فرهنگ و اندیشه ریاضی

فرهنگ و اندیشه ریاضی

جنبه‌های احتمالاتی دانش مالی

نوع مقاله : ترجمه

نویسنده
دانشگاه علامه طباطبائی، دانشکدۀ ریاضی، گروه ریاضی
چکیده
در دهه های گذشته روش های پیشرفتۀ احتمالاتی تأثیری مهم در زمینۀ مالی هم از جنبۀ نظری
و هم از نظر صنعت مالی برجای نهاده اند. درمقابل، مسائل مالی نیز انگیزه بخش جهت های تحقیقاتی
جدید در علم احتمال بوده اند. در این مقالۀ مروری، برخی از این پیشرفت ها را بررسی می کنیم و به
زمینه هایی اشاره می کنیم که ممکن است استحقاق تحقیقات بیشتری را داشته باشند. نخست مبانی
قیمت گذاری آربیتراژ را با تأکید ویژه بر بازارهای ناکامل و نقش های مختلفی که اندازۀ احتمال در
و اندازۀ مارتینگل معادل آن ایفا می کنند، مرور می کنیم. سپس بر مسئلۀ ابهام مدل، « دنیای واقعی »
که عدم قطعیت نایتی هم نامیده می شود، تمرکز می کنیم. دو مطالعۀ موردی را می آوریم که در آن ها
امکان کنار آمدن با عدم قطعیت نایتی  از طریق ابزارهای ریاضی وجود دارد. در مطالعۀ موردی
اول پوشش ریسک مشتقاتی چون سواپ های واریانس را به معنای اکیداً مسیری بررسی می کنیم. در
مطالعۀ موردی دوم با الزامات سرمایه و ترجیحاتی که سنجه های ریسک محدب و منسجم مشخص
می کنند، سروکار داریم. در دو بخش آخر مسائل ریاضی ناشی از افزایش چشم گیر دادوستد الگوریتمی
در بازارهای مالی مدرن را مورد بحث قرار می دهیم.
کلیدواژه‌ها

موضوعات


[1]    Alfonsi, A., Fruth, A., Schied, A., Optimal execution strategies in limit order books with general shape functions, Quant. Finance, 10 (2010), 143–157.
[2]    Alfonsi, A., Schied, A., Slynko, A., Order book resilience, price manipulation, and the positive portfolio problem, SIAM J. Financial Math., 3 (2012), 511–533.
[3]    Artzner, P., Delbaen, F., Eber, J. M., Heath, D., Coherent measures of risk, Math. Finance, 9 (1999), 203–228.
[4]    Avellaneda, M., Stoikov, S., High-frequency trading in a limit order book, Quant. Finance, 8 (2008), 217–224.
[5]    Bachelier, L., Theorie de la Speculation: Theorie Mathematique du Jeu, Les Grands Classiques, Gauthier-Villars, Paris, 1995.
[6]    Bernoulli, D., Specimen theoriae novae de mensura sortis, Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5 (1738), 175–1926.
[7]    Biagini, F., Föllmer, H., Nedelcu, S., Shifting martingale measures and the birth of a bubble as a submartingale, 2011 (unpublished manuscript).
[8]    Bick, A., Willinger, W., Dynamic spanning without probabilities, Stochastic Process. Appl., 50 (1994), 349–374.
[9]    Black, F., Scholes, M., The pricing of options and corporate liabilities, The Journal of Political Economy, (1973), 637–654.
[10]    Borkar, V. S., Konda, V. R., Mitter, S. K., On De Finetti coherence and Kolmogorov probability, Statist. Probab. Lett., 66 (2004), 417–421.
[11]    Bovier, A., Černý, J., Hryniv, O., The opinion game: Stock price evolution from microscopic market modeling, Int. J. Theor. Appl. Finance, 9 (2006), 91–111.
[12]    Brown, H., Hobson, D., Rogers, L. C. G., Robust hedging of barrier options, Math. Finance, 11 (2001), 285–314.
[13]    Brunnermeier M. K. Pedersen L. H. Predatory trading J. Finance 60 (2005) 1825–1863.
[14]    Bühler, H., Consistent variance curve models, Finance Stoch., 10 (2006), 178–203.
[15]    Bühler, H., Volatility markets: Consistent modeling, hedging and practical implementation, Ph.D. thesis, TU, Berlin, 2006.
[16]    Carlin, B. I., Lobo, M. S., Viswanathan, S., Episodic liquidity crises: Cooperative and predatory trading, J. Finance, 65 (2007), 2235–2274.
[17]    Cassidy, J., How Markets Fail: The Logic of Economic Calamities, Farrar, Straus & Giroux, New York, 2009.
[18]    CFTC-SEC, Findings regarding the market events of May 6, 2010, Technical report, 2010.
[19]    Cont, R., de Larrard, A., Linking volatility with order flow: Heavy traffic approximations and diffusion limits of order book dynamics, 2010 (unpublished manuscript).
[20]    Cont, R., de Larrard, A., Price dynamics in a Markovian limit order market, SIAM J. Financial Math. (to appear).
[21]    Cont, R., Fournie, D. A., Functional Ito calculus and stochastic integral representation of martingales, Ann. Probab., 41 (2013), 109–133.
[22]    Cont, R., Kukanov, A., Stoikov, S., The price impact of order book events, 2010 (unpublished manuscript), available at arXiv:1011.6402.
[23]    Cox, A. M. G., Oblòj, J., Robust hedging of double touch barrier options, SIAM J. Financial Math., 2 (2011), 141–182.
[24]    Davis, M., Oblòj, J., Raval, V., Arbitrage bounds for weighted variance swap prices, Math. Finance. (to appear).
[25]    de Finetti, B., Theory of Probability: A Critical Introductory Treatment, Vol. 1, Wiley Classics Library, Wiley, Chichester, 1990.
[26]    de Finetti, B., Theory of Probability: A Critical Introductory Treatment, Vol. 2, Wiley Classics Library, Wiley, Chichester, 1990.
[27]    Delbaen, F., Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463–520.
[28]    Delbaen, F., Schachermayer, W., The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann., 312 (1998), 215–250.
[29]    Delbaen, F., Schachermayer, W., The Mathematics of Arbitrage, Springer Berlin, 2006.
[30]    Dellacherie, C., Quelques commentaires sur les prolongements de capacités, in Séminaire de Prob-abilités, V (Univ. Strasbourg, Année Universitaire 1969–1970), Lecture Notes in Math., vol. 191, Springer, Berlin, 1971, 77–81.
[31]    Deprez, O., Gerber, H. U., On convex principles of premium calculation, Insurance Math. Econom., 4 (1985), 179–189.
[32]    Doeblin, W., Sur L’équation de Kolmogoroff, Par W. Doeblin, C. R. Acad. Sci. Paris Sér. I, Math., 3 31 (2000).
[33]    Dupire, B., Model art, Risk, 6 (1993), 118–124.
[34]    Dupire, B., Functional Itô Calculus, Bloomberg Portfolio Research paper, (2009).
[35]    Dynkin, E. B., Superdiffusions and parabolic nonlinear differential equations, Ann. Probab., 20 (1992), 942–962.
[36]    El Karoui, N., Jeanblanc-Picquè, M., Shreve, S. E., Robustness of the Black and Scholes formula, Math. Finance, 8 (1998), 93–126.
[37]    El Karoui, N., Quenez, M. C., Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optim., 33 (1995), 29–66.
[38]    Föllmer, H., Calcul d’Itô sans probabilités, in Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980), Lecture Notes in Math., vol. 850, Springer, Berlin, 1981, 143–150.
[39]    Föllmer, H., Probabilistic aspects of financial risk, in European Congress of Mathematics, (Barcelona, 2000), Progr. Math., vol. 201 Birkhäuser, Basel, 2001, 21–36.
[40]    Föllmer, H., Gundel, A., Robust projections in the class of martingale measures, Illinois J. Math., 50 (2006), 439–472.
[41]    Föllmer, H., Horst, U., Kirman, A., Equilibria in financial markets with heterogeneous agents: A probabilistic perspective, J. Math. Econom., 41 (2005), 123–155.
[42]    Föllmer, H., Kabanov, Y. M., Optional decomposition and Lagrange multipliers, Finance Stoch., 2 (1998), 69–81.
[43]    Föllmer, H., Leukert, P., Efficient hedging: Cost versus shortfall risk, Finance Stoch., 4 (2000), 117–146.
[44]    Föllmer, H., Schied, A., Convex measures of risk and trading constraints, Finance Stoch., 6 (2002), 429–447.
[45]    Föllmer, H., Schied, A., Stochastic Finance: An Introduction in Discrete Time, third revised and extended ed., de Gruyter, Berlin, 2011.
[46]    Föllmer, H., Schied, A., Weber, S., Robust preferences and robust portfolio choice, in Mathematical Modelling and Numerical Methods in Finance, P. Ciarlet, A. Bensoussan, Q. Zhang, eds., Elsevier/North-Holland, Amsterdam, 2009, 29–88.
[47]    Föllmer, H., Schweizer, M., Hedging of contingent claims under incomplete information, in Applied Stochastic Analysis (London, 1989), Stochastics Monogr., vol. 5, Gordon and Breach, New York, 1991, 389–414.
[48]    Föllmer, H., Schweizer, M., A microeconomic approach to diffusion models for stock prices, Math. Finance, 3 (1993), 1–23.
[49]    Föllmer, H., Schweizer, M., The minimal martingale measure, in Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley., Hoboken, NJ, 2010, 1200–1204.
[50]    Föllmer, H., Sondermann, D., Hedging of nonredundant contingent claims, in Contributions to Mathematical Economics, Amsterdam, 1986, 205–223.
[51]    Frittelli, M., Rosazza Gianin, E., Putting order in risk measures, Journal of Banking & Finance, 26 (2002), 1473–1486.
[52]    Friz, P. K., Victoir, N. B., Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Studies in Advanced Mathematics, vol. 120, Cambridge Univ. Press, Cambridge, 2010.
[53]    Gatheral, J., The Volatility Surface. A Practitioner ’s Guide, Wiley, Hoboken, NJ, 2006.
[54]    Gatheral, J., No-dynamic-arbitrage and market impact, Quant. Finance, 10 (2010), 749– 759.
[55]    Gatheral, J., Schied, A., Dynamical models of market impact and algorithms for order execution, in Handbook on Systemic Risk, J.-P. Fouque, J. Langsam, eds., Cambridge University Press, Cambridge, 2013.
[56]    Gilboa, I., Schmeidler, D., Maxmin expected utility with nonunique prior, J. Math. Econom., 18 (1989), 141–153.
[57]    Goovaerts, M. J., De Vylder, F., Haezendonck, J., Insurance Premiums: Theory and Applications, North-Holland, Amsterdam, 1984.
[58]    Harrison, J. M., Kreps, D. M., Martingales and arbitrage in multiperiod securities markets, J. Econom. Theory, 20 (1979), 381–408.
[59]    Heath, D., Back to the Future, Plenary Lecture in First World Congress of the Bachelier Finance Society, Paris, 2000.
[60]    Hellwig, M., Systemic risk in the financial sector: An analysis of the subprimemortgage financial crisis, De Economist, 157 (2009), 129–207.
[61]    Hernández-Hernández, D., Schied, A., A control approach to robust utility maximization with logarithmic utility and time-consistent penalties, Stochastic Process. Appl., 117 (2007), 980–1000.
[62]    Hobson, D. G., Robust hedging of the lookback option, Finance Stoch., 2 (1998), 329–347.
[63]    Huber, P. J., Robust Statistics, Wiley, New York, 1981.
[64]    Huber, P. J., Strassen, V., Minimax tests and the Neyman–Pearson lemma for capacities, Ann. Statist., 1 (1973), 251–263.
[65]    Huberman, G., Stanzl, W., Price manipulation and quasi-arbitrage, Econometrica, 72 (2004), 1247– 1275.
[66]    Jarrow, R. A., Protter, P., Shimbo, K., Asset price bubbles in complete markets, in Advances in Mathematical Finance, Birkh’́auser, Boston, MA, 2007, 97–121.
[67]    Jarrow, R. A., Protter, P., Shimbo, K., Asset price bubbles in incomplete markets, Math. Finance, 20 (2010), 145–185.
[68]    Kabanov, Y. M., On the FTAP of Kreps–Delbaen–Schachermayer, in Statistics and Control of Stochastic Processes (Moscow, 1995/1996), World Scientific, River Edge, NJ, 1997, 191–203.
[69]    Karatzas, I., Lehoczky, J. P., Shreve, S. E., Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557–1586.
[70]    Karatzas, I., Shreve, S. E., Methods of Mathematical Finance, Springer, New York, 1998.
[71]    Kirman, A., The economic crisis is a crisis for economic theory, CESifo Economic Studies, 56 (2010), 498–535.
[72]    Kl’́ock, F., Schied, A., Sun, Y., Price manipulation in a market impact model with dark pool, 2011 (unpublished manuscript).
[73]    Knight, F., Risk, Uncertainty, and Profit, Houghton Mifflin, Boston, 1921.
[74]    Kramkov, D., Schachermayer, W., The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), 904– 950.
[75]    Kramkov, D., Schachermayer, W., Necessary and sufficient conditions in the problem of optimal investment in incomplete markets, Ann. Appl. Probab., 13 (2003), 1504–1516.
[76]    Kramkov, D. O., Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probab. Theory Related Fields, 105 (1996), 459–479.
[77]    Kratz, P., Sch’́oneborn, T., Optimal liquidation in dark pools, 2010 (unpublished manuscript).
[78]    Kreps, D. M., Three Essays on Capital Markets, Stanford University, 1979.
[79]    Lehalle, C. A., Market microstructure knowledge needed to control an intra-day trading process, in Handbook on Systemic Risk, J.-P. Fouque, J. Langsam, eds., Cambridge University Press, Cambridge, 2013.
[80]    Lyons, T., Qian, Z., System Control and Rough Paths, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2002.
[81]    Maccheroni, F., Marinacci, M., Rustichini, A., Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447– 1498.
[82]    Merton, R. C., Theory of rational option pricing, Bell J. Econom. and Management Sci., 4 (1973), 141–183.
[83]    Mittal, H., Are you playing in a toxic dark pool? A guide to preventing information leakage, Journal of Trading, 3 (2008), 20–33.
[84]    Monroe, I., On embedding right continuous martingales in Brownian motion, Ann. Math. Statist., 43 (1972), 1293–1311.
[85]    Monroe, I., Processes that can be embedded in Brownian motion, Ann. Probab. 6 (1978), 42–56.
[86]    Musiela, M., Zariphopoulou, T., Portfolio choice under dynamic investment performance criteria, Quant. Finance, 9 (2009), 161–170.
[87]    Musiela, M., Zariphopoulou, T., Stochastic partial differential equations and portfolio choice, in Contemporary Quantitative Finance, Springer, Berlin, 2010, 195–216.
[88]    Neuberger, A., The log contract, The Journal of Portfolio Management, 20 (1994), 74–80.
[89]    Obizhaeva, A., Wang, J., Optimal trading strategy and supply/demand dynamics, J. Financial Markets, 16 (2013), 1–32.
[90]    Poincaré, H., Science et méthode, Revue Scient., (5) 10 (1908), 417–423.
[91]    Revuz, D., Yor, M., Continuous Martingales and Brownian Motion, 3rd ed., Grundlehren der Math-ematischen Wissenschaften, Springer, Berlin, 1999.
[92]    Samuelson, P. A., Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6 (1965).
[93]    Savage, L. J., The Foundations of Statistics, revised ed., Dover, New York, 1972.
[94]    Schervish, M. J., Seidenfeld, T., Kadane, J. B., The fundamental theorems of prevision and asset pricing, Internat. J. Approx. Reason., 49 (2008), 148–158.
[95]    Schied, A., Optimal investments for robust utility functionals in complete market models, Math. Oper. Res., 30 (2005), 750–764.
[96]    Schied, A., Optimal investments for risk- and ambiguity-averse preferences: A duality approach, Finance Stoch., 11 (2007), 107–129.
[97]    Schied, A., A control problem with fuel constraint and Dawson–Watanabe superprocesses. Ann. Appl. Probab. (to appear).
[98]    Schied, A., Stadje, M., Robustness of delta hedging for path-dependent options in local volatility models, J. Appl. Probab., 44 (2007), 865–879.
[99]    Schmeidler, D., Integral representation without additivity, Proc. Amer. Math. Soc., 97 (1986), 255– 261.
[100]    Schöneborn, T., Trade execution in illiquid markets. Optimal stochastic control and multi-agent equilibria, Ph.D. thesis, TU Berlin, 2008.
[101]    Schöneborn, T., Schied, A., Liquidation in the face of adversity: Stealth vs. sunshine trading, 2009 (unpublished manuscript).
[102]    Schweizer, M., Mean-variance hedging, in Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley, 2010, 1177–1181.
[103]    Sondermann, D., Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Lecture Notes in Economics and Mathematical Systems, vol. 579, Berlin, Springer, 2006.
[104]    Stoikov, S. F., Zariphopoulou, T., Dynamic asset allocation and consumption choice in incomplete markets, Australian Economic Papers, 44 (2005), 414–454.
[105]    Turner, A., The Turner Review: A regulatory response to the global banking crisis, FSA, March, 2009.
[106]    von Neumann, J., Morgenstern, O., Theory of Games and Economic Behavior, 3rd ed., Princeton Univ. Press, Princeton, NJ, 1980.
[107]    Weber, P., Rosenow, B., Order book approach to price impact, Quant. Finance, 5 (2005), 357–364.
[108]    Yan, J. A., A numeraire-free and original probability based framework for financial markets, in Proceedings of the International Congress of Mathematicians, vol. III, 2002.

  • تاریخ دریافت 24 اردیبهشت 1401
  • تاریخ بازنگری 11 آبان 1401
  • تاریخ پذیرش 12 آبان 1401
  • تاریخ انتشار 01 تیر 1402