We consider a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface constitutes the sole input to the model. We model demand using a two-parameter Brownian motion because (i) different points on the demand curve correspond to orders motivated by different information, and (ii) in general, the market price of risk equation of no-arbitrage theory has no solutions when the demand curve is driven by a finite number of factors, thus allowing for arbitrage. We prove that if the driving noise is infinite-dimensional, then there is no arbitrage in the model. Under the equivalent martingale measure, the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterizations of the model and show advantages of specifying the demand curve as a quantity that is a function of price, as opposed to price as a function of quantity. An online appendix presents a basic empirical analysis of the model: calibration using information from actual order books, computation of option prices using Monte Carlo simulations, and comparison with observed data.
{"title":"An infinite-dimensional model of liquidity in financial markets","authors":"S. Lototsky, H. Schellhorn, Ran Zhao","doi":"10.3934/puqr.2021006","DOIUrl":"https://doi.org/10.3934/puqr.2021006","url":null,"abstract":"We consider a dynamic market model of liquidity where unmatched buy and sell limit orders are stored in order books. The resulting net demand surface constitutes the sole input to the model. We model demand using a two-parameter Brownian motion because (i) different points on the demand curve correspond to orders motivated by different information, and (ii) in general, the market price of risk equation of no-arbitrage theory has no solutions when the demand curve is driven by a finite number of factors, thus allowing for arbitrage. We prove that if the driving noise is infinite-dimensional, then there is no arbitrage in the model. Under the equivalent martingale measure, the clearing price is a martingale, and options can be priced under the no-arbitrage hypothesis. We consider several parameterizations of the model and show advantages of specifying the demand curve as a quantity that is a function of price, as opposed to price as a function of quantity. An online appendix presents a basic empirical analysis of the model: calibration using information from actual order books, computation of option prices using Monte Carlo simulations, and comparison with observed data.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"104 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75151744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.
{"title":"Conditional coherent risk measures and regime-switching conic pricing","authors":"E. J. C. Dela Vega, R. Elliott","doi":"10.3934/puqr.2021014","DOIUrl":"https://doi.org/10.3934/puqr.2021014","url":null,"abstract":"This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"20 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75376680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.
{"title":"Explicit solutions for a class of nonlinear BSDEs and their nodal sets","authors":"Zengjing Chen, Shuhui Liu, Z. Qian, Xingcheng Xu","doi":"10.3934/puqr.2022017","DOIUrl":"https://doi.org/10.3934/puqr.2022017","url":null,"abstract":"In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"8 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78419011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.
{"title":"Stochastic ordering by g-expectations","authors":"S. Ly, Nicolas Privault","doi":"10.3934/PUQR.2021004","DOIUrl":"https://doi.org/10.3934/PUQR.2021004","url":null,"abstract":"We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"72 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73268778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-19DOI: 10.1186/s41546-020-00045-y
Jonathan Ansari, L. Rüschendorf
{"title":"Upper risk bounds in internal factor models with constrained specification sets","authors":"Jonathan Ansari, L. Rüschendorf","doi":"10.1186/s41546-020-00045-y","DOIUrl":"https://doi.org/10.1186/s41546-020-00045-y","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"19 1","pages":"1-30"},"PeriodicalIF":1.5,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89691815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-24DOI: 10.1186/s41546-020-00044-z
S. Singh
{"title":"Moderate deviation for maximum likelihood estimators from single server queues","authors":"S. Singh","doi":"10.1186/s41546-020-00044-z","DOIUrl":"https://doi.org/10.1186/s41546-020-00044-z","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"46 1","pages":"1-13"},"PeriodicalIF":1.5,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86260502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-19DOI: 10.1186/s41546-020-0043-5
D. Marushkevych, A. Popier
{"title":"Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting","authors":"D. Marushkevych, A. Popier","doi":"10.1186/s41546-020-0043-5","DOIUrl":"https://doi.org/10.1186/s41546-020-0043-5","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"12 1","pages":"1-24"},"PeriodicalIF":1.5,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89857594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we consider a type of stochastic differential equations driven by G-L'evy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
{"title":"Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes","authors":"H. Qiao, Jiang-Lun Wu","doi":"10.3934/puqr.2022007","DOIUrl":"https://doi.org/10.3934/puqr.2022007","url":null,"abstract":"In the paper, we consider a type of stochastic differential equations driven by G-L'evy processes. We prove that a kind of their additive functionals has path independence and extend some known results.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"16 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82695598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.
{"title":"An FBSDE approach to market impact games with stochastic parameters","authors":"Samuel Drapeau, Peng Luo, A. Schied, Dewen Xiong","doi":"10.3934/puqr.2021012","DOIUrl":"https://doi.org/10.3934/puqr.2021012","url":null,"abstract":"In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"36 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79144434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-01DOI: 10.1186/s41546-019-0040-8
C. Geiss, Alexander Steinicke
{"title":"Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting”","authors":"C. Geiss, Alexander Steinicke","doi":"10.1186/s41546-019-0040-8","DOIUrl":"https://doi.org/10.1186/s41546-019-0040-8","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"11 1","pages":"1-2"},"PeriodicalIF":1.5,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77921618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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