Similarities between Market Behavior and Network Traffic Shigeki Goto Waseda University
Black–Scholes http://en.wikipedia.org/wiki/Black-Scholes The Black–Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process. The Black–Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity. The Black–Scholes formula is the result obtained by solving the Black-Scholes PDE for European put and call options.
Black–Scholes dSt = μStdt+ σStdWt Merton and Scholes received the 1997 Nobel Prize in Economics for this and related work. Black was ineligible for the prize because of his death in 1995. dSt = μStdt+ σStdWt The price of the underlying instrument St follows a Wiener process Wt with constant drift μ and volatility σ, and the price changes are log-normally distributed: ブラック-ショールズモデルとは、1 種類の配当のない株と 1 種類の債券の 2 つが存在する証券市場のモデルで、時刻 t における株価 St と債券価格 Bt が dSt = St(σdWt + μdt), Bt = exp(rt) , r,σ,μは定数 , Wtは標準ブラウン運動, を満たすものをいう。 ja.wikipedia.org/wiki/ブラック-ショールズ方程式
Network Traffic Our diffusion equation is similar to that taken by Black and Scholes who modeled the option pricing behavior as a geometric Brownian motion. http://dspace.wul.waseda.ac.jp/dspace/bitstream/2065/4894/1/93038_391.pdf Yoshitaka Takahashi, A branching Poisson Process Input Finite-Capacity Queueing System for Telecommunication Networks, Waseda Commercial Review, vol.391, pp.105—116, Dec 2001.
Fractal and Multifractal Benoit B. Mandelbrot, A Multifractal Walk Down Wall Street, Scientific American, February 1999. http://www.elliottwave.com/education/SciAmerican/Mandelbrot_Article2.htm Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin, and Reward, 2004. (In Japanese, Econophysics) 高安秀樹「経済物理学の発見」 光文社新書１６７, 2004.
Random Walk vs. Fractal A Multifractal Walk Down Wall Street Burton G. Malkiel, Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing
Fractal Koch curve ja.wikipedia.org/wiki/コッホ曲線
Network Traffic histogram X (ti) Pr [X (ti) = x] asymmetric ti X (ti) Tatsuya Mori, PhD Thesis, Waseda Universtiy, 2005 histogram Captured Packets X (ti) Pr [X (ti) = x] asymmetric ti X (ti) histogram normal distribution (fGn model) X (ti) Pr [X (ti) = x] symmetric ti X (ti)
Power Function (High volume AS numbers) Naoki Arai and Shigeki Goto, Traffic Analysis Based on Autonomous System Numbers , IWS2000, Internet Workshp 2000 pp.121-126 , February 16, 2000.
Lessons It is not possible to predict stock prices by the new theory. The old model underestimates the risk because it uses random walk, or normal distribution. Network traffic: 95% rule of SLA is reasonable?