On Multilevel Quasi-Monte Carlo Methods

Transcription

1 On Mutieve Quasi-Monte Caro Methods Candidate Number University of Oxford A thesis submitted in partia fufiment of the MSc in Mathematica and Computationa Finance Trinity 2015

2 Acknowedgements I woud ike to express my gratitude to my supervisor Professor Mike Gies for his support and for the time he has devoted to me. Without his hep, this paper woud not have been the same.

3 Abstract In this paper, we show that the use of scrambed Sobo sequences in the context of mutieve quasi-monte Caro path simuation eads to a reduction of the order of compexity, that is the computationa cost required to attain a specific accuracy ε. More specificay, Sobo points ead to a cost of O(ε p ), where p ranges from p 1.12 to p 2.01 for the options we consider, which represents an improvement over the previous best known resuts obtained by Gies and Waterhouse ([GW09]) with the use of randomised rank-1 attice rues.

4 Contents 1 Introduction 1 2 Background Quasi-Monte Caro methods Notes on impementation Low-discrepancy sequences Main exampes Mutieve quasi-monte Caro The mutieve approach QMC component Numerica Resuts European ca option Asian option Lookback option Barrier option Digita option Dimension reduction Concusions and future work 42 Appendix 43 Bibiography 59

5 List of Figures 2.1 Two-dimensiona rank-1 attice rue (128 points) Two-dimensiona Sobo sequence (128 points) Two-dimensiona Haton sequence (128 points) Uniformity of ow-discrepancy sequences in high dimensions Orders of compexity for various options using MLQMC European ca option with rank-1 attice rue European ca option with Sobo sequence Regression on the order of compexity of a European ca Comparison of BB vs PCA for a European ca Asian ca option with rank-1 attice rue Asian ca option with Sobo sequence Regression on the order of compexity of an Asian option Comparison of BB vs PCA for an Asian option Lookback option with rank-1 attice rue Lookback option with Sobo sequence Regression on the order of compexity of a ookback option Comparison of BB vs PCA for a ookback option Barrier option with rank-1 attice rue Barrier option with Sobo sequence Regression on the order of compexity of a barrier option Comparison of BB vs PCA for a barrier option Digita option with rank-1 attice rue Digita option with Sobo sequence Regression on the order of compexity of a digita option Comparison of BB vs PCA for a digita option Cost in terms of threshod α

6 1 INTRODUCTION 1 Introduction Monte Caro methods are genera samping methods which are widey used to compute expectations arising in stochastic systems. In computationa finance, they are particuary usefu to evauate the expected vaue of a contract whose payoff depends on the soution of a stochastic differentia equation. However, a major drawback of this cass of agorithms is that they can be computationay expensive, and therefore inadequate for some appications. It is ater shown that to achieve a root-mean-square error of ε, the pain Monte Caro approach with an Euer Maruyama path discretisation incurs a cost of O(ε 3 ), which can be significant for sma ε. Buiding on mutigrid ideas from the theory of iterative soutions of discretised PDEs, Gies introduced a Mutieve Monte Caro (MLMC) agorithm ([Gi08b]). He showed that the mutieve approach reduced the computationa cost to O(ε 2 (og ε) 2 ) in most cases considered, amost reaching the ower bound derived by Creutzig et a. ([CDMK09]) for the Euer Maruyama discretisation ([Gi15], 5.1). The use of a Mistein discretisation in a subsequent paper ([Gi08a]) was shown to eiminate the ogarithm term by speeding up the convergence of the variance of the mutieve estimator, and therefore improving further the order of compexity. In 2009, Gies and Waterhouse pubished a paper ([GW09]) in which they combined the mutieve approach with ideas borrowed from cassica quasi- Monte Caro theory. By using a set of 32 independenty randomised rank-1 attice rues, they obtained a significant variance reduction on the coarsest eves, and further reduced the computationa cost to approximatey O(ε 1.5 ) in the case of a Lipschitz European payoff. In this paper, we show that the use of Sobo sequences eads to significant improvements. We impement a mutieve approach with a set of 32 independenty scrambed Sobo sequences together with a Mistein path discretisation, and we show that it achieves a computationa cost of O(ε 1.23 ) in the European option case. The paper is structured in three main parts. We first reca basic notions reated to quasi-monte Caro methods and its impementation in the context of financia engineering. We then give an introduction to the mutieve approach of Gies, make the ink with quasi-monte Caro theory, and give an outine for a MLQMC agorithm. Finay, we appy the theory to price a variety of financia options, and compare our resuts with those in ([GW09]). Page 1 of 60

7 2 BACKGROUND 2 Background In this section, we reca some ideas and concepts that wi be recurrent throughout the paper. In particuar, we briefy describe quasi-monte Caro methods and their impementation, and we give an introduction to the theory of owdiscrepancy sequences. 2.1 Quasi-Monte Caro methods In this paper, we fix d N, and we et I d = [0, 1] d be the cosed, d-dimensiona unit hypercube. Consider a random variabe X, uniformy distributed on I. For an integrabe function f defined on I, the expectation of f(x) is equa to its integra. In other words, E[f(X)] = f(x) dx, and simiary for I d when d > 1. Therefore, the probem of computing the expected vaue of a random variabe can be repaced by numerica integration. To be more specific, the estimator I θ N = 1 N N f(x i ), x i U(0, 1), i=1 defined for each N is unbiased, and by the strong aw of arge numbers, it converges amost surey to E[f(X)] as N. The variance of θ N is N 1 V[f], so that the root-mean-square error (RMSE) is O(N 1/2 ). To achieve a prespecified RMSE of ε therefore requires generating N = O(ε 2 ) sampes. This can prove to be computationay expensive, especiay if evauating each sampe is costy. For exampe, this is the case in option pricing, where one is interested in the expected vaue of a quantity which is a functiona of the soution to a given stochastic differentia equation (SDE). To fix ideas, suppose we want to estimate E[f(S T )], where f is a (scaar) payoff function which we assume to be Lipschitz continuous, and where S t satisfies the mutidimensiona SDE ds t = a(s t, t) dt + b(s t, t) dw t, 0 < t < T. Consider a time grid 0 = t 0 < t 1 < < t m = T with timestep h, and et S n = S(t n ) = S(nh). The simpest approximation to this SDE is the Euer Page 2 of 60

8 2 BACKGROUND 2.1 Quasi-Monte Caro methods Maruyama path discretisation, which eads to the scheme Ŝ n+1 = Ŝn + a(ŝn, t n )h + b(ŝn, t n ) W n, and our estimate for E[f(S T )] is then the mean of the payoff vaues f(ŝt/h) across N independent path simuations. Theoretica resuts ([BT95; KP92]) show that under certain conditions on a(s t, t) and b(s t, t), the error in the expected payoff due to using a finite timestep h (the weak error) is O(h), whie the expected error in the individua paths (the strong error) is O( h). This impies that the RMSE is of the form c 1 N 1/2 +c 2 h for constants c 1, c 2 R. To achieve a RMSE of ε as before therefore requires N = O(ε 2 ) sampes, and the use of a timestep h = O(ε) eads a tota computationa cost of C = O(ε 3 ). For sma ε, this cost can be significant, outining one of the weaknesses of Monte Caro methods. A common way of reducing this high cost is the use of quasi-monte Caro (QMC) methods. The basic idea behind QMC is to repace the d-dimensiona points uniformy samped from I d by we-chosen deterministic points. Indeed, by using a more reguary distributed set of points we can achieve a better samping of the function f, and therefore obtain faster convergence. As we wi see in Section 2.3, the uniformity of a set is formay measured by its discrepancy. Roughy speaking, a d-dimensiona sequence wi be referred to as ow-discrepancy if it fis I d more uniformy than uncorreated random points. The quasi-monte Caro method then estimates the integra of a function f over a d-dimensiona hypercube with an N-point equa-weight quadrature rue I d f(u) du 1 N N f(x i ), (2.1) i=1 where (x i ) 1 i N is a d-dimensiona ow-discrepancy sequence. As we have seen above, pain Monte Caro achieves an accuracy of ε = O(N 1/2 ). One advantage of QMC is that in the best cases, it eads to O(N 1 ) convergence: in fact, it can be shown ([KS05]) that under appropriate conditions, the error in a quasi-monte Caro approximation is O(N 1+δ ) for any δ > 0. However, the use of a deterministic sequence comes at a cost, as we ose the possibiity of constructing confidence intervas, and the estimate in (2.1) is biased. One way of regaining confidence intervas is by introducing randomised QMC. The basic idea is to repeat a QMC integration M times independenty, Page 3 of 60

9 2 BACKGROUND 2.2 Notes on impementation giving estimates I 1,..., I M. Using these M reaisations and the Centra Limit Theorem then aows us to construct confidence intervas in the cassica way. To be more specific, we consider M independenty randomised ow-discrepancy sequences (x (1) i ),..., (x (M) i ) and we define estimators I m = 1 N N i=1 f(x (m) i ), m = 1,..., M. We then construct the goba estimator I = M 1 M j=1 I j, which is unbiased. This aows the estimation of the variance of the estimator in the usua way, and therefore the computation of confidence intervas. Note that the choice of M is crucia: by increasing M, one can obtain a better variance estimate using the Centra Limit Theorem, but this comes at the cost of a poorer error. In this paper, we have chosen to use M = Notes on impementation In this section, we give more detais on the impementation of quasi-monte Caro integration in a financia context. We first describe how to approximate SDEs that we encounter when modeing underying assets, and we then outine the different steps in the impementation of a quasi-monte Caro method Path discretisation In many financia appications, part of the pricing probem is to mode the behaviour of the underying assets. In our appications, the assets wi foow SDEs of the form ds = a(s, t) dt + b(s, t) dw. (2.2) For exampe, we wi consider the case of Geometric Brownian Motion (GBM), where a(s, t) = µs and b(s, t) = σs for some µ, σ R. As mentioned in Section 2.1, an Euer Maruyama path discretisation eads to a weak error of O(h) and a strong error of O( h). Athough strong convergence is usuay not important, it proves to be key for mutieve Monte Caro methods. In particuar, the best order of compexity for a given RMSE cannot be achieved with an Euer Maruyama approximation, and instead one needs to use a Mistein discretisation (this is a direct consequence of Theorem 2.1 in [GW09]). For the Page 4 of 60

10 2 BACKGROUND 2.2 Notes on impementation SDE (2.2), a Mistein path discretisation eads to the scheme Ŝ n+1 = Ŝn + a n h + b n W n + 1 b n 2 S b n(( W n ) 2 h), where a n, b n and b n / S are a evauated at (Ŝn, t n ). Under some conditions on a(s, t) and b(s, t) in (2.2), it has been shown ([KP92]) that the Mistein scheme achieves a strong convergence of order one, which is why it is more suitabe for mutieve Monte Caro methods. For more detais on the use of the Mistein discretisation in the mutieve context, see ([GW09], 3) QMC impementation As described in Section 2.1, our aim is to estimate an expected payoff E[f(S(T ))]. Consider an SDE path simuation (as one described in Section 2.2.1) with M timesteps, giving a path Ŝ. Starting with a standard M-dimensiona norma random variabe Z, we can express this expected vaue as E[ f(ŝ(t ))] = f(ŝ(t ))φ(z) dz, (2.3) where φ(z) is the density function of Z, and where dz = M i=1 dz i, with Z i being standard one-dimensiona normas. Let Φ denote the corresponding distribution function of Z, and et U i U(0, 1) for i = 1,..., M. Then, putting Z i = Φ 1 (U i ) for each i turns the integra in (2.3) into E[ f(ŝ)] = I M f( Ŝ) du. This is then approximated as in (2.1), and so the QMC procedure can be summarised as foows: U (1) Z (2) W (3) Ŝ (4) f. Step (1) consists of generating quasi-normas from quasi-uniforms, and is achieved as described above by appying Φ 1 to the quasi-uniforms. Step (3) is described in Section above, and Step (4) is simpy a matter of evauating the payoff function given the discrete states Ŝi. As a consequence, the two aspects of a randomised QMC impementation on which we wi focus are the choice of the ow-discrepancy sequence and its randomisation method (which can be seen as Step (0)) as we as the generation Page 5 of 60

11 2 BACKGROUND 2.2 Notes on impementation of Brownian increments from norma random variabes, which is Step (2). We wi devote Section 2.3 to the theory of ow-discrepancy sequences, and we wi describe specific sequences and randomisation techniques in Section 2.4. We now focus on the atter aspect of quasi-monte Caro simuation, which is the generation of Brownian increments W. Let W (t) denote a scaar standard Brownian motion, and a for a timestep h, define W n = W (nh) where n N. Then, each W n is normay distributed, and for any i j we have E[W i W j ] = t j, so that the covariance matrix for W is Σ = (Σ ij ), where Σ ij = min{t i, t j }. We now need to find a matrix L such that LL = Σ. Ceary, L is not unique, and whie the choice of L does not matter for Monte Caro appications, it is very important for quasi-monte Caro methods ([Ga03], 3.1). A first approach is to take L to be the Choesky factor of Σ, which is easiy found to be L = h This gives W n = n m=1 h Zm, so W n = W n W n 1 = h Z n and we obtain a vector of norma random variabes with mean zero and variance h. A second method is known as Principa Component Anaysis. In this case, we set L = UΛ 1/2, where U, Λ are the matrices of eigenvectors and eigenvaues respectivey, and where the eigenvaues are arranged in descending order. In other words, the nth coumn of L is λ n u n, where λ n is the nth argest eigenvaue, and u n its corresponding eigenvector. In Section 4.1, we describe a hybrid PCA method which we use for our numerica resuts ater on. The third estabished method to compute L is known as the Brownian bridge construction. The idea behind Brownian bridge is to use the first component of Z to define the termina vaue W (M), the second vaue of Z to define W (M/2) conditiona on W (M), and so on. More specificay, we et W M = T Z 1. Conditiona on W M, the midpoint vaue W M/2 is normay distributed with mean 1W 2 M and variance T/4, and so W M/2 = 1 2 W M + T/4 Z 2. Page 6 of 60

12 2 BACKGROUND 2.3 Low-discrepancy sequences Repeating this procedure, the third iteration produces W M/4 and W 3M/4, and we can carry on recursivey to generate a Brownian vaues, and therefore generate W. Note that behind this construction, there is the impicit assumption that M is a power of two. If this is not the case, the method sti works but is sighty more compex. The advantage of Brownian bridge is its fexibiity, and furthermore it is we-known that this construction is suited for ow-discrepancy methods ([Ga03], 3.1.1). More specificay, for European payoffs, Brownian bridge amost reduces the generation of Brownian increments to a one-dimensiona probem. Under a Brownian bridge construction, and as opposed to a PCA approach for exampe, the main shape of the Brownian path is determined by the vaues of the first few norma random variabes Z i (see [Ga03], 5.5). This is particuary important in ight of a resut by Koksma and Hawka (Theorem 2.1 of this paper) which gives a bound for the quasi-monte Caro error in terms of the dimension of the probem. 2.3 Low-discrepancy sequences In this section, we give a forma introduction to discrepancy theory. We then make the ink with Monte Caro methods via the Koksma Hawka inequaity. Consider a set P = {x 1,..., x N } I d. For an arbitrary subset B I d, et A(B; P ) be the number of eements of P ying in B. If B denotes a nonempty famiy of Lebesgue-measurabe set, then a genera notion of discrepancy of the set P is given by D N (B; P ) = sup A(B; P ) λ d (B) N [0, 1], B B where λ d (B) is the d-dimensiona Lebesgue measure of B (in this context, one can think of λ d (B) as the voume of the d-box B). By specifying B, we obtain various definitions of discrepancy. For exampe, we wi refer to the discrepancy of P as D N (P ) = D N (B, P ) where B is the famiy of subintervas of [0, 1) d of the form d i=1 [u i, v i ). If a the u i are identicay zero in B, then we obtain the star discrepancy of P, denoted DN (P ). For our purposes, it wi be sufficient to consider the discrepancies defined above, but we point out that severa other notions of discrepancy exist ([Nie87], 2.1). We now expain the ink with Monte Caro methods. Let S be a sequence (i.e. a set) in I d. By cassica resuts in the theory of uniform distribution of sequences ([KN74], 2.1), S is uniformy distributed if and ony if D N (S) = 0, or equivaenty DN (S) = 0. In this sense, the Page 7 of 60

13 2 BACKGROUND 2.3 Low-discrepancy sequences discrepancy is an adequate measure of the reguarity of a set, as mentioned in Section 2.1. In addition to this appea, ow-discrepancy sequences pay a centra roe in bounding the error of the approximation (2.1). The centra resut in that direction is a bound obtained by Jurjen Koksma and generaised by Edmund Hawka. First, et us reca that for a sufficienty differentiabe function f, the Hardy Krause variation of f is defined as V (f) = I d d f x 1... x d dx. For an aternative definition, see ([KN74], 2.5). We can now state the main theorem. Theorem 2.1 (Koksma Hawka inequaity). If f has bounded Hardy Krause variation V (f) on I, then for any x 1,..., x N I, we have 1 N N f(x n ) n=1 1 0 f(u) du V (f)d N(x 1,..., x N ). In short, the error induced by the quadrature rue (2.1) is bounded by a product of two terms: first, a measure of the variation of the integrand, and second a measure of how the samped points deviate from being uniform. In genera, the Koksma Hawka inequaity is tight ([Nie87], Theorem 2.12) in the sense that it is the best possibe resut, even for C functions. A second observation is that this resut is a strict bound, whereas in the case of pain Monte Caro, the best we can achieve is a probabiistic bound (by using the Centra Limit Theorem). By constructing sequences with known asymptotic discrepancy, we can use this bound to show that QMC gives potentiay significant improvements over traditiona Monte Caro. To be more specific, we wi give exampes of sequences whose star discrepancy is O(N 1 (og N) d ), where d is the dimension of the probem. For d sufficienty sma, this eads to a much better error than the usua O(N 1/2 ) of pain Monte Caro. However, it has to be noted that there are severa imitations to using the Koksma Hawka inequaity in practice. First, both V (f) and D N (x 1,..., x N ) are expensive to compute, and in some cases even more costy than evauating the origina integra. Even worse, the Hardy Krause variation is ony rarey bounded in financia appications (which happens if f is unbounded for exampe), and even when it is, the bound given above often grossy overestimates the true integration error. Page 8 of 60

14 2 BACKGROUND 2.4 Main exampes 2.4 Main exampes We describe some of the ow-discrepancy sequences we use in ater parts of the paper. In particuar, we mention randomisation techniques and resuts on asymptotic discrepancy for each sequence Lattice rues Reca that a attice Λ is a Z-modue with a finite basis {w 1,..., w n } such that any u Λ can be written as u = λ 1 w λ n w n, λ i Z. In other words, Λ is a vector space over Z spanned by {w 1,..., w n } 1. Definition 2.2. A rank-1 attice rue is a set P n = {x 1,..., x n } I d with x k = kz n mod 1, k = 0,..., n 1. where z is a generating vector with integer components coprime with n. The definition above, together with ordinary addition, turns P n into a finite cycic group. To make the ink with attices, reca that the rank of a (free) modue is the cardinaity of any basis. Equivaenty, the rank of a attice is the smaest number of cycic groups into which it may be decomposed (this foows by the Chinese Remainder Theorem). In this case, rank-1 refers to the fact that P n can ony be decomposed as the product of one cycic group, namey P n itsef. The advantage of using rank-1 attice rues is that they have ow discrepancy, in the sense defined in Section 2.3. In particuar, it has been shown that for any d 2 and N 2, there are rank-1 attice rues who achieve a discrepancy D N (P ) CN 1 (og N) d for a constant C ([Nie78], 1). Since then, severa constructions of these attice rues have been outined (e.g. [Nuy07]). In the context of QMC, randomisation is achieved via the introduction of an offset vector, a technique initiay introduced by Craney and Patterson ([CP76]). More specificay, et 1,..., M be M independent random vectors, 1 Formay, et R be an integra domain with fraction fied Q, and et K be a finitedimensiona Q-inear vector space. An R-attice in K is an R-submodue M K such that M is finitey generated of rank equa to deg(k/q), and such that M spans K as a Q- vector space (e.g. [Sam08]). Here, taking R = Z, we recover the definition given above. Page 9 of 60

15 2 BACKGROUND 2.4 Main exampes Randomised rank-1 attice rue Standard Randomised 0.6 x x 1 Figure 2.1: Two-dimensiona rank-1 attice rue (128 points) uniformy distributed in I d, and for each m = 1,..., M, define a sequence ( ) x (m) kz k = n + m mod 1, k = 0,..., n 1. This offset operation appied to our rank-1 attice rue construction gives a set of M uniformy distributed sequences ([Ga03], 5.4). These M sequences give M independent, identicay distributed and unbiased estimates of the form (2.1) and therefore aow the computing of confidence intervas as required. We give an exampe of 128 points generated with a rank-1 attice rue in Figure 2.1 above, both with and without a Craney Patterson shift Sobo sequences Here, we describe the generation of a Sobo sequence (x n ) in one dimension, foowing the origina method by Sobo outined in [BF88]. The paper describes a method which generaises up to dimension 40, but a remark by Joe and Kuo ([JK03]), which is used in Matab, extends this to d = Page 10 of 60

16 2 BACKGROUND 2.4 Main exampes To start, we need a sequence (v n ) of direction numbers, where v i = m i /2 i for m i an odd integer satisfying 0 < m i < 2 i. To specify the sequence (m n ), choose a primitive poynomia P of order d in the poynomia ring Z 2 [x], where Z 2 = Z/2Z is the group of primitive residue casses moduo 2. Suppose that P takes the form P = x d + a 1 x d a d 1 x + 1 Z 2 [x]. We then define the sequence (m n ) recursivey, m i = 2a 1 m i a 2 m i d 1 a d 1 m i d+1 2 d m i d m i d, where denotes a bit-by-bit excusive-or operation. As noted in [BF88], if P is of degree d, then m 1,..., m d can be chosen arbitrariy so ong as each m i is odd and satisfies the condition 0 < m i < 2 i given above. Subsequent vaues m d+1,... are then determined by the recurrence reation. For reference, a working exampe is given in ([BF88], 2). Finay, the Sobo sequence (x n ) can be specified by x i = b 1 v 1 b 2 v , where... b 3 b 2 b 1 is the binary representation of i. The Sobo sequences described above are exampes of (t, d)-sequences (see [Ga03], 5.1.4), which were aso introduced by Sobo. These sequences are we-understood, and for exampe their star discrepancy is known to satisfy ) DN(P (og N) d 1 (og N) ) C 1 + O (C d 2 2, N N where C 1, C 2 are constants depending ony on d ([Nie87], Theorem 4.10). Sobo sequences can be randomised in various ways. A cassica technique is the use of digita scrambing, introduced by Owen ([Owe98]), which can be appied to more genera famiies of sequences. However, in our paper, we use a different scrambing, impemented in Matab, which was deveoped by Matou sek (see [Mat98] for more detais). Figure 2.2 shows that Sobo points are more uniformy distributed on I 2 than random points. It aso demonstrates a property of Sobo points: each square on the grid above contains exacty two Sobo points, whereas they contain Page 11 of 60

17 2 BACKGROUND 2.4 Main exampes anywhere from zero to five random points. For a compete description of this phenomenon, see ([Ga03], 5.1.4) Uniformity of Sobo and random points Sobo Random 0.6 x x 1 Figure 2.2: Two-dimensiona Sobo sequence (128 points) Haton sequences A third commony cited ow-discrepancy sequence is known as the Haton sequence. Let p 2 be an integer. For n N, et n = n 0 + n 1 p + n 2 p be the base p expansion of n. This expansion is ceary finite as n is an integer, and furthermore each n i {0,..., p 1}. Define the radica inverse function φ p : N [0, 1) in base p as φ p (n) = n s p 1 s. s=0 Now et p 1,..., p d 2 be integers. We can define a Haton sequence P of dimension d in the bases p 1,..., p d as the sequence (x n ) with x i = (φ p1 (i),..., φ pd (i)) [0, 1) d, i 0. Page 12 of 60

18 2 BACKGROUND 2.4 Main exampes For d = 1, we obtain a particuar case caed the Van der Corput sequence. It can be shown ([Nie87], Theorems ) that the star discrepancy of a Haton sequence as above satisfies ( ) DN(P (og N) d (og N) d 1 ) C 1 + O, N 2, N N where C 1 is a constant depending ony on the primes p 1,..., p d and which is known expicity. It is aso known that this constant can be minimised by choosing p 1,..., p d to be the first d prime numbers ([Nie87], 3.1). Figure 2.3 beow shows 128 points generated from a Haton sequence, together with pseudo-random points Uniformity of Haton and random points Haton Random 0.6 x x 1 Figure 2.3: Two-dimensiona Haton sequence (128 points) For Haton sequences, randomisation can aso be achieved in severa ways. The technique impemented in Matab is based on a permutation of the radica inverse coefficients, which is obtained by appying a reverse-radix operation to the possibe coefficients ([KW97]). Another method, described in ([KP15], 1), is based on appying a p-adic shift using the Monna map, a generaisation of the radica inverse function defined above. Page 13 of 60

19 2 BACKGROUND 2.4 Main exampes Higher dimensions The constructions described above can be used to generate d-dimensiona sequences for any vaue of d 1. However, in practice, many ow-discrepancy sequences suffer a oss of uniformity in higher dimensions ([DKS13; SAKK11]). Figure beow shows exampes where a cross-dimensiona study of dimensiona attice rues and Sobo points reveas a decine in uniformity. 1 Randomised rank-1 attice rue 1 Scrambed Sobo points x x x x 176 Figure 2.4: Uniformity of ow-discrepancy sequences in high dimensions Note that the behaviour described above is even more pronounced for the Haton sequence, as the decine in their uniformity is inherent to their construction ([Ga03], ). As an aside, we point out that an aternative to these Haton sequences was found by Faure, who deveoped an extension to the Van der Corput sequence who does not suffer such oss of uniformity. He showed that the Faure sequences are ow-discrepancy, and furthermore that the constant term in the expression for the discrepancy goes to zero quicker than it does for the Haton sequence. Note that a compete description of his construction can be found in ([Ga03], 5.2.2). In Section 4.6, we wi describe how to anayse the effect of this curse of dimensionaity in the context of a mutieve quasi-monte Caro integration. In particuar, we wi show that for high dimensions, it might be better to combine ow-discrepancy and pseudo-random sequences to achieve better samping of the unit hypercube and faster convergence of the MLQMC agorithm. Page 14 of 60

20 3 MULTILEVEL QUASI-MONTE CARLO 3 Mutieve quasi-monte Caro In this section, we describe the mutieve quasi-monte Caro theory and outine some detais of its impementation. Sections 3.1 and 3.2 are entirey based on the founding papers by Gies ([Gi08b]) and Gies and Waterhouse ([GW09]). 3.1 The mutieve approach The initia setting is the same as the one in Section 2.1. We consider Monte Caro path simuations with different timesteps h = 2 T, for = 0,..., L. Let W t denote a given Brownian path, P an option payoff, and et P denote its approximation using a numerica path discretisation with timestep h. The basis of the mutieve approach is the trivia observation that E[ P L ] = E[ P 0 ] + L E[ P P 1 ] =1 by inearity of the expectation operator. The idea is then to estimate each of the expectations in the sum in a way which minimises the overa variance of the estimator for a given computationa cost. Let Ŷ0 be an estimator for E[ P 0 ] using N 0 sampes, and et Ŷ (for > 0) be an estimator for E[ P P 1 ] using N paths. The estimator on eve is a mean of N independent sampes, which for > 0 is given by Ŷ = 1 N N i=1 ( P (i) ) (i) P 1. P (i) P (i) 1 To minimise the variance, it is important that comes from approximations on different eves, but with the same Brownian path. Under this assumption, the variance of the estimator is given by V[Ŷ] = N 1 V, (i) (i) where V = V[ P P 1 ] is the variance of a singe sampe. The variance of the goba estimator Y = L =0 Ŷ is therefore V[Y ] = L =0 N 1 V, for a computationa cost proportiona to L =0 N h 1. Using the method of Lagrange mutipiers and treating the N as continuous variabes, the variance is minimised for a fixed computationa cost by choosing N to be proportiona to V h. The anaysis is further refined in the main theorem beow. Theorem 3.1 ([Gi08b], Theorem 3.1). Let P denote a functiona of the soution of the SDE (2.2) for a given Brownian path W (t), and et P denote the cor- Page 15 of 60

21 3 MULTILEVEL QUASI-MONTE CARLO 3.1 The mutieve approach responding approximation using a numerica discretisation with timestep h = M T. If there exist independent estimators Ŷ based on N Monte Caro sampes, and positive constants α 1, β, c 2 1, c 2, c 3 such that 1. E[ P P ] c 1 h α, 2. E[Ŷ] = { 3. E[Ŷ] c 2 N 1 h β, E[ P 0 ], = 0, E[ P P 1 ], > 0, 4. C, the computationa compexity of Ŷ, is bounded by C c 3 N h 1, then there exists a positive constant c 4 such that for any ε < e 1, there are vaues L and N for which the mutieve estimator Ŷ = L =0 Ŷ has a mean-square-error with bound [ (Ŷ ) ] 2 MSE = E E[P ] < ε 2, with a computationa compexity C with bound c 4 ε 2, β > 1, C c 4 ε 2 (og ε) 2, β = 1, c 4 ε 2 (1 β)/α, 0 < β < 1. Proof. See ([Gi08b], 3). In the case of an Euer path discretisation, Theorem 3.1 above impies that achieving a RMSE of ε incurs a computationa cost of O(ε 2 (og ε) 2 ). This is an improvement compared to the O(ε 3 ) of pain Monte Caro, but sti eaves scope for improvement. The first step is the use of a Mistein path discretisation, which achieves β > 1 above, and therefore reduces the compexity to O(ε 2 ). A further improvement, which is the centra theme of this paper, is the appication of a quasi-monte Caro eement to Gies mutieve method. Page 16 of 60

22 3 MULTILEVEL QUASI-MONTE CARLO 3.2 QMC component 3.2 QMC component In this section, we wi foow a paper by Gies and Waterhouse [GW09] in which the authors deveop a MLQMC agorithm using a randomised rank-1 attice rue. Athough we have used another kind of ow-discrepancy sequence, namey Sobo points, the theory which we describe beow is simiar for a such sequences. At eve, we define the number of sampes N to be the number of QMC points, and Ŷ is the average of P (for = 0) or P P 1 (for > 0) over the 32 sets of N quasi-monte Caro points, each set being randomised appropriatey. Note here that some authors use fewer than 32 sets, but doing so might hinder the confidence interva obtained via randomisation of the QMC component. We then compute an (unbiased) estimate of the variance V of Ŷ from the 32 different averages. Assuming first order weak convergence, the remaining bias at the finest eve E[P P L ] is approximatey equa to ŶL. To aow for the possibiity that Ŷ changes sign as increases before setting into first order asymptotic convergence, we estimate the magnitude of the bias with max{ 1 ŶL 1, ŶL }. 2 As before, we know that the RMSE is the sum of the (combined) variance of the estimator and the square of the bias. To achieve a RMSE of ε, we therefore choose to make each term smaer than ε 2 /2. The mutieve QMC agorithm can then be summarised as foows: 1. start with L = 0 2. compute V L over the 32 sets of points, using N L = 1 3. whie the combined variance is greater than ε 2 /2, doube N on the eve with argest V /(2 N ) 4. if L < 2 or the bias estimate is greater than ε/ 2, set L = L + 1 and go to step 2. As expained in ([GW09]), on one hand, doubing N wi eiminate most of the variance V. On the other hand, it wi aso incur a cost proportiona to 2 N. We therefore choose to maximise V /(2 N ), as this wi ead to an optima reduction in the variance per unit cost. Page 17 of 60

23 4 NUMERICAL RESULTS 4 Numerica Resuts In this section, we start by giving a summary of our findings in terms of the order of compexity required by the MLQMC agorithm to achieve a prespecified accuracy. We then briefy describe the particuar impementation of the MLQMC agorithm for a variety of financia options, and we then give our resuts, comparing them with rank-1 attice rues. Summary of resuts Using independenty scrambed Sobo sequences with a Mistein path discretisation, we estimated the cost required to achieve an accuracy of ε. We then performed a regression to approximate this cost as O(ε p ), where p is the order of compexity, which is dependent on the type of the option and the choice of Brownian bridge or PCA construction, as described beow. Option Method European Asian Lookback Digita Barrier BB PCA Figure 4.1: Orders of compexity for various options using MLQMC In the simpest case of a Lipschitz European payoff, we obtain p 1.23, whie the previous best known resut reported by Gies and Waterhouse was approximatey p 1.5 ([Gi15], 5.2). Furthermore, significant improvements can be observed across a option types, except the digita option. This is consistent with the resuts obtained in ([GW09]) where the authors note that the use of a QMC component does not add benefits over a pain mutieve method for the digita option. Overa, it seems that Sobo sequences aow a better samping of the payoff functions for the options we considered, and therefore a faster convergence of the MLQMC agorithm. 4.1 European ca option We first appy our mutieve quasi-monte Caro technique to a European ca option, which has a payoff P = exp( rt )(S(T ) K) +. We use an initia stock price S(0) = 1, a strike K = 1, and parameters T = 1, r = 0.05, σ = 0.2. We first reca the resuts obtained by Gies and Waterhouse with attice rues in Page 18 of 60

24 4 NUMERICAL RESULTS 4.1 European ca option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost Std QMC MLQMC ε Figure 4.2: European ca option with rank-1 attice rue Figure 4.2 and we then present our resuts with Sobo points in Figure 4.3. The soid ines in the top eft pot of Figure 4.2 show how the variance of P varies with the eve when four different numbers N of ow-discrepancy points are used, and therefore iustrate the effect of using quasi-monte Caro methods (the case N = 1 corresponding to a standard Monte Caro method). On the other hand, the dashed ines show the variance of P P 1 on different eves, again for N = 1, 16, 256 and 4096 ow-discrepancy points, and therefore iustrate the effect of combining the mutieve approach with QMC methods. Reca that the quasi-monte Caro interpretation of the estimator Ŷ from Section 3.1 is an average over N ow-discrepancy points. When using a standard Monte Caro method, the variance of the average of M points with common variance σ 2 is σ 2 /M. For exampe, the variance of the estimator for N = 4096 wi approximatey be 1/256th of the corresponding variance when N = 16. Therefore, in order to obtain a fair comparison and usefu pots Page 19 of 60

25 4 NUMERICAL RESULTS 4.1 European ca option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost Std QMC MLQMC ε Figure 4.3: European ca option with Sobo sequence of the variances of P and P P 1 for various N, we have mutipied the variance estimates by the number of ow-discrepancy points to correct for this difference in the denominator. The soid ines resuts show that using quasi-monte Caro on its own aready gives significant improvements over pain Monte Caro. The dashed ines show that combining QMC with a mutieve approach gives additiona improvements. This is particuary cear on the coarsest eves, as the benefit decreases on the finest eves. However, most of the computationa cost of the mutieve method is on the coarsest eves and so overa we sti obtain a major reduction in the computationa cost. The top right pot of both figures shows that E[ P P 1 ] = O(h ), which demonstrates that we obtain first order weak convergence as expected. On the bottom eft pot, we can see the number of sampes N per eve, which decreases as the theory predicts. Page 20 of 60

26 4 NUMERICAL RESULTS 4.1 European ca option The bottom right pot of both figures shows the behaviour of ε 2 C m for various vaues of ε, where C m is the computationa cost of the mutieve QMC agorithm, defined as C m = 32 2 N, i.e. the tota number of fine grid timesteps on a eves. In the standard QMC case, the cost C s is the product of the number of sampes required to achieve the desired variance and the number of timesteps. For the standard QMC, we see that ε 2 C s is roughy constant. However, it is ceary increasing for the mutieve case, both for attice rues and Sobo points. To have a better idea of the order of compexity, we have run a simpe (ogarithmic) regression on the cost to find the power of ε which best describes the cost. Figure 4.4 presents our resuts in the case of Sobo points and a Brownian bridge construction Order of compexity of a European option Sobo and BB O(ε 1.23 ) Cost ε Figure 4.4: Regression on the order of compexity of a European ca Regression gives a cost which is cose to O(ε 1.23 ). This is near-optima, as in the best case the error is inversey proportiona to the number of points, and therefore at best inversey proportiona to the computationa cost. Furthermore, it amost gives an improvement of two orders of compexity over the pain Monte Caro method. Page 21 of 60

27 4 NUMERICAL RESULTS 4.1 European ca option Brownian bridge vs PCA For each option considered, we wi compare the Brownian bridge construction and a hybrid PCA method based on Fast Fourier Transforms, which we now briefy describe. Consider a scaar Brownian motion W (t) and times 0 t 1 < t 2. Cassica resuts in stochastic cacuus ([Ga03], 3.1.1) impy that for any t (t 1, t 2 ) and given W (t 1 ) and W (t 2 ), the random variabe W (t) is normay distributed with E[W (t)] = W (t 1 ) + t t 1 t 2 t 1 (W (t 2 ) W (t 1 )), V[W (t)] = (t 2 t)(t t 1 ) t 2 t 1. Now, we further specify that W satisfies W (0) = 0, we set W N = T Z 1, and we consider a time grid 0 = t 0 < t 1 < t 2 < < t N 1 < t N = 1, so that W n = W (n/n) = W (t n ). By the resuts above, it is easy to show that the covariance matrix Ω for the discrete Brownian vaues W n is given by Ω ij = min{t i, t j } t i t j. In particuar, we can compute Ω 1, and so the eigenvaues and unit eigenvectors of Ω are λ i = 1 ( ( iπ sin 4N 2N (v i ) j = 2 2N sin ( ijπ N )) 2, respectivey, where i, j = 1,..., N 1. Now in the hybrid construction, the discrete Brownian vaues W n are defined for each n as W n = n N W N + N 1 i=1 ), Z i+1 λi (v i ) n, where λ i is an eigenvaue of Ω, (v i ) n the nth component of the corresponding unit eigenvector, and the Z i are independent standard norma random variabes. The ink with Fast Fourier Transforms is made expicit if we define a i = 2 2N Z i+1 λi for each i. Indeed, this eads to W n = n N W N + N 1 i=1 a i sin ( ) inπ, n = 1,..., N 1. N Page 22 of 60

28 4 NUMERICAL RESULTS 4.1 European ca option We can then appy a discrete sine transform to the coefficients a i and produce the discrete Brownian vaues W 1,..., W N 1 as required. The advantage of this hybrid method is that when N is a power of 2, this resuts in a much faster agorithm than a standard PCA construction. In the case of a European ca option, we obtain the foowing resuts BB vs PCA comparison for a European ca 10 5 Lattice and BB Sobo and BB Lattice and PCA Sobo and PCA Cost ε Figure 4.5: Comparison of BB vs PCA for a European ca The first observation here is that Sobo sequences outperform rank-1 attice rues for this type of option. The second observation is that BB and PCA produce simiar resuts. The ony difference seems to be that Brownian bridge is sighty more efficient for smaer vaues of ε whie PCA is more performant on arger vaues of ε. In any case, it seems compicated to concude anything on that behaviour from these resuts aone. Page 23 of 60

29 4 NUMERICAL RESULTS 4.2 Asian option 4.2 Asian option We consider an Asian option with discounted payoff P = exp( rt )(S K) +, where S is the (continuous) arithmetic average of S. As described by Gies and Waterhouse ([GW09], 6.2), we have for the fine path Ŝ = 1 T n T 1 n=0 ( ) 1 2 h(ŝn + Ŝn+1) + b n I n, where I n = tn+1 t n (W (t) W (t n )) dt 1 2 h W is a N (0, h 3 /12) random variabe, independent of W. The approximation for the coarse path is simiar, except that the vaues for I n are derived from the fine path vaues. Noting that tn+2h t n (W (t) W (t n )) dt h(w (t n + 2h) W (t n )) = + tn+h t n (W (t) W (t n )) dt 1 2 h(w (t n + h) W (t n )) tn+2h t n+h (W (t) W (t n + h)) dt 1 2 h(w (t n + 2h) W (t n + h)) h(w (t n + h) W (t n )) 1 2 h(w (t n + 2h) W (t n + h)), we deduce that I c = I f1 + I f h( W f1 W f2 ), where I c is the vaue for the coarse timestep, I f1, W f1 are the vaues for the first fine timestep, and I f2, W f2 are the vaues for the second fine timestep. Again, we first mention the existing computations for rank-1 attice rues, and we then present our resuts for Sobo sequences. As in the European case, we take S(0) = 1, K = 1, and parameters T = 1, r = 0.05, σ = 0.2. Page 24 of 60

30 4 NUMERICAL RESULTS 4.2 Asian option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost Std QMC MLQMC ε Figure 4.6: Asian ca option with rank-1 attice rue The graphs of Figure 4.6 are simiar to those in obtained for European options. More specificay, the top eft pot shows that both the mutieve approach and the QMC eement reduce the variance of the estimator. Furthermore, the benefits of the mutieve approach are more significant on the coarsest eves. The top eft graph of Figure 4.7 shows that on the coarsest eves, Sobo sequences outperform rank-1 attice rues, with the benefits decreasing across the eves. In addition, we see that ε 2 Cost is increasing in the bottom right pots of both figures, but it is hard to quantify the exact computationa cost. Page 25 of 60

31 4 NUMERICAL RESULTS 4.2 Asian option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost Std QMC MLQMC ε Figure 4.7: Asian ca option with Sobo sequence Regression for the Asian option is given in Figure 4.8, and shows that the cost of our probem is approximatey O(ε 1.12 ). Again this is near-optima, and shows again the advantage of using QMC in the mutieve setting. The BB vs PCA graph given in Figure 4.9 confirms our resuts that Sobo sequences outperform rank-1 attice rues. Furthermore, there is very itte difference between the two methods described to generate Brownian vaues: indeed, Brownian bridge and PCA constructions ead to very simiar resuts. Page 26 of 60

32 4 NUMERICAL RESULTS 4.2 Asian option 10 5 Order of compexity of an Asian option Sobo and BB O(ε 1.12 ) Cost ε Figure 4.8: Regression on the order of compexity of an Asian option 10 6 BB vs PCA comparison for an Asian option 10 5 Lattice and BB Sobo and BB Lattice and PCA Sobo and PCA Cost ε Figure 4.9: Comparison of BB vs PCA for an Asian option Page 27 of 60

33 4 NUMERICAL RESULTS 4.3 Lookback option 4.3 Lookback option We now consider a ookback option, with discounted payoff ( ) P = exp( rt ) S(T ) min S(t). 0<t<T For the fine path cacuation, it is a standard resut ([Ga03], 6.4) that the minimum vaue can be given as ( Ŝn,min f = 1 ) ) 2 Ŝn f + 2 Ŝf n+1 (Ŝf n+1 Ŝf n 2b 2 nh og U n, where U n U(0, 1) is a uniform random variabe. We therefore obtain an approximation to min [0,T ] S(t) by taking the minimum over a timesteps, and therefore an approximation P to the payoff. For the coarse path cacuation, P 1 is defined in a simiar way, except that we used an interpoated midpoint in the Mistein discretisation ([GW09], 3). This eads to { ( ) Ŝm,min c 1 ) 2 = min Ŝm c + 2 Ŝc (Ŝc m+ 1 2 m+ 1 Ŝc m b 2 mh og U 2m 1, ( ) ) 2 Ŝ c + m+ 1 Ŝc m+1 (Ŝc m+1 Ŝc 2 m+ b 2 mh og U 1 2m }. 2 Note that the uniform random variabes U 2m 1 and U 2m in the coarse path cacuation are re-used from the fine path cacuation. This ensures that the minimum from the coarse path is cose to the minimum from the fine path, giving a ow variance for P P 1. Figures 4.10 and 4.11 give the numerica resuts for S(0) = 1, T = 1, r = 0.05, and σ = 0.2. Page 28 of 60

34 4 NUMERICAL RESULTS 4.3 Lookback option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost 10 0 Std QMC MLQMC ε Figure 4.10: Lookback option with rank-1 attice rue An interesting observation from the top eft graph in Figure 4.10 above is that the benefit of combining QMC with the mutieve method ony appears on the two or three coarsest eves (see dashed ines), as compared to the first five or six eves in the previous two cases. In addition, when ooking at QMC on its own (soid ines), the benefit aso decreases across the eves, whereas it was approximatey constant for European and Asian options. However, rank-1 attice rues sti provide an improvement over a pain MLMC agorithm, as wi be cear with Figures 4.12 and Page 29 of 60

35 4 NUMERICAL RESULTS 4.3 Lookback option 0 0 og 2 (variance) og 2 ( mean ) P P P N 10 5 ε = ε = ε = ε = ε = ε 2 Cost 10 0 Std QMC MLQMC ε Figure 4.11: Lookback option with Sobo sequence The resuts in Figure 4.11 are quaitativey simiar to those in Figure As in the previous cases, we perform a regression on the order of compexity, and we compare the BB and PCA constructions. Regression for the ookback option is given in Figure 4.12 beow, and shows that the cost of our probem is approximatey O(ε 1.58 ) for the ookback option. Athough this is not as ow as the European case, it is sti much ower than the O(ε 2 ) ower bound achieved without QMC component. The BB vs PCA graph given in Figure 4.13 confirms our resuts that Sobo sequences outperform rank-1 attice rues. Furthermore, Brownian bridge and PCA constructions once again ead to simiar resuts. Page 30 of 60

36 4 NUMERICAL RESULTS 4.3 Lookback option 10 6 Order of compexity of a Lookback option Sobo and BB O(ε 1.58 ) 10 5 Cost ε Figure 4.12: Regression on the order of compexity of a ookback option 10 6 BB vs PCA comparison for a Lookback option 10 5 Lattice and BB Sobo and BB Lattice and PCA Sobo and PCA Cost ε Figure 4.13: Comparison of BB vs PCA for a ookback option Page 31 of 60

37 4 NUMERICAL RESULTS 4.4 Barrier option 4.4 Barrier option For the barrier option case, we consider a down-and-out ca with discounted payoff P = exp( rt )(S(T ) K) + 1 τ>t, where 1 τ>t denotes an indicator function which is 1 if τ > T and 0 otherwise, and where τ is the barrier crossing time, i.e. τ = inf t>0 S(t) < B. For the fine path simuation, the conditiona expectation of the payoff ([Ga03], 6.4) can be expressed as exp( rt )(Ŝf n T n T 1 K) + n=0 p n, where p n denotes the probabiity that the interpoated path did not cross the barrier at timestep n, and which equas ( ) p n = 1 exp 2(Ŝf n B) + (Ŝf n+1 B) +. b 2 nh For the coarse path cacuation, we use the midpoint trick as for the ookback option case. Given the vaue Ŝc at each timestep, the probabiity that the m+ 1 2 Brownian interpoation path does not cross the barrier during the mth (coarse) timestep is p m c = 1 exp 2(Ŝc m B) + (Ŝc B) + m+ 1 2 b 2 mh 1 exp 2(Ŝc B) + m+ 1 (Ŝc m+1 B) + 2 b 2 mh. We give the numerica resuts with S(0) = 1, K = 1, B = 0.85 and the usua parameters T = 1, r = 0.05, σ = 0.2 in Figures 4.14 and Page 32 of 60