Absence of Arbitrage, equivalent measures and the Fundamental Theorem of asset pricing
Ausencia de arbitraje, medidas equivalentes y teorema fundamental de valoración
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Zapata Quimbayo, C.A. 2017. Absence of Arbitrage, equivalent measures and the Fundamental Theorem of asset pricing. Odeon. 13 (May 2017), 7–30. DOI:https://doi.org/10.18601/17941113.n13.02.
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Abstract
This paper identifies the main contributions of Stephen Ross to the definition of the basic principles of the fundamental theorem of asset pricing (FTAP), as well as its applications and extensions. By establishing the equivalence between the absence of arbitrage and the existence of a linear asset pricing rule, Ross formulates the principles of a valuation approach that preserves the essential characteristics of the Black-Scholes model, but from a more imple and intuitive framework.
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References (SEE)
Arrow, K. (1964). The role of securities in the optimal allocation of riskbearing. The Review of Economic Studies, 31(2), 91-96.
Artzner, P. y Heath, D. (1995). Approximate completeness with multiple martingale measures. Mathematical Finance, 5(1), 1-11.
Bachelier, L. (1900). Théorie de la Spéculation. Annales scientiques de l’ École Normale Supérieure, 17, 21-86. English translation in: The Random Character of stock market prices (P. Cootner, editor), MIT Press.
Back, K. y Pliska, S. (1991). On the fundamental theorem of asset pricing with an infinite state space. Journal of Mathematical Economics, 20(1), 1-18.
Balbás, A., Mirás, M. y Muñoz, M. (2002). Projective system approach to the martingale characterization of the absence of arbitrage. Journal of Mathematical Economics, 37(4), 311-323.
Brown, D. y Werner, J. (1995). Arbitrage and existence of equilibrium in infinite asset markets. The Review of Economic Studies, 62(1), 101-114.
Black, F. y Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-659.
Cox, J. y Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1), 145-166.
Cox, J., Ross, S. y Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263.
Dalang, R., Morton, A. y Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market model. International Journal of Probability and Stochastic Processes, 29(2), 185-201.
Delbaen, F. y Schachermayer, W. (1994). A general version of the Fundamental Theorem of asset pricing. Mathematische Annalen, 300(1), 463-520.
Delbaen, F. y Schachermayer, W. (1995). The no-arbitrage condition under a change of numéraire. Stochastics and Stochastic Reports, 53(3-4), 213-226.
Delbaen, F. y Schachermayer, W. (1998). The Fundamental Theorem of asset pricing for unbounded Stochastic processes. Mathematische Annalen, 312(1), 215-250.
Delbaen, F. y Schachermayer, W. (2006). The Mathematics of Arbitrage. Berlin: Springer Finance.
Dybvig, P. y Ross, S. (1987). Arbitrage. En: Eatwell, J., Milgate, M. y Newman, P. (eds.), The new Palgrave dictionary of economics, vol. 1. London: Macmillan.
Fernholz, E. y Karatzas, I. (2009). Stochastic portfolio theory: An overview. En Bensoussan, A. y Zhang, Q. (eds.), Handbook of numerical analysis, special volume: Mathematical Modelling and Numerical Methods in Finance. New York: Elsevier.
Fontana, C. (2015). Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance, 18(1), 1-34.
Fontana, C. y Runggaldier,W. (2013). Diffusion-based models for financial markets without martingale measures. En Risk Measures and Attitudes, 45-81. London: Springer.
Guasoni, P., R´asonyi, M. y Schachermayer,W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Annals of Finance, 6(2), 157-191.
Harrison, J. y Kreps, D. (1979). Martingales and Arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408.
Harrison, J. y Pliska, S. (1981). Martingales and Stochastic integrals in the Theory of continuous trading. Stochastic Processes and their Applications, 11(3), 215-260.
Harrison, J. y Pliska, S. (1983). A stochastic calculus model of continuous trading: Complete markets. Stochastic Processes and their Applications, 15(3), 313-316.
Jacod, J. y Shiryaev, A. (1998). Local martingales and the fundamental asset pricing theorems in the discrete time. Finance and Stochastics, 3(2), 259-273.
Johnson, T. (2017). Ethics in Quantitative Finance. Edinburgh: Palgrave Macmillan. Kabanov, Y. y Kramkov, D. (1994). No-arbitrage and equivalent martingale measures: An elementary proof of the Harrison-Pliska theorem. Theory of Probability and its Applications, 39(3), 523-527.
Kabanov, Y., Rásonyi, M. y Stricker, C. (2002). No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics, 6(3), 371-382.
Karatzas, I. y Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance and Stochastics, 11(4), 447-493.
Kreps, D. (1981). Arbitrage and equilibrium in Economics with infinitely many Commodities. Journal of Mathematical Economics, 8(1), 15-35.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics, 47(1), 13-37.
Lewis, K. (2013). A simple proof of the fundamental theorem of asset pricing. Documento de trabajo. Recuperado de http://kalx.net/ftapd.pdf
Merton, R. (1973). The theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica: Journal of the Econometric Society, 34(1), 768-783.
Platen, E. y Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Sidney: Springer.
Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341-360.
Ross, S. (1977). Return, risk and arbitrage. En: Rodney, L. Risk and Return in Finance, Vol. 1, 189-218. Pennyslvania: White Center for Financial Research, The Wharton School, University of Pennyslvania.
Ross, S. (1978). A simple approach to the valuation of risky streams. Journal of Business, 51(1), 453-475.
Ross, S. (2005). Neoclassical finance. New Jersey: Princeton University Press.
Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics and Management Science, 7(2), 407-425.
Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance: Mathematics and Economics, 11(4), 249-257.
Schachermayer,W. (1994). Martingale measures for discrete time processes with infinite horizon. Mathematical Finance, 4(1), 25-56.
Schwartz, E. y Trigeorgis, L. (2004). Real Options and Investment Under Uncertainty: Classical Readings and Recent Contributions. Cambridge: MIT press.
Artzner, P. y Heath, D. (1995). Approximate completeness with multiple martingale measures. Mathematical Finance, 5(1), 1-11.
Bachelier, L. (1900). Théorie de la Spéculation. Annales scientiques de l’ École Normale Supérieure, 17, 21-86. English translation in: The Random Character of stock market prices (P. Cootner, editor), MIT Press.
Back, K. y Pliska, S. (1991). On the fundamental theorem of asset pricing with an infinite state space. Journal of Mathematical Economics, 20(1), 1-18.
Balbás, A., Mirás, M. y Muñoz, M. (2002). Projective system approach to the martingale characterization of the absence of arbitrage. Journal of Mathematical Economics, 37(4), 311-323.
Brown, D. y Werner, J. (1995). Arbitrage and existence of equilibrium in infinite asset markets. The Review of Economic Studies, 62(1), 101-114.
Black, F. y Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-659.
Cox, J. y Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1), 145-166.
Cox, J., Ross, S. y Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263.
Dalang, R., Morton, A. y Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market model. International Journal of Probability and Stochastic Processes, 29(2), 185-201.
Delbaen, F. y Schachermayer, W. (1994). A general version of the Fundamental Theorem of asset pricing. Mathematische Annalen, 300(1), 463-520.
Delbaen, F. y Schachermayer, W. (1995). The no-arbitrage condition under a change of numéraire. Stochastics and Stochastic Reports, 53(3-4), 213-226.
Delbaen, F. y Schachermayer, W. (1998). The Fundamental Theorem of asset pricing for unbounded Stochastic processes. Mathematische Annalen, 312(1), 215-250.
Delbaen, F. y Schachermayer, W. (2006). The Mathematics of Arbitrage. Berlin: Springer Finance.
Dybvig, P. y Ross, S. (1987). Arbitrage. En: Eatwell, J., Milgate, M. y Newman, P. (eds.), The new Palgrave dictionary of economics, vol. 1. London: Macmillan.
Fernholz, E. y Karatzas, I. (2009). Stochastic portfolio theory: An overview. En Bensoussan, A. y Zhang, Q. (eds.), Handbook of numerical analysis, special volume: Mathematical Modelling and Numerical Methods in Finance. New York: Elsevier.
Fontana, C. (2015). Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance, 18(1), 1-34.
Fontana, C. y Runggaldier,W. (2013). Diffusion-based models for financial markets without martingale measures. En Risk Measures and Attitudes, 45-81. London: Springer.
Guasoni, P., R´asonyi, M. y Schachermayer,W. (2010). The fundamental theorem of asset pricing for continuous processes under small transaction costs. Annals of Finance, 6(2), 157-191.
Harrison, J. y Kreps, D. (1979). Martingales and Arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408.
Harrison, J. y Pliska, S. (1981). Martingales and Stochastic integrals in the Theory of continuous trading. Stochastic Processes and their Applications, 11(3), 215-260.
Harrison, J. y Pliska, S. (1983). A stochastic calculus model of continuous trading: Complete markets. Stochastic Processes and their Applications, 15(3), 313-316.
Jacod, J. y Shiryaev, A. (1998). Local martingales and the fundamental asset pricing theorems in the discrete time. Finance and Stochastics, 3(2), 259-273.
Johnson, T. (2017). Ethics in Quantitative Finance. Edinburgh: Palgrave Macmillan. Kabanov, Y. y Kramkov, D. (1994). No-arbitrage and equivalent martingale measures: An elementary proof of the Harrison-Pliska theorem. Theory of Probability and its Applications, 39(3), 523-527.
Kabanov, Y., Rásonyi, M. y Stricker, C. (2002). No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics, 6(3), 371-382.
Karatzas, I. y Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance and Stochastics, 11(4), 447-493.
Kreps, D. (1981). Arbitrage and equilibrium in Economics with infinitely many Commodities. Journal of Mathematical Economics, 8(1), 15-35.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics, 47(1), 13-37.
Lewis, K. (2013). A simple proof of the fundamental theorem of asset pricing. Documento de trabajo. Recuperado de http://kalx.net/ftapd.pdf
Merton, R. (1973). The theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica: Journal of the Econometric Society, 34(1), 768-783.
Platen, E. y Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Sidney: Springer.
Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341-360.
Ross, S. (1977). Return, risk and arbitrage. En: Rodney, L. Risk and Return in Finance, Vol. 1, 189-218. Pennyslvania: White Center for Financial Research, The Wharton School, University of Pennyslvania.
Ross, S. (1978). A simple approach to the valuation of risky streams. Journal of Business, 51(1), 453-475.
Ross, S. (2005). Neoclassical finance. New Jersey: Princeton University Press.
Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics and Management Science, 7(2), 407-425.
Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
Schachermayer, W. (1992). A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance: Mathematics and Economics, 11(4), 249-257.
Schachermayer,W. (1994). Martingale measures for discrete time processes with infinite horizon. Mathematical Finance, 4(1), 25-56.
Schwartz, E. y Trigeorgis, L. (2004). Real Options and Investment Under Uncertainty: Classical Readings and Recent Contributions. Cambridge: MIT press.
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