You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    The rough Bergomi (rBergomi) model, introduced recently in [@bayer2016pricing], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

    **Keywords** Rough volatility, Monte Carlo, Adaptive sparse grids, Quasi Monte Carlo, Brownian bridge construction, Richardson extrapolation.

    **2010 Mathematics Subject Classification** 91G60, 91G20, 65C05, 65D30, 65D32.
author:
- 'Christian Bayer[^1]'
- 'Chiheb Ben Hammouda[^2]'
- 'Raúl Tempone[^3] [^4]'
bibliography:
- 'smoothing\_rBergomi.bib'
title: Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
---

Introduction
============

Modeling volatility to be stochastic, rather than deterministic as in the Black-Scholes model, enables quantitative analysts to explain certain phenomena observed in option price data, in particular the implied volatility smile. However, this family of models has a main drawback in failing to capture the true steepness of the implied volatility smile close to maturity. Jumps can be added to stock price models to overcome this undesired feature, for instance by modeling the stock price process as an exponential Lévy process. However, the addition of jumps to stock price processes remains controversial [@christensen2014fact; @bajgrowicz2015jumps].

Motivated by the statistical analysis of realized volatility by Gatheral, Jaisson and Rosenbaum [@gatheral2018volatility] and the theoretical results on implied volatility [@alos2007short; @fukasawa2011asymptotic], rough stochastic volatility has emerged as a new paradigm in quantitative finance, overcoming the observed limitations of diffusive stochastic volatility models. In these models, the trajectories of the volatility have lower Hölder regularity than the trajectories of standard Brownian motion [@bayer2016pricing; @gatheral2018volatility]. In fact, they are based on fractional Brownian motion (fBm), which is a centered Gaussian process, whose covariance structure depends on the so-called Hurst parameter, $H$ (we refer to [@mandelbrot1968fractional; @coutin07introduction; @biagini2008stochastic] for more details regarding the fBm processes). In the rough volatility case, where $0<H<1/2$, the fBm has negatively correlated increments and rough sample paths. Gatheral, Jaisson, and Rosenbaum [@gatheral2018volatility] empirically demonstrated the advantages of such models. For instance, they showed that the log-volatility in practice has a similar behavior to fBm with the Hurst exponent $H \approx 0.1$ at any reasonable time scale (see also [@gatheral2014volatility_2]). These results were confirmed by Bennedsen, Lunde and Pakkanen [@bennedsen2016decoupling], who studied over a thousand individual US equities and showed that $H$ lies in $(0,1/2)$ for each equity. Other works [@bennedsen2016decoupling; @bayer2016pricing; @gatheral2018volatility] showed further benefits of such rough volatility models over standard stochastic volatility models, in terms of explaining crucial phenomena observed in financial markets.

The rough Bergomi (rBergomi) model, proposed by Bayer, Friz and Gatheral [@bayer2016pricing], was one of the first developed rough volatility models. This model, depending on only three parameters, shows remarkable fit to empirical implied volatility surfaces. The construction of the rBergomi model was performed by moving from a physical to a pricing measure and by simulating prices under that model to fit the implied volatility surface well in the case of the S&P $500$ index with few parameters. The model may be seen as a non-Markovian extension of the Bergomi variance curve model [@bergomi2005smile].

Despite the promising features of the rBergomi model, pricing and hedging under such a model still constitutes a challenging and time-consuming task due to the non-Markovian nature of the fractional driver. In fact, the standard numerical pricing methods, such as PDE discretization schemes, asymptotic expansions and transform methods, although efficient in the case of diffusion, are not easily carried over to the rough setting. Furthermore, due to the lack of Markovianity and affine structure, conventional analytical pricing methods do not apply. To the best of our knowledge, the only prevalent method for pricing options under such models is Monte Carlo (MC) simulation. In particular, recent advances in simulation methods for the rBergomi model and different variants of pricing methods based on MC under such a model have been proposed in [@bayer2016pricing; @bayer2017regularity; @bennedsen2017hybrid; @mccrickerd2018turbocharging; @jacquier2018vix]. For instance, in [@mccrickerd2018turbocharging], the authors employ a novel composition of variance reduction methods. When pricing under the rBergomi model, they achieved substantial computational gains over the standard MC method. Greater analytical understanding of option pricing and implied volatility under this model has been achieved in [@jacquier2017pathwise; @forde2017asymptotics; @bayer2018short]. It is crucial to note that hierarchical variance reduction methods, such as Multi-level Monte Carlo (MLMC), are inefficient in this context, because of the poor behavior of the strong error, that is of the order of $H$ [@neuenkirch2016order].

Despite recent advances in the MC method, pricing under the rBergomi model is still computationally expensive. To overcome this issue, we design novel fast option pricers for options whose underlyings follow the rBergomi model, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. To use these two deterministic quadrature techniques (ASGQ and QMC) for our purposes, we solve two main issues that constitute the two stages of our newly designed method. In the first stage, we smoothen the integrand by using the conditional expectation, as was proposed in [@romano1997contingent] in the context of Markovian stochastic volatility models, and in [@bayersmoothing] in the context of basket options. In a second stage, we apply the deterministic quadrature method, to solve the integration problem. In this stage, we apply two hierarchical representations before using the ASGQ or QMC method, to overcome the issue of facing a high-dimensional integrand due to the discretization scheme used for simulating the rBergomi dynamics. Given that ASGQ and QMC benefit from anisotropy, the first representation consists of applying a hierarchical path generation method, based on Brownian bridge (Bb) construction, with the aim of reducing the effective dimension. The second technique consists of applying Richardson extrapolation to reduce the bias, which in turn reduces the number of time steps needed in the coarsest level to achieve a certain error tolerance and consequently the maximum number of dimensions needed for the integration problem. We emphasize that we are interested in the pre-asymptotic regime (corresponding to a small number of time steps), and the use of Richardson extrapolation is justified by conjecture \[conj: Weak error structure\] and our observed experimental results in that regime, which suggest, in particular, that we have convergence of order one for the weak error and that the pre-asymptotic regime is enough to achieve sufficiently accurate estimates for the option prices. Furthermore, we emphasize that no proper weak error analysis has been performed in the rough volatility context.

Our first contribution is that we design a novel alternative approach based on a deterministic quadrature, in contrast to the aforementioned studies such as [@mccrickerd2018turbocharging]. Given that the only prevalent option in this context is to use different variants of the MC method, our work opens a new research direction in this field, i.e., to investigate the performance of methods other than MC for pricing and calibrating under the rBergomi model. Our second contribution is that we reduce the computational cost through bias reduction by using Richardson extrapolation. Finally, assuming one targets price estimates with a sufficiently small relative error tolerance, our proposed method demonstrates substantial computational gains over the standard MC method, even for very small values of $H$. We show these gains through our numerical experiments for different parameter constellations. However, we do not claim that these gains will hold in the asymptotic regime, which requires higher accuracy. Furthermore, in this work, we limit ourselves to comparing our novel proposed methods against the standard MC. A more systematic comparison with the variant of MC proposed in [@mccrickerd2018turbocharging] is left for future research.

The outline of this paper is as follows: We begin in Section \[sec:Problem setting\] by introducing the pricing framework that we are considering in this study. We provide some details about the rBergomi model, option pricing under this model and the simulation schemes used to simulate asset prices following the rBergomi dynamics. We also explain how we choose the optimal simulation scheme for an optimal performance of our approach. In Section \[sec:Weak error analysis\], we discuss the weak error in the context of the rBergomi. Then, in Section \[sec:Details our approach and error bounds\] we explain the different building blocks that constitute our proposed methods, which are basically ASGQ, QMC, Brownian bridge construction, and Richardson extrapolation. Finally, in Section \[sec:Numerical tests\] we show the results obtained through the different numerical experiments conducted across different parameter constellations for the rBergomi model. The reported results show the promising potential of our proposed methods in this context.

Problem setting {#sec:Problem setting}
===============

Weak error discussion {#sec:Weak error analysis}
=====================

Details of our hierarchical methods {#sec:Details our approach and error bounds}
===================================

Numerical experiments {#sec:Numerical tests}
=====================

Conclusions and future work
===========================

In this work, we propose novel, fast option pricers, for options whose underlyings follow the rBergomi model as in [@bayer2016pricing]. The new methods are based on hierarchical deterministic quadrature methods: i) ASGQ using the same construction as in [@haji2016multi], and ii) the QMC method. Both techniques are coupled with Brownian bridge construction and Richardson extrapolation.

Given that the only prevalent option, in this context, is to use different variants of the MC method, which is computationally expensive, our first contribution is that we uncover the available regularity in the rBergomi model and design novel approaches based on an ASGQ and QMC. These approaches open a new research direction in this field to investigate the performance of other methods besides MC, for pricing and calibrating under the rBergomi model. Our second contribution is that we reduce the computational cost through bias reduction by using Richardson extrapolation. Finally, assuming one targets price estimates with a sufficiently small relative error tolerance, our proposed method demonstrates substantial computational gains over the standard MC method, when pricing under the rBergomi model, even for very small values of the Hurst parameter. We show these gains through our numerical experiments for different parameter constellations. We clarify that we do not claim that these gains will hold in the asymptotic regime, i.e., for higher accuracy requirements. Furthermore, the use of Richardson extrapolation is justified in the pre-asymptotic regime, in which our observed experimental results suggest a convergence of order one for the weak error. We emphasize that, to the best of our knowledge, no proper weak error analysis has been done in the rough volatility context.

In this work, we limit ourselves to compare our novel proposed method against the standard MC. A more systematic comparison against the variant of MC proposed in [@mccrickerd2018turbocharging] can be carried out but this remains for a future study. Another future research direction is to provide a reliable method for controlling the quadrature error for ASGQ which is, to the best of our knowledge, still an open research problem. This is even more challenging in our context, especially for low values of $H$. We emphasize that the main aim of this work is to illustrate the high potential of deterministic quadrature, when coupled with hierarchical representations, for pricing options under the rBergomi model. Finally, accelerating our novel methods can be achieved by using better versions of the ASGQ or QMC methods.

**Acknowledgments** C. Bayer gratefully acknowledges support from the German Research Foundation (DFG, grant BA5484/1). This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. URF/1/2584-01-01 and the Alexander von Humboldt Foundation. C. Ben Hammouda and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. The authors would like to thank Joakim Beck, Eric Joseph Hall and Erik von Schwerin for their helpful and constructive comments that greatly contributed to improving the final version of the paper.

Appendix
========

[^1]: Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.

[^2]: King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), Thuwal $23955-6900$, Saudi Arabia ([<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}).

[^3]: King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), Thuwal $23955-6900$, Saudi Arabia ([<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}).

[^4]: Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Germany.