Posted by on 2017年3月11日

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The physics is the “trunk” of the research and ideology, while the financial mathematics is the “branch”. Today I want to briefly talk about some physicians in financial mathematics.

  1. Genetic Algorithm (GA, 遗传算法)

The problem of finding the optimal strategy of investment is an optimization problem (see Mathematical optimization or 最优化) and the general methods are convex programming, stochastic programming and so on. And compared to other methods, heuristic algorithms (启发式算法) can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical just like the financial market. Genetic algorithm is one of the heuristic algorithms.

Alan Turing ( see艾伦·图灵) proposed in 1950 a "learning machine" which would parallel the principles of evolution.[1] Then Nils Aall Barricelli (Mathematician, see 进化算法) [2] and Alex Fraser (Australian scientist, ) [3] published many papers on Computer simulation of evolution respectively. From these beginnings, computer simulation of evolution by biologists became more common in the early 1960s.

The work of Ingo Rechenberg [4] and Hans-Paul Schwefel in 1970s [5] made artificial evolution a widely recognized optimization method. Another approach was the evolutionary programming technique of Lawrence J. Fogel [6], which was proposed for generating artificial intelligence. Genetic algorithms in particular became popular through the work, which originated with studies of cellular automata (see 细胞自动机), of John Holland in the early 1970s. In his book Adaptation in Natural and Artificial Systems (1975) [7], Holland introduced a formalized framework for predicting the quality of the next generation, known as Holland's Schema Theorem. Genetic algorithms have increasingly been applied into economics since the pioneering work of John H. Miller (Professor of Economics and Social Science in CMU) in 1986.[8]

If you want to find the optimal condition to perform an experiment, the traditional method is to investigate the function mechanism and spend a long time doing research before you can find the best answer. However, genetic algorithm is an indirect method to investigate the problem: first of all, you can establish a pool of possible solutions; then you let all the answers and conditions in the pool compete with each other just like the biological competition; after a few generation of competition, you will get the optimal solution and can apply it into real experiment to get an optimal result. In a sense, the genetic algorithm is a method that is closest to the best answer.

Genetic algorithms (GAS) are versatile evolutionary computation techniques based on the principle of survival of the finest. After the first introduction as classifier systems by Holland and later developed by Goldberg [34] in search optimization and machine learning, GAs have found many good results in applications such as the time series forecasting [35-36],adaptive agents in stock markets [37], solving the crypto-arithmetic problem [38], computational finance [39], function optimization, traveling salesman problem [40], multi-stage logistic chain network [41], and airport scheduling [42].

  1. Stochastic process

In 1900, Louis Bachelier (路易·巴舍利耶, see 金融理论), a French mathematician, who is credited with being the first person to model the stochastic process now called Brownian motion, completed his PhD thesis The Theory of Speculation [9] that predated Einstein's celebrated study of Brownian motion [10] by five years. And in his thesis he discussed the use of Brownian motion to evaluate stock options. Thus, Bachelier is considered a pioneer in the study of financial mathematics and stochastic processes.

  1. Log-normal distribution of common-stock price

The American physicist M.F.M. Osborne’s work in 1960s [11-13] (more about Osborene see here), based on Weber-Fechner Law from psychology, showed the normal distribution of common-stock rate of return and log-normal distribution of common-stock price. And he also pointed out that the value of money could be regarded as an ensemble of decisions in statistical equilibrium, with properties quite analogous to an ensemble of particles in statistical mechanics.

  1. Lévy-stable distributions and heavy tails

French and American mathematician Benoit Mandelbrot (本华·曼德博 here is a TED talk about fractal given by Mandelbrot) who discovered the Mandelbrot set of intricate, never-ending fractal shapes, found in 1997 that price changes in financial markets did not follow a Gaussian distribution, but rather Lévy-stable distributions having theoretically infinite variance. [14-16] "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter. What’s more, distribution of rate of return has heavy tail but not belong to Levy-stable distribution.

  1. Delta hedging

American mathematics professor Edward Thorp (爱德华·索普,here is a talk with Thorp. He is genius.) applied information theory into research of certified equity and used his knowledge of probability and statistics in the stock market by discovering and exploiting a number of pricing anomalies in the securities markets. He adopted the strategy of delta hedging to make a significant fortune.[17]

  1. CAPM

The Capital Asset Pricing Model (CAPM, see famous资本资产定价模型) was introduced by Economists Jack Treynor [18-19], William Sharpe (Professor of Finance, Emeritus at Stanford. Come on, he creates Sharp Ratio! see 夏普比率)[20], John Lintner [21-22] and Jan Mossin [23] independently, building on the earlier work of American economist Harry Markowitz [24] on diversification and modern portfolio theory. The CAPM is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk. The Black–Scholes model was first published by Economists Fischer Black and Myron Scholes(here is a lecture by 1997 Nobel Prize Winner Scholes) in 1973, both of who had physical educational background and were fascinated by CAPM.[25] They derived a partial differential equation, now called the Black–Scholes equation, which governs dynamic hedging strategy and the price of the option over time.

  1. Volatility smile

Volatility smiles are implied volatility patterns that arise in pricing financial options, first proposed by financial mathematician and former physicist Emanuel Derman (recommend his book宽客人生).[26] This anomaly implies deficiencies in the standard Black-Scholes option pricing model which assumes constant volatility and log-normal distributions of underlying asset returns. Modeling the volatility smile is nowadays an active area of research in quantitative finance. Moreover, methods of modeling the volatility smile include stochastic volatility models [27] and local volatility models [28].

  1. Low-dimensional chaotic dynamical systems

American mathematician, meteorologist Edward N. Lorenz originally demonstrated that very simple low-dimensional systems could display “chaotic” or “turbulent behavior”.[29] In 1980, the work of Physicists James Doyne Farmer and Norman Packard showed how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally and used attractor applied as a theoretical tool to observe the behavior mode of chaotic systems.[30] After already familiar with investment strategies, Farmer and Packard started to use Nonlinear dynamic analysis and chaos theory to predict the behavior of financial market and founded Prediction Company in 1991, who uses statistical arbitrage such as pairs trading pioneered by Information Scientist Gerry Bamberger and later led by Astrophysicist Nunzio Tartaglia’s quantitative group at Morgan Stanley in the 1980s. (see Pairs Trade)[31], involving data mining and statistical methods.

  1. Predicting financial crisis

Automated trading systems become the main target of criticism of Quantitative Finance after the Financial Crisis in 2008 for the abuse and lack of supervision. We should keep in mind that effectiveness of mathematical models depends on the establishment of hypothesis and every model has its limit and will become out of order when extreme events happen. So is it possible to predict the extreme events? Didier Sornette (here is a TED talk by him), a professor both on Physics and Earth Sciences, proposed a theory about “King Effect” referring to the phenomenon where the top one or two members of a ranked set show up as outliers, used to predict economic bubbles, one kind of extreme events, and set up the "Financial Crisis Observatory".[32-33] His work is a good try using theory from physics to investigate the extreme events in economics and is very inspiring.

Part of the content is from the reference of book The Physics of Wall Street: A Brief History of Predicting. 部分内容参考自 《華爾街的物理學》

【欢迎订阅并关注微信公众号“饱蠹阁BaoDuGe”】
Read more:

  1. Turing, A. M. (1950). Computing machinery and intelligence. Mind, 433-460.
  2. Barricelli, N. A. (1954). Esempi numerici di processi di evoluzione. Methodos, 6(21-22), 45-68.
  3. Fraser, A. S. (1960). Simulation of genetic systems by automatic digital computers vi. epistasis. Australian Journal of Biological Sciences, 13(2), 150-162.
  4. Rechenberg, I. (1978). Evolutionsstrategien (pp. 83-114). Springer Berlin Heidelberg.
  5. Schwefel, H. P. (1977). Numerische Optimierung von Computer-Modellen mittels der Evolutionsstrategie: mit einer vergleichenden Einführung in die Hill-Climbing-und Zufallsstrategie. Birkhäuser.
  6. Fogel, L. J., Owens, A. J., & Walsh, M. J. (1966). Artificial intelligence through simulated evolution.
  7. Holland, J. H. (1975). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. U Michigan Press.
  8. Miller, J. H. (1986). A genetic model of adaptive economic behavior. University of Michigan. Mimeo.
  9. Bachelier, L. (1900). Théorie de la spéculation. Gauthier-Villars.
  10. Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der physik, 322(8), 549-560.
  11. Osborne, M. F. (1959). Brownian motion in the stock market. Operations research, 7(2), 145-173.
  12. Niederhoffer, V., & Osborne, M. F. M. (1966). Market making and reversal on the stock exchange. Journal of the American Statistical Association, 61(316), 897-916.
  13. Osborne, M. F. M. (1962). Periodic structure in the Brownian motion of stock prices. Operations Research, 10(3), 345-379.
  14. Mandelbrot, B. B. (1967). How long is the coast of Britain. Science, 156(3775), 636-638.
  15. Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM review, 10(4), 422-437.
  16. Mandelbrot, B. B. (1997). The variation of certain speculative prices (pp. 371-418). Springer New York.
  17. Thorp, E. O. (1971). Portfolio choice and the Kelly criterion. Stochastic models in finance, 599-619.
  18. Treynor, J. L. (1961). Toward a theory of market value of risky assets.
  19. Treynor, J. L. (1961). Market value, time, and risk. Unpublished manuscript, 95-209.
  20. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk*. The journal of finance, 19(3), 425-442.
  21. 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, 13-37.
  22. Lintner, J. (1965). Security Prices, Risk, and Maximal Gains from Diversification*. The Journal of Finance, 20(4), 587-615.
  23. Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica: Journal of the econometric society, 768-783.
  24. Markowitz, H. (1952). Portfolio selection*. The journal of finance, 7(1), 77-91.
  25. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 637-654.
  26. Derman, E., & Kani, I. (1994). Riding on a smile. Risk, 7(2), 32-39.
  27. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of financial studies, 6(2), 327-343.
  28. Dupire, B. (1997). Pricing and hedging with smiles (pp. 103-112). Mathematics of derivative securities. Dempster and Pliska eds., Cambridge Uni. Press.
  29. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2), 130-141.
    1. Packard, N. H., Crutchfield, J. P., Farmer, J. D., & Shaw, R. S. (1980). Geometry from a time series. Physical review letters, 45(9), 712.
  30. Bookstaber, R. (2007). A demon of our own design: Markets, hedge funds, and the perils of financial innovation.
  31. Sornette, D. (2006). Critical phenomena in natural sciences: chaos, fractals, selforganization and disorder: concepts and tools. Springer Science & Business.
  32. Sornette, D. (1998). Discrete-scale invariance and complex dimensions. Physics reports, 297(5), 239-270.
  33. Goldberg, D.E.(1989) Genetic Algorithm in Search, Optimization, andMachine Le-aming, Addison Wesley.
  34. Szeto, K.Y.,& Cheung, K.H.(1997). Proceedings of the World Mul- ticonference on Systemic, Cybernetics and Informatics, Caracas 3, 390–396.
  35. Szeto, K.Y., Chan, K.O.,& Cheung, K.H.(1997) Contributed paper
  36. of Proceedings of the Fourth International Conference on Neural Networks in the Capital Markets Progress in Neu- ral Processing, Decision Technologies for Financial Engi- neering, edited by A.S. Weigend, Y. Abu-Mostafa, A.P.N. Refenes, (World Scientific, NNCM-96), pp. 95–103.
  37. Fong, L. Y., & Szeto, K. Y. (2001). Rules extraction in short memory time series using genetic algorithms. The European Physical Journal B-Condensed Matter and Complex Systems, 20(4), 569-572.
  38. Li, S. P., & Szeto, K. Y. (1999). Crytoarithmetic problem using parallel Genetic Algorithms. In 5th International Conference on Soft Computing, Mendl (Vol. 99, pp. 9-12).
  39. Dempster, M. A. H., & Jones, C. M. (2001). A real-time adaptive trading system using genetic programming. Quantitative Finance, 1(4), 397-413.
  40. Jiang, R., Szeto, K. Y., Luo, Y. P., & Hu, D. C. (2000). Distributed parallel genetic algorithm with path splitting scheme for the large traveling salesman problems. In Proceedings of Conference on Intelligent Information Processing, 16th World Computer Congress (pp. 21-25).
  41. Syarif, A., Yun, Y., & Gen, M. (2002). Study on multi-stage logistic chain network: a spanning tree-based genetic algorithm approach. Computers & Industrial Engineering, 43(1), 299-314.
  42. Shiu, K.L.,& Szeto, K.Y.(2008) Self-adaptive Mutation Only Genetic Algorithm: An Application on the Optimization of Airport Capacity Utilization. In: Fyfe, C., Kim, D., Lee, S.-Y., Yin, H. (eds.) IDEAL 2008. LNCS, vol. 5326, pp. 428–435. Springer, Heidelberg

 

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