Appendix B:
Continuous Compounding
◼ Question B.1
Using a continuous rate of return, we have 67032 e = 100000. This implies e = 14918248.
r 5 r 5
Solving, we have r 5 = ln(14918248) = 040, hence the continuous return is r = 040 5 = 8%.
◼ Question B.2
The unknown in the continuous interest equation is the time to maturity, T. We must solve:
50 e010 T = 100. This implies e010 T = 2. Taking the natural log of each side, we have
0.10 T = ln(2) = 0693147 which implies T = 693147 years.
◼ Question B.3
In 7 years, at 7.5% continuous interest, $5 grows to 5 e = $84523.
0075 7
◼ Question B.4
Let S1 be the stock price in 1 year. The continuous return is
r = ln(S1 /S0 ) = ln(S1 /100) = ln(S1 ) − ln(100) = ln( S1 ) − 46052
Using this formula we have:
S0
r
5 −2.9957323
4 −3.2188758
3 −3.5065579
2 −3.912023
Note these are not percentages. For example, if S1 is 5, the return is −299.57%. For a −500%
return, the stock price next year must solve
ln(S1 ) = r + 46052 = −5 + 46052 = −03948
which implies S1 = e = $06738.
−03948
,◼ Question B.5
1. The arithmetic return for the first year is (200 − 100)/100 = 200% and, for the second
year, it is
(100 − 200)/200 = −50%.
2. The continuous return for the first year is ln(200/100) = 69.31% and, for the second
year, it is ln(100/200) = −69.31%.
3. We find that, for continuous returns, increases and decreases are symmetric. This
follows from
ln(x/y) = ln(x) − ln(y) = −ln(y/x).
◼ Question B.6
1. The arithmetic returns are given in the second column in the following table:
Stock Price Arithmetic Continuous
100
47 −0.5300 −0.7550
88 0.8723 0.6272
153 0.7386 0.5531
212 0.3856 0.3261
100 −0.5283 −0.7514
2. The continuous returns are given in the third column in the table above.
3. Given the arithmetic returns above, we must first take the cumulative product of the
gross return,
i.e., 1 plus the arithmetic return of the day. This is
04700 18723 17386 13856 04717 = 1
For the cumulative arithmetic return, we then subtract 1 from this amount. Using the
continuous return, we can simply sum continuous returns to arrive at the cumulative
continuous return.
,Chapter 1
Introduction to Derivatives
◼ Question 1.1
We will look at the CME
1. The CME trades derivatives (specifically futures and options on futures) on a wide
variety of assets and indices/variables. Assets include basic commodities such as
Cattle, Hogs, Pork Bellies, Lumber, Milk, and Butter. Derivatives are also traded on
weather (various measures of temperature), stock indices (such as the S&P, NASDAQ,
Nikkei, and Russell), interest rates (such as the Eurodollar interest rate), and
currencies.
2. At the time of this writing, monthly trading volume is available from:
http://www.cme.com/trading/dta/hist/monthly_volume.html.
Table 1.1 summarizes the trading volume for October 2007 for the major classes of
asset types.
Table 1.1 CME Trading Volume, October 2007
Volume Month Ago Year Ago Jan-Oct, 2007 Jan-Oct, 2006
Futures
Commodity & Others 1,370,451 1,528,494 1,408,670 15,877,224 14,528,206
Currency 11,832,907 11,710,211 8,723,870 116,996,045 87,876,871
Equity & Index 59,972,745 55,584,345 38,230,166 516,159,044 387,933,512
Interest Rate 46,012,514 49,812,941 46,537,725 528,881,778 419,626,901
Options
Commodity & Others 223,690 1,90,723 1,587,41 1,661,814 1,536,658
Currency 4,26,897 3,51,360 2,27,990 3,501,472 2,668,271
Equity & Index 31,97,318 2,643,714 2,848,116 34,983,389 22,773,974
Interest Rate 19,447,226 27,739,787 21,993,883 273,280,944 233,235,408
3. For notional value, consider the exchange rate contracts traded on the CME as shown
in Table 1.2.
As we can see, contract size (i.e., number of units of the currency) for currencies with a
lower
US dollar value (e.g., the Japanese Yen) are larger. If the exchange chose to cut the size
of the contract in half, traders would likely double their volume (all else equal) leaving
notional volume constant.
, Table 1.2 demonstrates that volume alone might not capture the relative economic
significance of the contracts. For example, the Euro contracts have approximately 3.8
times the trading volume of Australia, but the notational value is close to 7 times.
Continuous Compounding
◼ Question B.1
Using a continuous rate of return, we have 67032 e = 100000. This implies e = 14918248.
r 5 r 5
Solving, we have r 5 = ln(14918248) = 040, hence the continuous return is r = 040 5 = 8%.
◼ Question B.2
The unknown in the continuous interest equation is the time to maturity, T. We must solve:
50 e010 T = 100. This implies e010 T = 2. Taking the natural log of each side, we have
0.10 T = ln(2) = 0693147 which implies T = 693147 years.
◼ Question B.3
In 7 years, at 7.5% continuous interest, $5 grows to 5 e = $84523.
0075 7
◼ Question B.4
Let S1 be the stock price in 1 year. The continuous return is
r = ln(S1 /S0 ) = ln(S1 /100) = ln(S1 ) − ln(100) = ln( S1 ) − 46052
Using this formula we have:
S0
r
5 −2.9957323
4 −3.2188758
3 −3.5065579
2 −3.912023
Note these are not percentages. For example, if S1 is 5, the return is −299.57%. For a −500%
return, the stock price next year must solve
ln(S1 ) = r + 46052 = −5 + 46052 = −03948
which implies S1 = e = $06738.
−03948
,◼ Question B.5
1. The arithmetic return for the first year is (200 − 100)/100 = 200% and, for the second
year, it is
(100 − 200)/200 = −50%.
2. The continuous return for the first year is ln(200/100) = 69.31% and, for the second
year, it is ln(100/200) = −69.31%.
3. We find that, for continuous returns, increases and decreases are symmetric. This
follows from
ln(x/y) = ln(x) − ln(y) = −ln(y/x).
◼ Question B.6
1. The arithmetic returns are given in the second column in the following table:
Stock Price Arithmetic Continuous
100
47 −0.5300 −0.7550
88 0.8723 0.6272
153 0.7386 0.5531
212 0.3856 0.3261
100 −0.5283 −0.7514
2. The continuous returns are given in the third column in the table above.
3. Given the arithmetic returns above, we must first take the cumulative product of the
gross return,
i.e., 1 plus the arithmetic return of the day. This is
04700 18723 17386 13856 04717 = 1
For the cumulative arithmetic return, we then subtract 1 from this amount. Using the
continuous return, we can simply sum continuous returns to arrive at the cumulative
continuous return.
,Chapter 1
Introduction to Derivatives
◼ Question 1.1
We will look at the CME
1. The CME trades derivatives (specifically futures and options on futures) on a wide
variety of assets and indices/variables. Assets include basic commodities such as
Cattle, Hogs, Pork Bellies, Lumber, Milk, and Butter. Derivatives are also traded on
weather (various measures of temperature), stock indices (such as the S&P, NASDAQ,
Nikkei, and Russell), interest rates (such as the Eurodollar interest rate), and
currencies.
2. At the time of this writing, monthly trading volume is available from:
http://www.cme.com/trading/dta/hist/monthly_volume.html.
Table 1.1 summarizes the trading volume for October 2007 for the major classes of
asset types.
Table 1.1 CME Trading Volume, October 2007
Volume Month Ago Year Ago Jan-Oct, 2007 Jan-Oct, 2006
Futures
Commodity & Others 1,370,451 1,528,494 1,408,670 15,877,224 14,528,206
Currency 11,832,907 11,710,211 8,723,870 116,996,045 87,876,871
Equity & Index 59,972,745 55,584,345 38,230,166 516,159,044 387,933,512
Interest Rate 46,012,514 49,812,941 46,537,725 528,881,778 419,626,901
Options
Commodity & Others 223,690 1,90,723 1,587,41 1,661,814 1,536,658
Currency 4,26,897 3,51,360 2,27,990 3,501,472 2,668,271
Equity & Index 31,97,318 2,643,714 2,848,116 34,983,389 22,773,974
Interest Rate 19,447,226 27,739,787 21,993,883 273,280,944 233,235,408
3. For notional value, consider the exchange rate contracts traded on the CME as shown
in Table 1.2.
As we can see, contract size (i.e., number of units of the currency) for currencies with a
lower
US dollar value (e.g., the Japanese Yen) are larger. If the exchange chose to cut the size
of the contract in half, traders would likely double their volume (all else equal) leaving
notional volume constant.
, Table 1.2 demonstrates that volume alone might not capture the relative economic
significance of the contracts. For example, the Euro contracts have approximately 3.8
times the trading volume of Australia, but the notational value is close to 7 times.
