Probability and Statistics Basics

People who manage market risk sometimes use methods that estimate the size of a loss from the standard deviation. When the correlation between price moves changes, the overall risk changes, so they track that change as well. Option valuation uses volatility and the assumption that prices follow a lognormal distribution.

Random variables and expected value

The value of a derivatives position moves with changes in interest rates, stock prices, exchange rates, and the like¹. The basic tools for handling these changes with probability are the random variable and the expected value.

A random variable is a quantity whose value is determined by probability². Consider a hypothetical example in which an asset is worth 100 yen today, and tomorrow its price will be 103 yen, 101 yen, or 95 yen with probabilities of 25%, 50%, and 25%. Tomorrow's price is a random variable. The expected value of a random variable is the average you get by multiplying each possible value by its probability and adding the results, and in this example it is 100 yen³. The list of possible values together with their probabilities is called a probability distribution. The probabilities in a probability distribution always add up to 1 (100%)².

The expected value in this example is 103 × 0.25 + 101 × 0.5 + 95 × 0.25.

If you draw outcomes from the same probability distribution again and again, the average of the outcomes approaches the expected value³. Each individual outcome, however, differs from the expected value. In this example, the price can fall by 5 yen in a day or rise by 3 yen. Two distributions can have the same expected value and still spread their outcomes differently.

Variance and standard deviation

The standard deviation measures how widely the values of a random variable are spread around the expected value, and in the example above it is 3 yen. To measure the spread, you first take the difference between each possible value and the expected value (the deviation). If you simply average the deviations, the positive and negative ones cancel out and the result is 0. So you square the deviations and average them instead. The result is called the variance, and it serves as a measure of how large the spread is⁴.

The variance is 9. Because the variance is an average of squared values, its unit differs from that of the original values (yen squared). The positive square root of the variance is called the standard deviation, and it expresses the spread in the same unit as the original values⁴. The square root of 9 gives a standard deviation of 3 yen.

Deep diveFormulas for the expected value, variance, and standard deviation

If a random variable $X$ takes the values $x_1, x_2, \ldots, x_n$ with probabilities $p_1, p_2, \ldots, p_n$, its expected value $E[X]$ is given by the following formula³.

$$E[X] = \sum_{i=1}^{n} x_i \, p_i$$

Writing the expected value as $\mu$, the variance $V[X]$ is the expected value of the squared deviation, and the standard deviation $\sigma$ is its positive square root⁵.

$$V[X] = \sum_{i=1}^{n} (x_i - \mu)^2 \, p_i$$ $$\sigma = \sqrt{V[X]}$$

The normal distribution and the probability of falling within a range

The normal distribution is the premise of the variance-covariance method covered later⁶.

The normal distribution is a probability distribution with a symmetric bell shape centered on the mean, which sits at the highest point in the middle⁷. Once you know the mean and the standard deviation, the probability that a value falls within a given range is determined. Whatever the mean and the standard deviation are, the following relationships hold⁷.

Consider a hypothetical example in which the daily return follows a normal distribution with a mean of 0% and a standard deviation of 1%. The probability that the return stays between −1% and +1% is then about 68.3%, and the probability that it stays between −2% and +2% is about 95.4%. The probability that the return falls below −2% is half of the roughly 4.6% outside that range, or about 2.3%, because the distribution is symmetric⁸.

One way to measure market risk is the variance-covariance method⁶. The method assumes that changes in risk factors (the factors that move the value of positions)¹ follow a normal distribution, and it estimates the standard deviation from past observed data⁶. The method is used to calculate VaR (value at risk)⁹.

Returns and the lognormal distribution

Option valuation models use a lognormal distribution for prices¹⁰. The idea underlying it is to measure returns as log returns.

There are two ways to measure a price change: the simple return and the log return. Consider a hypothetical example in which a price rises from 100 yen to 110 yen and falls to 99 yen the next day.

With simple returns (the change in price ÷ the original price), +10% and −10% add up to 0%, yet the price has actually fallen 1%, from 100 yen to 99 yen. The log return is the natural logarithm of the later price ÷ the earlier price, and it follows the same idea as the continuously compounded rate on the Compounding and Discounting page¹⁰.

There are two reasons to measure with log returns¹⁰. First, adding the daily values gives the value for the whole period. In the table above, the sum of the two days' log returns matches the two-day value. Second, even if you assume a normal distribution for log returns, the price never turns negative.

If log returns follow a normal distribution, the price follows a lognormal distribution. A lognormal distribution is one in which the logarithm of the value follows a normal distribution, and its values are limited to the range above 0¹¹. The assumption that the price of the underlying asset follows a lognormal distribution is also used in option valuation models¹⁰.

Consider a hypothetical example in which the one-year log return on a price that is 100 yen today follows a normal distribution with a mean of 0% and a standard deviation of 20%. The probability that the log return falls between −20% and +20% is about 68.3%, which corresponds to a price range of about 81.87 yen to about 122.14 yen. The distance down is about 18.13 yen and the distance up is about 22.14 yen, so the distribution has a longer tail on the upside. The median (the value that the price is equally likely to end up above or below) is 100 yen¹². Because of the longer upside tail, the expected value is about 102.02 yen, higher than the median¹²¹³. The figure below shows this price distribution.

Figure description

This is the distribution of the price one year from now, when the price today is 100 yen and the one-year log return follows a normal distribution with a mean of 0% and a standard deviation of 20%. The curve lies only to the right of 0 yen, has its peak a little to the left of 100 yen, and its right tail extends further than its left tail. The vertical dashed lines mark 81.87 yen, 100 yen (the median), and 122.14 yen, and the shaded area from 81.87 yen to 122.14 yen is the probability of about 68.3%. The distance from the median down to the lower end is about 18.13 yen and up to the upper end is about 22.14 yen, so the upside is wider.

Deep diveFormulas for log returns and the lognormal distribution

Let $P_{t-1}$ be the price at time $t-1$ and $P_t$ the price at time $t$. The simple return $R_t$ and the log return $r_t$ are given by the following formulas¹⁰.

$$R_t = \frac{P_t}{P_{t-1}} - 1$$ $$r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)$$

Adding two days of log returns gives $\ln(P_{t+1}/P_t) + \ln(P_t/P_{t-1}) = \ln(P_{t+1}/P_{t-1})$, which is the log return over the two days. In the table's example, $\ln(1.1) + \ln(0.9) = \ln(0.99) \approx -0.0101$.

Let $P_0$ be today's price, and suppose the one-year log return $r$ follows a normal distribution with a mean of 0 and a standard deviation of $\sigma$. The price in one year, $P_1 = P_0 \, e^{r}$, then follows a lognormal distribution. Its median is $P_0$, and its expected value is given by the following formula¹³.

$$E[P_1] = P_0 \, e^{\sigma^2 / 2}$$

With $P_0 = 100$ and $\sigma = 0.2$, this is $e^{0.02} \times 100 \approx 102.02$.

Volatility and annualization

In option markets, the size of price moves is expressed as an annual figure,¹⁴ and a value calculated from daily price moves is also shown converted to an annual figure¹⁵.

Volatility expresses the size of price moves as the standard deviation of returns, and it is usually quoted as an annual percentage¹⁴¹⁵. Consider a hypothetical example in which the standard deviation of daily returns is 1%. With 250 business days in a year (an assumption for this example), the volatility is about 15.81% a year.

Volatility calculated from past prices is called historical volatility: you compute the percentage change in price for each business day, take the standard deviation, and convert it to an annual figure¹⁵. Volatility backed out from option market prices is called implied volatility¹⁴.

To convert to an annual figure, multiply the daily standard deviation by the square root of 250 (about 15.81)¹⁶. You multiply by the square root rather than by 250 because, if daily log returns are independent and follow the same distribution every day, the variance over a period grows in proportion to the number of days, and the standard deviation in proportion to its square root¹⁷.

Consider a hypothetical example in which the annual volatility is 20%. That is 20% ÷ about 15.81, or about 1.26%, per day. In the same way, a daily standard deviation of 1% becomes about 3.16% over 10 business days, since the square root of 10 is about 3.16.

Deep diveWhy the variance grows in proportion to the number of days

If the daily log returns $r_1, r_2, \ldots, r_n$ are independent and each has variance $\sigma_d^2$, the variance of their sum is the sum of their variances, as it is for any independent random variables⁵. The log return over $n$ days is the sum of the daily values, so its variance and standard deviation are as follows.

$$V[r_1 + r_2 + \cdots + r_n] = n \, \sigma_d^2$$ $$\sigma_n = \sigma_d \sqrt{n}$$

With $N$ business days in a year, the annual volatility is $\sigma_d \sqrt{N}$.

Correlation and the spread of a combined total

When you consider the spread of profit and loss on several positions combined, the correlation between price moves comes into play¹⁸.

The correlation coefficient measures how closely two price moves go in the same direction. It takes values from −1 to 1. The stronger the tendency to move in the same direction, the closer it gets to 1, and the stronger the tendency to move in opposite directions, the closer it gets to −1. A value near 0 means there is no correlation¹⁹.

Correlation changes the spread of two price moves combined. Consider a hypothetical example in which two positions each have a profit-and-loss standard deviation of 1 billion yen, and you combine them. In billions of yen, each variance is 1 squared, or 1, and the variance of the combined profit and loss is 1 + 1 + 2 × correlation coefficient × 1 × 1²⁰. With a correlation coefficient of 0.5, this is 3, and its square root, about 1.732 billion yen, is the standard deviation of the total.

Only when the correlation coefficient is 1 does the standard deviation of the total equal the sum of the two standard deviations (2 billion yen). When the correlation coefficient is below 1, it is smaller than the sum. When the correlation coefficient is −1, a gain on one position and a loss on the other cancel out, and the spread disappears. Correlations between risk factors change the overall amount of risk¹⁸. In one example, a negative correlation was observed between the rates of change in government bond prices and stock prices²¹.

Deep diveThe variance of the sum of two random variables, and the correlation coefficient

Let $\mathrm{Cov}(X, Y)$ be the covariance of the random variables $X$ and $Y$. The correlation coefficient $\rho$ is the covariance divided by the product of the two standard deviations¹⁹.

$$\rho = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \, \sigma_Y}$$

The variance of the sum is the sum of the two variances plus twice the covariance²⁰.

$$V[X + Y] = \sigma_X^2 + \sigma_Y^2 + 2 \rho \, \sigma_X \, \sigma_Y$$

With $\sigma_X = \sigma_Y = 1$ (in billions of yen), $V[X + Y] = 2 \times (1 + \rho)$.

Assumptions and limits

The calculations so far rest on assumptions such as a normal distribution and independence of the daily values. When you use them to estimate risk, you need to know the cases where those assumptions diverge from actual price moves.

Actual stock returns take extreme negative values more often than a normal distribution would suggest²². When extreme values occur more often than a normal distribution assumes, the distribution is said to have fat tails²³. One reason losses exceed the VaR estimate is that the actual probability distribution has fatter tails than the normal distribution²⁴. If price moves have a trend or are correlated with earlier moves (autocorrelation), approximating longer periods by multiplying by the square root of the number of days has limits²⁴.

The correlation coefficient measures only the strength of a proportional relationship between two values. A correlation coefficient of 0 does not necessarily mean the two values are unrelated, and a high correlation coefficient can also appear by chance between values that have nothing to do with each other¹⁹.

Models of how returns move over time are covered on the Random Walks and Brownian Motion page.

Notes

1. The same source, "（参考）リスクファクター：現在価値の変動をもたらすもの" ((Reference) Risk factors: what causes changes in present value), page 11, and "留意事項②" (Points to note 2), page 28. It states that changes in risk factors such as interest rates, stock prices, and exchange rates move the present value of financial assets and liabilities (in Japanese). ↩ ↩²

2. Statistics Bureau of Japan, Ministry of Internal Affairs and Communications, "推測統計" (Inferential statistics), in "なるほど統計学園" (Naruhodo Statistics Academy), the "（１）確率変数と確率分布" (Random variables and probability distributions) section (in Japanese). ↩ ↩²

3. "確率変数の期待値について" (On the expected value of a random variable), posted on the Chiba University website (version of June 10, 2009), section "1 賭けと期待値" (Bets and expected value), pages 1–2 (in Japanese). It explains that the average of the outcomes of a repeated game approaches the expected value (the law of large numbers), and gives the formula for the expected value as equation (1.2). ↩ ↩² ↩³

4. Statistics Bureau of Japan, "データの散らばりを見る" (Looking at the spread of data), in "なるほど統計学園," the "標準偏差と分散" (Standard deviation and variance) section (in Japanese). It defines the variance as the average of the squared deviations and the standard deviation as its positive square root. ↩ ↩²

5. National Institute of Advanced Industrial Science and Technology (AIST), "期待値・分散の定義および演算" (Definitions and operations of expected value and variance), the "分散・標準偏差" (Variance and standard deviation) and "平均値（期待値）・分散の演算" (Operations on the mean (expected value) and variance) sections (in Japanese). ↩ ↩²

6. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (II. Understanding and managing market risk) (June 2012), the "市場ＶａＲの計測手法" (Methods for measuring market VaR) and "Ａ．分散共分散法" (A. The variance-covariance method) slides, pages 22–24 (in Japanese). ↩ ↩² ↩³

7. Statistics Bureau of Japan, "推測統計," in "なるほど統計学園," the "正規分布" (Normal distribution) section (in Japanese). It gives the probabilities of falling within the ranges as 0.683, 0.954, and 0.997. ↩ ↩²

8. The same source, the figure "図　正規分布の性質" (Properties of the normal distribution): the one-sided probabilities beyond two standard deviations from the mean, 0.0214 and 0.0014 (in Japanese). ↩

9. Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応" (Measuring value at risk with the historical method: practical responses to the non-stationarity of market price movements), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), the "要旨" (Abstract), page 1 of the PDF (in Japanese). ↩

10. Tokuo Iwaisako, "Lecture 2: ランダム・ウォーク仮説のテスト" (Lecture 2: Testing the random walk hypothesis), Institute of Economic Research, Hitotsubashi University (revised October 2008), the "基本的な概念の復習" (Review of basic concepts) slides, slides 2–3 (in Japanese). ↩ ↩² ↩³ ↩⁴ ↩⁵

11. National Institute of Standards and Technology, "Lognormal Distribution," section 1.3.6.6.9 of the NIST/SEMATECH e-Handbook, "Probability Density Function" and "Range." ↩

12. National Institute of Standards and Technology, "Measures of Location," section 1.3.5.1 of the NIST/SEMATECH e-Handbook, "median" under "Definition of Location." ↩ ↩²

13. National Institute of Standards and Technology, "Lognormal Distribution," "Common Statistics," Mean (the source's m is the median, and σ is the σ in the text). ↩ ↩²

14. Osaka Exchange, "理論" (Theory), in "北浜投資塾" (Kitahama Investment School), the "ボラティリティ" (Volatility) and "インプライド・ボラティリティ" (Implied volatility) sections (in Japanese). ↩ ↩² ↩³

15. Japan Exchange Group, "ヒストリカル・ボラティリティ（HV）" (Historical volatility (HV)), in its glossary (in Japanese). ↩ ↩² ↩³

16. The annual figure is the daily standard deviation multiplied by the square root of 250, rounded to two decimal places. ↩

17. Tokuo Iwaisako, "Lecture 2: ランダム・ウォーク仮説のテスト," the "ランダム・ウォーク１" (Random walk 1) slide, equation (7) on slide 6 (in Japanese). It shows that when the disturbance terms are independent and identically distributed, the variance is $\sigma^2 t$, proportional to time. ↩

18. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (June 2012), "留意事項②" (Points to note 2), page 28 (in Japanese). ↩ ↩² ↩³ ↩⁴

19. Statistics Bureau of Japan, "複数の変数の関係性を見る" (Looking at relationships between several variables), in "なるほど統計学園," the "相関係数" (Correlation coefficient) section (in Japanese). ↩ ↩² ↩³

20. National Institute of Advanced Industrial Science and Technology (AIST), "期待値・分散の定義および演算," the "平均値（期待値）・分散の演算" section (in Japanese). ↩ ↩²

21. The same source, "国債価格変化率と株価変化率の相関関係" (Correlation between rates of change in government bond prices and stock prices), page 29 (in Japanese). ↩ ↩²

22. Tokuo Iwaisako, "Lecture 2: ランダム・ウォーク仮説のテスト," the "対数正規分布" (Lognormal distribution) slide, slide 4 (in Japanese). ↩

23. Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応," Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), the opening of the main text, page 2 of the PDF (in Japanese). It explains that the actual distribution of market prices has fatter tails than the normal distribution. ↩

24. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (June 2012), "ＶａＲ超過損失の発生原因・背景" (Causes and background of losses exceeding VaR), page 45 (in Japanese). ↩ ↩²