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Intuitive Method of Economic Mathematics

- Probability & Statistics -

Cat: ECO
Pub: 2016

Shinichiro Naganuma (長沼伸一郎)


Intuitive Method of Economic Mathematics


  1. Preface:
  2. Triangle deviation in a parallel world:
  3. Key of the probability & statistics:
  4. Stochastic process:
  5. Hedge by futures contract:
  6. Idea of Black & Scholes model:
  7. Ito's lemma and Stochastic differential equation::
  8. Actual Black & Scholes theory:
  9. Fourier Series:
  10. Lebesgue integral:
  1. 序文:
  2. パラレルワールドの三角形偏差:
  3. 確率統計の鍵:
  4. 確率過程論とは:
  5. 先物取引によるヘッジ:
  6. ブラック&ショールズモデルの考え方:
  7. 伊藤のレンまと確率微分方程式:
  8. 実際のブラック&ショールズ理論:
  9. フーリエ級数:
  10. ルベーグ積分:
  • The battle with Covid-19 is an issue of probability and statistics. The author explains the essence of probability and statistics as the God's fingerprint.
  • It is worth consider why the standard deviation is called 'standard'; is shape looks beautiful, but is moderately difficult to understand intuitively.
; ; CLT; Delta; Ito's lemma; Least square method; Linear-volatility type; Non-risk portfolio; Power law; Poisson distribution; Random walk; Standard deviation; Stochastic differential equation; Stochastic process; Taylor expansion; Trend; Volatility; Wiener process; ;

>Top 0. Preface:

  • Black-Scholes theory looks difficult but is a good example to know the essence of probability statistics; such as least-squares method, central limit theorem, non-risk bond, Ito's lemma, stochastic differential equation, etc.
  • It is also useful to treat anlog-digital value in a same consideration.

0. 序文:

  • least squares method: 最小二乗法
  • central limit theorem: 中心極限定理:
  • random walk (Brown movement): ランダムウォーク
  • Stochastic differential equation: 確率微分方程式

>Top 1. Triangle deviation in a parallel world:

  • If the standard deviation distributed in a shape of triangle (in a parallel world), the calculation would be much easier to understand.
    • The standard deviation of the parallel world is easily calculated by the average and basic width of the triangle.
    • $\frac{x-m}{d}$, where $m$ is the average, $d$ is basic width of the triangle, $x$ is deviation from the average.
    • The deviation value used in education will be: $\frac{x-m}{d}・10+50$; which is expressed in our world as: $\frac{x-m}{\sigma}・10+50$, where $\sigma$ is a standard deviation.
    • The deviation value is not less than 60 belongs within the top 1/8.
    • The standard deviation of our world shows that the deviation value (40-60) belongs to 68.3% of the total.
    • $d=\displaystyle\sum_i|x_i-m| \Longleftrightarrow \sigma^2=\displaystyle\sum_i(x_i-m)^2$ (↓Fig.)
    • dandsigma

1. パラレルワールドの三角形偏差:

  • Triange in a parallel world:triangle_parallelworlddeviationvalue
  • Standard deviation of our world:

>Top 2. Key of the probability & statistics:

  • The basic concept of probability and statistics is:
    1. Least square method:
      This is a standard approach in regression analysis to approximate position of the center line.
      Particularly, the square calculation can reflect more sensitively points located in the periphery from the center line.
    2. >Top Central limit theorem (CLT):
      when independent random variables are added, their properly normalized sum tends toward a normal distribution, even if original variables are not normally distributed.
      • in 1733, this concept first developed by de Moivre, it wasn't formally named until 1930 by George Polya.
      • average of the sample means and standard deviations will equal the population mean and standard deviation.
      • a sufficiently large sample size can predict the characteristics of a population accurately.
    3. Probability process and random walk:
      Generally speaking, errors or variations tend to appear having either bias, ① deformed position in a certain direction, which can be corrected by humans, or ② distributed in both (plus or minus) directions, which can be treated statistically only. (>Fig.)
      • Normal distribution: in 1733 discover by de Moivre, and in 1809 developed by Gauss.
      • >Top Poisson distribution:
        the probability of a given number of events occurring in a fixed interval of time or place, if these events occur with a known constant mean rate. It can be applied to a system with a large number of possible events, each of which is rare. (Eg.: the number of meteorites greater than 1m diameter strike Earth in a year.
      • Power law: in 1980s discovered by Pareto; a notable example of power laws are Pareto's law of income distribution, and structural self-similarity of fractals, etc.

2. 確率の鍵:

  • Offset portion and Normal distribution portion:
  • offset_ndistribution
  • Poisson Distribution:
  • poissondist
  • Powr Law:

>Top 3. Stochastic Process Theory:

  • >Top Stochastic process theory:
    • Random Walk: A random walker walks a distance $r$ in 360º direction per unit time.
    • The area of two portion of fans are different: the outer fan is a bit larger.: (>Fig.)
    • Vector ① goes away most, while vector ② comes close most.; vector ③ is the average, but seep out (=diffuses) from the original point.
    • The length of $R_1=\sqrt{R_0^2+r^2}$
    • $R_t^2=R_0^2+tr^2$, where $R_t$ is the distance from the original after time $t$.:
  • Range of an absolute value:
    • Case of coin toss: head $+a$, tail $-a$:
    • $A=a_1+a_2+a_3; \\
      The second item will be zero.
    • >Top $\therefore |A_n|\propto\sqrt{t}・a$ (Wiener process)
      • This means that SD of the sum of $n$-times SDs ($\sigma$) will be $\sqrt{n}・\sigma$.

3. 確率過程論とは:

  • stochastic: random probability distribution
  • Random Walk:
  • randomwalk
  • Wiener Process:

>Top 4. Hedge by Futures Contract:

  • A case of absolute value game:
  • Future contract-1:
    • Price of airplane parts and that of duralumin are changeable in conjunction.
    • When the future price of the parts declines, that of material also declines, while the former raises, the latter also raises.
    • Thus the future buying of the parts can be offset by the future selling of the material.
  • Future contract-2:
    • This case shows that the future price of the parts raises more than the material, while the former decline less than the latter.
    • >Top When we made future contract of buying the part at ¥1000 and future contract of selling the material at ¥1000.
      • What happens when the future price of the parts become ¥1200, and that of the material ¥1100.
      • By this future contract, we can gain ¥200 by selling the part at ¥1200, and we will lose ¥100 by buying the material, compared to the original price of both ¥1000.
      • This is a mechanism of 'Non-risk portfolio'.
      • The essence is the estimate of sensitivity of the future price; how to coordinate the future of price of the product and its material.; how to curve upward of the parts (=parabola curve with a concave top) is judged by expert of the maker.
  • >Top In the financial world, stable change of price to a certain direction is called 'trend', and random chance of price of fluctuations is called 'volatility'.
    • Commercial world tends to have more volatility than agricultural world; the former has more risks such as economic and pollical situation.

4. 先物取引によるヘッジ:

  • trend: a general direction in which sth is developing or changing
  • volatility: liabliity of change rapidly and unpredictably
  • Futures Contract: (Product & Material)

>Top 5. Idea of Black-Scholes Model:

  • This is a theoretical estimate of the price of European options showing the option has an unique price regardless of the risk of the security and its expected return.
    • This theory gives a strange impression, just like the agriculturist heard the merit of trade for the first time; i.e., a trader can always gain although the harvest is good or bad.
    • Black-Scholes theory utilizes most optimized selection of interrelated risks; adopting (=buying) preferable factors, and throwing away (=selling) unfavorable factors.
    • Commercial society has more options and is easier to sell out rapidly than agricultural society which is bound to the land.
  • Original capitalism or consumer society:
    • Manufacture-centric capitalism has a 'trend' of steady growth, but the consumer society has 'volatility' of random disturbance.
    • Black-Scholes theory has a new vision aiming to steady growth in modern capitalism. (Optimized selection and concentration)
      • IT technology and AI development tends to decrease labor force.
      • More money than needed for real economy tends to expand volatility.
      • Exponential growth economy vs. sustainable economy; causing expand disparity.
    • As the historical lessen: trend-pursuing capitalism can be curbed, but volatility-pursuing capitalism cannot. (In Edo period)
      • >Top Exponential-trend type growth vs. linear-volatility type growth:
      • Islamic financial system (=cosponsor type finance) vs. modern compound interest based finance.
      • Commercial economy pressures agricultural economy for a long time like a marathon race; which indicates that commercialism has not exponential growth, but rather linear growth.
  • The significance of Black-Scholes theory:
    • Expansion based-on volatility cannot be removed; particularly it is important to rebuild finance based on linear or sustainable growth.
    • Capitalism or post-capitalism should not aim an exponential growth, but linear growth which will be sustainable.

5. ブラック&ショールズモデルの考え方:

  • European option: a bon option to sell or buy at a certain date in the future for a predetermined price.
  • American option: ... on or before a certain date


>Top 6. Ito's lemma and Stochastic differential equation:

  • In 1940s, Ito's lemma is an identity used in the differential equation of a time-dependent function of a stochastic process. (Stochastic differential equation)
  • The lemma is widely employed in mathematical finance, particularly of Black-Scholes equation for option values.
    • $dx=Adt+Bdw$, where $Adt$ represents constant movement in $dt$, and $Bdw$ shows random movement ($B$ is strength of random, and $Bw(t)$ shows displacement magnitude.)
    • In applying the above equation in economics, the first term $Adt$ shows general trend of growth to a constant direction.
    • The above second term $Bdw$ shows volatility fluctuated by global market.
    • Ito's lemma indicates the clear separation of the first (=trend) and second term (=volatility).
  • >Top Taylor Expansion: (useful in making approximation)
    • $F(x_0+dx)=F(x_0)+\frac{dF(x_0)}{dx}dx+\frac{1}{2!}\frac{d^2F(x_0)}{dx^2}dx^2+
    • The second term (of $dx^2$) has more significant function.
  • Ito's lemma indicates:
    • $dx=Adt+Bdw; \longrightarrow dy=F(Adt+Bdw)$ (trying to express $dy=F(dt)$)
    • From Taylor Expansion:
    • Neglection of small amount:
      • Generally, $dw=\sqrt{dt}$ is assumed: [random walk expands $\propto\sqrt{t}$]
      • If $dt=0.01; dw=0.1; dtdw=0.001; dt^2=0.0001; dw^2=0.01; dw^3=0.001$
      • thus, we can leave $dt, dw, dw^2$.
      • the above equation can be simplified neglecting small terms:
      • as $dw^2=dt$
        $dy=\left(\frac{dF}{dx}A+\frac{1}{2}\frac{d^2F}{dx^2}B^2\right)dt+\frac{dF}{dx}Bdw$ [variable separation]
      • The above $w$ is the position changing part, and $dw^2$ cancels $+dw$ or $-dw$ position.]
      • $F(x)$ should be correctly $F(x, t)$, then the above Ito's lemma equation will be:
        $dy=\left(\frac{\partial F}{\partial x}A+\frac{\partial F}{\partial t}
        +\frac{1}{2}\frac{\partial^2F}{\partial x^2}B^2\right)dt+\frac{\partial F}{\partial x}Bdw$
  • Significance of Ito's lemma:
    • when $dx=A_1dt+B_1dw$, then if $dy=A_2dt+B_2dw$ [the first term shows the world 'trend', while the second term 'volatility'.]
    • from the the above equation:
      • $A_2=\frac{dF}{dx}・A_1+\frac{1}{2}\frac{d^2F}{dx^2}・B_1^2$
      • $B_2=\frac{dF}{dx}・B_1$
  • Application to non-risk portfolio:
    • when the price of duralumin bond is $x$, the price of airplane parts is $y$; $y=F(x)$
    • from Ito's lemma equation,
      • $dx=A_1dt+B_1dw$, then $B_2dw=\frac{dF}{dx}・B_1dw$
      • thus, the risk of future buying of 1 unit of the airplane parts bond $y$ can be covered by the future selling of $\frac{dF}{dx}$ unit of duralumin bond.

6. 伊藤のレンマと確率微分方程式:

  • Ito's lemma: 伊藤清の補題
  • Taylor Expansion:


  • >Top Future trade using Ito's lemma:
    Case of airplane parts & duralumin:
    camceling of the second term (volatility).
    $\frac{dF}{dx}$ is called 'delta'.
  • itolemma_futuretrade


>Top 7. Actual Black & Scholes Theory:

  • The actual Black & Scholes theory is applied to decide the 'option price' of a future bond, based on non-risk portfolio.
    • The price of non-risk portfolio tends to converge into the similar level of bank's non-risk interest rate.
    • Black & Scholes theory says two risk-hedging bonds should be at the ratio of $1:\frac{dF}{dx}$
    • When the price of non-risk portfolio is $P$, then: $P=y-\frac{dF}{dx}・x$ [$y$ will be price of the option]
    • Change of the price $P$ is described as $P+\Delta P$; This $P$ is expected constantly grow like a fixed interest rate; which will be written as $P+\Delta P=(1+qP・\Delta t)P$, thus,
      $\Delta P=qP・\Delta t$ , or $\Delta P=rP・\Delta t \;$ [q=r]
    • then, $\Delta P=r\left (y-\frac{dF}{dx}・x\right)・\Delta t$
    • herein, $\Delta P$ is a stable profit gained during $\Delta t$
      $\frac{1}{2}\frac{d^2F}{dx^2}・B^2=r\left (y-\frac{dF}{dx}・x\right)$
    • actually, (in the financial world-like expression), the above equation can be rewritten:
      $\frac{1}{2}\frac{d^2F}{dx^2}・\sigma^2・x^2=r(F(x)-r・\frac{dF}{dx}・x\;$ [$B=\sigma x\;; \sigma$ means 'volatility']
    • but actually again, $F(x)$ should be $F(x,t)$, then the above equation is:
      $\frac{1}{2}\frac{\partial^2F}{\partial x^2}・\sigma^2・x^2+\frac{\partial F}{\partial t}・x
      =r(F(x)-r・\frac{\partial F}{\partial x}・x\;$


7. 実際のブラック・ショールズ理論:

  • strike price: 権利行使価格
  • absolute value game: 絶対値ゲーム (-がない)



>Top 8. Fourier Series:

  • Fourier Series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. Such summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series.
  • Consider rectangle function:
  • The above rectangle function is: $F=\displaystyle\sum_{n=1}^4 a_nf_n$
  • $\displaystyle\int F/ f_1dx=\displaystyle\int(\displaystyle\sum_n a_nf_n)f_idx=
    \displaystyle\sum_n a_n\displaystyle\int f_nf_idx$
    • as, $\displaystyle\int f_i(x)f_j(x)dx=0 \;(i\ne j)$ [orthogonal relationship]
    • $\displaystyle\int F・f_idx=a_i\displaystyle\int(f_i)^2dx$
      $a_i=\frac{\displaystyle\int F・f_idx}{\displaystyle\int(f_i)^2dx}$
  • Area $S=-\displaystyle\int_0^{\frac{T}{2}}(F-a_1)dx+\displaystyle\int_{\frac{T}{2}}^T(F-a_1)dx$
    • Average $a_2=\displaystyle\frac{1}{T}\displaystyle\int_0^T F・f_2dx$
  • Fourier series:
    • $F(x)=\displaystyle\sum_{n=0}^{\infty}a_n e^{\frac{i\pi nx}{T}}$
      • $\displaystyle\frac{d^2F}{dx^2}=\displaystyle\sum_n a_n\displaystyle\frac{d^2}{dx^2}
        (e^{inx})=\displaystyle\sum_n a_n(-n^2)e^{inx} \;$ [second derivative]
    • If $a_n$ changes continuously: $F(x)=\displaystyle\int_0^{\infty}a(\nu)e^{i\nu x}d\nu$ [Fourier transformation]
  • Spectrum of light:
    • Wave of light can be expressed as $e^{i\omega t}$, where $\omega$=frequency
      then, $\displaystyle\int A(\omega)・e^{i\omega t}d\omega$ [Fourier transformation itself]
  • Inner product and orthogonal relationship:
    • $\displaystyle\int f・gdx\;$ [inner product of two functions]
    • $\displaystyle\sum_{i=1}^3 a_ib_i =0\;$ [inner product 0 = orthogonal relationship]
    • Eg. digital function $F=(1,1), \; G=(-1,1)$ [orthogonal relationship]

8. フーリエ級数::

  • Fourier series: フーリエ級数
  • Fourier transformation: フーリエ変換
  • sinusoid: 正弦曲線
  • innter product & orthogonal relationship: 内積と直交関係
  • differentiable: 微分可能
  • Deviation value $F-a_1$
  • deviationvalueF

>Top 9. Lebesgue integral:

  • developed by Henri Lebesgue (1875-1941)
  • Expected value (analog) and its integration: (>Fig.)
  • Idea of 'measure':
    • consider 'set function' instead of variable:
    • $P(A_1)+P(A_2)=P(A_1\cup A_2); \; A_1\cap A_2=\emptyset$
    • expected value can be: $S=\displaystyle\int_a^bfdp$
  • Lebesgue integral vs. Riemann integral:
    • $\displaystyle\int f(x)dx=0 (?)\;$ when $f(x)=0: \;x=$irrational number, $f(x)=1: \;x=$rational number
    • lebesgueintegral
  • In normal distribution, when $\sigma \rightarrow 0$: [analog →digital]
    • asigmazero

9. ルベーグ積分:

  • Lebesgue measure: ルベーグ測度
  • set function: 集合関数
  • Expected value:
  • expectedvalue
  • Divisiton width:
  • divisionwidth
  • God's fingerprint, and moderate difficiulty:
    It is remarkable that error and fluctuation of this world consist of 1) 'trend' or 'bias' to a certain direction, and 2) 'random walk' to + or - directions.
  • the latter 'random walk' portion can be treated by normal distribution; which is the essence that humans can handle various random phenomena.
  • 神の指紋と適度の難しさ:
    この世界の誤差やゆらぎは1) 一定方向へ進むトレンドやバイアスと、2)プラスマイナスいずれの方向に進むランダムウォークとがある。
  • ランダムウォークは正規分布によって扱える。これにより様々な振る舞いを扱えるようになる。

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