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41 pages, 2554 KiB

Open AccessArticle

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 1248

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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30 pages, 1080 KiB

Open AccessArticle

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 1148

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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18 pages, 794 KiB

Open AccessEditor’s ChoiceArticle

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 3117

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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18 pages, 5413 KiB

Open AccessArticle

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 2130

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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24 pages, 695 KiB

Open AccessArticle

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 1404

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

L_{1}

-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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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16 pages, 946 KiB

Open AccessArticle

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 2319

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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29 pages, 1138 KiB

Open AccessEditor’s ChoiceArticle

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 1371

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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20 pages, 582 KiB

Open AccessArticle

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 1460

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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10 pages, 462 KiB

Open AccessArticle

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 1797

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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27 pages, 586 KiB

Open AccessArticle

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 1437

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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6 pages, 283 KiB

Open AccessObituary

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 1303

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

(This article belongs to the Special Issue Emerging Topics in Finance and Risk Engineering—In Memory of Peter Carr)

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