- 1. Efficient Fourier Pricing of Multi-Asset Options: Quasi-Monte Carlo & Domain Transformation Approach Chiheb Ben Hammouda Christian Bayer Michael Samet Antonis Papapantoleon Raúl Tempone The International Conference on Computational Finance, CWI Amsterdam, April 3, 2024
- 2. 1 Motivation, Challenges and Framework 2 Quasi-Monte Carlo with Effective Domain transformation for Fast Fourier Pricing 3 Numerical Experiments and Results 4 Conclusion 0
- 3. Setting Pricing multi-asset options: compute E[P(XT )] P(⋅): payoff function (typically non-smooth), e.g., (K: the strike price) ▸ Basket put P(x) = max(∑ d i=1 ciexi − K,0), s.t. ci > 0,∑ d i=1 ci = 1; ▸ Rainbow (E.g., Call on min): P(x) = max(min(ex1 ,...,exd ) − K,0) ▸ Cash-or-nothing put: P(x) = ∏ d i=1 1[0,Ki](exi ). XT is a d-dimensional (d ≥ 1) vector of log-asset prices at time T, following a certain multivariate stochastic model with an affine structure (e.g., Lévy models). x1 4 2 0 2 4 x 2 4 2 0 2 4 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 (a) Basket put x1 0.0 0.2 0.4 0.6 0.8 1.0 x 2 0.0 0.2 0.4 0.6 0.8 1.0 P ( x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 (b) Call on min x1 0 1 2 3 4 5 6 7 x 2 0 1 2 3 4 5 6 7 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 (c) Cash-or-nothing put Figure 1.1: Payoff functions illustration
- 4. Setting Pricing multi-asset options: compute E[P(XT )] P(⋅) is a payoff function (typically non-smooth), e.g., (K: the strike price) ▸ Basket put P(x) = max(∑ d i=1 ciexi − K,0), s.t. ci > 0,∑ d i=1 ci = 1; ▸ Rainbow (E.g., Call on min): P(x) = max(min(ex1 ,...,exd ) − K,0) ▸ Cash-or-nothing put: P(x) = ∏ d i=1 1[0,Ki](exi ). XT is a d-dimensional vector of log-asset prices at time T, following a certain multivariate stochastic model with an affine structure (e.g., Lévy models). Challenges 1 Monte Carlo (MC) method (prevalent choice) has a rate of convergence independent of the problem’s dimension and regularity of the payoff but can be very slow. 2 P(⋅) is non-smooth ⇒ deteriorates convergence of deterministic quadrature. 3 The curse of dimensionality and other issues ⇒ Most proposed Fourier pricing approaches efficient for only 1D and 2D options (Carr et al. 1999; Lewis 2001; Fang et al. 2009; Hurd et al. 2010; Ruijter et al. 2012),. . . . Aim: Empower Fourier-based pricing methods of multi-asset options 1 C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models”. In: Journal of Computational Finance 27.3 (2024), pp. 43–86. (Michael’s talk) 2 C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options”. In: arXiv preprint arXiv:2403.02832 (2024). (Today’s talk)
- 5. Setting Pricing multi-asset options: compute E[P(XT )] P(⋅) is a payoff function (typically non-smooth), e.g., (K: the strike price) ▸ Basket put P(x) = max(∑ d i=1 ciexi − K,0), s.t. ci > 0,∑ d i=1 ci = 1; ▸ Rainbow (E.g., Call on min): P(x) = max(min(ex1 ,...,exd ) − K,0) ▸ Cash-or-nothing put: P(x) = ∏ d i=1 1[0,Ki](exi ). XT is a d-dimensional vector of log-asset prices at time T, following a certain multivariate stochastic model with an affine structure (e.g., Lévy models). Challenges 1 Monte Carlo (MC) method (prevalent choice) has a rate of convergence independent of the problem’s dimension and regularity of the payoff but can be very slow. 2 P(⋅) is non-smooth ⇒ deteriorates convergence of deterministic quadrature. 3 The curse of dimensionality and other issues ⇒ Most proposed Fourier pricing approaches efficient for only 1D and 2D options (Carr et al. 1999; Lewis 2001; Fang et al. 2009; Hurd et al. 2010; Ruijter et al. 2012),. . . . Aim: Empower Fourier-based pricing methods of multi-asset options 1 C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models”. In: Journal of Computational Finance 27.3 (2024), pp. 43–86. (Michael’s talk) 2 C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options”. In: arXiv preprint arXiv:2403.02832 (2024). (Today’s talk)
- 6. Numerical Integration Methods: Sampling in [0,1]2 E[P(X(T))] = ∫Rd P(x)ρXT (x)dx ≈ ∑ M m=1 ωmP (xm). Monte Carlo (MC) 0 0.2 0.4 0.6 0.8 1 u1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u2 Tensor Product Quadrature 0 0.2 0.4 0.6 0.8 1 u1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u2 Quasi-Monte Carlo (QMC) 0 0.2 0.4 0.6 0.8 1 u1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u2 Adaptive Sparse Grids Quadrature 0 0.2 0.4 0.6 0.8 1 u1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u2
- 7. Fast Convergence: When Regularity Meets Structured Sampling Monte Carlo (MC) (-) Slow convergence: O(M− 1 2 ). (+) Rate independent of dimension and regularity of the integrand. Tensor Product Quadrature Convergence: O(M− r d ) (Davis et al. 2007). r > 0 being the order of bounded total derivatives of the integrand. Quasi-Monte Carlo (QMC) Optimal Convergence: O(M−1 ) (Dick et al. 2013). Requires the integrability of first mixed partial derivatives of the integrand. Worst Case Convergence: O(M−1/2 ). Adaptive Sparse Grids Quadrature Convergence: O(M− p 2 ) (Chen 2018). p > 1 is related to the order of bounded weighted mixed (partial) derivatives of the integrand.
- 8. Challenge 1: Original problem is non smooth (low regularity) x1 4 2 0 2 4 x 2 4 2 0 2 4 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 (a) Basket Put x1 0.0 0.2 0.4 0.6 0.8 1.0 x 2 0.0 0.2 0.4 0.6 0.8 1.0 P ( x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 (b) Call on min x1 0 1 2 3 4 5 6 7 x 2 0 1 2 3 4 5 6 7 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 (c) Cash-or-nothing Solution: Uncover the available hidden regularity in the problem 1 Analytic smoothing (Bayer et al. 2018; Ben Hammouda et al. 2020): taking conditional expectations over subset of integration variables. / Good choice not always trivial. 2 Numerical smoothing (Ben Hammouda et al. 2022): / Additional computational work! Attractive when explicit smoothing or Fourier mapping not possible. 3 Mapping the problem to the Fourier space (Today’s talk) (Ben Hammouda et al. 2024b; Ben Hammouda et al. 2024c). " Fourier transform of the density function (characteristic function) available/cheap to compute.
- 9. Better Regularity in the Fourier Space x1 4 2 0 2 4 x 2 4 2 0 2 4 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 (a) Payoff: Basket put u1 20 10 0 10 20 u 2 20 10 0 10 20 |P(u 1 , u 2 )| 1e 9 0.0 0.5 1.0 1.5 2.0 2.5 (b) Fourier Transform x1 0 1 2 3 4 5 6 7 x 2 0 1 2 3 4 5 6 7 P(x 1 , x 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 (a) Payoff: Cash-or-nothing u1 15 10 5 0 5 10 15 u 2 15 10 5 0 5 10 15 |P(u 1 , u 2 )| 0.002 0.000 0.002 0.004 0.006 (b) Fourier Transform x1 2 1 0 1 2 3 4 5 x 2 2 1 0 1 2 3 4 5 P(x 1 , x 2 ) 0 20 40 60 80 100 120 140 (a) Payoff: Call on min u1 30 20 10 0 10 20 30 u 2 30 20 10 0 10 20 30 |P(u 1 , u 2 )| 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 (b) Fourier Transform 5
- 10. Fourier Pricing Formula in d Dimensions Notation Θm,Θp: the model and payoff parameters, respectively; ̂ P(⋅): the Fourier transform of the payoff P(⋅); XT : vector of log-asset prices at time T, with extended characteristic function ΦXT (⋅); R: vector of damping parameters ensuring integrability; δP : strip of regularity of ̂ P(⋅); δX: strip of regularity of ΦXT (⋅), Assumption 1.1 1 x ↦ P(x) is continuous on Rd (Can be replaced by more regularity assumptions on the model). 2 δP ∶= {R ∈ Rd ∶ x ↦ e−⟨R,x⟩ P(x) ∈ L1 bc(Rd ) and y ↦ ̂ P(y + iR) ∈ L1 (Rd )} ≠ ∅. 3 δX ∶= {R ∈ Rd ∶ y ↦∣ ΦXT (y + iR) ∣< ∞,∀ y ∈ Rd } ≠ ∅. Proposition (Ben Hammouda et al. 2024b) Under Assumptions 1, 2 and 3, and for R ∈ δV ∶= δP ∩ δX, the value of the option price on d stocks is V (Θm,Θp) = e−rT E[P(XT)] (1) = ∫ Rd (2π)−d e−rT R(ΦXT (y + iR) ̂ P(y + iR)) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ∶=g(y;R,Θm,Θp) dy. 6
- 11. Challenge 2: The choice of the damping parameters Damping parameters, R, ensure integrability and control the regularity of the integrand. Figure 1.6: Example of a strip of analyticity of the integrand of a 2D call on min option under VG model. Parameters: θ = (−0.3,−0.3),ν = 0.5,Σ = I2. -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 R1 30 25 20 15 10 5 0 5 10 R 2 V X P (a) σ = (0.2, 0.2) -30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 R1 30 25 20 15 10 5 0 5 10 R 2 V X P (b) σ = (0.2, 0.5) 7
- 12. Challenge 2: The choice of the damping parameters Damping parameters, R, ensure integrability and control the regularity of the integrand. Solution: (Ben Hammouda et al. 2024b) and Michael’s talk Based on contour integration error estimates: Parametric smoothing of the Fourier integrand via an (generic) optimization rule for the choice of damping parameters. Near-Optimal Damping Rule (Ben Hammouda et al. 2024b) We propose an optimization rule for choosing the damping parameters R∗ ∶= R∗ (Θm,Θp) = arg min R∈δV ∥g(u;R,Θm,Θp)∥∞ = arg min R∈δV g(0Rd ;R,Θm,Θp). (2) where R∗ ∶= (R∗ 1,...,R∗ d) denotes the optimal damping parameters.
- 13. Challenge 3: Curse of dimensionality 1 Most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. 2 Complexity of (standard) TP quadrature to solve (1) ↗ exponentially with the number of underlying assets (Recall Convergence: O(M− r d )). 2 3 4 5 6 7 8 dimension 10−1 100 101 102 103 104 Runtime TP TP Figure 1.7: Call on min option under Normal Inverse Gaussian model: Runtime (in sec) versus dimension for TP for a relative error TOL = 10−2 . Solution: Effective treatment of the high dimensionality 1 (Ben Hammouda et al. 2024b): Sparsification and dimension-adaptivity techniques to accelerate convergence (Michael’s talk). 2 (Ben Hammouda et al. 2024c): Quasi-Monte Carlo (QMC) with efficient domain transformation (Today’s talk).
- 14. Challenge 3: Curse of dimensionality 1 Most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. 2 Complexity of (standard) TP quadrature to solve (1) ↗ exponentially with the number of underlying assets (Recall Convergence: O(M− r d )). 2 3 4 5 6 7 8 dimension 10−1 100 101 102 103 104 Runtime TP TP Figure 1.7: Call on min option under Normal Inverse Gaussian model: Runtime (in sec) versus dimension for TP for a relative error TOL = 10−2 . Solution: Effective treatment of the high dimensionality 1 (Ben Hammouda et al. 2024b): Sparsification and dimension-adaptivity techniques to accelerate convergence (Michael’s talk). 2 (Ben Hammouda et al. 2024c): Quasi-Monte Carlo (QMC) with efficient domain transformation (Today’s talk).
- 15. 1 Motivation, Challenges and Framework 2 Quasi-Monte Carlo with Effective Domain transformation for Fast Fourier Pricing 3 Numerical Experiments and Results 4 Conclusion 9
- 16. Quasi-Monte Carlo (QMC): Need for Domain Transformation Recall: our Fourier integrand is: g (y;R) = (2π)−d e−rT R(ΦXT (y + iR) ̂ P(y + iR)), y ∈ Rd , R ∈ δV ∶= δP ∩ δX Our Fourier integrand is in Rd BUT QMC constructions are restricted to the generation of low-discrepancy point sets on [0,1]d . ⇒ Need to transform the integration domain Using an inverse cumulative distribution function, we obtain the value of the option price on d stocks: V (Θm,Θp) = ∫ Rd g(y)dy = ∫ [0,1]d g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ =∶g̃(u;Λ) du. ▸ ψ(⋅;Λ): a probability density function (PDF) with parameters Λ. ▸ Ψ(⋅;Λ): the corresponding cumulative distribution function (CDF). 10
- 17. Randomized Quasi-Monte Carlo (RQMC) The transformed integration problem reads now: V (Θm,Θp) = ∫ [0,1]d g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ =∶g̃(u;Λ) du. (3) Once the choice of ψ(⋅;Λ) (respectively Ψ−1 (⋅;Λ)) is determined, the RQMC estimator of (3) can be expressed as follows: QRQMC N,s [g̃] ∶= 1 S S ∑ i=1 1 N N ∑ n=1 g̃ (u(s) n ;Λ), (4) ▸ {un}N n=1 is the sequence of deterministic QMC points ▸ For n = 1,...,N, {u (s) n }S s=1: obtained by an appropriate randomization of {un}N n=1, such that {u (s) n }S s=1 i.i.d ∼ U([0,1]d ). Why Randomization? ▸ Practical error estimates based on the central limit theorem. 10
- 18. Randomized Quasi-Monte Carlo (RQMC) The transformed integration problem reads now: V (Θm,Θp) = ∫ [0,1]d g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ =∶g̃(u;Λ) du. (3) Once the choice of ψ(⋅;Λ) (respectively Ψ−1 (⋅;Λ)) is determined, the RQMC estimator of (3) can be expressed as follows: QRQMC N,s [g̃] ∶= 1 S S ∑ i=1 1 N N ∑ n=1 g̃ (u(s) n ;Λ), (4) ▸ {un}N n=1 is the sequence of deterministic QMC points ▸ For n = 1,...,N, {u (s) n }S s=1: obtained by an appropriate randomization of {un}N n=1, such that {u (s) n }S s=1 i.i.d ∼ U([0,1]d ). Why Randomization? ▸ Practical error estimates based on the central limit theorem. 10
- 19. Randomized Quasi-Monte Carlo (RQMC) The transformed integration problem reads now: V (Θm,Θp) = ∫ [0,1]d g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ =∶g̃(u;Λ) du. (3) Once the choice of ψ(⋅;Λ) (respectively Ψ−1 (⋅;Λ)) is determined, the RQMC estimator of (3) can be expressed as follows: QRQMC N,s [g̃] ∶= 1 S S ∑ i=1 1 N N ∑ n=1 g̃ (u(s) n ;Λ), (4) ▸ {un}N n=1 is the sequence of deterministic QMC points ▸ For n = 1,...,N, {u (s) n }S s=1: obtained by an appropriate randomization of {un}N n=1, such that {u (s) n }S s=1 i.i.d ∼ U([0,1]d ). Why Randomization? ▸ Practical error estimates based on the central limit theorem. 10
- 20. Challenge 4: Deterioration of QMC convergence if ψ or/and Λ are badly chosen Observe: The denominator of g̃(u) = g○Ψ−1(u;Λ) ψ○Ψ−1(u;Λ) decays to 0 as uj → 0,1 for j = 1,...,d. The transformed integrand may have singularities near the boundary of [0,1]d ⇒ Deterioration of QMC convergence. −20 −15 −10 −5 0 5 10 15 20 u 0.0 0.2 0.4 0.6 0.8 g ( u ) (a) Original Fourier integrand (1) for call option under GBM 0.0 0.2 0.4 0.6 0.8 1.0 u 0 10 20 30 40 50 60 70 ̃ g ( u ) ̃ ̃ σ = 1.0 ̃ ̃ σ = 5.0 ̃ ̃ σ = 9.0 (b) Domain transformation for the integrand (1) Questions Q1: Which density to choose? Q2: How to choose its parameters? 11
- 21. How to choose ψ(⋅;Λ) (respectively Ψ−1(⋅;Λ) ) and and its parameters, Λ? For u ∈ [0,1]d ,R ∈ δV , the transformed Fourier integrand reads: g̃(u) = g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) = e−rT (2π)d R ⎡ ⎢ ⎢ ⎢ ⎣ ̂ P(Ψ−1 (u) + iR) ΦXT (Ψ−1 (u) + iR) ψ (Ψ−1(u)) ⎤ ⎥ ⎥ ⎥ ⎦ . ⇒ Sufficient to design the domain transformation to control the growth at the boundaries of the term ΦXT (Ψ−1(u)+iR) ψ(Ψ−1(u)) (Conservative choice). The payoff Fourier transforms ( ̂ P(⋅)) decay at a polynomial rate. PDFs of the pricing models (light and semi-heavy tailed models), if they exist, are much smoother than the payoff ⇒ the decay of their Fourier transforms (charactersitic functions) is faster the one of the payoff Fourier transform (Trefethen 1996; Cont et al. 2003).
- 22. How to choose ψ(⋅;Λ) (respectively Ψ−1(⋅;Λ) ) and and its parameters, Λ? For u ∈ [0,1]d ,R ∈ δV , the transformed Fourier integrand reads: g̃(u) = g ○ Ψ−1 (u;Λ) ψ ○ Ψ−1(u;Λ) = e−rT (2π)d R ⎡ ⎢ ⎢ ⎢ ⎣ ̂ P(Ψ−1 (u) + iR) ΦXT (Ψ−1 (u) + iR) ψ (Ψ−1(u)) ⎤ ⎥ ⎥ ⎥ ⎦ . ⇒ Sufficient to design the domain transformation to control the growth at the boundaries of the term ΦXT (Ψ−1(u)+iR) ψ(Ψ−1(u)) (Conservative choice). The payoff Fourier transforms ( ̂ P(⋅)) decay at a polynomial rate. PDFs of the pricing models (light and semi-heavy tailed models), if they exist, are much smoother than the payoff ⇒ the decay of their Fourier transforms (charactersitic functions) is faster the one of the payoff Fourier transform (Trefethen 1996; Cont et al. 2003).
- 23. Model-dependent Domain Transformation Solution (Ben Hammouda et al. 2024c): Effective Domain Transformation 1 Choose the density ψ(⋅;Λ) to asymptotically follow the same functional form of the characteristic function. Table 1: Extended characteristic function: ΦXT (z) = exp(iz′ X0)exp(iz′ µT)ϕXT (z), and choice of ψ(⋅). ϕXT (z),z ∈ Cd , I[z] ∈ δX ψ(y;Λ),y ∈ Rd Gaussian (Λ = Σ̃): GBM model: exp(−T 2 z′ Σz) (2π)− d 2 (det(Σ̃))− 1 2 exp(−1 2 (y′ Σ̃ −1 y)) Generalized Student’s t (Λ = (ν̃,Σ̃)): VG model: (1 − iνz′ θ + 1 2 νz′ Σz) −T /ν Γ( ν̃+d 2 )(det(Σ̃))− 1 2 Γ( ν̃ 2 )ν̃ d 2 π d 2 (1 + 1 ν̃ (y′ Σ̃y)) − ν̃+d 2 NIG model: Laplace (Λ = Σ̃) and (v = 2−d 2 ): exp(δT ( √ α2 − β′ ∆β − √ α2 − (β + iz)′∆(β + iz))) (2π)− d 2 (det(Σ̃))− 1 2 (y′ Σ̃ −1 y 2 ) v 2 Kv ( √ 2y′Σ̃ −1 y) Notation: Σ: Covariance matrix for the Geometric Brownian Motion (GBM) model. ν,θ,σ,Σ: Variance Gamma (VG) model parameters. α,β,δ,∆: Normal Inverse Gaussian (NIG) model parameters. µ is the martingale correction term. Kv(⋅): the modified Bessel function of the second kind. 13
- 24. Model-dependent Domain Transformation: Case of Independent Assets Using independence: Observe ϕXT (Ψ−1 (u)+iR) ψ(Ψ−1(u)) = ∏ d j=1 ϕ X j T (Ψ−1 (uj )+iRj ) ψj (Ψ−1(uj )) Solution (Ben Hammouda et al. 2024c): Effective Domain Transformation 1 Choose the density ψ(⋅;Λ) in the change of variable to asymptotically follow the same functional form of the extended characteristic function. 2 Select the parameters Λ to control the growth of the integrand near the boundary of [0,1]d i.e limuj →0,1 g̃(uj) < ∞,j = 1,...,d. Table 2: Choice of ψ(u;Λ) ∶= ∏ d j=1 ψj(uj;Λ) and conditions on Λ for GBM, (ii) VG and (iii) NIG. See (Ben Hammouda et al. 2024c) for the derivation. Model ψj(yj;Λ) Growth condition on Λ GBM 1 √ 2σ̃j 2 exp(− y2 j 2σ̃j 2 ) (Gaussian) σ̃j ≥ 1 √ T σj VG Γ( ν̃+1 2 ) √ ν̃πσ̃j Γ( ν̃ 2 ) (1 + y2 j ν̃σ̃j 2 ) −(ν̃+1)/2 (t-Student) ν̃ ≤ 2T ν − 1, σ̃j = ( νσ2 j ν̃ 2 ) T ν−2T (ν̃) ν 4T −2ν NIG exp(− ∣yj ∣ σ̃j ) 2σ̃j (Laplace) σ̃j ≥ 1 δT " In case of equality conditions, the integrand still decays at the speed of the payoff transform.
- 25. Should Correlation Be Considered in the Domain Transformation? 104 NxS 10−4 10−3 10−2 10−1 Relative Statistical Error ρ= −0.7 N−0.99 ρ=0 N−1.48 ρ=0.7 N−0.69 Figure 2.2: Two-dimensional call on the minimum option under the GBM model: Effect of the correlation parameter, ρ, on the convergence of RQMC. For the domain transformation, we set σ̃j = 1 √ T σj = 5, j = 1, 2. N: number of QMC points; S = 32: number of digital shifts. 15
- 26. Model-dependent Domain Transformation: Case of Correlated Assets Challenge 5: Numerical Evaluation of the inverse CDF Ψ−1 (⋅) 1 We can not evaluate the inverse CDF componentwise using the univariate inverse CDF as in the independent case (Ψ−1 d (u) ≠ (Ψ−1 1 (u1),...,Ψ−1 1 (ud))). 2 The inverse CDF is not given in closed-form for most multivariate distributions, and its numerical approximation is generally computationally expensive. Observe: For GBM model: If Z ∼ N(0,Id) ⇒ X = L̃Z ∼ N(0,Σ̃) (L̃: Cholesky factor of Σ̃) ⇒ we have Ψ−1 nor,d(u;Σ̃) = L̃Ψ−1 nor,d(u;Id) = L̃(Ψ−1 nor,1(u1),...,Ψ−1 nor,1(ud)) Solution: Avoid the expensive computation of the inverse CDF 1 We consider multivariate transformation densities, ψ(⋅,Λ), which belong to the class of normal mean-variance mixture distributions; i.e., for X ∼ ψ(⋅,Λ), we can write X = µ + WZ, with Z ∼ Nd(0,Σ), and W ≥ 0, independent of Z. 2 We use the eigenvalue or Cholesky decomposition to eliminate the dependence structure.
- 27. Model-dependent Domain Transformation: Case of Correlated Assets Challenge 5: Numerical Evaluation of the inverse CDF Ψ−1 (⋅) 1 We can not evaluate the inverse CDF componentwise using the univariate inverse CDF as in the independent case (Ψ−1 d (u) ≠ (Ψ−1 1 (u1),...,Ψ−1 1 (ud))). 2 The inverse CDF is not given in closed-form for most multivariate distributions, and its numerical approximation is generally computationally expensive. Observe: For GBM model: If Z ∼ N(0,Id) ⇒ X = L̃Z ∼ N(0,Σ̃) (L̃: Cholesky factor of Σ̃) ⇒ we have Ψ−1 nor,d(u;Σ̃) = L̃Ψ−1 nor,d(u;Id) = L̃(Ψ−1 nor,1(u1),...,Ψ−1 nor,1(ud)) Solution: Avoid the expensive computation of the inverse CDF 1 We consider multivariate transformation densities, ψ(⋅,Λ), which belong to the class of normal mean-variance mixture distributions; i.e., for X ∼ ψ(⋅,Λ), we can write X = µ + WZ, with Z ∼ Nd(0,Σ), and W ≥ 0, independent of Z. 2 We use the eigenvalue or Cholesky decomposition to eliminate the dependence structure.
- 28. Illustration GBM model : Using L̃Ψ−1 nor,d(u;Id) = L̃(Ψ−1 nor,1(u1),...,Ψ−1 nor,1(ud)) (L̃: Cholesky factor of Σ̃), we obtain ∫ Rd g(y)dy = ∫ [0,1]d g (L̃Ψ−1 nor,d(u;Id)) ψnor (L̃Ψ−1 nor,d(u;Id)) du, VG model: Observe: If Z ∼ N(0,Σ̃),Y ∼ χ2 (ν̃) ⇒ X = Z × √ ν̃ √ Y ∼ td(ν̃,0,Σ̃), with Z,Y independent ⇒ we obtain (see Proposition 3.4 in (Ben Hammouda et al. 2024c)) ∫ Rd g(u)du = ∫ +∞ 0 ⎛ ⎜ ⎜ ⎜ ⎝ ∫ [0,1]d g ( L̃⋅Ψ−1 nor,d(u;Id) √ y ) ψstu ( L̃⋅Ψ−1 nor,d (u;Id) √ y ) du ⎞ ⎟ ⎟ ⎟ ⎠ ρY (y)dy ▸ td(ν̃,0,Σ̃): generalized t-student distribution. ▸ ρY (⋅): density of χ2 (ν̃) (chi-squared) distribution. ▸ L̃: Cholesky factor of ν̃ × Σ̃ 17
- 29. Model-dependent Domain Transformation: Case of Correlated Assets Solution (Ben Hammouda et al. 2024c): Effective Domain Transformation 1 Choose the density ψ(⋅;Λ) in the change of variable to asymptotically follow the same functional form of the extended characteristic function. 2 Select the parameters Λ to control the growth of the integrand near the boundary of [0,1]d i.e limuj→0,1 g̃(uj) < ∞,j = 1,...,d. Table 3: Choice of ψ(u;Λ) ∶= ∏ d j=1 ψj(uj;Λ) and conditions on Λ for GBM, (ii) VG and (iii) NIG. See (Ben Hammouda et al. 2024c) for the derivation. Model ψ(y;Λ) Growth condition on Λ GBM Gaussian: (2π)−d 2 (det(Σ̃))−1 2 exp(−1 2(y′ Σ̃ −1 y)) TΣ − Σ̃ −1 ⪰ 0 VG Generalized Student’s t: Γ(ν̃+d 2 )(det(Σ̃))− 1 2 Γ(ν̃ 2 )ν̃ d 2 π d 2 (1 + 1 ν̃ (y′ Σ̃y)) − ν̃+d 2 ν̃ = 2T ν − d, and Σ − Σ̃ −1 ⪰ 0 or ν̃ ≤ 2T ν − d, and Σ̃ = Σ−1 NIG Laplace (v = 2−d 2 ): (2π)− d 2 (det(Σ̃))− 1 2 (y′Σ̃ −1 y 2 ) v 2 Kv ( √ 2y′Σ̃ −1 y) δ2 T2 ∆ − 2Σ̃ −1 ⪰ 0
- 30. Illustration: Case of Correlated Assets 104 NxS 10−4 10−3 10−2 10−1 Relative Statistical Error ̃ Σ=1 TΣ N−1.69 ̃ σj = 1 √Tσj N−0.69 Figure 2.3: Two-dimensional call on the minimum option under the GBM model: Effect of the correlation parameter, ρ, on the convergence of RQMC. N: number of QMC points; S = 32: number of digital shifts. 19
- 31. 1 Motivation, Challenges and Framework 2 Quasi-Monte Carlo with Effective Domain transformation for Fast Fourier Pricing 3 Numerical Experiments and Results 4 Conclusion 19
- 32. Effect of Domain Transformation on RQMC Convergence 0.0 0.2 0.4 0.6 0.8 1.0 u 0 10 20 30 40 50 60 ̃ g ( u ) ̃ ̃ σ = 1.0 ̃ ̃ σ = 5.0 ̃ ̃ σ = 9.0 (a) 103 104 NxS 10−6 10−5 10−4 10−3 10−2 10−1 100 Relative Statistical Error ̃ σ=1.0 N−0.68 ̃ σ=5.0 N−1.42 ̃ σ=9.0 N−2.26 (b) Figure 3.1: Call option under the NIG model: Effect of the parameter σ̃ of the Laplace PDF on (a) the shape of the transformed integrand g̃(u) and (b) convergence of the relative statistical error of RQMC N: number of QMC points; S = 32: number of digital shifts. Boundary growth condition: σ̃ ≥ 1 T δ = 5. 20
- 33. Effect of Domain Transformation on RQMC Convergence 0.0 0.2 0.4 0.6 0.8 1.0 u −5 0 5 10 15 20 25 30 35 ̃ g ( u ) ̃ ̃ ν = 3.0 ̃ ̃ ν = 9.0 ̃ ̃ ν = 15.0 (a) 103 104 NxS 10−8 10−7 10−6 10−5 10−4 10−3 10−2 Relative Statistical Error ̃ ν = 3.0 N−3.05 ̃ ν = 9.0 N−1.35 ̃ ν = 15.0 N−0.66 (b) Figure 3.2: Call option under the VG model: Effect of the parameter ν̃ of the t-student PDF on (a) the shape of the transformed integrand g̃(u) and (b) convergence of the RQMC error N: number of QMC points; S = 32: number of digital shifts. Boundary growth condition: ν̃ ≤ 2T ν − 1 = 9 21
- 34. RQMC In Fourier Space vs MC in Physical Space Figure 3.3: Average runtime in seconds with respect to relative tolerance levels TOL: Comparison of RQMC in the Fourier space (with optimal damping parameters and appropriate domain transformation) and MC in the physical space. 10−2 10−1 TOL 100 101 102 Runtime MC TOL−1.97 RQMC TOL−0.98 (a) 6D-VG call on min 10−2 10−1 TOL 10−1 100 101 102 Runtime MC TOL−2.0 RQMC TOL−1.13 (b) 6D-NIG call on min 22
- 35. Comparison of the Different Methods Figure 3.4: Call on min option: Runtime (in sec) versus dimensions to reach a relative error, TOL = 10−2 . RQMC in the Fourier space (with optimal damping parameters and appropriate domain transformation), TP in the Fourier space with optimal damping parameters, and MC in the physical space. 2 3 4 5 6 7 8 9 10 12 15 dimension 10−1 101 103 105 107 109 1011 Runtime RQMC TP MC MC (a) NIG model with: α = 15, βj = −3, δ = 0.2, ∆ = Id, σ̃j = √ 2 δ2T 2 2 3 4 5 6 7 8 9 10 12 15 dimension 10−2 100 102 104 106 108 1010 Runtime RQMC TP MC MC (b) VG model with: σj = 0.2, θj = −0.3, ν = 0.2, Σ = Id, ν̃ = 2T ν − d, σ̃j = 1 σj . 23
- 36. 1 Motivation, Challenges and Framework 2 Quasi-Monte Carlo with Effective Domain transformation for Fast Fourier Pricing 3 Numerical Experiments and Results 4 Conclusion 23
- 37. Conclusion 1 We empower Fourier pricing methods of multi-asset options by employing QMC with an appropriate domain transformation. 2 We desing a practical (model dependent) domain transformation strategy that prevents singularities near boundaries, ensuring the integrand retains its regularity for faster QMC convergence in the Fourier space. 3 The designed QMC-based Fourier pricing approach outperforms the MC (in physical domain) and tensor product quadrature (in Fourier space) for pricing multi-asset options across up to 15 dimensions. 4 Accompanying code for the paper can be found here: Git repository: Quasi-Monte-Carlo-for-Efficient-Fourier-Pricing-of- Multi-Asset-Options 24
- 38. Related References Thank you for your attention! 1 C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options”. In: arXiv preprint arXiv:2403.02832 (2024) 2 C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models”. In: Journal of Computational Finance 27.3 (2024), pp. 43–86 3 C. Ben Hammouda et al. “Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing”. In: Quantitative Finance (2022), pp. 1–19 4 C. Ben Hammouda et al. “Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model”. In: Quantitative Finance 20.9 (2020), pp. 1457–1473 25
- 39. References I [1] Christian Bayer, Markus Siebenmorgen, and Rául Tempone. “Smoothing the payoff for efficient computation of basket option pricing.”. In: Quantitative Finance 18.3 (2018), pp. 491–505. [2] C. Ben Hammouda, C. Bayer, and R. Tempone. “Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model”. In: Quantitative Finance 20.9 (2020), pp. 1457–1473. [3] C. Ben Hammouda, C. Bayer, and R. Tempone. “Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation”. In: arXiv preprint arXiv:2003.05708, to appear in SIAM Journal on Scientific Computing (2024). 26
- 40. References II [4] C. Ben Hammouda, C. Bayer, and R. Tempone. “Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing”. In: Quantitative Finance (2022), pp. 1–19. [5] C. Ben Hammouda et al. “Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models”. In: Journal of Computational Finance 27.3 (2024), pp. 43–86. [6] C. Ben Hammouda et al. “Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options”. In: arXiv preprint arXiv:2403.02832 (2024). [7] Peter Carr and Dilip Madan. “Option valuation using the fast Fourier transform”. In: Journal of computational finance 2.4 (1999), pp. 61–73. 27
- 41. References III [8] Peng Chen. “Sparse quadrature for high-dimensional integration with Gaussian measure”. In: ESAIM: Mathematical Modelling and Numerical Analysis 52.2 (2018), pp. 631–657. [9] Rama Cont and Peter Tankov. Financial Modelling with Jump Processes. Chapman and Hall/CRC, 2003. [10] Philip J Davis and Philip Rabinowitz. Methods of numerical integration. Courier Corporation, 2007. [11] Josef Dick, Frances Y Kuo, and Ian H Sloan. “High-dimensional integration: the quasi-Monte Carlo way”. In: Acta Numerica 22 (2013), pp. 133–288. [12] Ernst Eberlein, Kathrin Glau, and Antonis Papapantoleon. “Analysis of Fourier transform valuation formulas and applications”. In: Applied Mathematical Finance 17.3 (2010), pp. 211–240. 28
- 42. References IV [13] Fang Fang and Cornelis W Oosterlee. “A novel pricing method for European options based on Fourier-cosine series expansions”. In: SIAM Journal on Scientific Computing 31.2 (2009), pp. 826–848. [14] Thomas R Hurd and Zhuowei Zhou. “A Fourier transform method for spread option pricing”. In: SIAM Journal on Financial Mathematics 1.1 (2010), pp. 142–157. [15] Alan L Lewis. “A simple option formula for general jump-diffusion and other exponential Lévy processes”. In: Available at SSRN 282110 (2001). [16] M. J. Ruijter and Cornelis W Oosterlee. “Two-dimensional Fourier cosine series expansion method for pricing financial options”. In: SIAM Journal on Scientific Computing 34.5 (2012), B642–B671. 29
- 43. References V [17] Lloyd Nicholas Trefethen. “Finite difference and spectral methods for ordinary and partial differential equations”. In: (1996). 30