Transcript Similarities between Market Behavior and Netw¥ ork Traffic

Similarities between
Market Behavior and
Network Traffic
Shigeki Goto
Waseda University
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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.
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Black–Scholes
• 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
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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.
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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.
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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
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Fractal
Koch curve
ja.wikipedia.org/wiki/コッホ曲線
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Network Traffic
normal
distribution
(fGn model)
X (ti)
ti
Pr [X (ti) = x]
ti
histogram
asymmetric
X (ti)
Pr [X (ti) = x]
Captured
Packets
X (ti)
Tatsuya Mori, PhD Thesis, Waseda Universtiy, 2005
histogram
symmetric
X (ti)
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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.
95.00%
R 2  0.9899
Cumulative Distribution
90.00%
85.00%
80.00%
75.00%
70.00%
Cumulative Distribution
65.00%
Approximated Curve (Power Function)
60.00%
1%
2%
3%
4%
6%
Distribution of ASes
7%
8%
9%
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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?
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