Research Article A Note on the Distribution of ...downloads.hindawi.com/journals/ijsa/2014/575270.pdf · Research Article A Note on the Distribution of Multivariate Brownian Extrema

Research ArticleA Note on the Distribution of Multivariate Brownian Extrema

Marcos Escobar1 and Julio Hernandez2

1 Ryerson University Toronto ON Canada M5B 2K32University of Toronto Toronto ON Canada M5S 2E4

Correspondence should be addressed to Marcos Escobar escobarryersonca

Received 12 June 2014 Accepted 29 October 2014 Published 16 November 2014

Academic Editor Enzo Orsingher

Copyright copy 2014 M Escobar and J HernandezThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wienerprocess for some correlation matrices This is the only explicit expressions found in the literature for this joint probability Theanalysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible sphericalsimplexes generated by reflections and the link to themethod of imagesThis joint distribution can be used in financial mathematicsto obtain prices of credit or market related products in high dimensionThe solution could be generalized to account for stochasticvolatility and other stylized features of the financial markets

1 Introduction

The paper finds closed-form expressions for the joint den-sitydistribution function of the endpoints and extrema ofa 119899-dimensional Wiener process This targeted functionrepresents a density function in terms of the endpoints oftheWiener process while it represents a distribution functionof the minimums of the processes underlying The resultsfound in this paper can be applied to processes that can bederived fromaWiener process using suitable transformationslike log-normal processes andOrnstein-Uhlenbeck processeswith suitable parameters The problem of finding such jointdensitydistribution has attracted research since the late nine-teenth century see for example [1] Closed-form solutionsare known for the cases of one and two dimensions see[2 3] with either a minimum or a maximum or both Theproblem was recently solved in three dimensions see [4] viathe method of images therefore it only applies to a restrictedset of correlation values

In this paper we generalize the results in [4] to anydimension 119899 The correlation structures for which a solutionvia the ImageMethod can be found depend on the dimension(119899) In principle there could be as many as 5 differentcorrelation structures without permutations in dimension4 or as little as three in dimensions 3 5 and higher than 8This is derived using the fundamental regions for irreducible

groups generated by reflections as presented in Table IV page297 in [5] together with the finding in [6 7]

The paper is organized as follows Section 2 introducesthe function of interest as the solution of a partial differentialequation (PDE)with specific initial and boundary conditionsThen it simplifies this PDE to a heat equation Section 3provides the main results described per dimension Section 4comments on applications as well as possible extensionsSection 5 concludes

2 Preliminaries and Simplificationto Heat Equation

Let (ΩF F Q) be a filtered probability space on the domainΩ with sigma algebra F filtration F = F

119905119905ge0

and aprobability measure Q on (ΩF) Consider 119894 = 1 119899where 119899 represents the dimension We introduce the follow-ing notation

(1) 119884119894(119905) stochastic processes with constant drift 120572

119894and

constant diffusion coefficient 120590119894

(2) 119882119894(119905) Brownian motion processes with mean 0 and

variance 119905(3) instantaneous correlation ⟨119889119882

119894(119905) 119889119882

119895(119905)⟩ = 120588

119894119895119889119905

119894 = 119895 covariance matrix Σ correlation matrix 119877 =(120588119894119895) and column vector drift 1205721015840 = (120572

1 120572

119899)

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2014 Article ID 575270 6 pageshttpdxdoiorg1011552014575270

2 International Journal of Stochastic Analysis

The underlying process 119884 can be expressed through thefollowing stochastic differential equation (SDE)

119889119884119894(119905) = 120572

119894119889119905 + 120590

119894119889119882119894 (1)

in order to simplify the notation we will assume 119884119894(0) = 0

Ourmain objective is to find the joint densitydistributionfunction for the minimum denoted as 119884

119894(119905) equiv min

0lt119904lt119905119884119894(119904)

and endpoints of 119884(119905) this function is defined below

119901 (1199101 119910

119899 1198981 119898

119899 119905) 119889119910

1sdot sdot sdot 119889119910119899

= 119875 (1198841(119905) isin 119889119910

1 119884

119899(119905) isin 119889119910

119899

1198841(119905) gt 119898

1 119884

119899(119905) gt 119898

119899)

(2)

where we also require 119884119894(0) gt 119898

119894and therefore119898

119894lt 0

In general 119884119894(119905) could be seen as a function of a stochastic

process 119878119894such that a transformation 119884

119894= 119891(119878

119894) leads to

a Wiener process For example 119878119894could be a log-normal

process (119891(sdot) = ln(sdot)) see [8] for the general type of transfor-mations 119891 and processes allowed

Whenever reasonable and in order to shorten the lengthof notation we will write 119901(119910119898 119905) = 119901(119910

1 119910

119899 119898 119905) =

119901(1199101 119910

119899 1198981 119898

119899 119905)The same conventionwill be used

for analogous intermediate functionsIt is important to realize that if (2) is known thenwe could

also derive the joint densitydistribution of endpoints andmaximums and all cases in between To see this first definethe maximum of a process by 119884

119894(119905) equiv max

0lt119904lt119905119884119894(119904) and let

119872119894gt 0 119894 = 1 119896 then note that in the case with no drift

119875 (1198841(119905) isin 119889119910

1 119884

119899(119905) isin 119889119910

119899 1198841(119905) lt 119872

1

119884119896(119905) lt 119872

119896 119884119896+1(119905) gt 119898

119896+1 119884

119899(119905) gt 119898

119899)

= 119875 (119884lowast

1(119905) isin 119889119910

lowast

1 119884

119899(119905) isin 119889119910

119899 119884lowast

1(119905) gt 119898

1

119884lowast

119896(119905) gt 119898

119896 119884119896+1(119905) gt 119898

119896+1 119884

119899(119905) gt 119898

119899)

= 119901 (1199101 119910

119899 1198981 119898

119899 119905) 119889119910

1sdot sdot sdot 119889119910119899

(3)

with 119884lowast119894= minus119884

119894 119898119894= minus119872

119894 119910lowast119894= minus119910

119894 119894 = 1 119896

and therefore correlation between 119884lowast119894and 119884

119895 Corr(119884lowast

119894 119884119895) =

minusCorr(119884119894 119884119895)

It is known (see [9]) that the function 119901(sdot) above satisfiesthe following PDE known as Fokker-Planck equation

120597119901

120597119905

(119910 119905) = minus

119899

sum

119894=1

120572119894

120597119901

120597119910119894

(119910 119905) +

1

2

119899

sum

119894=1

1205902

119894

1205972119901

1205971199102

119894

(119910 119905)

+sum

119894lt119895

120590119894120590119895120588119894119895

1205972119901

120597119910119894120597119910119895

(119910 119905)

(4)

Since 119884119894(0) = 0 119894 = 1 119899 the initial condition is

119901 (119910119898 119905 = 0) =

119899

prod

119894=1

120575 (119910119894) (5)

where 120575(119909) is the Dirac delta function with a spike at 119910119894= 0

The following 119899 boundary conditions are added to match theextrema

119901 (1199101 119910

119894= 119898119894 119910

119899 119898 119905) = 0 119894 = 1 119899 (6)

Themeaning of the above constraints in terms of theWienerprocess is clear as soon as any of the component119884

119894equals119898

119894

we constraint the function 119901(sdot) to be equal 0 Now noticingthe initial conditions 119884

119894(0) gt 119898

119894and the continuity of the

Wiener paths we conclude that the constraints (6) are equiv-alent to enforcing that each path of 119884

119894and therefore 119884

119894(119905)

remains above the barrier levels 119898119894before and up to time

119905 From the analytical point of view that is looking at thedensity 119901(119910119898 119905) as a function on 119877119899 the constraint (6)enforces that the support of the density is inside the set Ω =119910 isin 119877

119899 119910 ge 119898

Equations (4) (5) and (6) will be referred to as the PDEproblem associated to the density (2) This function 119901(sdot) willbe obtained for a set of special correlations by solving (4)(5) and (6) by means of the method of images the readeris referred to [6 7] for accounts of the method of images

To obtain the solution of the system (4) (5) and (6) weneed to simplify (4) to a heat equation The simplification isperformed in two steps first we use conventionalmethods tomodify the probability function to one associated to processeswithout drift unit volatility and barriers at zero then in asecond step we use a Cholesky transformation to generatea new set of independent processes hence uncorrelatedleading to a heat equation with new boundary conditions

Let us define

119889 = |Σ| 119886119894=

1

119889

1003816100381610038161003816Σ119894

1003816100381610038161003816 119894 = 1 119899 (7)

119887 = 1198861015840Σ119886 minus 120572119886 (8)

119901 (119910119898 119905) = 119902 (119910119898 119905) exp (119886 sdot 119910 + 119887119905) (9)

where Σ119894is the covariance matrix with column 119894 replaced by

column vector 120572 It follows that

120597119902

120597119905

=

119899

sum

119894=1

1205902

119894

2

1205972119902

1205971199102

119894

+sum

119894lt119895

120588119894119895120590119894120590119895

1205972119902

120597119910119894119894120597119910119895

(10)

with boundary conditions

119902 (1199101 119910

119894= 119898119894 119910

119899 119898 119905) = 0 119894 = 1 119899 (11)

and delta Dirac initial condition We continue to simplifythe above PDE by eliminating the parameters 120590

119894and 119898

119894

119894 = 1 119899 Consider the following change of variables andtransformation

119909119894=

119910119894minus 119898119894

120590119894

119894 = 1 119899

ℎ (119909 0 119905) = 119902 (119910119898 119905) (

119899

prod

119894=1

120590119894)

minus1

(12)

International Journal of Stochastic Analysis 3

leading to

120597ℎ

120597119905

=

119899

sum

119894=1

1

2

1205972ℎ

1205971199092

119894

+sum

119894lt119895

120588119894119895

1205972ℎ

120597119909119894120597119909119895

(13)

with boundary conditions and initial condition

ℎ (1199091 119909

119894= 0 119909

119899 119898 119905) = 0 119894 = 1 119899

ℎ (119909 0 119905 = 0) =

119899

prod

119894=1

120575 (119909119894minus 1199091198940)

(14)

Here 1199091198940= minus119898119894120590119894 119894 = 1 119899

The last step eliminates the correlations in matrix 119877 forthis we perform a Cholesky decomposition

Proposition 1 Let one define new variables 119911 as follows

119911 = 119909119871minus1 (15)

where 119871 is an upper triangular matrix in the Cholesky decom-position of 119877 (119877 = 1198711015840119871) Then the PDE associated to 119911 satisfies

120597ℎ

120597119905

=

119899

sum

119894=1

1

2

1205972ℎ

1205971199112

119894

(16)

with boundary and initial conditions given by

ℎ (119867119894 119905) = 0 119894 = 1 119899 (17)

ℎ (119911 0 119905 = 0) =

119899

prod

119894=1

120575 (119911119894minus 1199111198940) (18)

where 1199110= (11991110 119911

1198990) = 119909

0119871minus1 and 119867

119894= 119911 sdot 119871

119894= 0

119894 = 1 119899 where 119871119894represents column 119894 of 119871

Proof This follows directly from noticing that the processesassociated to 119911

119894are uncorrelated with zero drift

The variables 119911 are independent so (16) (18) and (17)represent the Fokker-Planck equation of an uncorrelatedBrownian motion with zero drift and constrained to a regionbounded by 119899 hyperplane (119867

119894) Moreover the vector 119871

119894

represents the normal vector to the hyperplane 119867119894 119894 =

1 119899

3 Solving the Heat Equation

The method of images (MofI) is now utilized to solve thesystem (16) (18) and (17) The nature of the method ofimages is to replace the boundary conditions by a set offictitious source points Then the solution of the originalequation satisfying the given boundary conditions reduces tothat of finding the solution without boundary conditions atthe source points In the case of linear differential equationsthe process of obtaining the final solution divides into threedistinct steps (see [7])

(1) checking that the differential equation is suitable forMofI and then solving this differential equation fora point source in an infinite medium but with noboundary conditions except that of good behavior atinfinity

(2) checking that the region of interest is suitable forMofIand then finding the set of image source at each of thereflecting regions

(3) summing the solution of (1) over the set of imagesobtained by step (2)

In general step (1) is the simplest one The Laplaceand heat equations are well known to fit into MofI (seeeg [6 7]) The solution in step (1) is usually known inclosed form For example in our case the solution withoutboundary conditions is well known Consider the followingheat equation for an arbitrary initial point 119887

0= (11988710 119887

1198990)

120597119891

120597119905

=

119899

sum

119894=1

1

2

1205972119891

1205971199112

119894

119891 (119911 0) =

119899

prod

119894=1

120575 (119911119894minus 1198871198940)

(19)

By means of an application of the Fourier transform thesolution of the above equation can be expressed as

119891 (119911 119905) =

1

(2120587119905)1198992

exp[minus 12119905

119899

sum

119894=1

(119911119894minus 1198871198940)2

] (20)

Wewill show next that the regions whereMofI applies areconnected to correlation matrices therefore feasible regionsimply feasible correlations For the issue of whether thebounded region allows for the method of images we rely on[6] Note hyperplanes 119867

119894pass through zero so they cut out

the surface of an 119899-dimensional sphere centered at the vertex0 in a convex polygonal domain also known as sphericalsimplexes These spherical simplexes divide the sphere intosymmetric parts where the method of images applies if andonly if these simplexes are irreducible fundamental regionsgenerated by reflections (see [5 6])

The next result relates these regions and in particular thedihedral angle between these hyperplanes with the correla-tion matrix 119877 for the multivariate process 119910(119905) Let us denoteby 120579119894119895the dihedral angle between hyperplanes 119867

119894and 119867

119895

with 120579 representing thematrix of anglesThenext propositionrelates these angles and the correlation matrix 119877

Proposition 2 The relationship between the dihedral anglebetween hyperplanes 119867

119894and 119867

119895 119894 119895 = 1 119899 and the

correlation matrix 119877 satisfies minus cos(120579) = 119877

Proof The dihedral angle between hyperplanes 119867119894and 119867

119895

can be obtained from the normal vectors to the plane as

cos (120587 minus 120579119894119895) = 1198711015840

119894sdot 1198711015840

119895 (21)

Here we use the fact that 119871119894 = 1 for all 119894 In matricial

form we have

minus cos (120579) = 1198711015840119871 = 119877 (22)

4 International Journal of Stochastic Analysis

Next we use the results in the seminal work of [5] inwhicha complete list of fundamental regions for irreducible groupsgenerated by reflections in the case of spherical simplexes isprovided (Table IV page 297)We use the notation in Coxeterfor irreducible groups for spherical simplexes in dimension119899 119860119899 119861119899 119862119899119863119899 119864119899 119865119899 and 119866

119899 In particular we extract the

feasible dihedral angles for each of these spherical simplexesand the dimension where they apply These results are givennext

Proposition 3 The type of correlation matrices 119877 where themethod of images can be used to solve the system (16) (18) and(17) is provided nextThe total number of source points needed119873 for an explicit expression of the solution is also provided

(i) For any 119899 gt 3 there are at least 3 cases of correlationmatrices

119860119899lArrrArr 119877(119894 119895) =

minus cos(1205873

) 119895 = 119894 + 1

0 119890119897119904119890

119873 = (119899 + 1)

119861119899lArrrArr 119877(119894 119895)

=

minus cos(1205873

) 119895 = 119894 + 1

119894 = 119899 minus 2 119895 = 119899

0 119890119897119904119890

119873 = 119899 sdot 2119899minus1

119862119899lArrrArr 119877(119894 119895)

=

minus cos(1205873

) 119895 = 119894 + 1

minus cos(1205874

) 119894 = 119899 minus 1 119895 = 119899

0 119890119897119904119890

119873 = 119899 sdot 2119899

(23)

(ii) If 119899 = 6 7 or 8 then there is one more case (4 total)

119864119899lArrrArr 119877(119894 119895) =

minus cos(1205873

) 119895 = 119894 + 1 119894 = 4

0 119894 = 4 119895 = 5

minus cos(1205873

) 119894 = 3 119895 = 5

0 119890119897119904119890

(24)

For 119899 = 6 we have119873 = 6 sdot 72 119899 = 7 implies119873 = 9 sdot 8and for 119899 = 8 then119873 = 10 sdot 192

(iii) If 119899 = 5 then there are only three cases1198605 1198615 and 119862

5

(iv) If 119899 = 4 then there are a total of 5 cases

1198654lArrrArr 119877(119894 119895)

=

minus cos(1205873

) 119895 = 119894 + 1 119894 = 2

minus cos(1205874

) 119894 = 2 119895 = 3

0 119890119897119904119890

119873 = 1152

1198664lArrrArr 119877(119894 119895)

=

minus cos(1205873

) 119895 = 119894 + 1 119894 = 3

minus cos(1205875

) 119894 = 3 119895 = 4

0 119890119897119904119890

119873 = 1202

(25)

(v) If 119899 = 3 then there are only three cases 1198603(119873 = 24)

1198623(119873 = 48) and 119866

3

1198663lArrrArr 119877(119894 119895)

=

minus cos(1205873

) 119895 = 119894 + 1 119894 = 2

minus cos(1205875

) 119894 = 2 119895 = 3

0 119890119897119904119890

119873 = 120

(26)

(vi) If 119899 = 2 then there are infinitely many cases (any 119901)

1198632lArrrArr 119877(1 2) = minus cos(120587

119901

) 119873 = 2119901 (27)

Proof This follows from the dihedral angles provided inTable IV page 297 in [5] together with Proposition 2 Inthis table the regions are represented by graphs Every noderepresents a bounding hyperplane and the branches indicatepairs of hyperplanes inclined at angles 120587119901 119901 gt 2 If avalue of 119901 is not provided then a 119901 = 3 is understoodPerpendicular hyperplanes are represented by nodes notjoined by a branchNote that all hyperplanes should intercepttherefore most dihedral angles are 1205872 The cases obtainedin this proposition follow easily from reading these graphsout

In particular for dimensions 2 and 3 the number offeasible cases and source points reproduces those found in[2 4] respectively

Corollary 4 Given a collection of source points in 119877119899119887119895119873

119895=1

where 119887119895= (1198871119895 119887

119899119895) and the sign associated to each point

119892(119895)119873

119895=1 where 119892(119895) = 0 1 then the solution to the system

(16) (18) and (17) can be found as follows

ℎ (119911 119905) =

119873

sum

119895=1

1

(2120587119905)1198992(minus1)119892(119895) exp[minus 1

2119905

119899

sum

119894=1

(119911119894minus 119887119894119895)

2

]

(28)

Note that (28) is basically a linear combination of 119899-dimensional gaussian density functions with mean zeroand (1119905)119868 covariance matrix Substituting (28) into (16)and further into (9) would lead to the targeted joint den-sitydistribution in (2) and this would be again a linearcombination of 119899-dimensional gaussian densities but nowwith nonzero means (depending on the image sources) andcovariance matrix (1119905)Σ as in [4]

International Journal of Stochastic Analysis 5

31 Finding Source Points In this sectionwe describe a quasi-analytical procedure to find all source points associated to aspherical simplex and therefore to a correlation matrix

Let us denote

(i) 119871119894 normal vector to hyperplane 119894 with 119894 = 1 119899

(ii) 1199090 initial point inside the region

(iii) 119873 number of source points(iv) 119871

119894119896(119871119894 1198711198941 119871

119894119896minus1) normal vector to an hyperplane

created after 119896 consecutive reflections each across thehyperplane associated to the normal vector in thesequence (119871

119894 1198711198941 119871

119894119896minus1)

(v) 119909(119871 119894 119871 1198941119871 119894119896minus1

)

0 119904119892(119909

(119871 119894 119871 1198941119871 119894119896minus1

)

0) source point and

sign of it created after 119896 consecutive reflections eachacross the hyperplane associated to the normal vectorin the sequence (119871

119894 1198711198941 119871

119894119896minus1)

We show next a method to create the image of a hyper-plane the image of a point reflected across a given hyperplane(passing through zero) and the sign of the new point asneeded by the method of images this uses standard conceptsfrom geometry

(1) New source point reflecting a point1199090across a hyper-

plane passing through zero defined by the normalvector 119871

119894leads to a new point with equation

119909(119871 119894)

0= 1199090+ (minus2119909

0sdot 119871119894) 119871119894 (29)

(2) New hyperplane reflecting a hyperplane with normalvector 119871

119895across a hyperplane with normal vector 119871

119894

leads to a new hyperplane with normal vector

119871119895(119871119894) = 119886119871

119894+ 119887119871119895

119886 = minus2119888 119887 = 1 119888 = 119871119894sdot 119871119895

(30)

Note the normal vector to both 119871119894and 119871

119895must also be

normal to 119871119895(119871119894) this is why the latter is in the same

hyperplane as the formers The equation is obtainedthen using two facts first the norm of 119871

119895(119871119894) is one

therefore (1198862 + 1198872 + 2119886119887119888 = 1) and second 119871119895(119871119894) sdot

(minus119871119894) = 1198712sdot 1198711 therefore (119888 = minus119886 minus 119887119888)

(3) Sign of newpoint assume original point has a positivesign ldquo+rdquo then

119904119892 (119909

(119871 119894 119871 1198941119871 119894119896minus1

)

0) = (minus1)

119896+1 (31)

The algorithms are based on doing reflections across alloriginals hyperplanes and then repeating the procedure forall new hyperplanes This method would lead to repeatedvalues therefore we also have to check if there are duplicates(hyperplanes or source points) The method stops after theknown number of different sources119873 has been detected butin principle few iterations should lead to a good approxima-tion

4 A Comment on Applications

One of the key application fields for our findings is mul-tidimensional financial derivatives (see [10] or [11] for anintroduction to financial markets and problems) Marketderivatives are contracts payable at future times called matu-rity deriving their value from the performance of potentiallyseveral underlying tradeable stocks There exists an extensivefamily of financial derivatives in the market which dependson first passage time (barriers) of the underlying stock priceprocesses In general as pointed out by [12] adding barriersis a convenient method for reducing the cost of a derivative

Two main families of such products are double lookbackoptions (see [2]) in particular double barrier options (see[13]) and mountain range derivatives (see [14]) The latterare high dimensional (119899 ge 3) and were created by SocieteGenerale in 1998 Examples of these products are AltiplanoAnnapurna and Atlas The payoff at maturity 119879 of anAnnapurna is of the form

119899

prod

119894=1

1119884119894(119879)gtlog119870119894 (32)

Here 119884 stands for the log stock price while 119870rsquos areprespecified strike pricesTherefore the product pays a dollarat time 119879 if and only if all stocks remain above given strikeprices (119870

119894 119894 = 1 119899) during the relevant period of the

product that is (0 119879] The price of such a product 119862(0) isthe expected value of this payoffTherefore the price is relatedto our function 1 in the following manner

119862 (0) = intint

Ω

119901 (119910 log119870119879) 119889119910 (33)

whereΩ = 119910 | 119910119894gt log119870

119894 119894 = 1 119899

In the past this expression could be evaluated eithervia Monte Carlo simulations or directly solving the PDEequations which are highly time consuming and inaccurateapproaches for dimensions higher than 3 This price underthe feasible correlations described in Proposition 3 couldbe now found as a linear combination of 119899-dimensionalGaussian cumulative distribution functions

Another family of products benefiting from this work arecredit derivatives in particular collateralized debt obligation(CDO) and a 119896th to default product (see [15]) These wereproducts at the very heart of the financial crisis in 2008A CDO has a payoff similar to that of a market productunder proper default assumptions For instance if a default isassumed to be triggered by the companyrsquos assets crossing itsconstant debt at any time prior to maturity of the companiesrsquodebt as proposed in the seminal work of [16] then the keyelement in the price of a CDO is the default of 119896 prespecifiedcompanies simultaneously which can be expressed as follows

119864[

119896

prod

119894=1

1119884119894(119879)ltlog119863119894] = intint

Ω119896

119901 (119910 log119863119879) 119889119910 (34)

where Ω119896= 119910 | 119910

119894gt log119863

119894 119894 = 1 119896 119884 stands for the

asset value of the company 119863 represents its constant debt

6 International Journal of Stochastic Analysis

and 119899 = 119896 This type of products was highly mispriceddue in part to the mathematical complexity of handlingmultidimensions and first passage time Simpler approacheslike those assuming default only at maturity (avoiding firstpassage time) or noneconomical approaches like modellingdefault as an exogenously given process were more prone tooversight

5 Conclusions and Possible Generalizations

This paper describes the correlation matrices for which aclosed-form solution for the joint densitydistribution of theendpoints and the minimum of a Wiener process can befound The results are also applicable to other processes likelog-normal The general solution requires a detailed geo-metrical analysis of certain partitions of the 119899-dimensionalsphere therefore the technique can only provide the closed-form solutions for a specific set of correlations The resultingdensities can be used in several applications in particularanalytical expressions for prices of financial products there-fore validating the accuracy of numerical simulations

The method developed in the present paper could alsobe extended to allow for stochastic volatility and randomcorrelation It can be applied to maximums and minimumscombined as long as one extreme per dimension is con-sidered Finally the solution could be the basis for furtherapproximations like those based on perturbation theory(see [17]) which currently work under the assumption ofindependence among the processes underlying

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Sommerfeld ldquoZur analytischen theorie der warmeleitungrdquoMathematische Annalen vol 45 no 2 pp 263ndash277 1894

[2] H He W P Keirstead and J Rebholz ldquoDouble lookbacksrdquoMathematical Finance vol 8 no 3 pp 201ndash228 1998

[3] A Metzler ldquoOn the first passage problem for correlated Brown-ian motionrdquo Statistics amp Probability Letters vol 80 no 5-6 pp277ndash284 2010

[4] M Escobar S Ferrando and X Wen ldquoThree dimensionaldistribution of Brownian motion extremardquo Stochastics vol 85no 5 pp 807ndash832 2013

[5] H S M Coxeter Regular Polytopes chapter 5 and 11 MethuenLondon UK 1948

[6] J B Keller ldquoThe scope of the image methodrdquo Communicationson Pure and Applied Mathematics vol 6 pp 505ndash512 1953

[7] G Rowlands ldquoThe method of images and the solutions ofcertain partial differential equationsrdquoApplied Scientific ResearchSection B vol 8 pp 62ndash72 1960

[8] G W Bluman ldquoOn the transformation of diffusion processesinto theWiener processrdquo SIAM Journal onAppliedMathematicsvol 39 no 2 pp 238ndash247 1980

[9] H RiskenThe Fokker-Planck Equation vol 18 of Springer Seriesin Synergetics Springer Berlin Germany 2nd edition 1989

[10] D Sondermann Introduction to Stochastic Calculus for FinanceSpringer 2006

[11] J Janssen O Manca and R Manca Applied Diffusion Processesfrom Engineering to Finance John Wiley amp Sons 2013

[12] D Pooley P Forsyth K Vetzal and R Simpson ldquoUnstructuredmeshing for two asset barrier optionsrdquo Applied MathematicalFinance vol 7 pp 33ndash60 2000

[13] B Goetz M Escobar and R Zagst ldquoClosed-form pricing oftwo-asset barrier options with stochastic covariancerdquo AppliedMathematical Finance vol 21 no 4 pp 363ndash397 2014

[14] M Overhaus ldquoHimalaya Options Risk Marchrdquo 2002[15] J Hull and A White ldquoValuation of a CDO and nth to default

CDS without Monte Carlo simulationrdquo Journal of Derivativesvol 12 no 2 pp 8ndash23 2004

[16] F Black and J C Cox ldquoValuing corporate securities someeffects of bond indenture provisionsrdquo Journal of Finance vol 31pp 351ndash367 1976

[17] V Bhansali and M Wise ldquoCorrelated random walks and thejoint survival probabilityrdquo httparxivorgabs08122000

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