Quiz: Mathematical Preliminaries

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6 questions to check your understanding of the chapter “Mathematical Preliminaries”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.

1. Question 1

   As a hypothetical example, 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%. Which statement about the median and the expected value of the price in one year is correct?

   1. A.Both the median and the expected value are 100 yen
   2. B.The expected value is 100 yen, and the median is about 102.02 yen
   3. C.The median is 100 yen, and the expected value is about 102.02 yen
   Answer and explanation

   Answer: C

   The log return follows a normal distribution that is symmetric around 0%, so the price is equally likely to end up above or below 100 yen, and the median is 100 yen. On the other hand, when the log return follows a normal distribution, the price follows a lognormal distribution with a longer tail on the upside. 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, and because of that difference the expected value is about 102.02 yen, higher than the median.

   If you reason that a mean log return of 0% keeps the expected price at 100 yen, you would choose one of the options with an expected value of 100 yen. However, 100 yen is the median. The expected value is above 100 yen because the distance on the upside is larger.

   Learn more: Probability and Statistics Basics

2. Question 2

   As a hypothetical example, you combine two positions whose profit and loss each have a standard deviation of 1 billion yen. When the correlation coefficient between the two is below 1, how does the standard deviation of the combined profit and loss compare with 2 billion yen, the sum of the two standard deviations?

   1. A.It is larger than 2 billion yen
   2. B.It equals 2 billion yen
   3. C.It is smaller than 2 billion yen
   Answer and explanation

   Answer: C

   In billions of yen, each variance is 1 squared, or 1. The variance of the combined profit and loss is 1 + 1 + 2 × correlation coefficient × 1 × 1 (in billion yen squared). With a correlation coefficient of 1, the variance is 4 and the standard deviation is 2 billion yen, the same as the sum of the two. The lower the correlation coefficient, the smaller the last term, so the combined variance shrinks and the standard deviation falls below 2 billion yen.

   With a correlation coefficient of 0, the variance is 2 and the standard deviation is about 1.414 billion yen. With a correlation coefficient of −1, a gain on one position and a loss on the other cancel out, and the standard deviation is 0. The combined standard deviation equals the sum of the two only when the correlation coefficient is 1. The correlation coefficient takes values from −1 to 1, so the combined standard deviation is never larger than 2 billion yen.

   Learn more: Probability and Statistics Basics

3. Question 3

   As a hypothetical example, a price moves up 1 yen or down 1 yen each day with equal probability, independently of the previous day, and its standard deviation 100 days after the starting point is 10 yen. What is the standard deviation 900 days after the starting point?

   1. A.30 yen
   2. B.90 yen
   3. C.900 yen
   Answer and explanation

   Answer: A

   The daily move has a mean of 0 and a variance of 1, and the variance of a sum of independent moves is the sum of their variances. After n days the variance is therefore n, and the standard deviation is its square root. After 100 days the variance is 100 and the standard deviation is 10 yen. After 900 days the variance is 900, and the standard deviation is its square root, 30 yen.

   90 yen assumes that the standard deviation grows 9 times because the number of days grows 9 times. What grows in proportion to the number of days is the variance, and the standard deviation grows in proportion to the square root of the number of days. 900 yen is the variance after 900 days. The variance is in yen squared, so the standard deviation is the square root, which returns the value to yen.

   Learn more: Random Walks and Brownian Motion

4. Question 4

   Consider a market in which daily price moves tend to go in the same direction as the previous day's move. If you take the daily standard deviation and multiply it by the square root of the number of days to get the standard deviation over a period, how does the result compare with the actual spread?

   1. A.It overstates the spread
   2. B.It understates the spread
   3. C.It equals the actual spread
   Answer and explanation

   Answer: B

   In a market where moves tend to follow the previous day's direction (the covariance with the previous day is positive), the variance over a period is larger than the sum of the daily variances by twice the sum of the covariances between days. A figure converted by the square root of time, treating the variance as proportional to the number of days, leaves out this covariance part, so it understates the actual spread.

   Overstating happens in a market where moves tend to reverse (the covariance is negative). The square-root conversion holds when daily moves are independent, so in a market where they are not independent, it does not equal the actual spread.

   Learn more: Random Walks and Brownian Motion

5. Question 5

   As a hypothetical example, a stock price $S_t$ follows $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$, with $\mu = 5\%$ and $\sigma = 30\%$, where $W_t$ is a Brownian motion. What is the coefficient of the $dt$ term of the log price $\ln S_t$ (the drift of the log)?

   1. A.0.5% per year
   2. B.5% per year
   3. C.−4% per year
   4. D.9.5% per year
   Answer and explanation

   Answer: A

   Applying Ito's lemma to $f = \ln S$, the coefficient of the $dt$ term is $\mu - \sigma^2/2$. The extra term $-\sigma^2/2$ comes from the second derivative of $\ln S$ being negative ($-1/S^2$). Here $\sigma^2 = 0.3^2 = 0.09$, so $\sigma^2/2 = 0.045$. Then $0.05 - 0.045 = 0.005$, or 0.5% per year.

   5% is $\mu$ taken as the drift of the log, which is the answer when the ordinary chain rule applies. For a Brownian motion the second-order term remains, so the drift is smaller than $\mu$. −4% subtracts $\sigma^2 = 0.09$ and forgets the factor $1/2$ of the extra term, computing $0.05 - 0.09$. 9.5% reverses the sign of the extra term and computes $0.05 + 0.045$.

   Learn more: Introduction to Stochastic Calculus (Ito's Lemma)

6. Question 6

   A stock price $S_t$ follows a geometric Brownian motion with volatility $\sigma$, and the interest rate $r$ is constant. Cash $x_t$ and a position $y_t$ in the underlying, with no funds added or withdrawn along the way, replicate the option price $C(S, t)$. When the $dW_t$ terms are set equal, what does $y_t$ equal?

   1. A.$\partial C/\partial t$ (the change in the option price with time)
   2. B.$\partial C/\partial S$ (the first-order change in the option price with the underlying price)
   3. C.$\partial^2 C/\partial S^2$ (the second-order change in the option price with the underlying price)
   Answer and explanation

   Answer: B

   The $dW_t$ term in the movement of the option price is $\sigma S\,\partial C/\partial S$, and the $dW_t$ term of the combination of cash and the underlying is $y_t \sigma S$. Setting the two equal gives $y_t = \partial C/\partial S$.

   $\partial C/\partial t$ and $\partial^2 C/\partial S^2$ appear in the $dt$ term. Setting the $dt$ terms equal gives an equation in which the growth of the cash at the interest rate $r$ ($e^{rt}$ is the continuous-compounding factor), with $dt$ coefficient $r x_t e^{rt}$, equals $\partial C/\partial t + \tfrac{1}{2}\sigma^2 S^2\,\partial^2 C/\partial S^2$, which fixes the cash side rather than the position $y_t$. Substituting this into the value of the combination and rearranging gives the partial differential equation.

   Learn more: Introduction to Stochastic Calculus (Ito's Lemma)