Monte Carlo methods are critical to many routines in quantitative finance such as derivatives pricing, hedging and risk metrics. Unfortunately, Monte Carlo methods are very computationally expensive when it comes to running simulations in high-dimensional state spaces where they are still a method of choice in the financial industry. Recently, Tensor Processing Units (TPUs) have provided considerable speedups and decreased the cost of running Stochastic Gradient Descent (SGD) in Deep Learning. After highlighting computational similarities between training neural networks with SGD and simulating stochastic processes, we ask in the present paper whether TPUs are accurate, fast and simple enough to use for financial Monte Carlo. Through a theoretical reminder of the key properties of such methods and thorough empirical experiments we examine the fitness of TPUs for option pricing, hedging and risk metrics computation. In particular we demonstrate that, in spite of the use of mixed precision, TPUs still provide accurate estimators which are fast to compute when compared to GPUs. We also show that the Tensorflow programming model for TPUs is elegant, expressive and simplifies automated differentiation.
Monte Carlo methods are critical to many routines in quantitative finance such as derivatives pricing, hedging and risk metrics. Unfortunately, Monte Carlo methods are very computationally expensive when it comes to running simulations in high-dimensional state spaces where they are still a method of choice in the financial industry. Recently, Tensor Processing Units (TPUs) have provided considerable speedups and decreased the cost of running Stochastic Gradient Descent (SGD) in Deep Learning. After highlighting computational similarities between training neural networks with SGD and simulating stochastic processes, we ask in the present paper whether TPUs are accurate, fast and simple enough to use for financial Monte Carlo. Through a theoretical reminder of the key properties of such methods and thorough empirical experiments we examine the fitness of TPUs for option pricing, hedging and risk metrics computation. In particular we demonstrate that, in spite of the use of mixed precision, TPUs still provide accurate estimators which are fast to compute when compared to GPUs. We also show that the Tensorflow programming model for TPUs is elegant, expressive and simplifies automated differentiation.
Monte Carlo methods are critical to many routines in quantitative finance such as derivatives pricing, hedging and risk metrics. Unfortunately, Monte Carlo methods are very computationally expensive when it comes to running simulations in high-dimensional state spaces where they are still a method of choice in the financial industry. Recently, Tensor Processing Units (TPUs) have provided considerable speedups and decreased the cost of running Stochastic Gradient Descent (SGD) in Deep Learning. After highlighting computational similarities between training neural networks with SGD and simulating stochastic processes, we ask in the present paper whether TPUs are accurate, fast and simple enough to use for financial Monte Carlo. Through a theoretical reminder of the key properties of such methods and thorough empirical experiments we examine the fitness of TPUs for option pricing, hedging and risk metrics computation. In particular we demonstrate that, in spite of the use of mixed precision, TPUs still provide accurate estimators which are fast to compute when compared to GPUs. We also show that the Tensorflow programming model for TPUs is elegant, expressive and simplifies automated differentiation.
Risk assessment and in particular derivatives pricing is one of the core areas in computational finance and accounts for a sizeable fraction of the global computing resources of the financial industry. We outline a quantum-inspired algorithm for multi-asset options pricing. The algorithm is based on tensor networks, which have allowed for major conceptual and numerical breakthroughs in quantum many body physics and quantum computation. In the proof-of-concept example explored, the tensor network approach yields several orders of magnitude speedup over vanilla Monte Carlo simulations. We take this as good evidence that the use of tensor network methods holds great promise for alleviating the computation burden of risk evaluation in the financial and other industries, thus potentially lowering the carbon footprint these simulations incur today.
Large-scale deep learning benefits from an emerging class of AI accelerators. Some of these accelerators' designs are general enough for compute-intensive applications beyond AI and Cloud TPU is one such example. In this paper, we demonstrate a novel approach using TensorFlow on Cloud TPU to simulate the two-dimensional Ising Model. TensorFlow and Cloud TPU framework enable the simple and readable code to express the complicated distributed algorithm without compromising the performance. Our code implementation fits into a small Jupyter Notebook and fully utilizes Cloud TPU's efficient matrix operation and dedicated high speed inter-chip connection. The performance is highly competitive: it outperforms the best published benchmarks to our knowledge by 60% in single-core and 250% in multi-core with good linear scaling. When compared to Tesla V100 GPU, the single-core performance maintains a ~10% gain. We also demonstrate that using low precision arithmetic---bfloat16---does not compromise the correctness of the simulation results.
Abstract Financial derivative pricing is a significant challenge in finance, involving the valuation of instruments like options based on underlying assets. While some cases have simple solutions, many require complex classical computational methods like Monte Carlo simulations and numerical techniques. However, as derivative complexities increase, these methods face limitations in computational power. Cases involving Non-Vanilla Basket pricing, American Options, and derivative portfolio risk analysis need extensive computations in higher-dimensional spaces, posing challenges for classical computers. Quantum computing presents a promising avenue by harnessing quantum superposition and entanglement, allowing the handling of high-dimensional spaces effectively. In this paper, we introduce a self-contained and all-encompassing quantum algorithm that operates without reliance on oracles or presumptions. More specifically, we develop an effective stochastic method for simulating exponentially many potential asset paths in quantum parallel, leading to a highly accurate final distribution of stock prices. Furthermore, we demonstrate how this algorithm can be extended to price more complex options and analyze risk within derivative portfolios.
Tensor processing units (TPUs) are specialized hardware accelerators developed by Google to support large-scale machine-learning tasks but they can also be leveraged to accelerate and scale other linearalgebra-intensive computations.In this paper, we demonstrate the usage of TPUs for massively parallel classical simulations of quantum many-body dynamics on long time scales.We apply our methods to study the phenomenon of Floquet prethermalization, i.e., exponentially slow heating in quantum spin chains subject to high-frequency periodic driving.We simulate the dynamics of L = 34 qubits for over 10 5 Floquet periods, corresponding to circuits with 4 × 10 6 nearest-neighbor two-qubit gates.The circuits simulated have no additional symmetries and represent a pure-state evolution in the full 2 L -dimensional Hilbert space.This is achieved by distributing the computation over 128 TPU cores.On that size TPU cluster, we find speed-ups in wall-clock run time of 230 times and 15 times when compared to reference CPU and single-graphics-processing-unit (GPU) simulations, respectively, for shorter-time 30-qubit simulations that can be handled by all three platforms.We study the computational cost of the simulations, as a function of both the number of qubits and the number of TPU cores used, up to our maximum capacity of L = 40 qubits, which requires a "full pod" of 2048 TPU cores with tens of terabytes of memory in total.For these simulations, an eight-TPU-core machine is comparable to a single A100 GPU and thus the full TPU pod is comparable to a machine with hundreds of top-of-the-line GPUs.However, the TPU pod is more energy and cost efficient and readily accessible (via Google Cloud), unlike such large many-GPU configurations.We also study the accumulation of numerical error as a function of circuit depth in very deep circuits.Our work demonstrates that TPUs can offer significant advantages for state-of-the-art simulations of quantum many-body dynamics.
Tensor Processing Units (TPUs) were developed by Google exclusively to support large-scale machine learning tasks. TPUs can, however, also be used to accelerate and scale up other computationally demanding tasks. In this paper we repurpose TPUs for the challenging problem of simulating quantum spin systems. Consider a lattice model made of N spin-21 quantum spins, or qubits, with a Hamiltonian H=∑ihi that is a sum of local terms hi and a wavefunction ∣Ψ⟩ consisting of 2N complex amplitudes. We demonstrate the usage of TPUs for both (i) computing the ground state ∣Ψgs⟩ of the Hamiltonian H, and (ii) simulating the time evolution ∣Ψ(t)⟩=e−itH∣Ψ(0)⟩ generated by this Hamiltonian starting from some initial state ∣Ψ(0)⟩. The bottleneck of the above tasks is computing the product H∣Ψ⟩, which can be implemented with remarkable efficiency utilising the native capabilities of TPUs. With a TPU v3 pod, with 2048 cores, we simulate wavefunctions ∣Ψ⟩ of up to N=38 qubits. The dedicated matrix multiplication units (MXUs), the high bandwidth memory (HBM) on each core, and the fast inter-core interconnects (ICIs) together provide performance far beyond the capabilities of general purpose processors.