Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0026
T. Björk
In this chapter we apply the general theory from Chapter 25 to the study of optimal consumption and investment problems. We solve the Merton problem and we derive the Merton mutual fund theorems.
{"title":"Optimal Consumption and Investment","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0026","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0026","url":null,"abstract":"In this chapter we apply the general theory from Chapter 25 to the study of optimal consumption and investment problems. We solve the Merton problem and we derive the Merton mutual fund theorems.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117238161","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 and Futures Contracts","authors":"T. Björk","doi":"10.1007/1-85233-846-6_6","DOIUrl":"https://doi.org/10.1007/1-85233-846-6_6","url":null,"abstract":"","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"56 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133356748","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}
Pub Date : 2019-12-05DOI: 10.1093/0198775180.003.0010
T. Björk
This chapter is an introduction to a series of chapters on incomplete markets. We present the general setting in terms of a Markov factor and we discuss how incompleteness comes into play in this model.
{"title":"Incomplete Markets","authors":"T. Björk","doi":"10.1093/0198775180.003.0010","DOIUrl":"https://doi.org/10.1093/0198775180.003.0010","url":null,"abstract":"This chapter is an introduction to a series of chapters on incomplete markets. We present the general setting in terms of a Markov factor and we discuss how incompleteness comes into play in this model.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133584030","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}
Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0005
T. Björk
In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.
{"title":"Stochastic Differential Equations","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0005","url":null,"abstract":"In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most important properties are derived. We then discuss the connection between SDEs and partial differential equations (PDEs). In particular we prove the Feynman–Kač representation theorem which provides the solution to a parabolic PDE in terms of an expected value connected to a certain SDE. We also discuss and derive the Kolmogorov forward and backward equations.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"38 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131829197","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}
Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0018
T. Björk
In this chapter we develop a theory for derivatives based on the exchange rate between two (or more) currencies. This is initially done using classical delta hedging methods, but the main part of the theory is developed using martingale methods. We discuss the foreign and the domestic martingale measures and the relations between these measures, and in particular we show that the likelihood ratio between the measures equals the ratio between the foreign and the domestic stochastic discount factors. Option pricing formulas are also derived, and we discuss the Siegel paradox.
{"title":"Currency Derivatives","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0018","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0018","url":null,"abstract":"In this chapter we develop a theory for derivatives based on the exchange rate between two (or more) currencies. This is initially done using classical delta hedging methods, but the main part of the theory is developed using martingale methods. We discuss the foreign and the domestic martingale measures and the relations between these measures, and in particular we show that the likelihood ratio between the measures equals the ratio between the foreign and the domestic stochastic discount factors. Option pricing formulas are also derived, and we discuss the Siegel paradox.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"289 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133964249","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}
Pub Date : 2019-12-05DOI: 10.1093/0198775180.003.0003
T. Björk
We introduce the Wiener process, the Itô stochastic integral, and derive the Itô formula. The connection with martingale theory is discussed, and there are several worked-out examples
我们介绍了维纳过程，Itô随机积分，并推导了Itô公式。讨论了其与鞅理论的联系，并给出了几个算例
{"title":"Stochastic Integrals","authors":"T. Björk","doi":"10.1093/0198775180.003.0003","DOIUrl":"https://doi.org/10.1093/0198775180.003.0003","url":null,"abstract":"We introduce the Wiener process, the Itô stochastic integral, and derive the Itô formula. The connection with martingale theory is discussed, and there are several worked-out examples","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131673037","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}
Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0006
T. Björk
We introduce the concept of a financial market and define the concept of a self-financing portfolio. The dynamics of a self-financing portfolio is then derived, both in discrete and continuous time. The theory includes dividend-paying assets and the concept of a cumulative dividend process is introduced and discussed.
{"title":"Portfolio Dynamics","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0006","url":null,"abstract":"We introduce the concept of a financial market and define the concept of a self-financing portfolio. The dynamics of a self-financing portfolio is then derived, both in discrete and continuous time. The theory includes dividend-paying assets and the concept of a cumulative dividend process is introduced and discussed.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131867914","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}
Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0023
T. Björk
This chapter presents the LIBOR market models. These models, which produce lognormal interest rates, are widely used in the industry, can be calibrated to the market cap curve and used to price and hedge exotic interest rate derivatives. They involve a highly nontrivial use of the change of numeraire technique discussed in Chapter 15.
{"title":"Libor Market Models","authors":"T. Björk","doi":"10.1093/oso/9780198851615.003.0023","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0023","url":null,"abstract":"This chapter presents the LIBOR market models. These models, which produce lognormal interest rates, are widely used in the industry, can be calibrated to the market cap curve and used to price and hedge exotic interest rate derivatives. They involve a highly nontrivial use of the change of numeraire technique discussed in Chapter 15.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134332885","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}
Pub Date : 2019-12-05DOI: 10.1093/oso/9780198851615.003.0007
Tomas Björk
The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.
{"title":"Arbitrage Pricing","authors":"Tomas Björk","doi":"10.1093/oso/9780198851615.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198851615.003.0007","url":null,"abstract":"The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126794203","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}
Pub Date : 2010-05-15DOI: 10.1002/9780470061602.EQF04010
T. Björk
In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.
{"title":"Change of Numeraire","authors":"T. Björk","doi":"10.1002/9780470061602.EQF04010","DOIUrl":"https://doi.org/10.1002/9780470061602.EQF04010","url":null,"abstract":"In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.","PeriodicalId":311283,"journal":{"name":"Arbitrage Theory in Continuous Time","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121056151","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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