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Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion 超越半鞅的投资组合优化:影子价格和分数布朗运动
arXiv: Mathematical Finance Pub Date : 2015-05-10 DOI: 10.1214/16-AAP1234
Christoph Czichowsky, W. Schachermayer
{"title":"Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion","authors":"Christoph Czichowsky, W. Schachermayer","doi":"10.1214/16-AAP1234","DOIUrl":"https://doi.org/10.1214/16-AAP1234","url":null,"abstract":"While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Ito process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127990934","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: 28
Super-replication with nonlinear transaction costs and volatility uncertainty 具有非线性交易成本和波动不确定性的超级复制
arXiv: Mathematical Finance Pub Date : 2014-11-05 DOI: 10.1214/15-AAP1130
P. Bank, Y. Dolinsky, Selim Gokay
{"title":"Super-replication with nonlinear transaction costs and volatility uncertainty","authors":"P. Bank, Y. Dolinsky, Selim Gokay","doi":"10.1214/15-AAP1130","DOIUrl":"https://doi.org/10.1214/15-AAP1130","url":null,"abstract":"We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim's payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [11] and [19] for the case of model uncertainty.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128978746","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
A system of quadratic BSDEs arising in a price impact model 在价格影响模型中出现的二次bsde系统
arXiv: Mathematical Finance Pub Date : 2014-08-05 DOI: 10.1214/15-AAP1103
D. Kramkov, Sergio Pulido
{"title":"A system of quadratic BSDEs arising in a price impact model","authors":"D. Kramkov, Sergio Pulido","doi":"10.1214/15-AAP1103","DOIUrl":"https://doi.org/10.1214/15-AAP1103","url":null,"abstract":"We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market maker's risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129466177","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: 38
The Gärtner-Ellis Theorem, Homogenization, and Affine Processes Gärtner-Ellis定理，均匀化和仿射过程
arXiv: Mathematical Finance Pub Date : 2014-06-14 DOI: 10.1007/978-3-319-11605-1_11
Archil Gulisashvili, J. Teichmann
{"title":"The Gärtner-Ellis Theorem, Homogenization, and Affine Processes","authors":"Archil Gulisashvili, J. Teichmann","doi":"10.1007/978-3-319-11605-1_11","DOIUrl":"https://doi.org/10.1007/978-3-319-11605-1_11","url":null,"abstract":"","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"150 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122188042","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
A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics 具有状态依赖价格动态的限价单的函数极限定理
arXiv: Mathematical Finance Pub Date : 2014-05-20 DOI: 10.1214/16-AAP1265
Christian Bayer, U. Horst, Jinniao Qiu
{"title":"A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics","authors":"Christian Bayer, U. Horst, Jinniao Qiu","doi":"10.1214/16-AAP1265","DOIUrl":"https://doi.org/10.1214/16-AAP1265","url":null,"abstract":"We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131920151","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: 34
Risk Neutral Option Pricing With Neither Dynamic Hedging nor Complete Markets 既不考虑动态套期保值也不考虑完全市场的风险中性期权定价
arXiv: Mathematical Finance Pub Date : 2014-05-11 DOI: 10.2139/ssrn.2435916
N. Taleb
{"title":"Risk Neutral Option Pricing With Neither Dynamic Hedging nor Complete Markets","authors":"N. Taleb","doi":"10.2139/ssrn.2435916","DOIUrl":"https://doi.org/10.2139/ssrn.2435916","url":null,"abstract":"Proof that under simple assumptions, such as constraints of Put-Call Parity, the probability measure for the valuation of a European option has the mean derived from the forward price which can, but does not have to be the risk-neutral one, under any general probability distribution, bypassing the Black-Scholes-Merton dynamic hedging argument, and without the requirement of complete markets and other strong assumptions. We confirm that the heuristics used by traders for centuries are both more robust, more consistent, and more rigorous than held in the economics literature. We also show that options can be priced using infinite variance (finite mean) distributions.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125194236","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
Optimal stopping under model uncertainty: randomized stopping times approach 模型不确定性下的最优停车:随机停车时间法
arXiv: Mathematical Finance Pub Date : 2014-05-09 DOI: 10.1214/15-AAP1116
D. Belomestny, Volker Krätschmer
{"title":"Optimal stopping under model uncertainty: randomized stopping times approach","authors":"D. Belomestny, Volker Krätschmer","doi":"10.1214/15-AAP1116","DOIUrl":"https://doi.org/10.1214/15-AAP1116","url":null,"abstract":"In this work we consider optimal stopping problems with conditional convex risk measures called optimised certainty equivalents. Without assuming any kind of time-consistency for the underlying family of risk measures, we derive a novel representation for the solution of the optimal stopping problem. In particular, we generalise the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. Finally, we develop several Monte Carlo algorithms and illustrate their power for optimal stopping under Average Value at Risk.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127189538","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: 20
POLYNOMIAL TERM STRUCTURE MODELS 多项式项结构模型
arXiv: Mathematical Finance Pub Date : 2014-04-24 DOI: 10.1142/S0219024921500096
Si Cheng, M. Tehranchi
{"title":"POLYNOMIAL TERM STRUCTURE MODELS","authors":"Si Cheng, M. Tehranchi","doi":"10.1142/S0219024921500096","DOIUrl":"https://doi.org/10.1142/S0219024921500096","url":null,"abstract":"In this article, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipovic's maximal degree theorem for exponential polynomial models, as well as an explicit characterisation of the set of feasible parameters in the case when the factor process is bounded. Extensions to time-inhomogeneous and multi-factor polynomial models are also considered.","PeriodicalId":385109,"journal":{"name":"arXiv: Mathematical Finance","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123608555","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: 4
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