How to tackle market challenges in quant finance? - QuantMinds 2021 Q4 eMagazine
Key insights from our contributors at QuantMinds International and beyond!
How to tackle market challenges in quant finance?
Insights from QuantMinds International and beyond
Introduction
After two years apart, we're thrilled to be back at the most significant event for quants. We look forward to welcoming you on Monday 6 Dec, but in the meantime, we are very happy to share some of our most valued collaborators' work in this eMagazine, including our partners' latest technological and strategic innovations. We hope you enjoy!
Expert insights:
Igor Halperin
Maxim Kartamyshev
Peter Quell
Andrey Itkin and Dmitry Muravey
Downloadable papers:
Jessica James, Michael Leister, and Christoph Rieger
Kathrin Glau, Christian Pötz, Mikhail Soloveitchik, and Linus Wunderlich
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Experts' quick takes
Interviews from Barcelona and beyond
Jessica James
Managing Director, Senior Quantitative Researcher, Commerzbank AG
What can we learn about super long bonds by studying a bit of history?
Kathrin Glau
Senior Lecturer in Financial Mathematics, Queen Mary University of London
Chebyshev methods: What are they and what can they do in practice?
Matthew Rooney
VP, Head of Quant Analytics Europe, Selby Jennings
Quant recruitment strategies: How can your firm stand out?
Arnaud de Chavagnac
Head of Cloud, Technology, and Services Marketing, Murex
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Towards non-equilibrium and non-perturbative finance
Igor Halperin, VP, AI Asset Management, Fidelity Investments
Igor Halperin, VP, AI Asset Management, Fidelity Investments
Financial markets are highly complex systems whose dynamics are driven by interactions of a large (or very large) number of market participants that pursue different objectives, as well as operate at different time scales. Furthermore, they are examples of open, rather than closed, systems. In particular, money is not a conserved quantity in finance because of new cash pumped into the market by retail investors, mostly via investment into pension funds or individual brokerage accounts.
As recent Robinhood meme stocks stories demonstrated, when ejected in large volumes over a short period of time, new cash inflows can produce very strong (and nonlinear) distortions of equity prices away from their fundamental values. Strong effects of new cash infusions as an origin of stock price fluctuations were proposed in a recent influential paper by X. Gabaix and R.S.J. Koijen [1], and links of such price impact mechanics with the market microstructure dynamics were proposed by J.F. Bouchaud [2] (see also the related Bloomberg story [3]).
This article deals with further effects caused by inflows of new money in the market. More specifically, I will argue that they result in non-linear market dynamics.
Recall that all complex systems in the natural sciences are systems with non-linear dynamics.
Non-linearity of dynamics implies, among other things, that small changes of the state of the system, due to either internal fluctuations or external control, can produce large changes in the state of the system. The meme stock craze can be viewed as an indication of an important role of non-linearity of market dynamics ensuing in the regime of large cash inflows. These events clearly demonstrated that the market often behaves in a highly non-equilibrium and non-linear regime.
On the other hand, most models employed by practitioners, such as factor models (CAMP, APT, etc.) for portfolio management, or derivatives pricing models that grew out of the celebrated Geometric Brownian Motion (GBM) model of Samuelson, are linear models. For example, the GBM model postulates that both the drift and diffusion functions of the price process are linear functions of the asset price. It can be considered as a particular specification of a stochastic process called Ito’s diffusion, with linear drift and diffusion functions. Its later extensions such as local or stochastic volatility, jump-diffusion, Levy models etc. enrich the ensuing price dynamics by considering progressively more complex models of the diffusion term, for example by making it dependent on additional risk factors, but they do not modify the linear state dependence of a drift term. Equivalently, a linear drift term for prices can be expressed as a constant drift for log-returns.
Of course, both the academics and practitioners are well aware that non-linear effects often arise in various financial settings, e.g. in models of market impact and transaction costs, as well as in certain models of incomplete markets and XVA models. On the other hand, it is also commonly believed that non-linear effects can usually be tackled using the means of perturbation theory methods, where nonlinear terms are considered as small perturbations around a linear model. The latter linear model (such as the GBM model) would then serve as a ‘zero-order’ approximation to the actual dynamics. Effects due to financial frictions and/or new money inflows would then be treated as corrections that could be obtained using methods of perturbation theory.
However, such an approach has a problem, as becomes apparent if we analyze the GBM model (or its later extensions) using insights provided by physics. More specifically, notice that the GBM model and its extensions can be interpreted as the so-called overdamped Langevin equations known in physics since the 1908 work of the French physicist Paul Langevin. Langevin extended the free Brownian motion theory of Einstein to the case of Brownian particles experiencing diffusion in an external potential field (which can be caused by other heavy molecules, electric or magnetic field, etc.). One of the simplest specifications of a potential in physics is a quadratic function of the particle position, which is known as a harmonic oscillator potential. Non-linear interactions in physical systems amenable to modeling within the Langevin approach typically give rise to higher order non-linearities, producing cubic or quartic potentials, or even non-polynomial potentials. Potentials of this sort commonly appear in many important problems in statistical and quantum physics. In particular, processes of a thermally-induced or a quantum mechanical escape (tunneling) from a potential well are often described using a quartic potential with two minima, called a double well potential in physics.
Back to finance, the Langevin dynamics of a particle in a potential is mathematically equivalent to Ito’s diffusion, but it gives a physics-provided interpretation to the drift term. More specifically, the drift term in the mathematical construction of Ito’s diffusion is interpreted as a negative gradient of the Langevin potential function. While this simple observation is based on comparison of two very famous equations, apparently it did not attract the due attention of the wider mathematical finance community. In the meantime, it produces a critically important observation, namely that the linear drift term of the classical GBM model and its offspring is equivalent to an inverted harmonic potential, or equivalently a harmonic potential with a negative mass!
The reason I believe that this observation is critically important is that an inverted harmonic potential describes a globally unstable system, whose (unstable) dynamics continue indefinitely. On the other hand, no systems in physics ever produce such globally unstable dynamics. Unstable systems in physics exist only for relatively short times, and arise in some models of the early universe, or in models of lasers, for example. On longer time horizons, non-linear effects in physical systems stabilize dynamics around some globally stable or metastable (i.e. very long-lived) states corresponding, respectively, to global or local minima of the potential.
What happens if we follow the logic of perturbation theory methods, and try to treat all possible non-linear effects in the price dynamics as small perturbations around the linear ‘GBM limit’ corresponding to a linear drift, or equivalently an inverted harmonic potential?
The problem is that such a ‘zero-order’ limit produces highly problematic behaviour in both the long time and small price limits. In the long-time limit, it implies an unlimited exponential (or average) growth of the stock price, while in the small price limit it implies a presence of a totally fictitious and counter-intuitive force that somehow ‘saves’ the firm from attaining the zero price – thus preventing the possibility to model corporate defaults within the same diffusion-based approach. While such linear dynamics can approximate the true non-linear dynamics over short time scales, the very range of time scales where such approximations are reasonably accurate should come from the underlying non-linear model itself. The main problem of linear models such as the GBM model is that when viewed as ‘fundamental’ models, they give no indication of their suggested range of applicability. In a way, this is analogous to how the equations of Newtonian mechanics give no indication that they have to be replaced by equations of quantum mechanics for very small distances, or equations of the special relativity theory for very high velocities.
The undesirable behaviour at very small or very large prices could be cured if the stock drift corresponded to a non-linear Langevin potential with global and local minima. But what physical mechanism could produce such non-linear potentials? A possible solution was proposed in Halperin & Dixon 2020 [4] (see also [5] for a non-technical presentation) as a composition of two effects. First, a new cash infusion produces a price impact in the spirit of the ‘dumb money’ effect of Frazzini and Lamont [6], and second, the amount of this new cash is itself driven by the current stock price (or return), among other driving factors. Together, this produces the Langevin potential for the stock price as a quartic polynomial in price. In [7], I proposed a similar but more tractable approach in the log-price space, with a non-linear potential that can smoothly vary from a single-well to a double well potential, and showed how parameters of this potential can be calibrated to available option quotes.
This produces a model that is capable of producing both small and large fluctuations of returns. In particular, in certain market regimes the resulting potential can be of a double well form. For such scenarios, the critical role in dynamics of large fluctuations is played by non-perturbative solutions which are named so because they cannot be established using a perturbation theory around a zero-friction limit. Such non-perturbative solutions arise in many problems in quantum and statistical physics, where they are known as instantons. In particular, instantons arising in statistical physics models are responsible for non-equilibrium dynamics of escape from a local potential minimum towards a global minimum. While non-perturbative solutions such as instantons are frequently encountered in physics, the framework developed in [4,5,7] based on the analysis of money flows and price impact suggests that similar mechanisms may be theoretically important in finance, and lead to tractable non-linear models that are suitable for practitioners. Extending this approach to multi-asset market models is a logical next step in this program, which is left here for future research.
- X. Gabaix and R.S.J. Koijen, “In Search of the Origins of Financial Fluctuations: The Inelastic Markets Hypothesis”, Swiss Finance Institute Research Paper Series (2020), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3686935#
- J.P. Bouchaud, “The Inelastic Market Hypothesis: A Microstructural Interpretation” (2021), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3896981
- https://www.bloomberg.com/news/articles/2021-10-01/stock-market-does-new-cash-make-shares-go-up
- I. Halperin and M. Dixon, “Quantum Equilibrium-Disequilibrium: Asset Price Dynamics, Symmetry Breaking, and Defaults as Dissipative Instantons“, Physica A 537, 122187, https://doi.org/10.1016/j.physa.2019.122187 (2020).
- I. Halperin, “The Inverted Potential World of Classical Quantitative Finance: a Non-Equilibrium and Non-Perturbative Finance Perspective", https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3724000 (2020).
- A. Frazzini and O.A. Lamont 2008. “Dumb Money: Mutual Fund Flows and the Cross-Section of Stock Returns”, Journal of Financial Economics, Elsevier, vol. 88(2), pages 299-322 (2008).
- I. Halperin, “Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry”, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3724000 (2020).
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Forecasting implications of the global energy transition
Maxim Kartamyshev, Senior Analyst, Norges Bank Investment Management
From the asset management perspective, the energy transition, i.e., substitution of fossil-based energy production with renewable energy sources, is a subject of significant interest. Energy transition policies lead, essentially, to structural and lasting changes in energy supply and demand. Estimating likely consequences of such changes for different sectors of the global industrialized economy is a non-trivial challenge.
Maxim Kartamyshev, Senior Analyst, Norges Bank Investment Management
At the macroeconomic level, the implications of an energy transition are investigated within the Shared Socioeconomic Pathways (SSP) research program[1]. The program aims at providing internally consistent integrated assessment scenarios, detailing hypothetical future developments in energy use, population, economic, climate change, and other relevant indicators. The online database of SSP scenarios serves as an excellent starting point for the exploration of investment risks and opportunities associated with different energy transition pathways.
It is apparent that the energy transition will affect all sectors of the global economy to varying degrees. Hence, the aftermath of changes in energy mix should be analysed with sufficient granularity, properly accounting for the economic interdependence of different sectors. This exercise requires a robust modelling framework.
To examine prospects for the sector-level energy transition analysis, we relied on the proprietary framework of the Economic Domains Model (EDM)[2]. In essence, EDM aims at modelling the world economy as an ensemble of interacting economic domains. The feasibility study considered only a dozen of such domains, corresponding to the largest sectors of the US economy. Using fundamentals of publicly listed companies to quantify domain-level economic activity, a collection of neural networks (NN) was trained to gauge impacts of energy transition on the target sectors.
Use of NNs as the primary modelling gear allowed us to consider a broad range of important transition variables. Of course, data augmentation techniques had to be employed to obtain large enough training datasets. Fortunately, the modelling context catered for a simple augmentation procedure, based on using clusters of companies within a domain to approximate domain-level economic indicators. With the models established, we constructed projections of sector-level fundamentals for energy transitions detailed by selected SSP scenarios.
Despite many approximations, the feasibility study produced compelling results. To identify the main transition variables influencing the projections, we employed Shapley Additive Explanations (SHAP) approach[3]. Enabling model-agnostic, unbiased feature attributions of ML forecasts, the approach also sparked interest in potential generic applications of interpretable machine learning for exploratory investigation of complex systems.
As universal approximators, NNs are well suited for quantitative descriptions of complex non-linear systems. Yet, success of the modelling is strongly dependent on the availability of sufficiently large datasets describing behaviour of the target system, alternatively on a realistic augmentation of available measurements. Within the data-driven approach, to improve generalization properties of NN models, one generally strives to obtain training datasets capturing essential system dynamics for a broad range of relevant regimes.
Equipped with generalizable NN model and adequate computational budget, it is tempting to apply SHAP to examine the inner workings of the target system. For example, one could compute explanations of NN outputs for a dynamic regime of interest, represented by a suitable grid in feature space. Observed relations between feature vectors and corresponding contributions to the forecasts could then be treated as phenomenological response functions, observed in a numerical experiment.
The extracted response functions, together with human expertise and intuition, are likely to assist in identifying hidden casual relationships, thus expanding available knowledge of the complex system in question. Serving as a numerical probe of dynamics captured by a black-box data-driven model, interpretable machine learning could hence provide a valuable supplement to explicit, analytical modelling toolbox.
Such numerical probes are particularly helpful for analysing the implications of extreme events. For example, it is straightforward to adapt an EDM methodology to estimate the impacts of lockdown policies on business activity in different domains. Yet, it is desirable to assess how well the resulting NN models generalize to the region of unprecedented changes in global economic activity and energy trends.
Recognizing that there is high uncertainty related to outcomes under the environment of ongoing lockdowns, we computed short-term estimates of sector-level developments for a large variety of hypothetical macro-economic futures[4]. Estimates of year-on-year (YoY) change rates in aggregated sales for selected sectors are presented on Figure 1.
Figure 1
Forecasted YoY change rates in sector level sales for hypothetical macroeconomic scenarios for year 2020. Each scenario is defined by corresponding YoY change rate in economic activity, energy consumption and electricity generation.
Oil & Gas Producers (US)
Automobiles & Parts (US)
Economic activity change rate [%]
Reviewing corresponding response functions (see Figure 2 for examples) revealed intuitively appealing relationships between changes in the macroeconomic indicators and forecasted levels of activity within different economic domains, encouraging further enhancements of the EDM framework.
Figure 2
Contributions to forecasted YoY change rates for year 2020 in sector level sales from selected feature groups, computed using SHAP methodology. Each scenario is defined by a corresponding YoY change rate in economic activity, energy consumption and electricity generation.
Oil & Gas Producers (US) - Contributions from macroeconomic features
Automobiles & Parts (US) - Contributions from macroeconomic features
Economic activity change rate [%]
Oil & Gas Producers (US) - Contributions from energy consumption features
Automobiles & Parts (US) - Contributions from energy consumption features
Clearly, to improve the realism of EDM forecasts, a much broader range of domains should be considered. Additional work is needed to properly account for interactions between domains of the interconnected global economy. It is also necessary - and possible - to increase the variety of relevant model features. For example, to accurately estimate the consequences of the inevitable changes in energy mix, we believe it will be helpful to utilize specialized measures such as energy return on investment[5].
Availability of a sufficiently realistic modelling framework could substantially improve prospects for the quantitative investigation of the ongoing energy transition. In addition to accurate and self-consistent analysis of potential consequences for hypothetical transition scenarios, the framework could possibly also be applied to identify transition pathways satisfying pre-defined optimality criteria.
Despite the numerous challenges associated with the process, we choose to remain cautiously optimistic on the prospects of enhanced EDM framework in attaining practical relevance for long-term investment management applications.
References
- Keywan Riahi et al, The Shared Socioeconomic Pathways and their energy, land use, and greenhouse gas emissions implications: An overview. Global Environmental Change, 42 (2017), pp. 153-168.
- Maxim Kartamyshev and Patrick du Plessis, Forecasting implications of energy transition pathways. Forthcoming.
- Scott M Lundberg and Su-In Lee, A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, (2017), pp. 4765–4774.
- Maxim Kartamyshev and Patrick du Plessis, Interpretable ML for fundamental analysis during extreme events. Forthcoming.
- Charles A.S. Hall et al, EROI of different fuels and the implications for society. Energy Policy 64, (2014), pp. 141–152.
Machine Learning: Deep Asymptotics
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The price of inflation uncertainty
Jessica James, Michael Leister, Christoph Rieger
Inflation and inflation risks – why are they relevant?
Inflation has rarely held the market's attention so closely as it does today. We seem to stand on the cusp of a regime change, from the lowflation of the past decade to an unknown higher range. Though there is talk of a 'return to normal', lowflation coupled with unprecedented liquidity injections by central banks has in fact become the 'new normal' and markets have become accustomed to life support. Should central banks react according to their mandates, a new, higher inflation regime would hit institutions who have been relying on low cost credit, and could cause many of the so called 'zombie' companies to go under – and also challenge the sustainability of ever rising public debt stocks. Savers, who are facing an erosion of the purchasing power of their savings would rejoice longer-term when rates normalise, and retail banking would have a chance to return to a sustainable 'borrow and lend' model.
The likely level of this higher inflation regime and inflation expectations are thus critically important, and widely discussed. If the spike in inflation does not trigger higher inflation expectations and second-round effects, policy makers and other economic agents can afford to look through it. Things will look very differently if rising inflation feeds through to longer-dated expectations, changing the underlying inflation dynamics.
In view of the 2% inflation targets by most major central banks, it will make a huge difference if underlying inflation settles again at levels closer to, say, 1% or 3%. And at times when inflation views diverge, so-called linear indications from inflation-linked bonds or swaps only tell half the story if the expected value is, say, at 2% while most people think the actual outcome is either below 1% or above 3%.
So indications of future levels are eagerly sought, and one popular method of obtaining probabilities of a rise to various levels has been the inflation options market. By using methods developed for liquidly traded option markets like equities or interest rates, it becomes possible to derive implied probabilities of inflation changes in the future from the prices of inflation options. We assess the value and robustness of these indications derived from inflation options, and show that while some parameters are stable and statistically valid, this is not the case for all.
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Efficient valuation of callable bonds: The dynamic Chebyshev method
Kathrin Glau, Christian Pötz, Mikhail Soloveitchik, and Linus Wunderlich
We investigate the application of the dynamic Chebyshev method to the pricing of callable bonds.
Callable corporate bonds belong to the most important fixed income instruments in financial markets. Acknowledging the fact that the academic literature on efficient pricing of these instruments has been less prioritized in the past, we aim to close this gap. In order to put the method under test in a model that is relevant for practice, we consider a two-factor rate/credit model. We formulate the pricing problem as a dynamic programming problem and present a dynamic Chebyshev algorithm for this problem in a simplified setting. Finally, we present some numerical results.
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Market risk: Optimal VaR adaptation & machine learning
Peter Quell, Head of Portfolio Modelling for Market & Credit Risk, DZ Bank
Peter Quell, Head of Portfolio Modelling for Market & Credit Risk, DZ Bank
Changing volatilities are constant companions in financial markets time series. Nevertheless, regulatory risk models usually rely on some kind of stationarity assumption that leads to problems in these transient environments. Practitioners have long found some ad hoc solutions for this problem, but could machine learning lead to sustainable improvement?
Due to the pandemic, many banks experienced large numbers of backtest outliers in the first quarter of 2020 when they compared actual profit and loss numbers to VaR estimates. The simple reason was that their regulatory VaR systems were not able to adapt to rapidly changing market conditions as volatility spiked to ever-higher levels. The problem lies with the amount of data required to “train” the VaR system.
In a nutshell, the amount of training data is a compromise between having enough observations to compute relevant statistical quantities (here we want lots of data) and the degree to which this data is still relevant for the current environment (here we only want recent data). Whereas in quiet times collecting many data to reduce measurement uncertainty seems to be a priority in some model risk management approaches, the situation changes once there is a rapid shift as experienced in March 2020. Obviously, this may call for some mechanism to “learn” how to spot a crisis environment.
My talk at QuantMinds in Focus in May 2021 contained a simple approach using a noisy random walk, i.e. a stochastic process for which we have observations that include some additional disturbances. This lead to some very easy optimality criterion that depends on the Signal-to-Noise Ratio, the ratio between the variance of the innovation term and the variance of the disturbance term. That relatively simple approach yielded a quite good performance (see diagram) in tracking the profit and loss profile.
More details as well as additional Monte Carlo Studies can be found in the third edition of the book on Risk Model Validation by my colleague Christian Meyer and me (riskbooks.com).
Of course, that could only be a first step to improve classical regulatory VaR systems, since that approach does not care about other deficiencies (e.g. tail events). Are there any machine learning approaches that could be used to improve the situation?
The first technique that comes to mind is a Hidden Markov Chain, and indeed the aforementioned noisy random walk is one of the simplest representatives of this kind. Hidden Markov Models have been around for quite a while. Their disadvantage of comparably large demands on computation time should not be too critical now. Nevertheless, in the context of market risk calculations, banks usually add market data from the current trading day and the algorithm should immediately integrate that information without re-running the lengthy calibration process. This calls for the usage of so-called online learning algorithms in contrast to batch learning approaches. Especially in the context of time series analysis, the sequential order of the data carries important information that should not be left out of the analysis.
What about methods based on neural networks?
Even though there are quite a number of success stories e.g. in image / speech recognition, the application of neural networks in time series analysis seems to be quite limited. The first issue relates to the low signal-to-noise ratio usually encountered in financial market data. Due to the large noise component, the algorithm might learn the noise pattern, even though (at least in theory) there is nothing relevant to learn there. A clear indication of overfitting in this case is a good performance of the algorithm applied to the training data versus a deteriorating performance when applied to new data. Because of their complexity, neural networks are prone to overfit training data, i.e. they fit spurious random aspects of the training data, as well as structure that is true for the entire population. There are techniques to handle overfitting, and all machine learning developers should be making use of these techniques.
Finally, what are the main challenges when it comes to the application of machine learning in a regulatory context
Explainability / interpretability: One should be in a position to explain how the algorithm makes a prediction or decision for one specific case at a time.
Robustness and transient environments: One should account for the fact that markets or environments can change, that calls for a good balance of adaptability and robustness.
Bias and adversarial attacks: Compared to classical statistics there is a much more prominent role for (training) data in machine learning applications.
Of course, some of these issues have been addressed within the machine learning community. What is needed now is the transfer to the banking industry without “reinventing the wheel”. For that reason, the Model Risk Managers’ International Association (mrmia.org) issued a white paper to discuss some (banking) industry best practices. That would only be a starting point since the applications are rapidly evolving.
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Semi-analytical pricing of barrier options in the time dependent lambda-SABR model
Andrey Itkin and Dmitry Muravey
The SABR model is one of the most popular models in mathematical finance. Introduced in (Hagan et al., 2002) (and then an arbitrage-free version in (Hagan et al., 2014)), it quickly became a standard tool among practitioners, especially in the interest rate derivative markets, due to its ability to capture skew and smile of implied volatilities. For European options it has an asymptotic solution, which allows closed form representation of the implied volatility. Later an extension for negative interest rates (the shifted SABR model) was developed in (Antonov, Konikov, and Spector, 2019). It can also be extended to its time-dependent version where all model parameters are deterministic functions of time. An advanced calibration method of the time dependent SABR model based on so-called "effective parameters" was developed in (Van der Stoep, Grzelak, and Oosterlee, 2015).
Another extension was proposed in (Hagan, Lesniewski, and Woodward, 2020). The authors consider the mean-reverting SABR model where the stochastic process for the instantaneous volatility has a mean-reverting drift, and coefficients of the model are functions of time. Then they develop asymptotic methods to obtain an effective forward equation for the marginal density of the forward price. This equation is not exact, but as accurate as the SABR implied volatility formulas and can be solved numerically to obtain the density.
In (Yang, Liu, and Cui, 2017) a closed form expression is obtained to approximate prices of various types of barrier option under the standard SABR model. Since the closed form expression for the barrier option price is not available (without approximations), various asymptotic methods have also been proposed assuming that the model contains a small parameter, see e.g., (Barger and Lorig, 2017; Kato, Takahashi, and Yamada, 2013). Alternatively, numerical methods have been used extensively to price options under the SABR model especially when those options have high maturities and, hence, the asymptotic solution of (Hagan et al., 2002) (for the European option) becomes inaccurate.
As mentioned in (Hagan, Lesniewski, and Woodward, 2020), the SABR model is effective at managing volatility smiles, volatility as a function of the strike K at a single expiry date T. Managing volatility surfaces requires a richer model, such as the dynamic SABR model where all the model coefficients are functions of time.
Therefore, our recent paper (Itkin and Muravey, 2021) is dedicated to pricing barrier options under the time dependent SABR model where the barrier level could also be a deterministic function of time. We propose a slightly modified version of the SABR model with the following changes:
- a) for the instantaneous stochastic volatility σt we assume a lognormal vol-of-vol (as in the original SABR model) but also add a mean-reverting drift;
- b) this drift has zero mean-reversion level and is linear in σt. The form of the mean-reversion term for σt is chosen by a tractability argument and is inspired by the λ-SABR model of (Henry-Labordere, 2005).
To price pricing barrier options under this model we develop a new method which is an extension of the generalized integral transform (GIT) method proposed in a series of the authors’ papers (some in cooperation with Peter Carr and Alex Lipton) and covered in detail in a recent book (Itkin, Lipton, and Muravey, 2021). Then in (Carr, Itkin, and Muravey, 2021) for the first time this approach was extended to stochastic volatility (Heston) model where the option price is expressed in a semi-analytical form as a two-dimensional integral. This integral depends on yet unknown function Ψ(t, σ2) which is the gradient of the solution at the moving boundary F = L(t) and solves a linear mixed Volterra-Fredholm (LMVF) equation of the second kind also derived in that paper.
However, here we do it in a slightly different way. We represent the solution of our problem as a weighted sum of the Bessel functions of the first kind, so actually this representation is a Fourier-Bessel series. The weights of this series solve a corresponding LMVF equation. Once they are found, the solution is expressed in closed form with no further integration. Also, in this way we are able to keep time dependence of the model coefficients with no approximation. Thus, this version of the GIT method is another new result of this paper.
Note, that since the SABR model is written for the forward price, the barrier H(t) is defined as a level of forward. However, since it is a function of time, we can set the barrier for the spot price as well.
We first consider an uncorrelated case and discuss in detail how the obtained LMVF equation can be solved numerically. In doing so we use the Radial Basis Functions (RBF) method and show that using Gaussian RBFs makes the problem tractable by reducing 2D integrals in the LVMF equation to the 1D ones. Then we consider the case of non-zero correlation by using asymptotic series expansions in small ρ. Numerical experiments prove that the speed and accuracy of our method are comparable with those of the finite-difference approach at small maturities, and outperform them at high maturities even by using a simplistic implementation of the RBF method.
About the authors
Andrey Itkin, Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY 11201, USA
Dmitry Muravey, Moscow State University, Moscow, Russia
Antonov, A., M. Konikov, and M. Spector (2019). Modern SABR Analytics. Springer Briefs in Quantitative Finance. Springer. ISBN: 978-3-030-10656-0.
Barger, W. and M. Lorig (2017). “Approximate pricing of European and Barrier claims in a local-stochastic volatility setting”. In: International Journal of Financial Engineering 4, 02n03, p. 1750017.
Carr, P., A. Itkin, and D. Muravey (2021). “Semi-analytical pricing of barrier options in the time-dependent Heston model”. in preparation. In preparation.
Hagan, P. et al. (2002). “Managing Smile Risk”. In: Wilmott magazine, pp. 84–108.
Hagan, P.S, A. Lesniewski, and D.E Woodward (2020). “Implied Volatilities for Mean Reverting SABR Models”. In: Wilmott (108), pp. 62–77.
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Henry-Labordere, P. (2005). A General Asymptotic Implied Volatility for Stochastic Volatility Models. url: https://arxiv.org/pdf/cond-mat/0504317.pdf.
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