{"title":"Risk Neutral Jump Arrival Rates Implied in Option Prices and Their Models","authors":"D. Madan, King Wang","doi":"10.1080/1350486X.2021.2007145","DOIUrl":"https://doi.org/10.1080/1350486X.2021.2007145","url":null,"abstract":"ABSTRACT Characteristic functions of risk neutral densities are constructed from the prices of options at a fixed maturity using well-known procedures. The logarithm of these characteristic functions are shown to synthesize the Fourier transform of jump arrival tails. The formal arrival rate tails are actual arrival rates if their derivatives have an appropriate sign. The derivatives of formal arrival rate tails embedded in option prices are observed on occasion to be negative, reflecting signed jump arrival rates. Although puzzling at first, we further observe that simple analytical cosine perturbations of the symmetric variance gamma Lévy density provides theoretical examples of such signed arrival rates consistent with a probability density. Additionally signed arrival rates also arise when models of signals perturbed by independent noise yield examples of characteristic functions for signal densities that are ratios of pure jump infinitely divisible characteristic functions. Such ratio characteristic functions can reflect signed arrival rates. Specific models using ratios of bilateral gamma and CGMY models are developed and calibrated to short maturity option prices. The ratio models provide significant improvements over their non-ratio counterparts. The models fall in the class of what have recently been termed to be quasi-infinitely divisible distributions.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"1 1","pages":"201 - 235"},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90420635","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}
{"title":"Static Replication of European Multi-Asset Options with Homogeneous Payoff","authors":"Sébastien Bossu","doi":"10.1080/1350486X.2022.2085122","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2085122","url":null,"abstract":"ABSTRACT The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by [Carr, Peter, and Dilip Madan. 1998. “Towards a Theory of Volatility Trading.” In Volatility: New Estimation Techniques for Pricing Derivatives, Vol. 29, edited by Robert A. Jarrow, 417–427. Risk books.] extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"3 1","pages":"381 - 394"},"PeriodicalIF":0.0,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82399131","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}
Giuliana Bordigoni, A. Figalli, A. Ledford, Philipp Ustinov
{"title":"Strategic Execution Trajectories","authors":"Giuliana Bordigoni, A. Figalli, A. Ledford, Philipp Ustinov","doi":"10.1080/1350486X.2023.2194658","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2194658","url":null,"abstract":"ABSTRACT We obtain the optimal execution strategy for two sequential trades in the presence of a transient price impact. We first present a novel and general solution method for the case of a single trade (a metaorder) that is executed as a sequence of sub-trades (child orders). We then analyze the case of two sequential metaorders, including the case where the size and direction of the second metaorder are uncertain at the time the first metaorder is initiated. We obtain the optimal execution strategy under two different cost functions. First, we minimize the total cost when each metaorder is benchmarked to the price at its initiation, the total separate costs approach widely used by practitioners. Although simple, we show that optimizing total separate costs can lead to a significant understatement of the real costs of trading whilst also adversely impacting order scheduling. We overcome these issues by introducing a new cost function that splits the second metaorder into two parts, one that is predictable when the first metaorder is initiated and a residual that is not. The predictable and residual parts of the second metaorder are benchmarked using the initiation prices of the first and second metaorders, respectively. We prove existence of an optimal execution trajectory for linear instantaneous price impact and positive definite decay, and derive the explicit form of the minimizer in the special case of exponentially decaying impact, however uniqueness in general remains unproven. Various numerical examples are included for illustration.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"138 1","pages":"288 - 330"},"PeriodicalIF":0.0,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86515699","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}
{"title":"Heterogeneity and Competition in Fragmented Markets: Fees Vs Speed","authors":"Jose S. Penalva, Mikel Tapia","doi":"10.1080/1350486X.2021.1960574","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1960574","url":null,"abstract":"ABSTRACT This paper provides an integrated overview of the effects of the implementation of the SEC’s Tick Pilot program on liquidity and competition in U.S. markets, separated into three groups by tick size. We confirm the standard effects of tick size changes on quoted spreads, realized spreads, and depth, as well as the role of the size of the quoted spread prior to the change in tick size. We add that the increase in the tick size leads to a significant reduction in the frequency and magnitude of price changes, primarily driven by a reduction in the frequency of aggressive limit orders. The major effect of the tick size is to alter competition by driving trading volume to inverted fee and off-exchange venues. We find that traders prefer a larger price improvement rather than lower latency for the smallest tick stocks while the reverse is true for largest tick stocks. Overall, the effect of the tick change has an insignificant effect on volume except for stocks with the smallest tick sizes subject to the trade-at rule, who see a substantial drop in volume.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"13 1","pages":"143 - 177"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78987678","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}
M. Avellaneda, T. Li, A. Papanicolaou, Gaozhan Wang
{"title":"Trading Signals in VIX Futures","authors":"M. Avellaneda, T. Li, A. Papanicolaou, Gaozhan Wang","doi":"10.1080/1350486X.2021.2010584","DOIUrl":"https://doi.org/10.1080/1350486X.2021.2010584","url":null,"abstract":"ABSTRACT We propose a new approach for trading VIX futures. We assume that the term structure of VIX futures follows a Markov model. Our trading strategy selects a position in VIX futures by maximizing the expected utility for a day-ahead horizon given the current shape and level of the term structure. Computationally, we model the functional dependence between the VIX futures curve, the VIX futures positions, and the expected utility as a deep neural network with five hidden layers. Out-of-sample backtests of the VIX futures trading strategy suggest that this approach gives rise to reasonable portfolio performance, and to positions in which the investor will be either long or short VIX futures contracts depending on the market environment.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"319 1","pages":"275 - 298"},"PeriodicalIF":0.0,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72555438","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}
{"title":"On Regularized Optimal Execution Problems and Their Singular Limits","authors":"M. Souza, Yuri Thamsten","doi":"10.1080/1350486X.2022.2148115","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2148115","url":null,"abstract":"We investigate the portfolio execution problem under a framework in which volatility and liquidity are both uncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both of them. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law to relate it to the agent's turnover rate. We first analyse the regularized setting, in which the admissible strategies do not ensure complete execution of the initial inventory. We prove the existence and uniqueness of a continuous and bounded viscosity solution of the Hamilton–Jacobi–Bellman equation, whence we obtain a characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm. Then, we analyse the constrained problem, in which admissible strategies must guarantee complete execution to the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singular limit of the regularized counterparts.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"68 1","pages":"79 - 109"},"PeriodicalIF":0.0,"publicationDate":"2021-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75635695","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}
{"title":"A Structural Approach to Default Modelling with Pure Jump Processes","authors":"Jean-Philippe Aguilar, N. Pesci, V. James","doi":"10.1080/1350486X.2021.1957956","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1957956","url":null,"abstract":"ABSTRACT We present a general framework for the estimation of corporate default based on a firm’s capital structure, when its assets are assumed to follow a pure jump Lévy processes; this setup provides a natural extension to usual default metrics defined in diffusion (log-normal) models, and allows to capture extreme market events such as sudden drops in asset prices, which are closely linked to default occurrence. Within this framework, we introduce several pure jump processes featuring negative jumps only and derive practical closed formulas for equity prices, which enable us to use a moment-based algorithm to calibrate the parameters from real market data and to estimate the associated default metrics. A notable feature of these models is the redistribution of credit risk towards shorter maturity: this constitutes an interesting improvement to diffusion models, which are known to underestimate short-term default probabilities. We also provide extensions to a model featuring both positive and negative jumps and discuss qualitative and quantitative features of the results. For readers convenience, practical tools for model implementation and GitHub links are also included.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"33 1","pages":"48 - 78"},"PeriodicalIF":0.0,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76400031","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}
{"title":"Deep Learning for Market by Order Data","authors":"Zihao Zhang, Bryan Lim, S. Zohren","doi":"10.1080/1350486X.2021.1967767","DOIUrl":"https://doi.org/10.1080/1350486X.2021.1967767","url":null,"abstract":"ABSTRACT Market by order (MBO) data – a detailed feed of individual trade instructions for a given stock on an exchange – is arguably one of the most granular sources of microstructure information. While limit order books (LOBs) are implicitly derived from it, MBO data is largely neglected by current academic literature, which focuses primarily on LOB modelling. In this paper, we demonstrate the utility of MBO data for forecasting high-frequency price movements, providing an orthogonal source of information to LOB snapshots and expanding the universe of alpha discovery. We provide the first predictive analysis on MBO data by carefully introducing the data structure and presenting a specific normalization scheme to consider level information in order books and to allow model training with multiple instruments. Through forecasting experiments using deep neural networks, we show that while MBO-driven and LOB-driven models individually provide similar performance, ensembles of the two can lead to improvements in forecasting accuracy – indicating that MBO data is additive to LOB-based features.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"1 1","pages":"79 - 95"},"PeriodicalIF":0.0,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82332955","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}
{"title":"Explicit Representations for Utility Indifference Prices","authors":"M. Hess","doi":"10.2139/ssrn.3751344","DOIUrl":"https://doi.org/10.2139/ssrn.3751344","url":null,"abstract":"ABSTRACT In this paper, we apply stochastic maximum principles to derive representations for exponential utility indifference prices. We also obtain the related optimal portfolio processes and utility indifference hedging strategies. To illustrate our theoretical results, we present several concrete examples and study the limit behaviour of utility indifference prices for vanishing and infinite risk aversion. We further investigate how the optimal trading strategies and utility indifference prices alter if one assumes that an investor has some additional information on the future behaviour of the underlying stock price process available. In this regard, we propose a customized enlarged filtration approach and deduce a formula for the utility indifference price in this extended setup. We finally provide a representation for the information premium in our utility indifference pricing framework.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"24 1","pages":"23 - 47"},"PeriodicalIF":0.0,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87105032","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}
{"title":"A Bivariate Normal Inverse Gaussian Process with Stochastic Delay: Efficient Simulations and Applications to Energy Markets","authors":"M. Gardini, P. Sabino, E. Sasso","doi":"10.1080/1350486X.2021.2010106","DOIUrl":"https://doi.org/10.1080/1350486X.2021.2010106","url":null,"abstract":"ABSTRACT Using the concept of self-decomposable subordinators introduced by Gardini, Sabino, and Sasso, we build a new bivariate Normal Inverse Gaussian process that can capture stochastic delays. In addition, we also develop a novel path simulation scheme that relies on the mathematical connection between self-decomposable Inverse Gaussian laws and Lévy-driven Ornstein–Uhlenbeck processes with Inverse Gaussian stationary distribution. We show that our approach provides an improvement to the existing simulation scheme detailed in Zhang and Zhang, because it does not rely on an acceptance–rejection method. Eventually, these results are applied to the modelling of energy markets and to the pricing of spread options using the proposed Monte Carlo scheme and Fourier techniques.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"12 1","pages":"178 - 199"},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74364370","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}
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