) (cid:19) , where d = log ( S / K ) + ( r + σ /2 ) t − ln E (cid:104) exp ( σ σ (cid:82) t ρ u du ) (cid:105) σ √ t , d = d − σ √ t . Note that ln E (cid:20) exp ( σ σ (cid:90) t ρ u du ) (cid:21) = ln E [ exp ( σ σ ( T t − t ))] , then Quantos’ price is actually a function of T t and determined by Laplace transform of T t . .3 Pricing of 2-Color Rainbow Options In this section, we focus on a class of multi-asset options, the 2-color rainbow option which is written on themaximum or minimum of two risky assets. This kind of option was first studied in Margrabe 1978, and the mostwell known Stulz 1982 where the author showed its extensive applications in valuing many financial instrumentssuch as foreign currency bonds, option-bonds, risk-sharing contracts in corporate finance, secured debt, etc.In this part we use the same asset-price models as in Section 4.1.The payoff of a rainbow option with maturity τ may have the forms listed in table 4 (see Ouwehand and West2006). We will demonstrate that all these types of rainbow options could be valuated through a unified approach.Table 4: Types of rainbow optionOption Style Payoff Best of assets or cash max ( S τ , S τ , K ) Put 2 and Call 1 max ( S τ − S τ , 0 ) Call on max max ( max ( S τ , S τ ) − K , 0 ) Call on min max ( min ( S τ , S τ ) − K , 0 ) Put on max max ( K − max ( S τ , S τ ) , 0 ) Put on min max ( K − min ( S τ , S τ ) , 0 ) Define a 2-dimensional process M t = ( X T t , Y S t ) (cid:62) . Similar to the cases studied in Carr and Wu 2004, the payoffsin Table 4 could be reformulated as ( a + b e θ (cid:62) M τ ) { c (cid:62) M τ ≤ k } { c (cid:62) M τ ≤ k } + ( a + b e θ (cid:62) M τ ) { c (cid:62) M τ ≤ k } { c (cid:62) M τ ≥ k } ,with some proper parameters.For example, consider the Call-on-max option, whose payoff is max ( max ( S τ , S τ ) − K , 0 ) , the parameters are(for i =
1, 2) a i = − K , b i = S i e ( r − σ i ) τ , θ = (cid:18) σ σ (cid:19) , θ = (cid:18) σ − σ (cid:19) , c i = − θ i , k i = − ln Kb i , c = θ − θ , k = ln b b .It is easy to check that { c (cid:62) i M τ ≤ k i } = { S i τ ≥ K } , { c (cid:62) M τ ≤ k } = { S τ ≥ S τ } .Now we can present a unified valuation approach for options with payoffs in Table 4 trough priocess M . First,for given parameters γ , γ ∈ R , γ , γ , γ ∈ R , define an intermediate valuation function G : R → R as G ( x , x ; γ , γ , γ , γ , γ ) (cid:44) E Q (cid:104) ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } { γ (cid:62) M τ ≤ x } (cid:105) , (22)where E Q indicates the expectation under the risk-neutral measure Q . It is obvious that the initial price of arainbow option could be given by G as e − r τ (cid:104) G ( k , k ; a , b , θ , c , c ) + G ( k , − k ; a , b , θ , c , − c ) (cid:105) . (23)The following proposition gives a general rule to calculate function G .18 roposition 4.3. Let G ( x , x ) , ( x , x ) ∈ R be given as in (22) , and L t represents the fourier transform of T t . Define Φ M τ ( z , z ) = e − tz L t ( − ( z − z )) , (24) then the generalized fourier transform of G ( x , x ) , denoted by ˆ G ( λ , λ ) , is given as ˆ G ( λ , λ ) = − γ λλ Φ M τ ( λ γ + λ γ ) − γ λλ Φ M τ ( λ γ + λ γ − i γ ) , (25) where Im λ , Im λ > . In particular, if ρ t = ρ is a constant, then L t ( z ) = exp ( + ρ tz ) and ˆ G ( λ , λ ) can be obtained from (24) and (25) . Given Proposition 4.3, the function G ( x , x ; γ , γ , γ , γ , γ ) could be calculated by the inversion formula andthen the prices of rainbow options are obtained from (23). Remark 4.4.
For general cases where the payoffs can not be represented as before, Proposition 4.3 no longer applies. But wecan still apply the Fourier-transform method directly to pricing functionals. For given parameters ( S , τ , r , σ , σ ) , rewritethe option payoffs as V ( y + B τ , y + W τ ) , where y i = ( r σ i − σ i ) τ , i =
1, 2 . Denote by f ( b , w ) the joint probability densityof B τ and W τ under Q, then the price of V ( y + B τ , y + W τ ) isC ( y , y ) = (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ V ( y + b , y + w ) f ( b , w ) dbdw . According to Leentvaar and Oosterlee 2008, the Fourier transform of C ( y , y ) is ˆ C ( λ , λ ) = (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ e i λ y + i λ y V ( y + b , y + w ) f ( b , w ) dbdwdy dy = F V ( λ , λ ) E Q (cid:104) e − i λ B τ − i λ W τ (cid:105) where F V denotes the Fourier transform of V. In general, F V has no explicit expression and thus usually be calculatednumerically.Leentvaar and Oosterlee 2008 have put forward a numerical method for the cases when the correlation coefficient of B andW is constant. In fact, if this is the case, E Q (cid:2) e − i λ B τ − i λ W τ (cid:3) could be calculated explicitly,E Q (cid:104) e − i λ B τ − i λ W τ (cid:105) = exp (cid:16) − ( λ + λ + ρλ λ ) τ (cid:17) . When the correlation coefficient of B and W is not constant, we can still use similar approaches as in Leentvaar andOosterlee 2008 by means of common decomposition. Continuing to use the notions as before, we haveE Q (cid:104) e − i λ B τ − i λ W τ (cid:105) = Φ M τ ( − λ − λ , − λ + λ ) = e − ( λ − λ ) τ L τ ( − λ λ ) , Consequently, ˆ C ( λ , λ ) = F V ( λ , λ ) e − ( λ − λ ) τ L τ ( − λ λ ) . (26) Hence when the Fourier transform of T t , L t , is known, the price will be obtained by inverse Fourier transform formula. For simplicity, we omit the parameters γ i , i =
1, . . . , 5, in the function expressions when there is no confusion.
19n the last few pages, we considered how to calculate the price of a rainbow option. Actually, following similarapproach outlined in Proposition 4.3, we could give a Fourier-transform method for calculating Greeks. The nextcorollary set forth an example of this.
Corollary 4.5.
Consider the Delta of S for a Call-on-Max option listed in Table 4 ∆ ( S ) = S (cid:18) ∂ G ∂ x ( k , k ; a , b , θ , c , c ) + ∂ G ∂ x ( k , k ; a , b , θ , c , c ) − ∂ G ∂ x ( k , − k ; a , b , θ , c , − c ) (cid:19) + e ( r − σ ) τ ∂ G ∂γ ( k , k ; a , b , θ , c , c ) : = g ( k , k ) + g ( k , k ) , where g ( k , k ) = (cid:32) S ( ∂ G ∂ x + ∂ G ∂ x ) + e ( r − σ ) τ ∂ G ∂γ (cid:33) ( k , k ; a , b , θ , c , c ) , g ( k , k ) = (cid:32) − S ∂ G ∂ x (cid:33) ( k , k ; a , b , θ , c , c ) . The fourier transform of g has an explicit expression asia S ( λ + λ ) Φ M τ ( λ c + λ c ) + ( ib S λ − e ( r − σ ) τ λλ ) Φ M τ ( λ c + λ c − i θ ) , and the expression of Fourier transform of g is − ia S λ Φ M τ ( λ c − λ c ) − ib S λ Φ M τ ( λ c − λ c − i θ ) . And Delta will be obtained by the inverse Fourier transform formula.
Remark 4.6.
Other Greeks can be derived along the same procedures.
From the foregoing content of this section, we know that, thanks to the common decomposition method, tocalculate the price and Greeks of a rainbow option, we only need to find out the Fourier transform of T t . Weconsider some specific models of T t in the following examples to give the readers more intuitive insights.The first example is under the regime switching model, which is widely used in financial modelling. Example 4.7.
Consider the regime switching model given in Example 3.3, by Lemma A.1 in Buffington and Elliott 2002, theFourier trans form of T t is L t ( z ) = Ee zT t = (cid:62) e ( A + z diag ω ) t Q , where diag ω = ω · · · ω · · · ... ... . . . ... · · · ω n , A = ( a ij ) n × n is the generator of Q t . Then by Proposition 4.3, we can get the optionprice from L t ( z ) . For example, if the option style is Call-on-max, then ˆ G ( λ , λ ; a , b , θ , c , c ) = K λλ e − ( λ σ − λσ ) τ (cid:62) e ( A − λ λσ σ diag ω ) τ Q − S λλ e ( r − σ − ( λ σ + i σ − λσ ) ) τ (cid:62) e ( A − ( λ + i ) λσ σ diag ω ) τ Q .20n the next example, { T t } t ≥ has a specific modelling through a bounded function of some stochastic processesand the Fourier transform of T t is given by a PDE. Example 4.8.
Suppose that h is a bounded function with values in [
0, 1 ] and ν is a diffusion process satisfying the followingSDE d ν t = µ ( t , ν t ) dt + σ ( t , ν t ) dZ t , where { Z t } t ≥ is a Brownian motion and µ ( t , x ) , σ ( t , x ) are determined functions such that the SDE have an unique solution.Let T t = (cid:82) t h ( ν s ) ds. By Feynman-Kac formula, the Laplace transform of T t − T s for fixed t under the condition ν s , whichis denoted by L ( s , ν s ; t , z ) , satisfies the following PDE: ∂ L ∂ s + µ ( t , ν ) ∂ L ∂ν + σ ( t , ν ) ∂ L ∂ν + z f ( ν ) L =
0, (27) with terminal condition L ( t , ν t ; t , z ) = The solution of (27) are related with Sturm-Liouville problem, see Polyanin 20021.8.6.5 and 1.8.9 for more details.Particularly, Teng et al. 2016 considered f ( x ) = + tanh ( x ) for modelling stochastic correlation. Ma 2009 discussedf ( x ) = + x and ν t as a bounded Jacobi processd ν t = κ ( θ − ν t ) dt + σ ν (cid:113) ( h − ν t )( ν t − l ) dZ t , the boundary for bounded Jacobi process is [ l , h ] when κ ( θ − l ) > σ ν ( h − l ) , κ ( h − θ ) > σ ν ( h − l ) .Sometimes, there is no closed-form solution of financial derivatives, so Monte Carlo method is needed. Andwe have illustrated simulation steps in Section 3. In literatures that study the pricing of two-asset derivatives with models driven by two Brownian motions, B and W ,it is a commonly used assumption that the local correlation of B and W is a constant, i.e., d [ B , W ] t = ρ dt for some ρ ∈ R . However, as we have mentioned before, this assumption is inconsistent with empirical studies. For ex-ample, based on data from different markets around the world, Chiang et al. 2007, Syllignakis and Kouretas 2011and Junior and Franca 2012 all found that the correlation coefficients changed as time and economic situationschanged. Then it is natural to ask, when the actual correlation coefficient is dynamic and stochastic, how much itwould influent the pricing error if we still applied the constant-correlation model?In this part, we consider as an example the pricing of two-color rainbow options. We investigate the differ-ence of option prices under constant and dynamic stochastic correlations by numerical experiments and try tosummarize when this difference is negligible or nonnegligible.Regime switching model is widely considered in financial modelling, thus we apply the regime switchingmodel in this section which has been introduced in Example 3.3 and 4.7. Suppose that the market has threedifferent states described by a finite-state-space Markov process { Q t } t ≥ with an initial value Q and a transitionrate matrix A that A = − − − . (28)21ince our concern is in correlations, we assume for simplicity that all coefficients of the underlying assets, exceptfor the local correlation, are constants. Thus the underlying prices are assumed to satisfy (under the risk neutralprobability) dS t S t = rdt + σ dB t , dS t S t = rdt + σ dW t ,where r = S = S = σ = σ = B and W , ρ t = ω (cid:62) Q t − ω ∈ R + which indicates the switching states for local correlation coefficient of log prices, d [ log S , log S ] t / dt σ σ : = ρ t . For example, if ω = [ ] (cid:62) , at any time t , ρ t switches among − ρ is consideredas a constant, the option prices can be given in closed form as in Stulz 1982. While for the actual case with aregime-switching ρ , we can apply Proposition 4.3 to derive the true prices. Following the notations in Proposition4.3, by the inversion fourier formula, we have G ( x , x ) = (cid:90) ∞ + i λ i − ∞ + i λ i (cid:90) ∞ + i λ i − ∞ + i λ i e − i λ x − i λ x ˆ G ( λ , λ ) d λ d λ , (29)where λ i , λ i denote the imaginary part of λ and λ .In the subsequent numerical experiment, we choose λ i = λ i =
1, since ˆ G ( λ , λ ) is well defined only for λ , λ with strictly positive imaginary. And we approximate (29) by G ( x , x ) ≈ N ∑ j = − N N ∑ k = − N e λ i x + λ i x − i ( j η x + k η x ) ˆ G ( j η + i λ i , k η + i λ i ) η η ,where we set N = N = η = η = τ = K =
90. Let ω = [ ] (cid:62) , then theregime switching model degenerates to the constant correlation model. We verified the group of parameters areaccurate enough and the difference of option price obtained from Stulz 1982 and Proposition 4.3 is smaller than10 − .In the following two sections, 5.1 and 5.2, we compare the option prices induced by the constant- ρ models inStulz 1982 to the prices given by the regime-switching- ρ models (through (23)). Since we have assumed the regime-switching case to be actual, the latter could be regarded as the “true” prices. And thus the comparison results willindicate how large the pricing error would be when we substituted a constant for the original nonconstant ρ . Forclarity, we make comparison in an ideal situation that the investor knows exactly the other coefficients except for ρ . In Section 5.1, we compare the two cases in a more theoretical way. We assume that the investor estimates ρ historically from the observed stock prices. The numerical results in this section show that there may be bigdifferences between the prices. In Section 5.2, we adopt an approach more close to the practical procedure. Wesuppose the investor calibrate the constant correlation model to option prices he observed (which were calculatedfrom the regime-switching model). And then the calibrated model is used for pricing. And it shows that there willbe a big pricing error by using constant correlation model, especially for those options deep out of the money. Thisis in line with the results given in Costin et al. 2016 for CDS options. In empirical, the risk free interest r can be observed and σ , σ can be calibrated precisely from vanilla options. .1 Constant and Nonconstant Correlation in Pricing Rainbow Options In this section, we estimate a constant correlation coefficient ˆ ρ from the historical data which are given by theregime switching model, and then calculate the option prices derived from this ˆ ρ . By comparing these optionprices with those deriving directly from the regime switching model, we can get a general idea of the error wewould make when applying constant correlation model in the situations where the actual correlation coefficientsare dynamic and stochastic. For the robustness of the results, we consider the comparisons in 5 different caseswith different vector ω s.Since we have assumed that all the other parameters can be obtained precisely, the investor actually could getthe data of ( B , W ) by observing prices of the underlying assets. Suppose that he has got these historical data ofa long term and with a relatively high frequency as ( B t i , W t i ) , i =
0, 1, . . . , n , where 0 = t < t < . . . , < t n = t .According to definition, the estimated constant correlation based on data till time t isˆ ρ (cid:44) ∑ n − i = ∆ B t i ∆ W t i t .Note that, setting ∆ t = max { t i + − t i | i =
0, . . . , n } , we have ∑ n − i = ∆ B t i ∆ W t i t P −−−→ ∆ t → [ B , W ] t t = T t − S t t = t (cid:90) t ( ω − ) (cid:62) Q s ds ,and according to the Ergodic Theorem of Markov processes,lim t → ∞ t (cid:90) t Q s ds = π ,where π denotes the stationary distribution of Markov process Q t .Thus, as long as we assume these data to be long-term and with a relatively high frequency, we always haveˆ ρ ≈ t (cid:90) t ( ω − ) (cid:62) Q s ds ≈ ω (cid:62) π − . (30)In this case, no matter how violently the correlation coefficient switches over time, the investors may havesimilar estimates from long-term historical data. And thus the option prices calculated along these estimates maydeviate a lot from the “true” prices. We will show these price deviations by the relative error defined asRelative error = Price with constant ˆ ρ − Price with regime switching ρ Price with regime switching ρ . (31)In the numerical experiments, for each case, we simulate a path of ( B , W ) to present the historical data, wherewe choose t =
20 and ∆ t i = ∀ i . In order to make consistent comparison, we randomly choose 5 different ω , which all satisfy the condition 2 ω (cid:62) π − = We have illustrated in Remark 2.7 that it is feasible to apply directly the estimated ˆ ρ from historical data into option pricing. Note that the stationary distribution π satisfies the following equations A (cid:62) π = π (cid:62) = A denotes the generator of Q . In our numerical experiments, A = − − − , and then π = [ ] (cid:62) . ρ ≈ ρ estimated from the ”historical data”, the forthcolumn shows the prices obtained by constant correlation model with ˆ ρ , while the last column shows the relativeerrors defined as in (31). Table 5: Option pricing with all history data ω True Prices ˆ ρ Prices with ˆ ρ Relative errors [ ] (cid:62) [ ] (cid:62) [ ] (cid:62) [ ] (cid:62) [ ] (cid:62) It is obviously from Table 5 that there may be big pricing errors when using constant correlation coefficientestimated from historical data. In this numerical example, although all the other coefficients were assumed toinduce zero error, the relative errors for pricing can mount to unacceptable levels. It is almost certain that thesehigh errors come from the substitution of ˆ ρ s for the real dynamic stochastic ρ s. As a verification, we considered thecase of ω = [ ] (cid:62) , where the regime switching model degenerates to the constant correlation model. Theresults were shown in the last row of the table. We can see that there is only a small relative error, 0.19%, whichpresents the technical error other than substitution of constant correlations to dynamic ones.More specifically, we can see that in all cases the estimated ˆ ρ s are around 0.2, and thus the resulting optionprices are around 35.3, while the true prices deviate from as high as 38.2 to as low as 33.8. There would be a bigunexpected loss if the investor applied the constant correlation model to value these options and used these pricesas a guidance of his investments. In this section, we investigate the difference between option prices under constant correlation model and dynamicstochastic correlation model through a more practical way. First, in practice, when considering derivatives’ pricing,investors do not use coefficients estimated from historical data commonly. More often, they observe the marketprices of a class of derivatives, and calibrate the theoretical model to the observed prices. In our case, the ”marketprices” are supposed to be given by the regime switching model, and the ” theoretical model” held by investorsis supposed to be the constant correlation model. And ”calibration of the theoretical model” reduces to ” findingthe optimum ρ to fit the market prices” since this is assumed to be the only unknown parameter for the theoreticalmodel. On the other hand, just like the idea of ”implied volatility”, each observed option price can deduce an” implied correlation ”, ρ imp . The change of ρ imp with strikes can also indicate the deviation of option prices given byconstant correlation model from actual prices based on dynamic correlation.The numerical simulations are carried out along the following procedure.First, we give the prices for options with a maturity τ = K =
80, 90, . . . , 140 under regimeswitching model by the Fourier transform method. These will play the part of ”initial market data” in our numer-ical experiment. 24hen based on these data, we calibrate the constant correlation model to a proper ρ . This is done by mini-mizing the following cumulative square error function by Gradient Descent method, L ( ρ ) = ∑ n (cid:18) Price constant n ( ρ ) − Price dynamic n (cid:19) .And then, the calibrated correlation coefficients are applied to the constant correlation model for pricing op-tions with strikes K =
82, 84, . . . , 88, 92, 94, . . . , 98, . . . , 132, 134, . . . , 138. The resulting prices will be compared withthe prices under regime switching model.To see the variations of implied correlation, we apply the definition of ρ imp given by Da Fonseca et al. 2007which satisfies Price = Price constant ( ρ imp ) ,to the prices given by regime switching models with more strikes K =
80, 82, 84, . . . , 140.In the following, We run through the calibrating-pricing procedure for Call on Min, Call on Max, Put on Maxand Put on Min options, consider their relative errors defined as in (31), and calculate the implied correlationsrespectively. We show the results in Figures 2-5. In each figure, the dotted line separates the curve into two parts,the out-of-the-money cases (in figures, the left part for puts or the right for calls) and the in-the-money cases. Theintersection is at-the-money case. (a) Relative error (Calibrated ρ = − Figure 2: Call on Min option with Q = [
1, 0, 0 ] (cid:62) , ω = [ ] (cid:62) On the first try, we choose parameters Q = [
1, 0, 0 ] (cid:62) and ω = [ ] (cid:62) to generate the regime switchingmodel. The immediate observation is the huge pricing error for deep-out-of-the-money options of Put on Max andCall on Min. The relative error reaches more than 70%, which are shown in Figure 2(a) and 4(a). While for Call onMax option, the relative error is no more than 0.1%, as shown in Figure 3(a). And it is also small for Put on Minoption whose figure is omitted here since the relative error always lies below the level 0.5%. Just as before, all the other coefficients are supposed to be known exactly. The initial value is taken as ρ =
0. The step size is set as | L (cid:48) ( ) | where L (cid:48) denotes the first derivative of L . The gradient descentmethod terminates when | L (cid:48) ( ρ ) | is smaller than 10 − . a) Relative error (Calibrated ρ = − Figure 3: Call on Max option with Q = [
1, 0, 0 ] (cid:62) , ω = [ ] (cid:62) (a) Relative error (Calibrated ρ = − Figure 4: Put on Max option with Q = [
1, 0, 0 ] (cid:62) , ω = [ ] (cid:62) a) Relative error (Calibrated ρ = Figure 5: Put on Min option with Q = [ ] (cid:62) , ω = [ ] (cid:62) To see whether this is a common property or not, we change the initial regime switching model to a new onewith parameters Q = [ ] (cid:62) and ω = [ ] (cid:62) , and repeat the calibrating-pricing procedure. For Callon Max, Call on Min and Put on Max, the results are really similar with the previous group of parameters and weomit the figures. But for Put on Min, the result is different from former one, relative error could be more than 10%for out-of-the-money options as shown in Figure 5 which is also nonnegligible. We will try to give a reasonableexplanation for this in the next section, Section 5.3. And also there in addition, we will explain why the calibratedoption prices perform well for in-the-money and at-the-money options but terribly bad for deep-out-of-the-moneyoptions and why the Call on Max seems different from the other options.On the other side, for implied correlation, we can see in Figure 2(b)-5(b), the implied correlation always changessharply for out-of-the money cases and mildly for in-the-money cases, which is similar with the calibrated ρ . ForCall on Max options, though there are only tiny pricing errors, the implied correlations change a lot with differentstrikes.Figure 6 investigate Put on Max options again and the maturity considered as τ = The pricing errors coming from setting the dynamic stochastic correlation of underlying log prices to be constant,as shown in the last section, are further analyzed in this part. This analysis is from a theoretical view but with thehelp of numerical simulations. Through this analysis, we try to explain the phenomenon discovered in Section 5.2.Now we consider options with payoffs V ( S τ , S τ , τ , K ) , then the price of the option is E Q (cid:104) e − r τ V ( S τ , S τ , τ , K ) (cid:105) = E Q (cid:104) e − r τ V (cid:16) S e ( r − σ ) τ + σ B τ , S e ( r − σ ) τ + σ W τ , τ , K (cid:17)(cid:105) ,where Q denotes the risk neutral probability measure. Note that all the payoffs considered in previous numerical simulations are in this way a) Relative error (Calibrated ρ = − Figure 6: Put on Max option with τ = ρ , since ( B τ , W τ ) ∼ N (cid:18) (
0, 0 ) , τ (cid:18) ρρ (cid:19)(cid:19) , the option price is afunction of ρ which will be denoted as Price c ( ρ ) in the following.For more general case where ρ is a stochastic process, we first recall the term of average correlation coefficient¯ ρ t = t (cid:82) t ρ u du , which by common decomposition, can be rewritten as¯ ρ t = T t − S t t . (32)Since under the condition of F T τ , ( B τ , W τ ) = ( X T τ + Y S τ , X T τ − Y S τ ) ∼ N (cid:18) (
0, 0 ) , τ (cid:18) ρ τ ¯ ρ τ (cid:19)(cid:19) , following thediscussions in the constant- ρ case, the option price ( denoted by Price d ) equals Price d = E Q (cid:104) E Q [ e − r τ V ( S τ , S τ , τ , K ) |F T τ ] (cid:105) = E Q [ Price c ( ¯ ρ τ )] .If Price c is an affine function of ρ , i.e., ∃ a , b ∈ R , Price c ( ρ ) = a ρ + b , we have Price d = E Q [ Price c ( ¯ ρ τ )] = E Q [ a ¯ ρ τ + b ] = Price c ( E Q [ ¯ ρ τ ]) . (33)In other words, when the option price under constant- ρ model is linear in ρ , the price under a general dynamiccorrelation model is exactly the same as that with a constant correlation coeeficient E Q [ ¯ ρ τ ] .Otherwise, for general Price c , by Taylor’s expansion, we can get the following approximation formula, Price d = E Q [ Price c ( ¯ ρ τ )] ≈ Price c ( E Q [ ¯ ρ τ ]) +
12 Var Q ( ¯ ρ τ ) ∂ Price c ∂ρ ( E Q [ ¯ ρ τ ]) . (34)In the following, based on the above analysis, we try to explore causes for the two phenomena found in theformer section: 1) the pricing errors seem more remarkable for out-the-money options when applying constantcorrelation model; 2) the pricing errors for Call-on-Max options seem relatively small than other kind of options.28e first consider relations between Price c ( ρ ) and ρ in the cases of in-the-money, at-the-money and out-of-the-money for Put-on-Max options. Choosing parameters as r = τ = S = S = σ = σ = Price c ( ρ ) when Strike =
150 (in the money), Strike =
120 (at the money) and Strike = Price c ( ρ ) reveals a strong linearity on ρ except for when ρ is near to 1. But it is quite nonlinear for out the moneycase. We conduct similar diagraming with different parameters for Put-on-Max option as well as Put-on-Min,Call-on-Min and Call-on-Max options, and get similar results. Recall the approximations (33) and (34), the aboveresults give an explanation for why constant correlation model performs well on the whole for in-the-money andat-the-money options but poorly for out-of-the-money options. (a) Strike =
150 (b) Strike =
120 (c) Strike = Figure 7: Put on max option price in constant correlation modelTable 6: Expectation of ¯ ρ τ τ = τ = ω = [ ] , Q = [
1, 0, 0 ] -0.3177 -0.2488 ω = [ ] , Q = [ ] E ¯ ρ τ ; on the contrary, when strike is out-of-the-money, the impliedcorrelation changes sharply and far away from E ¯ ρ τ . This is coincident with the conclusion in previous and (33),that is why calibrating constant correlation model into dynamic correlation model perform well when option isin-the-money and at-the-money.We now turn to the Call-on-Max option whose performance in calibration in Section 5.2 seemed quite differentfrom the others that the calibrated constant correlation model always performs well, even for out-the-money case.Note that as mentioned before, we have already got diagrams for this kind of option which have similar linear ornonlinear shapes like other options and we did not include them in the main text. A interesting question is, nowthat the shape of Price c ( ρ ) for out-the-money case looks apparently nonlinear, why does it still approximate thetrue price well? We choose the same parameters as before except for τ = Price c ( ρ ) for Call-on-Max option for the case Strike =
130 (out-of-the-money) in Figure 8. The diagram looks still quitenonlinear, but it is worth noting that in the fifure
Price c ( ρ ) just changes from 3.97 to 3.995. In other words, when ρ changes in its full range, the price changes only about 0.6% which implies that, for Call-on-max option, thecorrelation between underlying assets has only a small, almost negligible, impact on the option price. While onthe contrary, think about calibrating ρ from option prices, a small deviation in the price may cause great changes29igure 8: Call on max option price in constant correlation model with K = τ = ρ . This result on one hand explains why the implied correlation of Call-on-Max option is volatilebut the calibrated constant correlation model always performs well and on the other indicates that when the dataare from out-of-the-money Call-on-Max options, correlation-coefficient calibrating may be unsuitable since theimplied correlation is too sensitive with the price. Proof of Theorem 2.3.
First note that B + W and B − W are continuous martingales and (cid:20) B + W B − W (cid:21) t = ([ B ] t − [ W ] t ) = τ and ς , τ t = inf (cid:26) u : (cid:20) B + W (cid:21) u > t (cid:27) , ς t = inf (cid:26) u : (cid:20) B − W (cid:21) u > t (cid:27) .Then according to Revuz and Yor 2013[Chapter V, Theorem 1.10], { X t } t ≥ and { Y t } t ≥ are two independent Brow-nian motions.Before going further, we first prove three lemmas as preparations. Lemma 6.1.
Suppose the conditions in Theorem 2.3 and the following condition (E) hold, then { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ in the common decomposition are mutually independent. (E) For any F T -progressively measurable processes { φ t } t ≥ and { φ t } t ≥ that guarantee E (cid:20) exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) < ∞ ,30e have E (cid:20) exp (cid:18) (cid:90) ∞ φ u dX T u + (cid:90) ∞ φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19) . Proof.
For ∀ n , m ∈ N , and 0 = t < t < · · · < t n ; 0 = s < s < · · · < s m ; we consider the joint distribution of { X t , . . . , X t n , Y s , . . . , Y t m } conditional on F T ∞ by calculating E (cid:34) exp ( n ∑ i = θ i ( X t i − X t i − ) + m ∑ j = θ j ( Y s j − Y s j − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:35) (35)where θ i , θ j ∈ R , i =
1, 2, . . . , n ; j =
1, 2, . . . , m .Define Φ u = n ∑ i = θ i { t i − ≤ u < t i } , Φ u = m ∑ j = θ j { s j − ≤ u < s j } ,it is easy to verify (cid:82) ∞ Φ u dX u = ∑ ni = θ i ( X t i − X t i − ) , (cid:82) ∞ Φ u dY u = ∑ mj = θ j ( Y s j − Y s j − ) and E (cid:20) exp (cid:18) (cid:90) ∞ ( Φ u ) dT u + (cid:90) ∞ ( Φ u ) dS u (cid:19)(cid:21) < ∞ .By definitions of X and Y (for simplicity, we set (cid:82) ∞ T ∞ Φ u dX u = T ∞ = ∞ .), (cid:90) ∞ Φ u dX u = (cid:90) T ∞ Φ u dX u + (cid:90) ∞ T ∞ Φ u dX u = (cid:90) ∞ Φ T u dX T u + (cid:90) ∞ Φ u + T ∞ d ˜ X u , (cid:90) ∞ Φ u dY u = (cid:90) S ∞ Φ u dY u + (cid:90) ∞ S ∞ Φ u dY u = (cid:90) ∞ Φ S u dY S u + (cid:90) ∞ Φ u + S ∞ d ˜ Y u .Thus E (cid:20) exp (cid:18) (cid:90) ∞ Φ u dX u + (cid:90) ∞ Φ u dY u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) ∞ Φ T u dX T u + (cid:90) ∞ Φ u + T ∞ d ˜ X u + (cid:90) ∞ Φ S u dY S u + (cid:90) ∞ Φ u + S ∞ d ˜ Y u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = E (cid:20) E (cid:104) exp (cid:18) (cid:90) ∞ Φ T u dX T u + (cid:90) ∞ Φ u + T ∞ d ˜ X u + (cid:90) ∞ Φ S u dY S u + (cid:90) ∞ Φ u + S ∞ d ˜ Y u (cid:19) (cid:12)(cid:12)(cid:12) F X T ∞ (cid:95) F Y S ∞ (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) ∞ Φ T u dX T u + (cid:90) ∞ Φ S u dY S u (cid:19) E (cid:104) exp ( (cid:90) ∞ Φ u + T ∞ d ˜ X u + (cid:90) ∞ Φ u + S ∞ d ˜ Y u ) (cid:12)(cid:12)(cid:12) F X T ∞ (cid:95) F Y S ∞ (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) . (36)Note that F T ∞ ⊂ F X T ∞ (cid:87) F Y S ∞ , ˜ X and ˜ Y are martingales with respect to F ˜ Xt (cid:87) F ˜ Yt , F ˜ X ∞ and F ˜ Y ∞ are independentwith respect to F X T ∞ , F Y S ∞ and F T ∞ , so exp (cid:16) (cid:82) t Φ u + T ∞ d ˜ X u + (cid:82) t Φ u + S ∞ d ˜ Y u − (cid:82) t ( Φ u + T ∞ ) du − (cid:82) t ( Φ u + S ∞ ) du (cid:17) is a local martingale with F T ∞ (cid:87) F X T ∞ (cid:87) F Y S ∞ (cid:87) F ˜ Xt (cid:87) F ˜ Yt by It ˆo’s lemma. Since any positive local martingale is asupermartingale, and according to Karatzas and Shreve 2012[Chapter 3, Proposition 5.12], we have E (cid:20) exp (cid:18) (cid:90) t Φ u + T ∞ d ˜ X u + (cid:90) t Φ u + S ∞ d ˜ Y u − (cid:90) t ( Φ u + T ∞ ) du − (cid:90) t ( Φ u + S ∞ ) du (cid:19)(cid:21) = ∀ t (cid:110) exp (cid:16) (cid:82) t Φ u + T ∞ d ˜ X u + (cid:82) t Φ u + S ∞ d ˜ Y u − (cid:82) t ( Φ u + T ∞ ) du − (cid:82) t ( Φ u + S ∞ ) du (cid:17)(cid:111) t ≥ is a martingale. Hence, E (cid:104) exp ( (cid:90) ∞ Φ u + T ∞ d ˜ X u + (cid:90) ∞ Φ u + S ∞ d ˜ Y u ) (cid:12)(cid:12)(cid:12) F T ∞ (cid:95) F X T ∞ (cid:95) F Y S ∞ (cid:105) = exp (cid:18) (cid:90) ∞ ( Φ u + T ∞ ) du + (cid:90) ∞ ( Φ u + S ∞ ) du (cid:19) . (37)Substitute (37) into (36), E (cid:20) exp (cid:18) (cid:90) ∞ Φ u dX u + (cid:90) ∞ Φ u dY u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( Φ u + T ∞ ) du + (cid:90) ∞ ( Φ u + S ∞ ) du (cid:19) E (cid:20) exp (cid:18) (cid:90) ∞ Φ T u dX T u + (cid:90) ∞ Φ S u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( Φ u + T ∞ ) du + (cid:90) ∞ ( Φ u + S ∞ ) du (cid:19) exp (cid:18) (cid:90) ∞ ( Φ T u ) dT u + (cid:90) ∞ ( Φ T u ) dS u (cid:19) = exp (cid:18) { (cid:90) T ∞ ( Φ u ) du + (cid:90) S ∞ ( Φ u ) du + (cid:90) ∞ T ∞ ( Φ u ) du + (cid:90) ∞ S ∞ ( Φ u ) du } (cid:19) = exp (cid:18) (cid:90) ∞ ( Φ u ) du + (cid:90) ∞ ( Φ u ) du (cid:19) , (38)i.e. E (cid:34) exp ( n ∑ i = θ i ( X t i − X t i − ) + m ∑ j = θ j ( Y s j − Y s j − )) |F T ∞ (cid:35) = exp ( n ∑ i = ( θ i ) ( t k − t k − ) + m ∑ j = ( θ j ) ( s j − s j − )) ,which implies X and Y are independent and F T ∞ does not affect the distribution of { X t , Y t } t ≥ . Hence, { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ are mutually independent. Lemma 6.2.
Suppose ( X t , Y t ) t ≥ is a 2-dimensional standard Brownian motion and { T t } t ≥ , { S t } t ≥ are two increasingprocesses with T t + S t = t , ∀ t. If X, Y and T are mutually independent, then we have the following consequences:(1) the condition (E) holds;(2) F X T ∞ ⊥ F Y S t |F X T t and F X T t ⊥ F Y S ∞ |F Y S t ;(3) { X T t } t ≥ and { Y S t } t ≥ are martingales with respect to {F B , Wt (cid:87) F T ∞ } t ≥ .Proof. (1) Set S t = t − T t , let τ and ς be the inverse of T and S as defined in (2). Define Φ u = φ τ u { u ≤ T ∞ } , Φ u = φ ς u { u ≤ S ∞ } ,then (cid:90) ∞ ( Φ u ) du = (cid:90) T ∞ ( φ τ u ) du = (cid:90) ∞ ( φ u ) dT u , (cid:90) ∞ ( Φ u ) du = (cid:90) S ∞ ( φ ς u ) du = (cid:90) ∞ ( φ u ) dS u , (39)and E (cid:20) (cid:90) ∞ ( Φ u ) du + (cid:90) ∞ ( Φ u ) du (cid:21) = E (cid:20) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:21) ≤ E exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19) < ∞ .32ence (cid:82) ∞ Φ u dX u and (cid:82) ∞ Φ u dY u are well defined and (cid:90) ∞ Φ u dX u = (cid:90) T ∞ φ τ u dX u = (cid:90) ∞ φ u dX T u , (cid:90) ∞ Φ u dY u = (cid:90) S ∞ φ ς u dY u = (cid:90) ∞ φ u dY S u . (40)Since X , Y , T are independent, X and Y are martingales with respect to {F Xt (cid:87) F Yt (cid:87) F T ∞ } t ≥ . Observe that E exp (cid:16) (cid:82) ∞ ( Φ u ) du + (cid:82) ∞ ( Φ u ) du (cid:17) < ∞ , thus by Karatzas and Shreve 2012[Chapter 3, Proposition 5.12],exp (cid:16) (cid:82) t Φ u dX u + (cid:82) t Φ u dY u − (cid:82) t ( Φ u ) du − (cid:82) t ( Φ u ) du (cid:17) is a martingale. Consequently E (cid:20) exp (cid:18) (cid:90) ∞ Φ u dX u + (cid:90) ∞ Φ u dY u − (cid:90) ∞ ( Φ u ) du − (cid:90) ∞ ( Φ u ) du (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) ∞ φ u dX T u + (cid:90) ∞ φ u dY S u − (cid:90) ∞ ( φ u ) dT u − (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = (cid:16) (cid:82) ∞ ( φ u ) dT u + (cid:82) ∞ ( φ u ) dS u (cid:17) is measurable with F T ∞ , and the desired result holds immedi-ately.(2) First note that, when X , Y , T are independent, by the former result, (E) is true. As a direct consequence of (E), F X T t and F Y S t is conditional independent given F Tt , ∀ t ∈ [ + ∞ ] .Thus for every F Y S t -measurable random variable η , E [ η |F X T t (cid:87) F Tt ] = E [ η |F Tt ] . Furthermore, since F Tt ⊂ F X T t , ∀ t ∈ [ + ∞ ] , we have E [ η |F Tt ] = E [ η |F X T t ] , ∀ t ∈ [ + ∞ ] . (41)To prove the result of this part, i.e., F Y S t and F X T ∞ are conditional independent given F X T t , it is sufficient toprove that for any F T -progressively measurable process φ satisfying conditions in (E), E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F X T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F X T t (cid:21) .By (41), E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F X T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F T ∞ (cid:21) = exp (cid:18) (cid:90) t ( φ u ) dS u (cid:19) .where the second equality comes from (E) immediately.Since exp (cid:16) (cid:82) t ( φ u ) dS u (cid:17) ∈ F Tt , and applying (41) again, we have E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F X T ∞ (cid:21) = E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F Tt (cid:21) = E (cid:20) exp (cid:18) (cid:90) t φ u dY S u (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F X T t (cid:21) ,which is the desired conclusion. By similar proofs, we have F X T t ⊥ F Y S ∞ |F Y S t .333) Given F T ∞ , we can see X T u + t − X T t , F X T t and F Y S t are mutually independent according to (E). Hence, E (cid:104) X T u + t − X T t |F X T t (cid:95) F Y S t (cid:95) F T ∞ (cid:105) = E (cid:104) X T u + t − X T t |F T ∞ (cid:105) = F X T t (cid:87) F Y S t = F B , W , X T t is a martingale with F B , Wt (cid:87) F T ∞ . The same argument for Y S . Lemma 6.3.
Suppose { X t } t ≥ is a Brownian motion and { T t } t ≥ is a nondecreasing stochastic process independent with { X t } t ≥ . Given { φ t } t ≥ and { θ t } t ≥ , which are progressively measurable with {F Tt } t ≥ andE (cid:20) exp (cid:18) (cid:90) t ( φ u ) dT u (cid:19)(cid:21) < ∞ , E (cid:20) exp (cid:18) (cid:90) t ( θ u ) dT u (cid:19)(cid:21) < ∞ , ∀ t , let X φ t = X t − (cid:90) t φ τ u du , τ t = inf { u : T t ≥ u } , then we haveE (cid:20) exp (cid:18) (cid:90) t ∧ T t θ τ u dX φ u (cid:19) exp (cid:18) (cid:90) t ∧ T t φ τ u dX u − (cid:90) t ∧ T t ( φ τ u ) du (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) t ∧ T t ( θ τ u ) du (cid:19) . Proof.
Fix t , from E exp (cid:18) (cid:90) s ∧ T t ( φ τ u ) du (cid:19) ≤ E exp (cid:18) (cid:90) T t ( φ τ u ) du (cid:19) = E exp (cid:18) (cid:90) t ( φ u ) dT u (cid:19) < ∞ ,and Karatzas and Shreve 2012[Chapter 3, Proposition 5.12] we have (cid:110) exp (cid:16) (cid:82) s ∧ T t φ τ u dX u − (cid:82) s ∧ T t ( φ τ u ) du (cid:17)(cid:111) s ≥ is a martingale with respect to {F Xs (cid:87) F T ∞ } s ≥ . Let d ˜ QdP (cid:12)(cid:12)(cid:12) F Xs (cid:95) F T ∞ = exp (cid:18) (cid:90) s ∧ T t φ τ u dX u − (cid:90) s ∧ T t ( φ τ u ) du (cid:19) ,note that X is a Brownian motion with respect to {F Xs (cid:87) F T ∞ } s ≥ , then by Girsanov theorem,˜ X φ s (cid:44) X s − (cid:90) s ∧ T t φ τ u du is a Brownian motion with {F Xs (cid:87) F T ∞ } s ≥ under probability measure ˜ Q . Hence (cid:110) exp (cid:16) (cid:82) s θ τ u d ˜ X φ u − (cid:82) s ( θ τ u ) du (cid:17)(cid:111) s ≥ is a martingale under ˜ Q , by optional stopping theorem , we obtain E ˜ Q (cid:20) exp (cid:18) (cid:90) t ∧ T t θ τ u d ˜ X φ u − (cid:90) t ∧ T t ( θ τ u ) du (cid:19) |F T ∞ (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t ∧ T t θ τ u d ˜ X φ u (cid:19) exp (cid:18) (cid:90) t ∧ T t φ τ u dX u − (cid:90) t ∧ T t ( φ τ u ) du (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) t ∧ T t ( θ τ u ) du (cid:19) .Note that ˜ X φ s = X φ s , ∀ s ∈ [ t ∧ T t ] , we get desired result immediately. In Girsanov theorem, we need to determine an upper bound t in advance, then ˜ X φ is a Brownian motion with {F Xs (cid:87) F T ∞ } ≤ s ≤ t in [ t ] .Thanks to 0 ≤ t ∧ T t ≤ t , optional stopping theorem for t ∧ T t remains valid. roof of Theorem 2.4. For the “if” part: since F B ∞ ⊥ F T ∞ |F B , Wt and F B , Wt ⊂ F t , E (cid:104) B t − B s |F T ∞ (cid:95) F B , Ws (cid:105) = E (cid:104) B t − B s |F B , Ws (cid:105) =
0, (42)therefore the process B is a martingale with respect to {F B , Wt (cid:87) F T ∞ } t ≥ , so is the process W by similar analysis.As a consequence, X T t = B t + W t , Y S t = B t − W t are martingales with respect to the same filtration. So for any F T -progressively measurable process φ , φ satisfying the conditions in (E), D φ t (cid:44) exp (cid:18) (cid:90) t φ u dX T u + (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dT u − (cid:90) t ( φ u ) dS u (cid:19) , t ∈ [ + ∞ ) is a local martingale with respect to {F B , Wt (cid:87) F T ∞ } t ≥ by It ˆo’s lemma. From Karatzas and Shreve 2012[Chapter 3,Proposition 5.12], and E (cid:20) exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) < ∞ ,we have ED φ t = ∀ t , consequently, D φ t is a martingale. Moreover, E (cid:20)(cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) < E (cid:20) exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) < ∞ ,implies D φ ∞ exists. And thus E [ D φ ∞ |F B , W (cid:95) F T ∞ ] = D φ = E (cid:20) exp (cid:18) (cid:90) ∞ φ u dX T u + (cid:90) ∞ φ u dY S u (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19) .According to Lemma 6.1, the desired result is obtained.For the“only if” part: if { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ are independent, by Lemma 6.2, { X T t } t ≥ and { Y S t } t ≥ are martingales with respect to F B , Wt (cid:87) F T ∞ .Consequently, B t = X T t + Y S t , W t = X T t − Y S t , are martingales with respect to F B , Wt (cid:87) F T ∞ . Since [ B ] t = [ W ] t = t , B t and W t are Brownian motions with respect to F B , Wt (cid:87) F T ∞ according to L´evy characterisation.On the other hand, B t and W t are Brownian motions with respect to F B , Wt as well. That is to say, for any t ≥ B given F B , Wt (cid:87) F T ∞ is coincident with its conditional distribution given F B , Ws . Then we can conclude that F B ∞ ⊥ F T ∞ |F B , Wt . Similarly, F W ∞ ⊥ F T ∞ |F B , Wt . Proof of Proposition 2.5.
We prove that the independency of X , Y and T is equivalent with condition (C2), then from Theorem 2.4, wehave condition (C1) is equivalent with condition (C2).For the ” ⇒ ” part: It is obvious that D φ t is a positive local martingale according to It ˆo’s lemma, hence D φ t is asupermartingale. From Karatzas and Shreve 2012[Chapter 3, Proposition 5.12], and E (cid:20) exp (cid:18) (cid:90) t ( φ u ) dT u + (cid:90) t ( φ u ) dS u (cid:19)(cid:21) < ∞ ,we have ED φ t = ∀ t , consequently, D φ t is a martingale. 35uppose θ it , i =
1, 2 are bounded determined processes, then E Q (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u + (cid:90) t θ u dY φ S u (cid:19)(cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u + (cid:90) t θ u dY φ S u (cid:19) D φ t (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u (cid:19) exp (cid:18) (cid:90) t φ u dX T u − (cid:90) t ( φ u ) dT u (cid:19) E P (cid:20) exp (cid:18) (cid:90) t θ u dY φ S u (cid:19) exp (cid:18) (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dS u (cid:19) |F T ∞ (cid:95) F X ∞ (cid:21)(cid:21) .(43)According to the independency of X , Y , T , we have E P (cid:20) exp (cid:18) (cid:90) t θ u dY φ S u (cid:19) exp (cid:18) (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dS u (cid:19) |F T ∞ (cid:95) F X ∞ (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dY φ S u (cid:19) exp (cid:18) (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dS u (cid:19) |F T ∞ (cid:21) = E P (cid:20) exp (cid:18) (cid:90) S t θ ς u dY φ u (cid:19) exp (cid:18) (cid:90) S t φ ς u dY u − (cid:90) S t ( φ ς u ) du (cid:19) |F T ∞ (cid:21) , (44)where Y φ t = Y t − (cid:82) ς t φ u dS u = Y t − (cid:82) t φ ς u du . Observe that t ∧ S t = S t , then from Lemma 6.3 we have E P (cid:20) exp (cid:18) (cid:90) S t θ ς u dY φ u (cid:19) exp (cid:18) (cid:90) S t φ ς u dY u − (cid:90) S t ( φ ς u ) du (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) S t ( θ ς u ) du (cid:19) = E P (cid:20) exp (cid:18) (cid:90) S t θ ς u dY u (cid:19) |F T ∞ (cid:21) = E P (cid:20) exp (cid:18) (cid:90) S t θ ς u dY u (cid:19) |F T ∞ (cid:95) F X ∞ (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dY S u (cid:19) |F T ∞ (cid:95) F X ∞ (cid:21) (45)Substitute (44) and (45) into (43), E Q (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u + (cid:90) t θ u dY φ S u (cid:19)(cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u (cid:19) exp (cid:18) (cid:90) t φ u dX T u − (cid:90) t ( φ u ) dT u (cid:19) E P [ exp (cid:18) (cid:90) t θ u dY S u (cid:19) |F T ∞ (cid:95) F X ∞ ] (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u (cid:19) exp (cid:18) (cid:90) t φ u dX T u − (cid:90) t ( φ u ) dT u (cid:19) exp (cid:18) (cid:90) t θ u dY S u (cid:19)(cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dY S u (cid:19) E P [ exp (cid:18) (cid:90) t θ u dX φ T u (cid:19) exp (cid:18) (cid:90) t φ u dX T u − (cid:90) t ( φ u ) dT u (cid:19) |F T ∞ (cid:95) F Y ∞ ] (cid:21) apply Lemma 6.3 to the former equation again, we obtain E Q (cid:20) exp (cid:18) (cid:90) t θ u dX φ T u + (cid:90) t θ u dY φ S u (cid:19)(cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dY S u (cid:19) E P [ exp (cid:18) (cid:90) t θ u dX T u (cid:19) |F T ∞ (cid:95) F Y ∞ ] (cid:21) = E P (cid:20) exp (cid:18) (cid:90) t θ u dX T u + (cid:90) t θ u dY S u (cid:19)(cid:21) .36f θ it , i =
1, 2 are complex, the proof remains valid, hence we have ( ˜ X φ , ˜ Y φ ) Q d = ( X T , Y S ) P immediately.For the ” ⇐ ” part: Suppose { φ t } t ≥ and { φ t } t ≥ satisfy the conditions in (E) (note that the range of { φ t } t ≥ and { φ t } t ≥ in (E) is smaller than (C2)), then E (cid:20)(cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) ≤ E (cid:20) exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19)(cid:21) < ∞ ,accordingly D φ ∞ exists. We first claim that E P [ D φ ∞ |F T ∞ ] = a . s ..To see this, we only need to prove for any A ∈ F T ∞ , E P [ D φ ∞ A ] = P ( A ) .Let D (cid:44) { A ∈ F | E P [ D φ ∞ A ] = P ( A ) } , P (cid:44) { n (cid:92) i = A t i | A t i ∈ σ ( T t i ) , n ≥ t < t < · · · < t n } ,note that E P [ D φ ∞ ] =
1, so D is a λ -system and obviously P is a π -system, moreover, σ ( P ) = F T ∞ . Then for any A = (cid:84) ni = A t i ∈ P , suppose A t i = { T t i ∈ B i } , where B i is a Borel set, we have E P [ D φ ∞ A ] = E P [ A E P [ D φ ∞ |F t n ]] = E P [ D φ t n A ] . (46)Since ( ˜ X φ , ˜ Y φ ) Q d = ( X T , Y S ) P , and [ X T ] t = X T t − (cid:90) t X T u dX T u ,so we have ([ ˜ X φ ] , [ ˜ Y φ ]) Q d = ([ X T ] , [ Y S ]) P , i.e. ( T , S ) Q d = ( T , S ) P . Consequently, P ( A ) = P ( T t i ∈ B i , i =
1, 2, . . . , n ) = Q ( T t i ∈ B i , i =
1, 2, . . . , n ) = E Q [ A ] = E P [ D φ t n A ] , (47)combine (46) and (47) we know that P ⊂ D . According to π − λ theorem we can conclude F T ∞ = σ ( P ) ⊂ D ,hence, we have proved our claim. E P [ D φ ∞ |F T ∞ ] = E (cid:20) exp (cid:18) (cid:90) ∞ φ u dX T u + (cid:90) ∞ φ u dY S u (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19) ,we complete proof by Lemma 6.1. Proof of Proposition 2.9.
We prove it by the equivalence of condition (C3) and (C1).”(C1) ⇒ (C3)”: According to F B ∞ ⊥ F T ∞ |F B , Wt and (42), we have { B t } t ≥ is a martingale with respect to F B , Wt (cid:87) F T ∞ .Because F ˜ Z ∞ ⊥ F B , W ∞ (cid:87) F T ∞ (actually, F T ∞ ⊂ F B , W ∞ ), then for any ξ ∈ F ˜ Zt , E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , W ∞ (cid:95) F T ∞ (cid:105) = E [ ξ ] = E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , Wt (cid:95) F T ∞ (cid:105) ,37hich is equivalent with F ˜ Zt ⊥ F B , W ∞ |F B , Wt (cid:87) F T ∞ . Hence, E (cid:104) B t − B s (cid:12)(cid:12)(cid:12) F B , Ws (cid:95) F T ∞ (cid:95) F ˜ Zs (cid:105) = E (cid:104) B t − B s (cid:12)(cid:12)(cid:12) F B , Ws (cid:95) F T ∞ (cid:105) = { B t } t ≥ is a martingale with respect to {F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt } t ≥ . With the same arguments, { W t } t ≥ is a martingale with respect to F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt as well. Obviously, { ˜ Z t } t ≥ is a martingale with respect to F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt , so from the definition of Z t , we know { Z t } t ≥ is a martingale with respect to F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt and [ Z ] t = t , [ B , Z ] t =
0. According to L´evy characterisation (see Shreve 2004), { B t } t ≥ and { Z t } t ≥ are twoindependent Brownian motions with respect to F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt . Since { B t } t ≥ and { Z t } t ≥ are adapted with F B , Wt (cid:87) F ˜ Zt ⊂ F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt , so { B t } t ≥ and { Z t } t ≥ are also two independent Brownian motions with re-spect to F B , Wt (cid:87) F ˜ Zt . Consequently, the joint distribution of { B t } t ≥ and { Z t } t ≥ is same under the condition of F B , Wt (cid:87) F T ∞ (cid:87) F ˜ Zt and F B , Wt (cid:87) F ˜ Zt which implies F Z ∞ (cid:95) F B ∞ ⊥ F T ∞ |F B , Wt (cid:95) F ˜ Zt . (48)Let t = F Z ∞ (cid:87) F B ∞ ⊥ F T ∞ and note that { B t } t ≥ is independent with { Z t } t ≥ we can conclude { B t } t ≥ , { Z t } t ≥ and { ρ t } t ≥ are mutually independent.”(C3) ⇒ (C1)”: Note that F B , Wt ⊂ F Bt (cid:87) F Zt (cid:87) F Tt , and obviously F B ∞ , F T ∞ and F Zt are mutually independentunder the condition of F Bt by (C3). Then for any ξ ∈ F B ∞ , E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , Wt (cid:105) = E (cid:104) E [ ξ |F Bt (cid:95) F Zt (cid:95) F Tt ] (cid:12)(cid:12)(cid:12) F B , Wt (cid:105) = E (cid:104) E [ ξ |F Bt ] (cid:12)(cid:12)(cid:12) F B , Wt (cid:105) = E (cid:104) ξ (cid:12)(cid:12)(cid:12) F Bt (cid:105) ,with similar approach we can prove E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , Wt (cid:87) F T ∞ (cid:105) = E (cid:2) ξ (cid:12)(cid:12) F Bt (cid:3) as well, immediately E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , Wt (cid:105) = E (cid:104) ξ (cid:12)(cid:12)(cid:12) F B , Wt (cid:95) F T ∞ (cid:105) , ∀ ξ ∈ F B ∞ ,which is equivalent to F B ∞ ⊥ F T ∞ |F B , Wt .As for F W ∞ ⊥ F T ∞ |F B , Wt , we first observe that { B t } t ≥ and { Z t } t ≥ are martingales with respect to F B , Zt (cid:87) F T ∞ by the independecy of { ρ t } t ≥ , { B t } t ≥ and { Z t } t ≥ . So according to W t = (cid:90) t ρ s dB s + (cid:90) t (cid:113) − ρ s dZ s , { W t } t ≥ is a martingale with respect to F B , Zt (cid:87) F T ∞ . Since { W t } t ≥ is adapted to F B , Wt ( ⊂ F B , Zt (cid:87) F T ∞ ) and F B , Wt (cid:87) F T ∞ ( ⊂ F B , Zt (cid:87) F T ∞ ) respectively, so { W t } t ≥ is a martingale with respect to F B , Wt and F B , Wt (cid:87) F T ∞ respectively. Hence,by L´evy characterisation, { W t } t ≥ is a Brownian motion with respect to F B , Wt and F B , Wt (cid:87) F T ∞ respectively. Thus,the distribution of { W t } t ≥ is same under the condition of F B , Wt (cid:87) F T ∞ and F B , Wt , which result in F W ∞ ⊥ F T ∞ |F B , Wt . Proof of Proposition 2.12.
Let W Π s = (cid:90) s ρ Π u dB u + (cid:90) s (cid:113) − ( ρ Π u ) dZ u = i ∑ k = ( ρ t k ∆ B t k + (cid:113) − ρ t k ∆ Z t k ) + ρ t i ( B s − B t i ) + (cid:113) − ρ t i ( Z s − Z t i ) , t i ≤ s < t i + ∆ B t k = B t k + − B t k , ∆ Z t k = Z t k + − Z t k .Observe that given F T ∞ , the conditional distribution of ( ∆ B t i , ∆ W Π t i ) is ( ∆ B t i , ∆ W Π t i ) ∼ N (cid:18)(cid:18) (cid:19) , (cid:18) ∆ t i ρ t i ∆ t i ρ t i ∆ t i ∆ t i (cid:19)(cid:19) .which is just the same as the conditional distribution of ( ∆ B t i , ∆ ˜ W Π t i ) . If condition (C3) holds, { B t } t ≥ and { Z t } t ≥ are independent Brownian motions with respect to F B , Zt (cid:87) F T ∞ . Hence, by the independent property of increments,given F T ∞ , we have ( ∆ B t , ∆ B t , . . . , ∆ B t n − , ∆ W Π t , ∆ W Π t , . . . , ∆ W Π t n − ) d = ( ∆ B t , ∆ B t , . . . , ∆ B t n − , ∆ ˜ W Π t , ∆ ˜ W Π t , . . . , ∆ ˜ W Π t n − ) .Consequently, ( B t , B t , . . . , B t n − , W Π t , W Π t , . . . , W Π t n − ) d = ( B t , B t , . . . , B t n − , ˜ W Π t , ˜ W Π t , . . . , ˜ W Π t n − ) . (49)Next, for any given u k , v l , k =
1, 2, . . . , K , l =
1, 2, . . . , L , we consider the difference between the distribution of ( B u , B u , . . . , B u K , W Π v , W Π v , . . . , W Π v L ) and ( B u , B u , . . . , B u K , ˜ W Π v , ˜ W Π v , . . . , ˜ W Π v L ) . Let i k = sup { z ∈ Z : t z < u k } , j l = sup { z ∈ Z : t z < v l } , k =
1, 2, . . . , K , l =
1, 2, . . . , L .Then, for all δ > P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , W Π t j ≤ b − δ , . . . , W Π t jL ≤ b L − δ , B u k − B t ik ≤ δ , W Π v l − W Π t jl ≤ δ , k =
1, . . . , K , l =
1, . . . , L ) ≤ P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) ≤ P ( B t i ≤ a + δ , . . . , B t iK ≤ a K + δ , W Π t j ≤ b + δ , . . . , W Π t jL ≤ b L + δ )+ K ∑ k = P ( B u k − B t ik ≤ − δ ) + L ∑ l = P ( W Π v l − W Π t jl ≤ − δ ) ,similar inequality holds for P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) . Let H = { B u k − B t ik ≤ δ , ˜ W Π v l − ˜ W Π t jl ≤ δ , k =
1, . . . , K , l =
1, . . . , L } , then P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) − P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) ≤ P ( B t i ≤ a + δ , . . . , B t iK ≤ a K + δ , W Π t j ≤ b + δ , . . . , W Π t jL ≤ b L + δ ) + K ∑ k = P ( B u k − B t ik ≤ − δ )+ L ∑ l = P ( W Π v l − W Π t jl ≤ − δ ) − P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , ˜ W Π t j ≤ b − δ , . . . , ˜ W Π t jL ≤ b L − δ , H ) . (50)Note that (49) implies P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , ˜ W Π t j ≤ b − δ , . . . , ˜ W Π t jL ≤ b L − δ ) = P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , W Π t j ≤ b − δ , . . . , W Π t jL ≤ b L − δ ) , and compared the first term and last term in the right hand39f (50), we have P ( B t i ≤ a + δ , . . . , B t iK ≤ a K + δ , W Π t j ≤ b + δ , . . . , W Π t jL ≤ b L + δ ) − P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , ˜ W Π t j ≤ b − δ , . . . , ˜ W Π t jL ≤ b L − δ , H )= P ( B t i ≤ a + δ , . . . , B t iK ≤ a K + δ , W Π t j ≤ b + δ , . . . , W Π t jL ≤ b L + δ ) − P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , ˜ W Π t j ≤ b − δ , . . . , ˜ W Π t jL ≤ b L − δ )+ P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , ˜ W Π t j ≤ b − δ , . . . , ˜ W Π t jL ≤ b L − δ , H c ) ≤ P ( B t i ≤ a + δ , . . . , B t iK ≤ a K + δ , W Π t j ≤ b + δ , . . . , W Π t jL ≤ b L + δ ) − P ( B t i ≤ a − δ , . . . , B t iK ≤ a K − δ , W Π t j ≤ b − δ , . . . , W Π t jL ≤ b L − δ ) + P ( H c ) ≤ K ∑ k = P ( | B t ik − a k | ≤ δ ) + L ∑ l = P ( | W Π t jl − b l | ≤ δ ) + P ( H c ) . (51)Substitute (51) into (50), we obtain P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) − P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) ≤ K ∑ k = P ( B u k − B t ik ≤ − δ ) + L ∑ l = P ( W Π v l − W Π t jl ≤ − δ ) + K ∑ k = P ( | B t ik − a k | ≤ δ ) + L ∑ l = P ( | W Π t jl − b l | ≤ δ ) + P ( H c ) ≤ K ∑ k = P ( B u k − B t ik ≤ − δ ) + L ∑ l = P ( W Π v l − W Π t jl ≤ − δ ) + K ∑ k = P ( | B t ik − a k | ≤ δ ) + L ∑ l = P ( | W Π t jl − b l | ≤ δ )+ K ∑ k = P ( B u k − B t ik ≥ δ ) + L ∑ l = P ( ˜ W Π v l − ˜ W Π t jl ≥ δ )= K ∑ k = Φ ( − δ (cid:112) u k − t i k ) + L ∑ l = Φ ( − δ (cid:112) v l − t j l ) + K ∑ k = P ( | B t ik − a k | ≤ δ ) + L ∑ l = P ( | W Π t jl − b l | ≤ δ ) ,where Φ denotes the standard normal distribution. For any (cid:101) >
0, we first give a δ small enough such that for any a k , k =
1, 2, . . . , K and b l , l =
1, 2, . . . , L , K ∑ k = P ( | B t ik − a k | ≤ δ ) + L ∑ l = P ( | W Π t jl − b l | ≤ δ ) < (cid:101) δ and let || Π || small enough such that2 K ∑ k = Φ ( − δ (cid:112) u k − t i k ) + L ∑ l = Φ ( − δ (cid:112) v l − t j l ) ≤ ( K + L ) Φ ( − δ (cid:112) || Π || ) < (cid:101) P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) − P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) ≤ (cid:101) ,40imilarly, P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) − P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) ≤ (cid:101) ,i.e. | P ( B u ≤ a , . . . , B u K ≤ a K , W Π v ≤ b , . . . , W Π v L ≤ b L ) − P ( B u ≤ a , . . . , B u K ≤ a K , ˜ W Π v ≤ b , . . . , ˜ W Π v L ≤ b L ) | ≤ (cid:101) ,(52)From the definition of It ˆo’s integral, we have ( B u , B u , . . . , B u K , W Π v , W Π v , . . . , W Π v L ) d −→ ( B u , B u , . . . , B u K , W v , W v , . . . , W v L ) . (53)Combine (52) and (53), as || Π || → ( B u , B u , . . . , B u K , ˜ W Π v , ˜ W Π v , . . . , ˜ W Π v L ) d −→ ( B u , B u , . . . , B u K , W v , W v , . . . , W v L ) . Remark 6.4.
Suppose { M t } t ≥ , { N t } t ≥ are two local martingales with respect to F t and [ M ] t = [ N ] t , ∀ t, then Theorem2.3, Lemma 6.1, Proposition 2.5, Proposition 2.9 can be generalized directly. SetT t = [ M ] t + [ M , N ] t S t = [ M ] t − [ M , N ] t and define X t , Y t similarly with Section 2.1. Then we have ( M t , N t ) = ( X T t + Y S t , X T t − Y S t ) , and { X t } t ≥ and { Y t } t ≥ are two independent Brownian motions. { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ are mutually independentis equivalent with following statements: (E’) for any F T , S ∞ measurable processes { φ t } t ≥ and { φ t } t ≥ which satisfy (cid:82) ∞ ( φ u ) dT u and (cid:82) ∞ ( φ u ) dS u are bounded, wehave E (cid:20) exp (cid:18) (cid:90) ∞ φ u dX T u + (cid:90) ∞ φ u dY S u (cid:19) |F T ∞ (cid:21) = exp (cid:18) (cid:90) ∞ ( φ u ) dT u + (cid:90) ∞ ( φ u ) dS u (cid:19) . (D1) F M ∞ ⊥ F T , S ∞ |F M , Nt and F N ∞ ⊥ F T , S ∞ |F M , Nt . (D2) suppose { φ t } t ≥ and { φ t } t ≥ are two bounded predictable processes which is measurable with F T , S ∞ , thenD φ t = exp (cid:18) (cid:90) t φ u dX T u + (cid:90) t φ u dY S u − (cid:90) t ( φ u ) dT u − (cid:90) t ( φ u ) dS u (cid:19) is a martingale and dQdP |F t = D φ t defines a new probability satisfy ( X φ T , Y φ S ) Q d = ( X T , Y S ) P , where ˜ X φ T t = X T t − (cid:82) t φ u dT u , ˜ Y φ S t = Y S t − (cid:82) t φ u dS u .41 oreover, if [ M ] t = [ N ] t is absolute continuous with t, then according to martingale representation theorem, we can rewrite ( M t , N t ) as ( M t , N t ) = ( (cid:90) t θ u dB u , (cid:90) t ξ u dB u + (cid:90) t η u dZ u ) , where B and Z are two independent Brownian motions. It is evident that F T , St ⊂ F θ , ξ , η t . Then { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ are mutually independent is equivalent with (D3) F θ , ξ , η t , { B t } t ≥ and { Z t } t ≥ are mutually independent.Particularly, the equivalence condition (D2), (D3) of { X t } t ≥ , { Y t } t ≥ and { T t } t ≥ are mutually independent in 1-dimension situation, i.e. Ocone martingale, illustrated in Kallsen 2006 and Vostrikova and Yor 2000 is a special case ofM t = N t . Ocone martingale has been widely used in financial mathematics, such as Carr et al. 2005 and Geman et al. 2001. Proof of Theorem 3.1.
The martingale properties of X T and Y S are easy to check, since immediately by optionalstopping theorem, X T , Y S are martingales under F X T , F Y S respectively and F Y S t ⊥ F X T ∞ |F X T t , F X T t ⊥ F Y S ∞ |F Y S t guarantee that E (cid:104) X T u | F X T , Y S t (cid:105) = E (cid:104) X T u | F X T t (cid:105) , E (cid:104) Y S u | F X T , Y S t (cid:105) = E (cid:104) Y S u | F Y S t (cid:105) , u ≥ t .When T t + S t = t , X T and Y S are continuous martingales under F X T , Y S . By direct calculation, for u ≥ t ≥ E (cid:104) X T u Y S u | F X T , Y S t (cid:105) = E (cid:104) E (cid:104) X T u Y S u | F X T ∞ (cid:95) F X T , Y S t (cid:105)(cid:12)(cid:12)(cid:12) F X T , Y S t (cid:105) = E (cid:104) X T u E (cid:104) Y S u | F Y S t (cid:105)(cid:12)(cid:12)(cid:12) F X T , Y S t (cid:105) = Y S t E (cid:104) X T u | F X T , Y S t (cid:105) = X T t Y S t .Accordingly, [ X T , Y S ] t = < X T , Y S > t =
0. And thus, [ B ] t = [ X T + Y S ] t = [ X T ] t + [ Y S ] t + [ X T , Y S ] t = T t + S t = t .Similarly, [ W ] t = [ X T − Y S ] t = t .Hence B and W are Brownian motions with respect to F B , W (which is equal to F X T , Y S ). And [ B , W ] t = [ X T + Y S , X T − Y S ] t = T t − S t , t ≥ Proof of Corollary 3.2.
Let ˜ F t (cid:44) σ { X u , Y u , { T v ≤ u } , { S v ≤ u } : u ≤ t , ∀ v } ,then { X t } t ≥ , { Y t } t ≥ are two standard Brownian motions with respect to ˜ F t result from the independency of { X t } t ≥ , { Y t } t ≥ , { T t } t ≥ . By definition of ˜ F , { T u ≤ t } , { S u ≤ t } ∈ ˜ F t for any u >
0, hence T u , S u are stoppingtimes, and { T t } t ≥ , { S t } t ≥ are time changes of ˜ F .Then by Lemma 6.2, the conditions in Theorem 3.1 are satisfied, and we get the desired result.42 roof of Proposition 4.3. By definition,ˆ G ( λ , λ ) = (cid:90) ∞ − ∞ (cid:90) ∞ − ∞ e i λ x + i λ x G ( x , x ) dx dx .According to Fubini theorem, (cid:90) ∞ − ∞ e i λ x G ( x , x ) dx = E (cid:20) (cid:90) ∞ − ∞ e i λ x ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } { γ (cid:62) M τ ≤ x } dx (cid:21) = E (cid:20) ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } (cid:90) ∞ − ∞ e i λ x { γ (cid:62) M τ ≤ x } dx (cid:21) = E (cid:20) ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } i λ e i λ x (cid:12)(cid:12) ∞ γ (cid:62) M τ (cid:21) = i λ E (cid:104) e i λ γ (cid:62) M τ ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } (cid:105) , (54)where the last equality comes from the fact that the imaginary part of λ is positive, which deduces that e i λ x | x = ∞ =
0. With similar calculation for x , we haveˆ G ( λ , λ ) = (cid:90) ∞ − ∞ i λ e i λ x E (cid:104) e i λ γ (cid:62) M τ ( γ + γ e γ (cid:62) M τ ) { γ (cid:62) M τ ≤ x } (cid:105) dx = − λλ E (cid:104) e i λ γ (cid:62) M τ + i λ γ (cid:62) M τ ( γ + γ e γ (cid:62) M τ ) (cid:105) = − γ λλ Φ M τ ( λ γ + λ γ ) − γ λλ Φ M τ ( λ γ + λ γ − i γ ) , Φ M τ denotes the characteristic function of M τ . In our model, it can be calculated by conditional expectation Φ M τ ( z , z ) = Ee iz X Tt + iz Y St = E (cid:104) E [ e iz X Tt + iz Y St | T t , S t ] (cid:105) = Ee − T t z − S t z = e − tz Ee − ( z − z ) T t = e − tz L t ( − ( z − z )) ,where L t means the generalized fourier transform of T t at time t . References [1] Ankirchner, S., Heyne, G., 2012. Cross hedging with stochastic correlation. Finance and Stochastics 16 (1),17–43.[2] Bahmani-Oskooee, M., Saha, S., 2015. On the relation between stock prices and exchange rates: a reviewarticle. Journal of Economic Studies 42 (4), 707–732.[3] Balcilar, M., Uwilingiye, J., Gupta, R., 2018. Dynamic relationship between oil price and inflation in southafrica. The Journal of Developing Areas 52 (2), 73–93.[4] Ballotta, L., Bonfiglioli, E., 2016. Multivariate asset models using l´evy processes and applications. The Euro-pean Journal of Finance 22 (13), 1320–1350. 435] Barndorff-Nielsen, O. E., Pedersen, J., Sato, K.-i., 2001. Multivariate subordination, self-decomposability andstability. Advances in Applied Probability 33 (1), 160–187.[6] Barndorff-Nielsen, O. E., Shiryaev, A., 2015. Change of time and change of measure. Vol. 21. World ScientificPublishing Company.[7] Benhamou, E., Gobet, E., Miri, M., 2010. Time dependent heston model. SIAM Journal on Financial Mathe-matics 1 (1), 289–325.[8] Billingsley, P., 1968. Convergence of Probability Measures, Second Edition. Wiley.[9] Bosc, D., Jun. 2012. Three essays on modeling the dependence between financial assets. Theses, Ecole Poly-technique X.URL https://pastel.archives-ouvertes.fr/pastel-00721674https://pastel.archives-ouvertes.fr/pastel-00721674