Financial Mathematics Flashcards
What is Financial Mathematics?
Exercise(s)1:Sums and products
Exercise(s) 2
Exercise(s) 4
Sum symbol
Formula for easy calculation of the sum of the first N integers (Gauss summation):
Formula for easy calculation of a geometric series
products symbol
binomial coefficient：二项式系数，example中的最后一个
当下面的k=0时，二项式系数=1. n为自然数，k为整数。二项式系数对组合数学很重要，因它的意义是从n件物件中，不分先后地选取k件的方法总数，因此也叫做组合数。
Binomial formula二项式
• Describes the algebraic expansion of powers of a binomial
Elementary functions：基本功能
Exponential function：指数函数
Logarithm：对数
Trigonometric function：三角函数
Definition vector
Definition matrix
Calculation rules for matrices
Determinant行列式
行列式展开的不同的方法
Cramer’s Rule：克莱默规则
Gaussian elimination：高斯消除
用最后的结果，代替变量的那一列来计算
Linear algebra加减
共轭
Linear algebra定义
-The slope of the tangent at point X0 .
- Measures the sensitivity to change of the function value
- Fundamental tool of calculus
- Process of finding a derivative is called differentiation
- The reverse process is called antidifferentiation resp. integration
- Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century
Basic derivatives
• Derivatives for some of the most common (/elementary) functions:
Basic differentiation rules Examples
Basic differentiation rules Examples
求导公式
Integral calculus 积分微积分Definition
• Signed area bounded by the graph of f the x-axis and the vertical lines x=a and x=b.
Indefinite Integral不定积分
中文定义
Integrals of (some) elementary functions
• Indefinite integrals for some of the most common (/elementary) functions:
Calculation rules
Exercise(s) 7：Integral calculus 积分
Basic statistics基本统计
Univariate Statistics单变量统计
odd：奇数
Mean Squared deviation：均方偏差
(Unbiased) Standard deviation： （无偏）标准偏差
Bivariate Statistics双变量统计
Variations and combinations变化和组合
• How many ways of choosing elements out of a set of elements are there for different setups?
Probability space
A probability space models a process consisting of states that occur randomly. It consists of three parts （Ω，f，P):
- The set of all possible outcomes, called Ω
- A set of events f where each event is a set containing zero or more outcomes.
• A function P from events to probabilities, the so-called probability measure.
example 2
Probability theory
- Some basic rules (can be generalized for an indexed set of sets)
Random variable definition随机变量定义
Random walk definition
- Random walk on the integer number line
- starts at 0 and at each step moves +1 or −1 with equal probability.
- This walk can be illustrated as follows. A marker is placed at zero on the number line and a fair coin is flipped. If it lands on heads, the marker is moved one unit to the right. If it lands on tails, the marker is moved one unit to the left
Problem with random walk hypothesis
- Stock prices can not only go up or down by predefined ticks,…
- Meaning stock prices are not only discrete states
- Instead they are continuous numbers, i.e. the price of a stock can not only go up by e.g. an integer number like 1% but also by e.g. 1.2345 %….
But beforehand le us shortly revisit stock price time series and go into more detail with respect to its hypothetical behavior…
Financial time series
Transformations：
Sometimes it makes sense to transform the data you are analyzing. E.g. for financial time series it makes sense to analyze so called log-returns:
计算长期收益率，用log returns，fit best to normal returns in life.
Financial time series
Are Stock returns normally distributed?
Answer: Obviously absolute returns are not normally distributed but log-returns (or relative returns) are. —————Log-Normal Distribution
Definition I
For a discrete random space we had that P(A) was defined for every single event in A∈Ω. Now let us assume that Ω is no longer discrete (remember: stock prices can, of course, not only take certain discrete states), then we need a probability density function.
.Definition II
For a discrete random space we had that P(A) was defined for every single event in A∈Ω. Now let us assume that Ω is no longer discrete (remember: stock prices can, of course, not only take certain discrete states), then we need a probability density function.
Example
Exponential distribution
Normal distribution
- Due to the so called central limit theorem (under some conditions) the average of independent random variables tends toward a normal distribution (“bell curve”) even if the original variables themselves are not normally distributed
- If quantities are expected to be the sum of many independent processes (e.g. measurement errors), then they often have distributions that are nearly normal.
- Furthermore , many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
- 具有均值和标准差的正态分布的概率密度函数
标准正态变量
Gauss Bell Shape不同的概率密度函数参数化（高斯钟形）
Normal distribution cumulative distribution function
Remark
Sampling for normal variable
• Numerical practice of generating pseudo-random numbers that are distributed according to a given probability distribution.
Cholesky 分解是把一个对称正定的矩阵表示成一个下三角矩阵L和其转置的乘积的分解。它要求矩阵的所有特征值必须大于零，故分解的下三角的对角元也是大于零的。Cholesky分解法又称平方根法，是当A为实对称正定矩阵时，LU三角分解法的变形。
The empirical distribution function 经验分布函数
example
Fat-tail/Heavy tail distributions
肥尾效应：
肥尾效应（Fat tail）是指极端行情发生的机率增加，可能因为发生一些不寻常的事件造成市场上大震荡。如2008年雷曼兄弟倒闭、2010年的南欧主权债信危机，皆产生肥尾效应。
Fat tail definition
Pareto distribution帕累托分布
6δ是mean的范围，记住就行了
Definition & Background
Originally applied to the distribution of wealth in a society (fitting the trend that a large portion of wealth is held by a small fraction of the population) by Vilfredo Pareto
Pareto principle, “80-20 rule“ or “Matthew principle“ states that, for example, 80% of sales come from 20% of clients or by fixing top 20% of the most-reported bugs, 80% of the related errors and crashes in a given system would be eliminated.
The Pareto distribution with parameters λ and x<strong>m</strong> is defined by the following probability density function. Xm as minimum value of x with a positive probability
Interest
Central question:
Assume you start with an initial capital of K0 and you are getting a constant interest rate of r for N periods. How much money do you have after N periods?
Definition Future Value
Example
Definition Present Value
Example
Annuity
- Asset that pays fixed sum each year for specified number of years
- The present value of an annuity which pays a fixed sum from year 1 to year is given as follows (under the assumption of a risk free rate )
定义
变量
二叉树概率计算
Example
Example解释
结果
We already tried to give adequate answers to the type of question:
Assume you have a portfolio of certain stocks. How likely is it that a loss of X% might occur in the next Y days?
But what if you want to answer the question:
How high can a loss of my portfolio be such that only with a probability of X% I would be
losing more than that amount?
value at risk风险价值模型
事后检验是指将市场风险计量方法或模型的估算结果与实际发生的损益进行比较，以检验计量方法或模型的准确性、可靠性，并据此对计量方法或模型进行调整和改进的一种方法。
事后检验的目的，就是看实际观测到的结果与所定义风险度量的置信水平是否一致，如模型中定义了99％置信度下的风险值，那么，就要考察这个风险值是否真的覆盖了真实损失的99％。
事后检验一般采用一种移动窗口的方法进行计算。以1天的事后检验为例，先采用某种方法计算出给定头寸该交易日的VaR值，接着计算出该头寸在本交易日的实际损失额，进而判断计算出来的VaR值是否覆盖了实际损失额。然后，将VaR的计算窗口、待考察的交易日不断后移，计算并记录各交易日的超出情况。
Kolmogorov Smirnov Test background
- Nonparametric test of the equality of probability distributions (can be used to compare a sample with a reference probability distribution)
- Quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution
- Named after Andrei Nikolajewitsch Kolmogorow and Nikolai Smirnov, two famous Russian mathematicians.
How to apply？
Behavioral finance
- Remember: Due to the work of Fama and Mandelbrot (log-)returns of stocks are not normally distributed (in the tails). Furthermore, behavioral finance tries to explain why market participants make “irrational systematic errors” contrary to the standard assumption of the rational agent.
- Therefore, behavioral finance highlights inefficiencies, such as under- or over-reactions to information, as causes of market trends and, in extreme cases, of bubbles and crashes.
- Explanations are e.g. limited investor attention, overconfidence, overoptimism, mimicry (herding instinct), loss aversion and noise trading.
网上例题：积分1
网上例题：积分2
网上例题：积分常熟的确定
网上例题：积分3
Ch-3：Univariate Statistics单变量统计
odd：奇数
Mean Squared deviation：均方偏差
(Unbiased) Standard deviation： （无偏）标准偏差
Bivariate Statistics双变量统计
Excercise 8：Basic Statistics
Calculate mean, mean squared deviation & autocorrelation (lag 1) for the following DOW time series
Exercise 10、11、12：variables and combinations
Excercise 13：Financial time series
Calculate absolute, relative & log-returns for the previous times series of the DOW:
Exercise 16：反导练习
Exercise 17：换底求导
Exponential distribution
Exercise 18：exponential distribution
Are the log-returns really normally distributed (in the tails)?
- Remember: The probability for the highest (log-)returns of the DOW Jones and the DAX was…
- The tail of a distribution isn’t a precisely defined term
- Benoit Mandelbrot and Eugene Fama showed that the log-returns of stocks are not normally distributed (in the tails) but indeed are fat-tailed
Empirical distribution function vs. normal cumulative distribution function for DOW Jones
Remark: Heavy tail distributions
Remark
Apart from the fat tailed distributions, there are heavy tailed distributions. Whereas fat-tailed distributions are always heavy tailed the opposite case is not always true. The probability density function for heavy tailed distributions does converge faster to zero (fx(x)~e-x ) than the ones for fat tailed distributions do.
Examples
- Exponential distribution
- Log-normal distribution
- Cauchy-distribution
- t-distribution
- Weibull-distribution
- Levy distribution
Exercise 25: Pareto distribution
Remark
As you will see, the results of the above Exercise are quite stunning and the assumptions regarding the distribution of the returns of stocks are absolutely critical!
不会做
Exercise 26：Interest
Assume you start with an initial capital of and get an constant annual interest rate of 1.5% over a period of 5 years. Hoch much money do you have after these 5 years?
Exercise 27： 求r
Suppose you set the financial goal of having 10.000€ in three years and you start with an initial capital of 8.000€. How high does the annual interest have to be at least for you to achieve your goal?
* Perpetuity
- An annuity which has no end, i.e. stream of cash flow is theoretically received forever
- The present value of an annuity which pays a fixed sum forever is is given as follows (under the assumption of a risk free rate )
* Exercise 28
Example： Kolmogorov-Smirnov-Test
不会，excel
Exercise 31：Check if the DOW Jones log-return times series of 2017 passes the Kolmogorov Smirnov test with respect to a statistical significance of 5%.
Further distribution tests
- Anderson–Darling test (tests if data is drawn from a given probability distribution, does not only work for normal distributions).
- Kuiper-Test (tests if data is drawn from a given probability distribution, does not only work for normal distributions).
- Cramér–von Mises criterion (esp. Watson test; tests if data is drawn from a given probability distribution, does not only work for normal distributions).
- Shapiro–Wilk test (test for normality)
- Jarque–Bera test (tests if skewness and kurtosis are matching the ones of a normal distribution)
Test for autocorrelation
- Due to the assumptions of the arbitrage free pricing theory, price processes should be martingales
- This especially implies that there should be no autocorrelation in the data, otherwise you could make predictions with respect to future prices based on the current price
- Therefore, one should test, if the log-returns show significant autocorrelation as we already did beforehand
