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Solving High-Dimensional Optimal Stopping Problems Using Optimization Based Model Order Reduction 基于优化的模型降阶方法求解高维最优停车问题
Applied Mathematical Finance Pub Date : 2022-03-04 DOI: 10.1080/1350486X.2022.2154682
Martin Redmann
{"title":"Solving High-Dimensional Optimal Stopping Problems Using Optimization Based Model Order Reduction","authors":"Martin Redmann","doi":"10.1080/1350486X.2022.2154682","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2154682","url":null,"abstract":"Solving optimal stopping problems by backward induction in high dimensions is often very complex since the computation of conditional expectations is required. Typically, such computations are based on regression, a method that suffers from the curse of dimensionality. Therefore, the objective of this paper is to establish dimension reduction schemes for large-scale asset price models and to solve related optimal stopping problems (e.g., Bermudan option pricing) in the reduced setting, where regression is feasible. The proposed algorithm is based on an error measure between linear stochastic differential equations. We establish optimality conditions for this error measure with respect to the reduced system coefficients and propose a particular method that satisfies these conditions up to a small deviation. We illustrate the benefit of our approach in several numerical experiments, in which Bermudan option prices are determined.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81527808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
The Impact of Stochastic Volatility on Initial Margin and MVA for Interest Rate Derivatives 随机波动率对利率衍生品初始保证金和MVA的影响
Applied Mathematical Finance Pub Date : 2022-03-04 DOI: 10.1080/1350486X.2022.2156900
J. H. Hoencamp, J. P. de Kort, B. D. Kandhai
{"title":"The Impact of Stochastic Volatility on Initial Margin and MVA for Interest Rate Derivatives","authors":"J. H. Hoencamp, J. P. de Kort, B. D. Kandhai","doi":"10.1080/1350486X.2022.2156900","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2156900","url":null,"abstract":"ABSTRACT In this research we investigate the impact of stochastic volatility on future initial margin (IM) and margin valuation adjustment (MVA) calculations for interest rate derivatives. An analysis is performed under different market conditions, namely during the peak of the Covid-19 crisis when the markets were stressed and during Q4 of 2020 when volatilities were low. The Cheyette short-rate model is extended by adding a stochastic volatility component, which is calibrated to fit the EUR swaption volatility surfaces. We incorporate the latest risk-free rate benchmarks (RFR), which in certain markets have been selected to replace the IBOR index. We extend modern Fourier pricing techniques to accommodate the RFR benchmark and derive closed-form sensitivity expressions, which are used to model IM profiles in a Monte Carlo simulation framework. The various results are compared to the deterministic volatility case. The results reveal that the inclusion of a stochastic volatility component can have a considerable impact on nonlinear derivatives, especially for far out-of-the-money swaptions. The effect is particularly pronounced if the market exhibits a substantial skew or smile in the implied volatility curve. This can have severe consequences for funding cost valuation and risk management.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78205748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solvability of Differential Riccati Equations and Applications to Algorithmic Trading with Signals 微分Riccati方程的可解性及其在信号算法交易中的应用
Applied Mathematical Finance Pub Date : 2022-02-15 DOI: 10.1080/1350486X.2023.2241130
Fayçal Drissi
{"title":"Solvability of Differential Riccati Equations and Applications to Algorithmic Trading with Signals","authors":"Fayçal Drissi","doi":"10.1080/1350486X.2023.2241130","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2241130","url":null,"abstract":"ABSTRACT We study a differential Riccati equation (DRE) with indefinite matrix coefficients, which arises in a wide class of practical problems. We show that the DRE solves an associated control problem, which is key to provide existence and uniqueness of a solution. As an application, we solve two algorithmic trading problems in which the agent adopts a constant absolute risk-aversion (CARA) utility function, and where the optimal strategies use signals and past observations of prices to improve their performance. First, we derive a multi-asset market making strategy in over-the-counter markets, where the market maker uses an external trading venue to hedge risk. Second, we derive an optimal trading strategy that uses prices and signals to learn the drift in the asset prices.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75154727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Pricing the Excess Volatility in Foreign Exchange Risk Premium and Forward Rate Bias 外汇风险溢价的超额波动定价与远期汇率偏差
Applied Mathematical Finance Pub Date : 2022-01-02 DOI: 10.1080/1350486X.2022.2108857
T. T. Swan, Bruce Q. Swan, Xinfu Chen
{"title":"Pricing the Excess Volatility in Foreign Exchange Risk Premium and Forward Rate Bias","authors":"T. T. Swan, Bruce Q. Swan, Xinfu Chen","doi":"10.1080/1350486X.2022.2108857","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2108857","url":null,"abstract":"ABSTRACT We present the pricing of the documented excess volatility of the foreign exchange risk premium, relative to the interest rate differential. By specifying a term structure of interest rate model, the physical probability measure along with the pricing kernels or discount factors are used to derive a system for the expected future spot rate and the forward rate. The theoretical loads are found by solving the Riccati ordinary differential equations, and dynamic factors are captured to set up the global factors for both currencies. It shows that we prove the interest-rate surfaces are almost identical to the empirical ones, and the theoretical interest rates are guaranteed to be positive.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87520689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Expected Utility Theory on General Affine GARCH Models 一般仿射GARCH模型的期望效用理论
Applied Mathematical Finance Pub Date : 2021-11-02 DOI: 10.1080/1350486X.2022.2101010
M. Escobar-Anel, Ben Spies, R. Zagst
{"title":"Expected Utility Theory on General Affine GARCH Models","authors":"M. Escobar-Anel, Ben Spies, R. Zagst","doi":"10.1080/1350486X.2022.2101010","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2101010","url":null,"abstract":"ABSTRACT Expected utility theory has produced abundant analytical results in continuous-time finance, but with very little success for discrete-time models. Assuming the underlying asset price follows a general affine GARCH model which allows for non-Gaussian innovations, our work produces an approximate closed-form recursive representation for the optimal strategy under a constant relative risk aversion (CRRA) utility function. We provide conditions for optimality and demonstrate that the optimal wealth is also an affine GARCH. In particular, we fully develop the application to the IG-GARCH model hence accommodating negatively skewed and leptokurtic asset returns. Relying on two popular daily parametric estimations, our numerical analyses give a first window into the impact of the interaction of heteroscedasticity, skewness and kurtosis on optimal portfolio solutions. We find that losses arising from following Gaussian (suboptimal) strategies, or Merton's static solution, can be up to and 5%, respectively, assuming low-risk aversion of the investor and using a five-years time horizon.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89015222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the Valuation of Discrete Asian Options in High Volatility Environments 高波动环境下离散亚洲期权的估值研究
Applied Mathematical Finance Pub Date : 2021-11-02 DOI: 10.1080/1350486X.2022.2108858
Sascha Desmettre, J. Wenzel
{"title":"On the Valuation of Discrete Asian Options in High Volatility Environments","authors":"Sascha Desmettre, J. Wenzel","doi":"10.1080/1350486X.2022.2108858","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2108858","url":null,"abstract":"ABSTRACT In this paper, we are concerned with the Monte Carlo valuation of discretely sampled arithmetic and geometric average options in the Black-Scholes model and the stochastic volatility model of Heston in high volatility environments. To this end, we examine the limits and convergence rates of asset prices in these models when volatility parameters tend to infinity. We observe, on the one hand, that asset prices, as well as their arithmetic means converge to zero almost surely, while the respective expectations are constantly equal to the initial asset price. On the other hand, the expectation of geometric means of asset prices converges to zero. Moreover, we elaborate on the direct consequences for option prices based on such means and illustrate the implications of these findings for the design of efficient Monte-Carlo valuation algorithms. As a suitable control variate, we need among others the price of such discretely sampled geometric Asian options in the Heston model, for which we derive a closed-form solution.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86052992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a Neural Network to Extract Implied Information from American Options 基于神经网络的美式期权隐含信息提取
Applied Mathematical Finance Pub Date : 2021-09-03 DOI: 10.1080/1350486X.2022.2097099
Shuaiqiang Liu, Álvaro Leitao, A. Borovykh, C. Oosterlee
{"title":"On a Neural Network to Extract Implied Information from American Options","authors":"Shuaiqiang Liu, Álvaro Leitao, A. Borovykh, C. Oosterlee","doi":"10.1080/1350486X.2022.2097099","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2097099","url":null,"abstract":"Extracting implied information, like volatility and dividend, from observed option prices is a challenging task when dealing with American options, because of the complex-shaped early-exercise regions and the computational costs to solve the corresponding mathematical problem repeatedly. We will employ a data-driven machine learning approach to estimate the Black-Scholes implied volatility and the dividend yield for American options in a fast and robust way. To determine the implied volatility, the inverse function is approximated by an artificial neural network on the effective computational domain of interest, which decouples the offline (training) and online (prediction) stages and thus eliminates the need for an iterative process. In the case of an unknown dividend yield, we formulate the inverse problem as a calibration problem and determine simultaneously the implied volatility and dividend yield. For this, a generic and robust calibration framework, the Calibration Neural Network (CaNN), is introduced to estimate multiple parameters. It is shown that machine learning can be used as an efficient numerical technique to extract implied information from American options, particularly when considering multiple early-exercise regions due to negative interest rates.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86108984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Fragmentation, Price Formation and Cross-Impact in Bitcoin Markets 比特币市场的碎片化、价格形成和交叉影响
Applied Mathematical Finance Pub Date : 2021-08-21 DOI: 10.1080/1350486X.2022.2080083
Jakob Albers, Mihai Cucuringu, S. Howison, Alexander Y. Shestopaloff
{"title":"Fragmentation, Price Formation and Cross-Impact in Bitcoin Markets","authors":"Jakob Albers, Mihai Cucuringu, S. Howison, Alexander Y. Shestopaloff","doi":"10.1080/1350486X.2022.2080083","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2080083","url":null,"abstract":"In the light of micro-scale inefficiencies due to the highly fragmented bitcoin trading landscape, we use a granular data set comprising orderbook and trades data from the most liquid bitcoin markets, to understand the price formation process at sub-1-second time scales. To this end, we construct a set of features that encapsulate relevant microstructural information over short lookback windows. These features are subsequently leveraged, first to generate a leader–lagger network that quantifies how markets impact one another, and then to train linear models capable of explaining between 10% and 37% of total variation in 500 ms future returns (depending on which market is the prediction target). The results are then compared with those of various PnL calculations that take trading realities, such as transaction costs, into account. The PnL calculations are based on natural taker strategies (meaning they employ market orders) associated with each model. Our findings emphasize the role of a market's fee regime in determining both its propensity to lead or lag, and the profitability of our taker strategy. We further derive a natural maker strategy (using only passive limit orders) which, due to the difficulties associated with backtesting maker strategies, we test in a real-world live trading experiment, in which we turned over 1.5 M USD in notional volume. Lending additional confidence to our models, and by extension to the features they are based on, the results indicate a significant improvement over a naive benchmark strategy, which we also deploy in a live trading environment with real capital, for the sake of comparison.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87225404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
The Role of Binance in Bitcoin Volatility Transmission 币安在比特币波动传输中的作用
Applied Mathematical Finance Pub Date : 2021-07-01 DOI: 10.1080/1350486X.2022.2125885
C. Alexander, Daniel F. Heck, Andreas Kaeck
{"title":"The Role of Binance in Bitcoin Volatility Transmission","authors":"C. Alexander, Daniel F. Heck, Andreas Kaeck","doi":"10.1080/1350486X.2022.2125885","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2125885","url":null,"abstract":"ABSTRACT We analyse high-frequency realized volatility dynamics and spillovers between centralized crypto exchanges that offer spot and derivative contracts for bitcoin against the US dollar or the stable coin tether. The tether-margined perpetual contract on Binance is clearly the main source of volatility, continuously transmitting strong flows to all other instruments and receiving very little volatility from other sources. We also find that crypto exchanges exhibit much higher interconnectedness when traditional Western stock markets are open. Especially during the US time zone, volatility outflows from Binance are much higher than at other times, and Bitcoin traders are more attentive and reactive to prevailing market conditions. Our results highlight that market regulators should pay more attention to the tether-margined derivatives products available on most self-regulated exchanges, most importantly on Binance.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73705164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Valuation of European Options Under an Uncertain Market Price of Volatility Risk 波动性风险市场价格不确定下的欧式期权估值
Applied Mathematical Finance Pub Date : 2021-05-20 DOI: 10.1080/1350486X.2022.2125884
Bartosz Jaroszkowski, Max Jensen
{"title":"Valuation of European Options Under an Uncertain Market Price of Volatility Risk","authors":"Bartosz Jaroszkowski, Max Jensen","doi":"10.1080/1350486X.2022.2125884","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2125884","url":null,"abstract":"We propose a model to quantify the effect of parameter uncertainty on the option price in the Heston model. More precisely, we present a Hamilton–Jacobi–Bellman framework which allows us to evaluate best and worst-case scenarios under an uncertain market price of volatility risk. For the numerical approximation, the Hamilton–Jacobi–Bellman equation is reformulated to enable the solution with a finite element method. A case study with butterfly options exhibits how the dependence of Delta on the magnitude of the uncertainty is nonlinear and highly varied across the parameter regime.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88501871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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