Numerical valuation of Bermudan basket options via partial differential equations
aa r X i v : . [ m a t h . NA ] S e p Numerical valuation of Bermudan basket optionsvia partial differential equations
Karel J. in ’t Hout ∗ and Jacob Snoeijer ∗ September 4, 2019
Abstract
We study the principal component analysis (PCA) based approach introduced by Reisinger& Wittum [6] for the approximation of Bermudan basket option values via partial differentialequations (PDEs). This highly efficient approximation approach requires the solution of only alimited number of low-dimensional PDEs complemented with optimal exercise conditions. It isdemonstrated by ample numerical experiments that a common discretization of the pertinentPDE problems yields a second-order convergence behaviour in space and time, which is asdesired. It is also found that this behaviour can be somewhat irregular, and insight into thisphenomenon is obtained.
Key words:
Bermudan basket options, principal component analysis, finite differences, ADI scheme,convergence.
This paper deals with the valuation of Bermudan basket options. Basket options have a payoffdepending on a weighted average of different assets. Semi-closed analytic valuation formulas aregenerally lacking in the literature for these options. Consequently, research into efficient andstable methods for approximating their fair values is of much interest. The valuation of basketoptions gives rise to multidimensional time-dependent partial differential equations. Here thespatial dimension d ≥ d is large,it is well-known that this leads to a highly challenging numerical task. In the present paperwe shall consider Bermudan-style basket options and investigate a principal component analysisbased approach introduced by Reisinger & Wittum [6] and subsequently studied in e.g. [3, 4, 5]that renders this task feasible.A European-style basket option is a financial contract that provides the holder the right, butnot the obligation, to buy or sell a given weighted average of d assets at a specified future date T for a specified price K . Parameter T is called the maturity time and K the strike price ofthe option. In this paper we assume the well-known Black–Scholes model. Then the asset prices S iτ ( i = 1 , , . . . , d ) follow a multidimensional geometric Brownian motion, under the risk-neutralmeasure, given by the system of stochastic differential equations dS iτ = rS iτ dτ + σ i S iτ dW iτ (0 < τ ≤ T, ≤ i ≤ d ) . Here τ is time, with 0 being the time of inception of the option, r ≥ σ i > i = 1 , , . . . , d ) are the given volatilities and W iτ ( i = 1 , , . . . , d ) is a multidimensionalstandard Brownian motion with given correlation matrix ( ρ ij ) di,j =1 . Further, initial asset prices S i ∗ Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp,Belgium. Email: { karel.inthout,jacob.snoeijer } @uantwerpen.be . i = 1 , , . . . , d ) are given. Let u ( s, t ) = u ( s , s , . . . , s d , t ) be the fair value of a European basketoption if at time τ = T − t the i -th asset price equals s i ( i = 1 , , . . . , d ), where t is the timeremaining till maturity of the option. From financial mathematics theory it follows that u satisfiesthe d -dimensional time-dependent partial differential equation (PDE) ∂ u∂t ( s, t ) = 12 d X i =1 d X j =1 σ i σ j ρ ij s i s j ∂ u∂s i ∂s j ( s, t ) + d X i =1 rs i ∂ u∂s i ( s, t ) − ru ( s, t ) (1.1)for ( s, t ) ∈ (0 , ∞ ) d × (0 , T ]. PDE (1.1) is also fulfilled whenever s i = 0 for any given i , defining anatural boundary condition. In almost all financial applications, the correlation matrix has nonzerooff-diagonal entries, and hence, (1.1) contains mixed spatial derivative terms. At maturity time ofthe option its fair value is known and determined by the particular option contract. This yieldsthe initial condition u ( s,
0) = φ ( s ) (1.2)for s ∈ (0 , ∞ ) d . Here function φ is the given payoff of the option.A Bermudan-style basket option is a financial contract that provides the holder the right tobuy or sell a given weighted average of d assets for a specified price K at one from a specified finiteset of exercise times τ < τ < · · · < τ E = T with τ >
0. Let α e = T − τ E − e for e = 0 , , . . . , E − α E = T . Then the fair value function u of a Bermudan basket option satisfies the PDE (1.1),with the natural boundary condition, on each time interval ( α e − , α e ) for e = 1 , , . . . , E . Next,the initial condition (1.2) holds and for e = 1 , , . . . , E − u ( s, α e ) = max (cid:18) φ ( s ) , lim t ↑ α e u ( s, t ) (cid:19) (1.3)whenever s ∈ (0 , ∞ ) d . Condition (1.3) stems from the early exercise feature of Bermudan optionsand represents the optimal exercise condition. Notice that it is nonlinear. In the present paperwe shall consider the class of Bermudan basket put options. These have a payoff function of theform φ ( s ) = max K − d X i =1 ω i s i , ! (1.4)with given fixed weights ω i > P di =1 ω i = 1.An outline of the rest of this paper is as follows.Following Reisinger & Wittum [6], we first apply in Section 2.1 a useful coordinate transfor-mation to (1.1) by using a spectral decomposition of the pertinent covariance matrix. This leadsto a d -dimensional time-dependent PDE for a transformed option value function w in which eachcoefficient is directly proportional to one of the eigenvalues. In Section 2.2 this feature is employedto define a principal component analysis (PCA) based approximation e w to w . The key property of e w is that it is defined by only a limited number of one- and two-dimensional PDEs. In Section 2.3a note on the optimal exercise condition is given. Section 2.4 describes a common discretization ofthe one- and two-dimensional PDE problems by means of finite differences on a suitable nonuni-form spatial grid followed by the Brian and Douglas ADI scheme on a uniform temporal grid. Inview of the nonsmoothness of the payoff function, cell averaging and backward Euler damping areapplied.The main contribution of our paper is given in Section 3. Extensive numerical experimentsare presented where we study in detail the error of the discretization described in Section 2.4of the PCA-based approximation e w defined in Section 2.2 for Bermudan basket options. Threefinancial parameter sets from the literature are considered, with number of assets d = 5 , , Approximation approach
In this section the PDE (1.1) for a Bermudan basket option is converted into a form that is thestarting point for the solution approach discussed in the subsequent sections. In the following, theelementary functions ln, exp, tan and arctan are to be taken componentwise whenever they areapplied to vectors.Consider the covariance matrix Σ = (Σ ij ) ∈ R d × d given by Σ ij = σ i ρ ij σ j for i, j = 1 , , . . . , d .Let Q ∈ R d × d be an orthogonal matrix of eigenvectors of Σ and Λ = diag( λ , λ , . . . , λ d ) ∈ R d × d a diagonal matrix of eigenvalues of Σ such that Σ = Q Λ Q T . As in [6], consider the coordinatetransformation x ( s, t ) = Q T (ln( s/K ) − b ( t )) , (2.1)with b ( t ) = ( b ( t ) , b ( t ) , . . . , b d ( t )) T and b i ( t ) = ( σ i − r ) t for 1 ≤ i ≤ d . Define the function v by u ( s, t ) = v ( x ( s, t ) , t ) . A straightforward calculation shows that v satisfies ∂ v∂t ( x, t ) = 12 d X k =1 λ k ∂ v∂x k ( x, t ) − rv ( x, t ) (2.2)whenever x ∈ R d , t ∈ ( α e − , α e ), 1 ≤ e ≤ E . The PDE (2.2) is a pure diffusion equation, withoutmixed derivatives, and with a simple reaction term.It is convenient to perform a second coordinate transformation [6], which maps the spatialdomain R d onto the d -dimensional open unit cube, y ( x ) = 1 π arctan( x ) + 12 . (2.3)Define the function w by v ( x, t ) = w ( y ( x ) , t ) . Then it is readily verified that ∂ w∂t ( y, t ) = d X k =1 λ k (cid:20) p ( y k ) ∂ w∂y k ( y, t ) + q ( y k ) ∂w∂y k ( y, t ) (cid:21) − rw ( y, t ) (2.4)whenever y ∈ (0 , d , t ∈ ( α e − , α e ), 1 ≤ e ≤ E with p ( η ) = 12 π sin ( πη ) , q ( η ) = 1 π sin ( πη ) cos ( πη ) for η ∈ R . Clearly, the PDE (2.4) is a convection-diffusion-reaction equation without mixed derivative terms.Define the function ψ by ψ ( y, t ) = φ ( K exp [ Qx + b ( t )]) with x = tan (cid:2) π ( y − ) (cid:3) (2.5)whenever y ∈ (0 , d , t ∈ [0 , T ]. Then for (2.4) one has the initial condition w ( y,
0) = ψ ( y,
0) (2.6)together with the optimal exercise condition w ( y, α e ) = max (cid:18) ψ ( y, α e ) , lim t ↑ α e w ( y, t ) (cid:19) (2.7)3or y ∈ (0 , d and e = 1 , , . . . , E −
1. At the boundary ∂D of the spatial domain D = (0 , d we shall consider a Dirichlet condition. In Appendix A the details of its derivation are provided,where the minor Assumption A.1 on the matrix Q is made. For any given k ∈ { , , . . . , d } suchthat the entries of the k -th column of Q are all strictly positive there holds w ( y, t ) = Ke − r ( t − α e − ) (2.8)whenever y ∈ ∂D with y k = 0 and t ∈ ( α e − , α e ), 1 ≤ e ≤ E . On the complementary part of ∂D a homogeneous Dirichlet condition is valid. Let the eigenvalues of Σ be ordered such that λ ≥ λ ≥ · · · ≥ λ d ≥
0. In financial applications itoften holds that λ is significantly larger than the other eigenvalues. Motivated by this observation,Reisinger & Wittum [6] proposed a principal component analysis (PCA) based approximation ofthe exact solution w to the multidimensional PDE (2.4). To this purpose, consider w also as afunction of the eigenvalues and write w ( y, t ; λ ) with λ = ( λ , λ , . . . , λ d ) T . Set b λ = ( λ , , . . . , T and δλ = λ − b λ = (0 , λ , . . . , λ d ) T . Assuming sufficient smoothness of w , a first-order Taylor expansion at b λ yields w ( y, t ; λ ) ≈ w ( y, t ; b λ ) + d X l =2 δλ l ∂ w∂λ l ( y, t ; b λ ) . (2.9)The partial derivative ∂w/∂λ l (for 2 ≤ l ≤ d ) can be approximated by a forward finite difference, ∂ w∂λ l ( y, t ; b λ ) ≈ w ( y, t ; b λ + δλ l e l ) − w ( y, t ; b λ ) δλ l , (2.10)where e l denotes the l -th standard basis vector in R d . Combining (2.9) and (2.10), gives w ( y, t ; λ ) ≈ w ( y, t ; b λ ) + d X l =2 h w ( y, t ; b λ + δλ l e l ) − w ( y, t ; b λ ) i . Write w (1) ( y, t ) = w ( y, t ; b λ ) and w (1 , l ) ( y, t ) = w ( y, t ; b λ + δλ l e l ) . Then the PCA-based approximation reads w ( y, t ) ≈ e w ( y, t ) = w (1) ( y, t ) + d X l =2 h w (1 , l ) ( y, t ) − w (1) ( y, t ) i (2.11)whenever y ∈ (0 , d , t ∈ ( α e − , α e ), 1 ≤ e ≤ E . By definition, w (1) satisfies the PDE (2.4)with λ k being set to zero for all k = 1 and w (1 , l ) satisfies (2.4) with λ k being set to zero forall k
6∈ { , l } , which is completed by the same initial condition, optimal exercise condition andboundary condition as for w , discussed above.In financial applications one is often interested in the option value at inception in the singlepoint s = S , where S = ( S , S , . . . , S d ) T is the vector of known asset prices. Let Y = y ( x ( S , T )) ∈ (0 , d denote the corresponding point in the y -domain. Then w (1) ( Y , T ) can be obtained by solving aone-dimensional PDE on the line segment L in the y -domain that is parallel to the y -axis andpasses through y = Y . In other words, y k can be fixed at the value Y ,k whenever k = 1. Next,4 (1 , l ) ( Y ,T ) (for 2 ≤ l ≤ d ) can be obtained by solving a two-dimensional PDE on the planesegment P l in the y -domain that is parallel to the ( y , y l )-plane and passes through y = Y . Thus,in this case, y k can be fixed at the value Y ,k whenever k
6∈ { , l } .In view of the above key observation, computing the PCA-based approximation (2.11) for( y, t ) = ( Y , T ) requires solving just 1 one-dimensional PDE and d − d -dimensionalPDE whenever d is large. Moreover, the different terms in (2.11) can be computed in parallel.Reisinger & Wissmann [4] have given a rigorous analysis of the error in the PCA-based approx-imation relevant to European basket options. Under a mild assumption on the payoff function φ ,they proved that e w − w = O (cid:0) λ (cid:1) in the maximum norm. Let 1 ≤ e ≤ E − ψ e ( y ) = ψ ( y, α e ). Let y ∈ L , which forms the intersection of L and P , . . . , P d . By the optimal exercise condition (2.7), the natural approximation to w ( y, t ) at t = α e based on e w is w ( y, α e ) ≈ max (cid:16) ψ e ( y ) , lim t ↑ α e e w ( y, t ) (cid:17) = lim t ↑ α e max (cid:16) ψ e ( y ) , e w ( y, t ) (cid:17) = lim t ↑ α e max (cid:16) ψ e ( y ) , w (1) ( y, t ) + d X l =2 h w (1 , l ) ( y, t ) − w (1) ( y, t ) i (cid:17) . On the other hand, by construction of w (1) and w (1 , l ) (2 ≤ l ≤ d ), we have w ( y, α e ) ≈ e w ( y, α e )= w (1) ( y, α e ) + d X l =2 h w (1 , l ) ( y, α e ) − w (1) ( y, α e ) i = lim t ↑ α e max (cid:16) ψ e ( y ) , w (1) ( y, t ) (cid:17) + d X l =2 h max (cid:16) ψ e ( y ) , w (1 , l ) ( y, t ) (cid:17) − max (cid:16) ψ e ( y ) , w (1) ( y, t ) (cid:17)i! . It may hold that e w ( y, α e ) = max (cid:16) ψ e ( y ) , lim t ↑ α e e w ( y, t ) (cid:17) , and hence, the PCA-based approximation e w does not satisfy the optimal exercise condition. Afurther investigation into this will be the subject of future research. To arrive at the values w (1) ( Y , T ) and w (1 , l ) ( Y , T ) (for 2 ≤ l ≤ d ) in the approximation e w ( Y , T )of w ( Y , T ) we perform finite difference discretization of the pertinent one- and two-dimensionalPDEs on a (Cartesian) nonuniform spatial grid, followed by a suitable implicit time discretization.Let κ = and κ >
0. Note that the point ( κ , κ , . . . , κ ) T in the y -domain corresponds to thepoint ( K, K, . . . , K ) T in the s -domain if t = 0. For any given k ∈ { , , . . . , d } a nonuniform mesh0 = y k, < y k, < . . . < y k,m +1 = 1 in the k -th spatial direction is defined by (see e.g. [1]) y k,i = ϕ ( ξ i ) with ξ i = ξ min + i ∆ ξ, ∆ ξ = ξ max − ξ min m + 1 ( i = 0 , , . . . , m + 1) , with ϕ ( ξ ) = κ + κ sinh( ξ ) ( ξ min ≤ ξ ≤ ξ max )5nd ξ min = − sinh − ( κ /κ ) and ξ max = sinh − ((1 − κ ) /κ ) . Remark that ξ max = − ξ min since κ = . The parameter κ controls the fraction of mesh pointsthat lie in the neighborhood of κ . We make the heuristic choice κ = . The above mesh issmooth in the sense that there exist constants C , C , C > i , m ) such that themesh widths ∆ y k,i = y k,i − y k,i − satisfy C ∆ ξ ≤ ∆ y k,i ≤ C ∆ ξ and | ∆ y k,i +1 − ∆ y k,i | ≤ C (∆ ξ ) . The spatial derivatives in (2.4) are discretized using central finite difference schemes. Let f : R → R be any given smooth function, let · · · < η i − < η i < η i +1 < · · · be any given smoothmesh and denote the mesh widths by h i = η i − η i − . Then second-order approximations to thefirst and second derivatives are given by f ′ ( η i ) ≈ β i, − f ( η i − ) + β i, f ( η i ) + β i, f ( η i +1 ) ,f ′′ ( η i ) ≈ γ i, − f ( η i − ) + γ i, f ( η i ) + γ i, f ( η i +1 ) , with β i, − = − h i +1 h i ( h i + h i +1 ) , β i, = h i +1 − h i h i h i +1 , β i, = h i h i +1 ( h i + h i +1 ) , and γ i, − = 2 h i ( h i + h i +1 ) , γ i, = − h i h i +1 , γ i, = 2 h i +1 ( h i + h i +1 ) . The above two finite difference formulas are applied with η i = y k,i for 1 ≤ i ≤ m and 1 ≤ k ≤ d .Semidiscretization of the PDE for w (1 , l ) on the plane segment P l leads to a system of ordinarydifferential equations (ODEs) W ′ ( t ) = ( λ A + λ l A l ) W ( t ) (2.12)for t ∈ ( α e − , α e ), 1 ≤ e ≤ E . Here W ( t ) is a vector of dimension m and A , A l are given m × m matrices that are tridiagonal (possibly up to permutation) and correspond to, respectively, thefirst and the l -th spatial direction. The ODE system (2.12) is completed by an initial condition W (0) = Ψ and, for 1 ≤ e ≤ E −
1, an optimal exercise condition W ( α e ) = max (cid:18) Ψ e , lim t ↑ α e W ( t ) (cid:19) . Here the vector Ψ e is determined by the function ψ ( · , α e ) on P l for 0 ≤ e ≤ E −
1. The maximumof any given two vectors is to be taken componentwise.The payoff function φ given by (1.4) is continuous but not everywhere differentiable, andhence, this also holds for the function ψ given by (2.5). It is well-known that the nonsmoothnessof the payoff function can have an adverse impact on the convergence behaviour of the spatialdiscretization. To alleviate this, we employ cell averaging near the points of nonsmoothness indefining the initial vector Ψ , see e.g. [1].For the temporal discretization of the ODE system (2.12) a standard Alternating DirectionImplicit (ADI) method is applied. Consider a given step size ∆ t = T /N with integer N ≥ E anddefine temporal grid points t n = n ∆ t for n = 0 , , . . . , N . Assume that α e = t n e for some integer n e whenever e = 1 , , . . . , E −
1. Let W = Ψ and N = { n , n , . . . , n E − } . W n ≈ W ( t n ) that is successively defined for n = 1 , , . . . , N by Z = W n − + ∆ t ( λ A + λ l A l ) W n − ,Z = Z + ∆ t λ A ( Z − W n − ) ,Z = Z + ∆ t λ l A l ( Z − W n − ) ,W n = Z (if n
6∈ N ) and W n = max(Ψ e , Z ) (if n = n e ∈ N ) . (2.13)In the scheme (2.13) a forward Euler predictor stage is followed by two implicit but unidirectionalcorrector stages, which serve to stabilize the predictor stage. The two linear systems in each timestep can be solved very efficiently by using a priori LU factorizations of the pertinent matrices.As for the spatial discretization, also the convergence behaviour of the temporal discretization canbe adversely effected by the nonsmooth payoff function. To remedy this, backward Euler damping(or Rannacher time stepping) is applied at initial time as well as at each exercise date, that is,with n = 0, the time step from t n e to t n e +1 , is replaced by two half steps of the backward Eulermethod for e = 0 , , . . . , E − w (1) on the line segment L is performed completelyanalogously to the above. Then a semidiscrete system W ′ ( t ) = λ A W ( t ) is obtained with W ( t )a vector of dimension m and A an m × m tridiagonal matrix. Temporal discretization is doneusing the Crank–Nicolson scheme with backward Euler damping. In this section we investigate by ample numerical experiments the error of the discretizationdescribed in Section 2.4 of the PCA-based approximation e w ( Y , T ) defined in Section 2.2. Weconsider three parameter sets for the basket option and the underlying asset price model.Set A is given by Reisinger & Wittum [6] and has d = 5, K = 1, T = 1, r = 0 .
05 and( ρ ij ) di,j =1 = .
00 0 .
79 0 .
82 0 .
91 0 . .
79 1 .
00 0 .
73 0 .
80 0 . .
82 0 .
73 1 .
00 0 .
77 0 . .
91 0 .
80 0 .
77 1 .
00 0 . .
84 0 .
76 0 .
72 0 .
90 1 . , ( σ i ) di =1 = (cid:0) .
518 0 .
648 0 .
623 0 .
570 0 . (cid:1) , ( ω i ) di =1 = (cid:0) .
381 0 .
065 0 .
057 0 .
270 0 . (cid:1) . The eigenvalues of the corresponding covariance matrix Σ are( λ i ) di =1 = (cid:0) . . . . . (cid:1) . Hence, λ is clearly dominant.Sets B and C are taken from Jain & Oosterlee [2] and have dimensions d = 10 and d = 15,respectively. Here K = 40, T = 1, r = 0 .
06 and ρ ij = 0 . σ i = 0 .
20 and ω i = 1 /d whenever1 ≤ i = j ≤ d . Sets B and C have λ = 0 .
13 and λ = 0 .
18, respectively, and λ = . . . = λ d = 0 . λ is also dominant for these parameter sets. It can be shown that for all three Sets A, B, Cthe matrix of eigenvectors Q of Σ satisfies Assumption A.1.We consider a Bermudan basket option with E = 10 exercise times τ i = i T /E ( i = 1 , , . . . , E )and study the absolute error in the discretization of e w ( Y , T ) at the point Y = y ( x ( S , T ))with S = ( K, K, . . . , K ) T . For comparison, also the European basket option is included inthe experiments. The number of time steps is taken as N = m for the European option and N = 2 E ⌈ m/E ⌉ for the Bermudan option. 7uropean BermudanSet A 0.17577 0.18041Set B 0.83257 1.05537Set C 0.77065 0.99277Table 1: Reference values for e w ( Y , T ).Table 1 provides reference values for e w ( Y , T ), which have been computed by choosing m =1000. In the case of Set A, Reisinger & Wittum [6] obtain the approximation w ( Y , T ) ≈ . w ( Y , T ) ≈ .
06 and w ( Y , T ) ≈ .
00, respec-tively, for the Bermudan basket option. Clearly, these three approximations from the literatureagree well with our corresponding values for e w ( Y , T ) given in Table 1.Figure 1 displays the absolute error in the discretization of e w ( Y , T ) versus 1 /m for all m =10 , , , . . . , cm − with a moderate constant c , which is as desired.For the European option and Set A, the error drop in the (less important) region m ≤
20 issomewhat surprising, but it is easily explained from a change of sign in the error. Except for this,in the case of the European basket option, the error behaviour is always found to be regular andsecond-order.For the Bermudan basket option the observed error behaviour is less regular, in particularin the interesting region of large values m . To gain more insight into this phenomenon, we havecomputed separately the discretization error for the leading term w (1) ( Y , T ) and for the correctionterm P dl =2 (cid:2) w (1 , l ) ( Y , T ) − w (1) ( Y , T ) (cid:3) in e w ( Y , T ), see (2.11). Reference values for the leadingterm are given in Table 2. European BermudanSet A 0.18061 0.18407Set B 1.00043 1.17792Set C 0.94368 1.11902Table 2: Reference values for w (1) ( Y , T ).The obtained result is shown in Figure 2, where dark squares indicate the error e (1) ( m ) for theleading term and light circles the error P dl =2 (cid:2) e (1 , l ) ( m ) − e (1) ( m ) (cid:3) for the correction term. It isclear that in all six cases the error for the leading term behaves regularly and the error for thecorrection term is small compared to this (for Set A if m ≥
20, as above). For the Bermudanbasket option, however, the behaviour of the discretization error for the correction term is ratherirregular. A subsequent study shows that for any given l the error e (1 , l ) ( m ) is always very closeto the error e (1) ( m ), which is as expected, but the difference can be both positive and negative,leading to an irregular behaviour of e (1 , l ) ( m ) − e (1) ( m ). This is exacerbated when summing thesedifferences up over l = 2 , , . . . , d . Hence, the irregular behaviour of the error for the correctionterm can adversely affect the regular behaviour of the error for the leading term. We remark thatthis has been observed in many other experiments we performed for the Bermudan basket option,for example for other points Y , for other numbers of exercise times E ≥
2, for other dimensions d ≥ λ ≫ λ > · · · > λ d >
0. It is our aim of futureresearch to find a remedy for this phenomenon. 8
Conclusions
In this paper we have investigated the PCA-based approach by Reisinger & Wittum [6] for thevaluation of Bermudan basket options. This approximation approach is highly effective as it re-quires the solution of only a limited number of low-dimensional PDEs, supplemented with optimalexercise conditions. By numerical experiments the favourable result is shown that a common dis-cretization of these PDE problems leads to a second-order convergence behaviour in space andtime. It is also observed that this convergence behaviour can be somewhat irregular. Insight intothis phenomenon is obtained by regarding the total discretization error as a superposition of dis-cretization errors for the leading term and the correction term. Our aim for future research is todetermine a suitable remedy for it. Another topic for future research concerns a rigorous analysisof the error in the PCA-based approximation for Bermudan basket options. The results obtainedby Reisinger & Wissmann [4], relevant to European basket options, will be important here.
References [1] K.J. in ’t Hout.
Numerical Partial Differential Equations in Finance Explained . FinancialEngineering Explained. Palgrave Macmillan UK, 2017.[2] S. Jain and C.W. Oosterlee. The stochastic grid bundling method: efficient pricing of Bermudanoptions and their Greeks.
Appl. Math. Comp. , 269:412–431, 2015.[3] C. Reisinger and R. Wissmann. Numerical valuation of derivatives in high-dimensional settingsvia partial differential equation expansions.
J. Comp. Finan. , 18(4):95–127, 2015.[4] C. Reisinger and R. Wissmann. Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs.
ESAIM: M2AN , 51(6):2435–2463, 2017.[5] C. Reisinger and R. Wissmann. Finite difference methods for medium- and high-dimensionalderivative pricing PDEs. In
High-Performance Computing in Finance: Problems, Methods,and Solutions , pages 175–196. 2018.[6] C. Reisinger and G. Wittum. Efficient hierarchical approximation of high-dimensional optionpricing problems.
SIAM J. Sci. Comp. , 29(1):440–458, 2007.9 -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r Figure 1: Discretization error for e w ( Y , T ) in all cases of Table 1. Left: European basket option.Right: Bermudan basket option. Top: Set A. Middle: Set B. Bottom: Set C10 -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r -2 -1 -6 -5 -4 -3 -2 -1 e rr o r Figure 2: Discretization error for the leading term (dark squares) and the correction term (lightcircles) in e w ( Y , T ). Left: European basket option. Right: Bermudan basket option. Top: Set A.Middle: Set B. Bottom: Set C 11 Dirichlet boundary condition for (2.4)
Consider the following minor assumption on the matrix Q of eigenvectors of the covariance matrix Σ. Assumption A.1
Each column of Q satisfies one of the following two conditions:(a) all its entries are strictly positive;(b) it has both a strictly positive and a strictly negative entry. Then we have
Lemma A.2
Let the function ψ be given by (2.5) with φ defined by (1.4) . Let k ∈ { , , . . . , d } , t ∈ [0 , T ] and y = ( y , y , . . . , y d ) T with fixed y j ∈ (0 , whenever j = k . If the k -th column of Q satisfies (A.1.a), then ψ ( y, t ) → K as y k ↓ . If the k -th column of Q satisfies (A.1.b), then ψ ( y, t ) → as y k ↓ . Finally, ψ ( y, t ) → as y k ↑ .Proof Let x = tan (cid:2) π ( y − ) (cid:3) and s = K exp [ Qx + b ( t )], so that ψ ( y, t ) = φ ( s ).Suppose first y k ↓
0. Then x k → −∞ . If the k -th column of Q satisfies condition (A.1.a), thenall entries of Qx tend to −∞ . Consequently, all entries of s tend to zero and φ ( s ) → K . If the k -th column of Q satisfies condition (A.1.b), then the entries of Qx go to either −∞ or + ∞ withat least one entry that tends to + ∞ . It follows that the entries of s go to either zero or + ∞ withat least one entry that tends to + ∞ , and therefore φ ( s ) → y k ↑
1. Then x k → + ∞ and the entries of Qx go to either + ∞ or −∞ with atleast one entry that tends to + ∞ . Hence, φ ( s ) → (cid:4) For any given k ∈ { , , . . . , d } the diffusion and convection coefficients p ( y k ) and q ( y k ) in (2.4)vanish as y k ↓ y k ↑
1. Accordingly, (2.4) is also satisfied on each boundary part { y : y = ( y , y , . . . , y d ) T with y k = δ and y j ∈ (0 ,