Quadratic Hedging Research Articles

Deep learning for quadratic hedging in incomplete jump market

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

Numeraire-Invariant Quadratic Hedging and Mean–Variance Portfolio Allocation

The paper investigates quadratic hedging in a semimartingale market that does not necessarily contain a risk-free asset. An equivalence result for hedging with and without numeraire change is established. This permits direct computation of the optimal strategy without choosing a reference asset and/or performing a numeraire change. New explicit expressions for optimal strategies are obtained, featuring the use of oblique projections that provide unified treatment of the case with and without a risk-free asset. The analysis yields a streamlined computation of the efficient frontier for the pure investment problem in terms of three easily interpreted processes. The main result advances our understanding of the efficient frontier formation in the most general case in which a risk-free asset may not be present. Several illustrations of the numeraire-invariant approach are given.

Digital marketing of the global economic landscape and the financial sustainability of the economy

The article considers the theoretical foundations for assessing and forecasting the sustainability of the development of the Russian economy using artificial intelligence systems. The relevance of the study lies in the fact that in modern conditions the question of the formation of methodological approaches for a balanced sustainable development of the economy is particularly acute. The novelty of the study lies in the fact that an approach is proposed that involves the use of a certain system of macroeconomic indicators that reflect the dynamics of development of both the real sector of the economy and the parameters of the development of the fi nancial sector to predict the GDP of the Russian Federation using the "Perceptron" AI system. An analysis was made of both the dynamics of macroeconomic indicators of the real sector of the economy and the parameters of the development of the fi nancial sector of the Russian Federation. The authors considered the advantages of using the VaR model indicator as a measure of risk to assess the fi nancial risk of reducing the GDP of the Russian Federation. The "Perceptron" program was developed using the Deductor platform to predict the dynamics of the RF GDP. A wide range of fi nancial mathematics tools proposed by individual authors for assessing and minimizing fi nancial risk, including quantile hedging, defi cit hedging with minimal risk, and optimal quadratic hedging, are considered. The architecture of the neural network, in addition to the input layer, has two hidden layers, and an output layer with one parameter. The neural network "Perceptron" shows a high accuracy of the forecast.

Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging

Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging

Deep Learning for Quadratic Hedging in Incomplete Jump Market

Deep Learning for Quadratic Hedging in Incomplete Jump Market

Evaluation of Deep Learning Algorithms for Quadratic Hedging

1. Zhiwen Dai 1. is a PhD candidate in the Department of Systems Engineering and Engineering Management at The Chinese University of Hong Kong in Hong Kong SAR. (zwdai{at}se.cuhk.edu.hk) 2. Lingfei Li 1. is an associate professor in the Department of Systems Engineering and Engineering Management at The Chinese University of Hong Kong in Hong Kong SAR. (lfli{at}se.cuhk.edu.hk) 3. Gongqiu Zhang 1. is an assistant professor at the School of Science and Engineering at The Chinese University of Hong Kong in Shenzhen, China. (zhanggongqiu{at}cuhk.edu.cn) We solve the quadratic hedging problem by deep learning in discrete time. We consider three deep learning algorithms corresponding to three architectures of neural network approximation: approximating controls of different periods by different feedforward neural networks (FNNs) as proposed by [Han and Weinan (2016)][1], using a single FNN with decision time as an input to approximate controls of different periods, and using a recursive neural network (RNN) to utilize historical information. We evaluate these algorithms under the discrete-time Black-Scholes model and the DCC-GARCH model for hedging basket options on portfolios of up to 100 assets with time to maturity up to one year. We compare them in terms of their hedging error on the test data, extent of overlearning, learned hedging strategy, training speed, and scalability. Our results favor the single FNN and RNN approximations overall; the multiple FNN approximation can fail for a large portfolio and a long maturity. We also evaluate the performance of the single FNN and RNN algorithms in a data-driven framework, where data is generated by resampling without assuming any parametric model. [1]: #ref-20

Quadratic hedging of risk neutral values

Risk neutral valuation determines no arbitrage values for financial or real assets, including ones that are exposed to energy price risk. It is always uniquely associated with a hedging strategy if and only if markets are complete, which is the exception in theory and never the case in practice. We apply quadratic hedging, which is both conceptually simple and partially analytically tractable, in a nonstandard fashion to approximately offset the change in the value of an asset obtained from using any chosen risk neutral measure when markets are incomplete. Consistency between valuation and hedging conditional on this value is thus ensured. Achieving this goal with standard quadratic hedging requires employing the so called variance optimal martingale measure for valuation, which can be problematic in general because this measure can fail to be a risk neutral one. Heuristics that rely on a complete market assumption are compatible with the proposed conditional quadratic hedging approach. Simple examples suggest that such techniques can perform near optimally. The methodology put forth therein applies to fully risk neutral valuation of assets with cash flows that depend on both market and private risks, reducing to quadratic hedging if markets are partially complete, which we show provides a novel justification for this valuation strategy in this case. It can be extended beyond the single and fixed-date cash flow purview of this research.

An artificial intelligence system for forecasting financial risk in the banking sphere in the conditions of market uncertainty

The article discusses the theoretical foundations of the analysis and forecasting of financial risk in the banking sector in conditions of market uncertainty. The relevance of the study lies in the fact that the growth of problem debts of commercial banks on loans to legal entities, individual entrepreneurs and individuals is currently the most relevant and debated issue in the banking community. The analysis of the dynamics of assets and the share of overdue loans in 2010–2021 has been carried out, trends in portfolio changes have been identified. The authors considered the advantages of using the VaR indicator as a measure of risk, noting that its weak side is the inability to assess extreme losses (in the tails) if the risk is realized in the range above the confi dence interval. The Perseptron program has been developed for forecasting the dynamics of the share of overdue loans in the portfolio of a commercial bank, which is formed on the Deductor platform. Quantization (grouping) of data was carried out using a neural network on the Deductor platform, which made it possible to identify certain patterns in the change in portfolio quality. It was revealed that the value of the share of overdue loans of commercial banks is influenced by many factors, including factors included in the AI-system, so four parameters are placed on the input layer of the perceptron (%): growth in assets, market share, change in the loan portfolio, dynamics of overdue loans. The output layer has one parameter: forecast of the share of overdue loans (%). In addition to the input layer, the neural network architecture has two hidden layers, and an output layer with one parameter. The neural network shows high forecast accuracy. The error rate for Post Bank was 1.2159 %. The authors consider a wide range of financial mathematics tools proposed by the authors in order to assess and minimize financial risks, including quantile hedging, deficit hedging with minimal risk, and optimal quadratic hedging.

Evaluation of Deep Learning Algorithms for Quadratic Hedging

Evaluation of Deep Learning Algorithms for Quadratic Hedging

INSURANCE VALUATION: A TWO-STEP GENERALISED REGRESSION APPROACH

AbstractCurrent approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

Equal risk pricing and hedging of financial derivatives with convex risk measures

In this paper, we consider the problem of equal risk pricing and hedging in which the fair price of an option is the price that exposes both sides of the contract to the same level of risk. Focusing for the first time on the context where risk is measured according to convex risk measures, we establish that the problem reduces to solving independently the writer and the buyer's hedging problems with zero initial capital. By further imposing that the risk measures decompose in a way that satisfies a Markovian property, we provide dynamic programming equations that can be used to solve the hedging problems for both European and American options. All of our results are general enough to accommodate situations where the risk is measured according to a worst-case risk measure, as is typically done in robust optimization. Our numerical study illustrates the advantages of equal risk pricing over schemes that only account for a single party, pricing based on quadratic hedging (i.e. ϵ-arbitrage pricing), or pricing based on a fixed equivalent martingale measure (i.e. Black–Scholes pricing). In particular, the numerical results confirm that when employing an equal risk price both the writer and the buyer end up being exposed to risks that are more similar and on average smaller than what they would experience with the other approaches.

A discrete-time hedging framework with multiple factors and fat tails: On what matters

This article presents a quadratic hedging framework for a general class of discrete-time affine multi-factor models and investigates the extent to which multi-component volatility factors, fat tails, and a non-monotonic pricing kernel can improve the hedging performance. A semi-explicit hedging formula is derived for our general framework which applies to a myriad of the option pricing models proposed in the discrete-time literature. We conduct an extensive empirical study of the impact of modelling features on the hedging effectiveness of S&P 500 options. Overall, we find that fat tails can be credited for half of the hedging improvement observed, while a second volatility factor and a non-monotonic pricing kernel each contribute to a quarter of this improvement. Moreover, our study indicates that the added value of these features for hedging is different than for pricing. A robustness analysis shows that a similar conclusion can be reached when considering the Dow Jones Industrial Average. Finally, the use of a hedging-based loss function in the estimation process is investigated in an additional robustness test, and this choice has a rather marginal impact on hedging performance.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $H>1/2$, and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

Worst-Case Valuation of Equity-Linked Products Using Risk-Minimizing Strategies

The impact of model risk when hedging equity-linked products and other investment guarantees is significant. We propose a model to determine the worst-case value of an equity-linked product through partial hedging. Risk control strategies based on conditional Value at Risk measures are used. The model integrates both mortality and financial risk associated with these products to find the worst-case value. We adopt robust optimization techniques to compute an optimal hedging strategy. To demonstrate versatility of the framework, numerical examples of point-to-point equity-indexed annuities are presented in multinomial lattice dynamics. We compare robustness of the model to super-replicating and quadratic hedging strategies by computing their capital requirements.

A Discrete-Time Hedging Framework with Multiple Factors and Fat Tails: On What Matters

This article presents a quadratic hedging framework for a general class of discrete-time affine multi-factor models and investigates the extent to which multi-component volatility factors, fat tails, and a non-monotonic pricing kernel can improve the hedging performance. A semi-explicit hedging formula is derived for our general framework which applies to a myriad of the option pricing models proposed in the discrete-time literature. We conduct an extensive empirical study of the impact of modelling features on the hedging effectiveness of S&P 500 options. Overall, we find that fat tails can be credited for half of the hedging improvement observed, while a second volatility factor and a non-monotonic pricing kernel each contribute to a quarter of this improvement. Moreover, our study indicates that the added value of these features for hedging is different than for pricing. A robustness analysis further shows that a similar conclusion can be reached when considering the Dow Jones Industrial Average. Finally, the use of a hedging-based loss function in the estimation process is investigated in an additional robustness test, and this choice has a rather marginal impact on hedging performance.

Quadratic hedging for sequential claims with random weights in discrete time

We study a quadratic hedging problem for a sequence of contingent claims with random weights in discrete time. We obtain the optimal hedging strategy explicitly in a recursive representation, without imposing the non-degeneracy (ND) condition on the model and square integrability on hedging strategies. We relate the general results to hedging under random horizon and fair pricing in the quadratic sense. We illustrate the significance of our results in an example in which the ND condition fails.

Quadratic Hedging of Futures Term Structure Risk in Merchant Energy Trading Operations

Quadratic Hedging of Futures Term Structure Risk in Merchant Energy Trading Operations

Numeraire-Invariant Quadratic Hedging and Mean-Variance Portfolio Allocation

The paper investigates quadratic hedging in a general semimartingale market that does not necessarily contain a risk-free asset. An equivalence result for hedging with and without numeraire change is established. This permits direct computation of the optimal strategy without choosing a reference asset and/or performing a numeraire change. New explicit expressions for optimal strategies are obtained, featuring the use of oblique projections that provide unified treatment of the case with and without a risk-free asset. The main result advances our understanding of the efficient frontier formation in the most general case where a risk-free asset may not be present. Several illustrations of the numeraire-invariant approach are given.

Lattice-Based Hedging Schemes Under GARCH Models

This paper proposes an efficient way to implement quadratic hedging schemes for European options when the asset return process follows an asymmetric non-affine GARCH model driven by Gaussian innovations. More specifically, using a lattice approximation for the underlying, we construct locally risk-minimizing (LRM) hedge ratios under both physical and risk-neutral measures, as well as standard delta strategies. We investigate the convergence of option prices and hedges resulting from the LRM strategies relative to the number of intra-daily periods used in the lattice. Several numerical experiments are conducted to assess the sensitivity of the hedge ratios to the equity risk premium and leverage effect parameters, and to compare their performance by computing the corresponding one-period and terminal hedging errors. Our results suggest that the LRM scheme under the physical measure consistently outperforms competing hedging strategies.

On the risk management of demand deposits: quadratic hedging of interest rate margins

This paper examines the problem of hedging banks interest rate margins. We assume that the demand’s deposits follow an exponential Levy process with potential jumps. The forward market rates are assumed to follow the standard market model introduced by Brace et al. (Math Finance 7(2):127–155, 1997). As Adam et al. (Hedging interest rate margins on demand deposits, Universite Paris 1 Pantheon-Sorbonne working paper, 2012), we consider that deposit rates depend linearly on market rates. Face to incompleteness, the liability manager must hedge both interest rate and demand deposit risks. For this purpose, we introduce various quadratic hedging criteria, allowing us to provide explicit hedging strategies that we further analyze. We illustrate in particular the impact of both the trends and the volatilities of interest rates and demand deposits.