Random Walks and Brownian Motion

Market risk staff use this when converting standard deviations between periods, and valuation model developers use it when generating price paths with random numbers. If you build up daily moves, why does the spread over a period grow with the square root of time?

Mean and variance of a symmetric random walk

A stochastic process is a sequence of quantities whose values are determined by probability as time passes. A symmetric random walk is one such process: it starts at 0 and, at each step, moves +1 or −1 with probability 1/2 each¹. The steps are independent of one another. After just two steps, the value is +2, 0, or −2, with a mean of 0 and a variance of 2.

The value ends at 0 in two ways, up then down or down then up, so its probability is 0.25 + 0.25 = 0.50. Because the mean is 0, the sum of each squared value times its probability, 2, is the variance.

As the number of steps grows to n, the mean stays at 0 and the variance becomes n. Each step has a mean of 0 and a variance of 1 (the average of 1 squared and −1 squared), and the variance of a sum of independent random variables is the sum of their variances². Writing $X_n$ for the value after n steps and $Y_i$ for the move at step i, the relationships are as follows.

$$X_n = Y_1 + Y_2 + \cdots + Y_n$$ $$E[X_n] = 0$$ $$V[X_n] = n$$

The standard deviation is the square root of the variance, $\sqrt{n}$. Consider a hypothetical example in which a price each day rises 1 yen or falls 1 yen with equal probability, independently of the previous day. Its standard deviation is 10 yen after 100 days and 20 yen after 400 days. Four times as many days only doubles the spread. The basics of variance and the normal distribution are covered on the Probability and Statistics Basics page.

The same kind of relationship holds even if each step is not just +1 or −1, as long as the steps are independent and follow the same distribution. One example is a random walk in which the change in price over one period is the sum of a mean $\mu$ and a disturbance term (a random move with mean 0 and variance $\sigma^2$)³. In that case, the expected price after t periods moves away from the starting point by $\mu t$, and the variance is $\sigma^2 t$. A component such as the average move $\mu$, which shifts the value in proportion to time, is called the drift⁴. Adding a drift does not change the fact that the variance grows in proportion to the number of steps (time).

Brownian motion as the limit of finer steps

Brownian motion is a stochastic process in continuous time, obtained as the limit of a symmetric random walk when the time per step is shortened and the step size is reduced to match⁵. If you divide one year into n steps and set the step size to the square root of the time step, the variance after one year stays at 1 whatever n is.

The variance after one year is the number of steps × the step size squared (for 100 steps, 100 × 0.1 × 0.1 = 1). If you set the step size to 0.01, equal to the time step rather than its square root, the variance after 100 steps is 100 × 0.01 × 0.01 = 0.01, and the spread shrinks toward zero as the time step gets finer. With the square root, the spread after one year is preserved however fine the time step. As the number of steps grows, the distribution of the value after n steps divided by $\sqrt{n}$ approaches the normal distribution with mean 0 and variance 1, by the central limit theorem¹. In the same way, the distribution of the value at time t approaches the normal distribution with mean 0 and variance t⁶.

Brownian motion $W_t$, the limit, has the following properties⁷.

• It starts at 0 ($W_0 = 0$).
• Increments over non-overlapping periods (the changes in value over those periods) are independent of one another.
• The increment from time s to t, $W_t - W_s$, follows a normal distribution with mean 0 and variance t − s.
• Its path (the value traced over time) is continuous and never jumps.

The path is continuous but cannot be differentiated at any point⁸. As a result, changes in quantities that involve Brownian motion cannot be calculated with the ordinary rules of differentiation, and Ito's lemma is used instead. That calculation is covered on the Introduction to Stochastic Calculus (Ito's Lemma) page.

The first person to consider this process was L. Bachelier, who used it to value stocks and options in his 1900 doctoral thesis⁹. Brownian motion is also called the Wiener process, after Norbert Wiener, who was a professor at MIT⁵.

Spread proportional to the square root of time

The standard deviation of the value of Brownian motion is proportional to the square root of elapsed time. Because the variance of the value at time t is t, the standard deviation is $\sqrt{t}$. In geometric Brownian motion, covered in the next section, the spread of a price's log return (the continuously compounded return) is expressed as $\sigma$ times Brownian motion. Here $\sigma$ is the standard deviation per year, in other words the annual volatility¹⁰. The standard deviation of the log return over T years is given by the following formula.

$$\sigma_T = \sigma \sqrt{T}$$

Consider a hypothetical example in which the annual volatility is 20%. Converting it into the standard deviation for each period gives the following table¹¹.

The 3-month standard deviation of 10% is 20% multiplied by 0.5, the square root of 0.25. Going the other way, if you know the 3-month standard deviation is 10%, one year is 4 times 3 months, so the annual figure is 10% multiplied by 2, the square root of 4, or 20%. Converting from a daily standard deviation to an annual one uses the same square-root relationship.

Plotting this conversion as curves gives Figure 1.

Figure description

These curves overlay the distributions over 3 months, 1 year, and 4 years, assuming an annual volatility of 20% and a log return (the logarithm of how many times the price has multiplied) that follows a normal distribution with mean 0. The mean is set to 0 to remove the drift and compare only the spread for each period. The standard deviation is 10% for 3 months, 20% for 1 year, and 40% for 4 years, so each time the period is multiplied by 4, the standard deviation doubles. Every curve is a symmetric bell shape centered on 0, and the longer the period, the lower and wider it is. The area under a curve is the probability, and the area is 1 for every curve. You can read off that even a 16-fold longer period makes the spread only 4 times larger.

Stock price models and geometric Brownian motion

Geometric Brownian motion is a stochastic process in which the logarithm of the price follows Brownian motion with drift, and it is used to model stock prices. If you move the price itself by normally distributed increments (this is called arithmetic Brownian motion³), the price can turn negative. In modeling a stock price, it is more reasonable to assume that the percentage change in price follows a normal distribution¹². If you think in terms of log returns (continuously compounded returns), the price never turns negative even under a normal distribution, returns over successive periods can simply be added, and the approach is consistent with lognormal models such as Black-Scholes¹³.

Writing $S_t$ for the price at time t, geometric Brownian motion is expressed by the following formula¹⁴.

$$\frac{dS_t}{S_t} = \mu \, dt + \sigma \, dW_t$$

Here $\mu$ is the expected growth of the price per year (the drift), $\sigma$ is the annual volatility, $dt$ is a very short time, and $dW_t$ is the increment of Brownian motion over that time. A price that satisfies this formula is obtained from today's price $S_0$ as follows¹⁵.

$$S_T = S_0 \exp\left( \left( \mu - \frac{\sigma^2}{2} \right) T + \sigma W_T \right)$$

The log return $\ln(S_T / S_0)$ follows a normal distribution with mean $(\mu - \sigma^2/2)T$ and standard deviation $\sigma \sqrt{T}$¹⁴. The expected price grows at an annual rate of $\mu$, but the mean log return is smaller by $\sigma^2/2$. This is because Brownian motion cannot be differentiated, which adds one term to the ordinary rules of differentiation¹⁵.

Consider a hypothetical example in which today's price is 100 yen, the drift is 5% a year, the annual volatility is 20%, and the period is one year. The log return after one year follows a normal distribution with a mean of 0.05 − 0.2 × 0.2 ÷ 2 = 0.03 (3%) and a standard deviation of 20%.

The price is 100 × $e^{x}$, where x is the log return (at the mean, 100 × $e^{0.03}$ ≈ 103.05)¹⁶. The expected price, on the other hand, is 100 × $e^{0.05}$ ≈ 105.13 yen, today's price grown at 5% a year ($\mu$)¹⁷. The price of 103.05 yen, reached when the log return equals its mean, is smaller than this expected value. This is because, in a lognormal distribution, the expected value is $e^{\sigma^2/2}$ times the median (103.05 yen).

Assumptions and limitations

Random walk and Brownian motion models assume that price moves are independent of the past and that the size of the spread is constant. A random walk can also be used as a model of stock prices, but the MIT lecture notes add that it is "rather inaccurate"¹. Geometric Brownian motion assumes normally distributed returns. Actual stock returns have fatter tails than the normal distribution, making extreme values more likely¹⁸.

If each day's move is not independent, square-root scaling breaks down. In a market where moves tend to go in the same direction as the previous day (a positive covariance with the previous day), the variance over the period exceeds the sum of the daily variances by twice the sum of the covariances between days¹⁹. So a figure scaled by the square root, on the assumption that the variance is proportional to the number of days, understates the spread. In a market where moves tend to reverse (a negative covariance), it overstates the spread. When the size of the spread changes over time, one model used is the ARCH model, which lets the variance vary from period to period¹⁸.

Notes

1. Massachusetts Institute of Technology, "Lecture 5: Stochastic Processes I" (18.S096 Topics in Mathematics with Applications in Finance, Fall 2013), section 1 (page 1) and section 2 (page 2). ↩ ↩² ↩³

2. 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), "平均値（期待値）・分散の演算" section (in Japanese). When random variables are independent, the variance of their sum is the sum of their variances. ↩

3. 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), equation (4) on slide 5 and equations (6) and (7) on slide 6 (in Japanese). The special case in which the disturbance term follows a normal distribution is called arithmetic Brownian motion; $\mu$ is the average change per period. ↩ ↩²

4. Massachusetts Institute of Technology, "Lecture 17: Stochastic Processes II" (18.S096 Topics in Mathematics with Applications in Finance, Fall 2013), Example 2.6 (page 5). The notes write Brownian motion as B; this site writes it as W. ↩

5. Massachusetts Institute of Technology, "Lecture 17: Stochastic Processes II" (18.S096 Topics in Mathematics with Applications in Finance, Fall 2013), section 2 (page 2). It describes Brownian motion as a limit of random walks and also calls it the Wiener process. ↩ ↩²

6. Massachusetts Institute of Technology, "Lecture 17," section 2 (page 2), the paragraph after Theorem 2.1. Setting the step size to the square root of the time step is the same relationship as $X_n / \sqrt{n}$ in Lecture 5. ↩

7. Massachusetts Institute of Technology, "Lecture 17," Theorem 2.1 (page 2). The notes write Brownian motion as B; this site writes it as W. ↩

8. Massachusetts Institute of Technology, "Lecture 17," the paragraph after Proposition 2.4 (page 4). The path is continuous but nowhere differentiable. ↩

9. Massachusetts Institute of Technology, "Lecture 17," section 2 (page 2) and reference [3] (page 6). The paper is L. Bachelier, "Théorie de la spéculation" (Annales Scientifiques de l'École Normale Supérieure). ↩

10. Japan Exchange Group, "ヒストリカル・ボラティリティ（HV）" (Historical volatility (HV)), glossary (in Japanese). The figure is the standard deviation of the daily rate of change, converted to an annual rate. ↩

11. The number of years T for one month is 1/12 (0.0833 is rounded to four decimal places). The square root of 1/12 is 0.288675, and 20% multiplied by it, 5.7735%, is rounded to two decimal places. ↩

12. Massachusetts Institute of Technology, "Lecture 17," page 6. The formula $dS_t = \sigma S_t \, dB_t$ is in the same place; B is replaced with W. ↩

13. Tokuo Iwaisako, "Lecture 2," "基本的な概念の復習" (Review of basic concepts), slide 2 (in Japanese). It gives three reasons why log returns are preferable. ↩

14. Takashi Kawakami, "ヘッジ取引におけるデリバティブ信用評価調整の影響についての考察" (A study of the effect of credit valuation adjustments for derivatives on hedging transactions), Kinyu Kenkyu, Bank of Japan (April 2018), Appendix 2, equations (A-25) and (A-26) (page 160) (in Japanese). The paper's firm value $V(t)$ is replaced with the price $S$ and $t$ with 0. ↩ ↩²

15. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus" (18.S096, Fall 2013), Example 1.3 (iii) (page 3). The formula in the text replaces the notes' $\mu$ with $\mu - \sigma^2/2$ and multiplies by $S_0$. ↩ ↩²

16. The prices are 100 multiplied by $e^{-0.17} = 0.843665$, $e^{0.03} = 1.030455$, and $e^{0.23} = 1.258600$, rounded to two decimal places. ↩

17. National Institute of Standards and Technology, "Lognormal Distribution," section 1.3.6.6.9 of the NIST/SEMATECH e-Handbook of Statistical Methods, "Common Statistics." The expected value is $e^{0.5\sigma^2}$ times the median $m$. ↩

18. Tokuo Iwaisako, "Lecture 2," "対数正規分布" (Lognormal distribution), slide 4 (in Japanese). It shows the fat tails of stock returns and the ARCH model. ↩ ↩²

19. AIST, "期待値・分散の定義および演算," "平均値（期待値）・分散の演算" section (in Japanese). The variance of a sum is the sum of the variances plus twice the sum of the covariances. ↩