# Intuitive Method of Economic Mathematics

## - Probability & Statistics -

### 経済数学の直観的方法

##### Why
• 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.
##### Key
; ; 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.)
• #### 1. パラレルワールドの三角形偏差:

• Triange in a parallel world:  • 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:
• • Poisson Distribution:
• • 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 $t$.:
$R_t=\sqrt{t}･r$
• Range of an absolute value:
• Case of coin toss: head $+a$, tail $-a$:
• $A=a_1+a_2+a_3; \\ |A|=\sqrt{(a_1+a_2+a_3)^2}=\sqrt{a_1^2+a_2^2+a_3^2+2(a_1a_2+a_2a_3+a_3a_1)}$
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:
• • 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)
• $F(x_0+dx)=F(x_0)+\frac{dF(x_0)}{dx}dx+\frac{1}{2!}\frac{d^2F(x_0)}{dx^2}dx^2+ \frac{1}{3!}\frac{d^3F(x_0)}{dx^3}dx^3+\dots$
• 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:
$dy=\frac{dF}{dx}(Adt+Bdw)+\frac{1}{2}\frac{d^2F}{dx^2}(Adt+Bdw)^2+\dots)$
$dy=\frac{dF}{dx}Adt+\frac{dF}{dx}Bdw+\frac{1}{2}\frac{d^2F}{dx^2} (A^2dt^2+2ABdtdw+B^2dw^2)+\dots$
• 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:
$dy=\frac{dF}{dx}Adt+\frac{dF}{dx}Bdw+\frac{1}{2}\frac{d^2F}{dx^2}B^2dw^2$
• 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'.
• #### >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:
$f_1=\pmatrix{1\\1\\1\\1}\;f_2=\pmatrix{1\\1\\-1\\-1}\;f_3=\pmatrix{1\\-1\\-1\\1} \;f_4=\pmatrix{1\\-1\\1\\-1}$
• 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 $F-a_1$
• #### >Top 9. Lebesgue integral:

• developed by Henri Lebesgue (1875-1941)
• Expected value (analog) and its integration: (>Fig.)
$\displaystyle\sum_{i=1}^5Y_ip_i$
• 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
• • In normal distribution, when $\sigma \rightarrow 0$: [analog →digital]
• a #### 9. ルベーグ積分:

 Lebesgue measure: ルベーグ測度 set function: 集合関数
• Expected value:
• • Divisiton width:
• ##### Comment
• 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。