{"title":"Multidimensional Models: Martingale Approach","authors":"T. Björk","doi":"10.1093/0199271267.003.0014","DOIUrl":"https://doi.org/10.1093/0199271267.003.0014","url":null,"abstract":"In this chapter we study a very general multidimensional Wiener-driven model using the martingale approach. Using the Girsanov Theorem we derive the martingale equation which is used to find an equivalent martingale measure. We provide conditions for absence of arbitrage and completeness of the model, and we discuss hedging and pricing. For Markovian models we derive the relevant pricing PDE and we also provide an explicit representation formula for the stochastic discount factor. We discuss the relation between the market price of risk and the Girsanov kernel and finally we derive the Hansen–Jagannathan bounds for the Sharpe ratio.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126535902","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":"Martingale Models for the Short Rate","authors":"T. Björk","doi":"10.1093/0198775180.003.0017","DOIUrl":"https://doi.org/10.1093/0198775180.003.0017","url":null,"abstract":"This chapter is devoted to an overview and analysis of the most common short rate models, such as the Vasiček, Dothan, Hull–White, and CIR models. These models are analyzed and classified from the point of view of positive short rates, normal distribution, mean reversion, and computability. In particular we present the theory of affine term structures, and discuss the inversion of the yield curve. Analytical results for bond prices and bond options are presented for all the affine models.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128904556","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":"Stochastic Optimal Control","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0025","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0025","url":null,"abstract":"We study a general stochastic optimal control problem within the framework of a controlled SDE. This problem is studied using dynamic programming and we derive the Hamilton–Jacobi–Bellman PDE. By stating and proving a verification theorem we show that solving this PDE is equivalent to solving the control problem. As an example the theory is then applied to the linear quadratic regulator.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125796415","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":"The Binomial Model","authors":"T. Björk","doi":"10.1093/OSO/9780198851615.003.0002","DOIUrl":"https://doi.org/10.1093/OSO/9780198851615.003.0002","url":null,"abstract":"The binomial model is introduced. We discuss the concept of pricing by no arbitrage and derive pricing formulas for financial derivatives within the binomial model, and the market is shown to be complete. The concept of a martingale measure is introduced and related to the pricing formulas.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"545 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130389965","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":"Forward Rate Models","authors":"T. Björk","doi":"10.1093/0198775180.003.0018","DOIUrl":"https://doi.org/10.1093/0198775180.003.0018","url":null,"abstract":"In this chapter we study the Heath–Jarrow–Morton framework for forward rate models. Building on results from the previous chapter, the HJM drift condition is derived, some examples are studied, and the general Gaussian HJM model is analyzed in detail. The Musiela parameterization of forward rates is introduced and discussed in the context of infinite dimensional SDEs.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127892217","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":"Short Rate Models","authors":"T. Björk","doi":"10.1093/OSO/9780198851615.003.0020","DOIUrl":"https://doi.org/10.1093/OSO/9780198851615.003.0020","url":null,"abstract":"The simplest Markovian short rate model is analyzed using classical and martingale methods, and the term structure equation for the determination of zero coupon bond prices is derived.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122548530","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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