Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr

A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 20412

Special Issue Editors

School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA
Interests: derivatives pricing; risk management; applied mathematics

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Guest Editor
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
Interests: applied probability; financial engineering; operations research; data science

Special Issue Information

Dear Colleagues,

We would like to invite contributions to this issue of Risks honoring the memory of Peter Carr (1958–2022). After receiving a Ph.D. in finance from UCLA, Peter Carr worked for eight years as an Assistant Professor of Finance at Cornell University before joining Morgan Stanley as a Vice President in 1996 and later at Bank of America Securities as a Principal in 1999. He was the head of the Quantitative Financial Research Group at Bloomberg from 2003–2010, a Managing Director and Global Head of Market Modeling at Morgan Stanley from 2010 to 2016, before returning to academia in 2016 as the Chair of the Department of Finance and Risk Engineering at NYU’s Tandon School of Engineering.

Peter was a colleague, friend, and mentor to many of us. His contributions to financial engineering and financial mathematics span a wide range, touching on many topics, such as derivatives pricing, asset price modeling with martingales, uncovering symmetries and exploiting them in derivatives valuation, volatility modeling, and much more.

A recurring theme in Peter’s work is the application of novel concepts and methods to financial problems. Some examples are the study of derivatives with randomly distributed maturity, the use of algebraic methods for derivatives pricing, and exploring new modeling ideas using artificial intelligence and machine learning. In the same spirit, we would like to invite contributions proposing new methods and applications of emerging methods to problems of financial and risk engineering.

Dr. Dan Pirjol
Dr. Lingjiong Zhu
Guest Editors

Manuscript Submission Information

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Keywords

  • derivatives pricing
  • asset price modeling
  • risk management
  • novel methods in finance

Published Papers (11 papers)

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41 pages, 2554 KiB  
Article
A Semi-Static Replication Method for Bermudan Swaptions under an Affine Multi-Factor Model
by Jori Hoencamp, Shashi Jain and Drona Kandhai
Risks 2023, 11(10), 168; https://doi.org/10.3390/risks11100168 - 26 Sep 2023
Cited by 1 | Viewed by 1230
Abstract
We present a semi-static replication algorithm for Bermudan swaptions under an affine, multi-factor term structure model. In contrast to dynamic replication, which needs to be continuously updated as the market moves, a semi-static replication needs to be rebalanced on just a finite number [...] Read more.
We present a semi-static replication algorithm for Bermudan swaptions under an affine, multi-factor term structure model. In contrast to dynamic replication, which needs to be continuously updated as the market moves, a semi-static replication needs to be rebalanced on just a finite number of instances. We show that the exotic derivative can be decomposed into a portfolio of vanilla discount bond options, which mirrors its value as the market moves and can be priced in closed form. This paves the way toward the efficient numerical simulation of xVA, market, and credit risk metrics for which forward valuation is the key ingredient. The static portfolio composition is obtained by regressing the target option’s value using an interpretable, artificial neural network. Leveraging the universal approximation power of neural networks, we prove that the replication error can be arbitrarily small for a sufficiently large portfolio. A direct, a lower bound, and an upper bound estimator for the Bermudan swaption price are inferred from the replication algorithm. Additionally, closed-form error margins to the price statistics are determined. We practically study the accuracy and convergence of the method through several numerical experiments. The results indicate that the semi-static replication approaches the LSM benchmark with basis point accuracy and provides tight, efficient error bounds. For in-model simulations, the semi-static replication outperforms a traditional dynamic hedge. Full article
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30 pages, 1080 KiB  
Article
Pricing of Pseudo-Swaps Based on Pseudo-Statistics
by Sebastian Franco and Anatoliy Swishchuk
Risks 2023, 11(8), 141; https://doi.org/10.3390/risks11080141 - 03 Aug 2023
Viewed by 1110
Abstract
The main problem in pricing variance, volatility, and correlation swaps is how to determine the evolution of the stochastic processes for the underlying assets and their volatilities. Thus, sometimes it is simpler to consider pricing of swaps by so-called pseudo-statistics, namely, the pseudo-variance, [...] Read more.
The main problem in pricing variance, volatility, and correlation swaps is how to determine the evolution of the stochastic processes for the underlying assets and their volatilities. Thus, sometimes it is simpler to consider pricing of swaps by so-called pseudo-statistics, namely, the pseudo-variance, -covariance, -volatility, and -correlation. The main motivation of this paper is to consider the pricing of swaps based on pseudo-statistics, instead of stochastic models, and to compare this approach with the most popular stochastic volatility model in the Cox–Ingresoll–Ross (CIR) model. Within this paper, we will demonstrate how to value different types of swaps (variance, volatility, covariance, and correlation swaps) using pseudo-statistics (pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance). As a result, we will arrive at a method for pricing swaps that does not rely on any stochastic models for a stochastic stock price or stochastic volatility, and instead relies on data/statistics. A data/statistics-based approach to swap pricing is very different from stochastic volatility models such as the Cox–Ingresoll–Ross (CIR) model, which, in comparison, follows a stochastic differential equation. Although there are many other stochastic models that provide an approach to calculating the price of swaps, we will use the CIR model for comparison within this paper, due to the popularity of the CIR model. Therefore, in this paper, we will compare the CIR model approach to pricing swaps to the pseudo-statistic approach to pricing swaps, in order to compare a stochastic model to the data/statistics-based approach to swap pricing that is developed within this paper. Full article
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18 pages, 794 KiB  
Article
A Diversification Framework for Multiple Pairs Trading Strategies
by Kiseop Lee, Tim Leung and Boming Ning
Risks 2023, 11(5), 93; https://doi.org/10.3390/risks11050093 - 16 May 2023
Cited by 1 | Viewed by 3023
Abstract
We propose a framework for constructing diversified portfolios with multiple pairs trading strategies. In our approach, several pairs of co-moving assets are traded simultaneously, and capital is dynamically allocated among different pairs based on the statistical characteristics of the historical spreads. This allows [...] Read more.
We propose a framework for constructing diversified portfolios with multiple pairs trading strategies. In our approach, several pairs of co-moving assets are traded simultaneously, and capital is dynamically allocated among different pairs based on the statistical characteristics of the historical spreads. This allows us to further consider various portfolio designs and rebalancing strategies. Working with empirical data, our experiments suggest the significant benefits of diversification within our proposed framework. Full article
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18 pages, 5413 KiB  
Article
BeVIXed: Trading Fear in the Volatility Complex
by Chakravarthy Varadarajan and Klaus R. Schenk-Hoppé
Risks 2023, 11(5), 86; https://doi.org/10.3390/risks11050086 - 04 May 2023
Viewed by 2074
Abstract
We explain the evolution of the volatility market and present the infamous day of ‘Volmageddon’ as an insightful case study. Our survey focuses on the pricing and trading of volatility-linked assets, highlighting the impact of mechanical hedging in markets for futures and higher-order [...] Read more.
We explain the evolution of the volatility market and present the infamous day of ‘Volmageddon’ as an insightful case study. Our survey focuses on the pricing and trading of volatility-linked assets, highlighting the impact of mechanical hedging in markets for futures and higher-order derivatives. We supplement the vast statistical analysis of volatility derivatives with a financial economist’s perspective. Full article
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24 pages, 695 KiB  
Article
Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities
by Daniel Guterding
Risks 2023, 11(5), 83; https://doi.org/10.3390/risks11050083 - 28 Apr 2023
Viewed by 1378
Abstract
We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations [...] Read more.
We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an L1-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases. Full article
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16 pages, 946 KiB  
Article
Backward Deep BSDE Methods and Applications to Nonlinear Problems
by Yajie Yu, Narayan Ganesan and Bernhard Hientzsch
Risks 2023, 11(3), 61; https://doi.org/10.3390/risks11030061 - 16 Mar 2023
Cited by 1 | Viewed by 2277
Abstract
We present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent financial interest. The [...] Read more.
We present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent financial interest. The nonlinear equation for the backward time-step is solved exactly or by a Taylor-based approximation. This is the first application of such a pathwise backward time-stepping deep BSDE approach for problems with nonlinear generators. We extend the method to the case when the initial value of the forward components X can be a parameter rather than fixed and similarly to also learn values at intermediate times. We present numerical results for a call combination and for a straddle, the latter comparing well to those obtained by Forsyth and Labahn with a specialized PDE solver. Full article
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29 pages, 1138 KiB  
Article
Optimal Investment in a Dual Risk Model
by Arash Fahim and Lingjiong Zhu
Risks 2023, 11(2), 41; https://doi.org/10.3390/risks11020041 - 09 Feb 2023
Cited by 1 | Viewed by 1346
Abstract
Dual risk models are popular for modeling a venture capital or high-tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrates on the optimal dividend strategies. In [...] Read more.
Dual risk models are popular for modeling a venture capital or high-tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrates on the optimal dividend strategies. In this paper, we propose to study the optimal investment strategy on research and development for the dual risk models to minimize the ruin probability of the underlying company. We will also study the optimization problem when, in addition, the investment in a risky asset is allowed. Full article
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20 pages, 582 KiB  
Article
Spectral Expansions for Credit Risk Modelling with Occupation Times
by Giuseppe Campolieti, Hiromichi Kato and Roman N. Makarov
Risks 2022, 10(12), 228; https://doi.org/10.3390/risks10120228 - 30 Nov 2022
Viewed by 1436
Abstract
We study two credit risk models with occupation time and liquidation barriers: the structural model and the hybrid model with hazard rate. The defaults within the models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) [...] Read more.
We study two credit risk models with occupation time and liquidation barriers: the structural model and the hybrid model with hazard rate. The defaults within the models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) of the U.S. Bankruptcy Code. The models assume that credit events trigger as soon as the occupation time (the cumulative time the firm’s value process spends below some threshold level) exceeds the grace period (time allowance). The hazard rate model extends the structural occupation time models and presumes that other random factors may also lead to credit events. Both approaches allow the firm to fulfill its obligations during the grace period. We derive new closed-from pricing formulas for credit derivatives containing the (risk-neutral) probability of defaults and credit default swap (CDS) spreads as special cases, which are derived analytically via a spectral expansion methodology. Our method works for any solvable diffusion, such as the geometric Brownian motion (GBM) and several state-dependent volatility processes, including the constant elasticity of variance (CEV) model. It allows us to write the pricing formulas explicitly as infinite series that converges rapidly. We then calibrate our models (assuming that GBM governs the firm’s value) to market CDS spreads from the Total Energy company. Our calibration results show that the computations are fast, and the fit is near-perfect. Full article
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10 pages, 462 KiB  
Article
Modeling Momentum and Reversals
by Harvey J. Stein and Jacob Pozharny
Risks 2022, 10(10), 190; https://doi.org/10.3390/risks10100190 - 02 Oct 2022
Cited by 2 | Viewed by 1746
Abstract
Stock prices are well known to exhibit behaviors that are difficult to model mathematically. Individual stocks are observed to exhibit short term price reversals and long term momentum, while their industries only exhibit momentum. Here we show that individual stocks can be modeled [...] Read more.
Stock prices are well known to exhibit behaviors that are difficult to model mathematically. Individual stocks are observed to exhibit short term price reversals and long term momentum, while their industries only exhibit momentum. Here we show that individual stocks can be modeled by simple mean reverting processes in such a way that these behaviors are captured, the model is arbitrage free, and market informational efficiency is preserved. Simulation shows that in such a market, when mean reversion is sufficiently high, strategies which use reversals would substantially outperform buy and hold strategies. Full article
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27 pages, 586 KiB  
Article
Probability Density of Lognormal Fractional SABR Model
by Jiro Akahori, Xiaoming Song and Tai-Ho Wang
Risks 2022, 10(8), 156; https://doi.org/10.3390/risks10080156 - 02 Aug 2022
Cited by 1 | Viewed by 1409
Abstract
Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied [...] Read more.
Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation of joint density often leads to a heuristic derivation of the large deviations principle for joint density in a small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients. Full article
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6 pages, 283 KiB  
Obituary
In Memory of Peter Carr (1958–2022)
by Giuseppe Campolieti, Arash Fahim, Dan Pirjol, Harvey Stein, Tai-Ho Wang and Lingjiong Zhu
Risks 2024, 12(2), 39; https://doi.org/10.3390/risks12020039 - 18 Feb 2024
Viewed by 1210
Abstract
The editors of this special issue and several of the contributing authors have known Peter for a long time. We thought that the special issue will be enriched by adding a few personal notes and recollections about our interactions with Peter. Full article
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