Volatility index and derivative contracts based thereon

• US 20050102214A1
• Filed: 10/06/2004
• Published: 05/12/2005
• Est. Priority Date: 11/12/2003
• Status: Abandoned Application

• Current Assignee: Chicago Board Options Exchang
• Original Assignee: Chicago Board Options Exchange Incorporated (Cboe Global Markets, Inc.)
• Inventors: Speth, William, Shah, Devesh, Levin, Joseph, Rattray, Sandy
• Application Number:
  US10/959,528
• Publication Number:
  US 20050102214A1
• Time in Patent Office: Days
• US Class Current: 705/35
• CPC Class Codes: G06Q 40/00 Finance; Insurance; Tax strategies; Processing of corporate or income taxes
• CPC Class Codes: G06Q 40/04 Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

An improved volatility index and related futures contracts are provided. An index in accordance with the principals of the present invention estimates expected volatility from the prices of stock index options in a wide range of strike prices, not just at-the-money strikes. Also, an index in accordance with the principals of the present invention is not calculated from the Black/Scholes or any other option pricing model: the index of the present invention uses a newly developed formula to derive expected volatility by averaging the weighted prices of out-of-the money put and call options. In accordance with another aspect of the present invention, derivative contracts such as futures and options based on the volatility index of the present invention are provided.

Claims

1. A method of estimating expected volatility in financial markets comprising:

averaging weighted prices of out-of-the money put and call options based on a financial instrument.

2. The method of estimating expected volatility in financial markets of claim 1 further including determining the average weighted prices of out-of-the money put and call options in accordance with:

${\mathrm{\sigma <br><br>}}^2=\frac{2}{T}<br><br>\underset{i}{\overset{\text{\hspace{1em}<br><br>}}{\sum <br><br>}}<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_i}{K_i^2}<br><br>{\mathrm{e<br><br>}}^{\mathrm{RT}}<br><br>Q<br><br>\left(K_i\right)-{\frac{1}{T}<br><br>\left[\frac{F}{K_0}-1\right]}^2$

where;

T is the time to expiration;

F is the forward index level;

K_i is the strike price of i^th out-of-the-money option—

a call if K_i>

F and a put if K_i<

F;

Δ

K_i is the interval between strike prices;

K₀ is the first strike below the forward index level, F;

R is the risk-free interest rate to expiration; and

Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i.

3. The method of estimating expected volatility in financial markets of claim 2 further wherein the time to expiration is calculated in minutes.

4. The method of estimating expected volatility in financial markets of claim 3 further wherein the time to expiration T is calculated in accordance with the following:

T={M_{Current day}+M_{Settlement day}+M_{Other days}}/Minutes in a year;

where;

M_{Current day} is the number of minutes remaining until midnight of the current day;

M_{Settlement day} is the number of minutes from midnight until the target time on the settlement day; and

M_{Other days} is the Total number of minutes in the days between current day and settlement day.

5. The method of estimating expected volatility in financial markets of claim 1 further including determining a forward index level based on at-the-money option prices and selecting out-of-the-money call options that have a strike price greater than the forward index level.

6. The method of estimating expected volatility in financial markets of claim 1 further including determining a forward index level based on at-the-money option prices and selecting out-of-the-money put options that have a strike price less than the forward index level.

7. The method of estimating expected volatility in financial markets of claim 1 further including determining a forward index level based on at-the-money option prices and adding both put and call options with strike prices equal to a strike price immediately below the forward index level.

8. The method of estimating expected volatility in financial markets of claim 1 further including using options that have non-zero bid prices.

9. The method of estimating expected volatility in financial markets of claim 8 further including determining a forward index level based on at-the-money option prices and selecting options that have a strike price greater than the forward index level.

10. The method of estimating expected volatility in financial markets of claim 8 further including determining a forward index level based on at-the-money option prices and selecting options that have a strike price less than the forward index level.

11. The method of estimating expected volatility in financial markets of claim 8 further including determining a forward index level based on at-the-money option prices and adding options with strike prices equal to a strike price immediately below the forward index level.

12. The method of estimating expected volatility in financial markets of claim 1 further including selecting put and call options in the two nearest-term expiration months in order to bracket a calendar period selected from the group consisting of 30 to 365 days.

13. The method of estimating expected volatility in financial markets of claim 1 further including rolling the put and call options to subsequent contract months in order to minimize pricing anomalies that might occur close to expiration.

14. The method of estimating expected volatility in financial markets of claim 13 further wherein the options used have between and including 8 to 68 days to expiration.

15. The method of estimating expected volatility in financial markets of claim 14 further wherein the options used have 16 days and 44 days to expiration.

16. The method of estimating expected volatility in financial markets of claim 1 further wherein the same number of options is used for each contract month.

17. The method of estimating expected volatility in financial markets of claim 1 further wherein the interval between strike prices is uniform.

18. The method of estimating expected volatility in financial markets of claim 1 further wherein the contribution of a single option is proportional to the price of that option and inversely proportional to the square of a strike price of that option.

19. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is a security.

20. The method of estimating expected volatility in financial markets of claim 19 further wherein the security is a stock.

21. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is a stock index.

22. The method of estimating expected volatility in financial markets of claim 21 further wherein the stock index is the S&

P 500®

index.

23. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is a bond.

24. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is a basket of stocks.

25. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is an exchange-traded fund.

26. The method of estimating expected volatility in financial markets of claim 1 further wherein the financial instrument is a commodity.

27. The method of estimating expected volatility in financial markets of claim 1 further including interpolating near and future term options volatility to arrive at a single value.

64. The method of estimating expected volatility in financial markets of claim 3 further wherein the future term options are the next term options.

65. The method of estimating expected volatility in financial markets of claim 64 further including selecting put and call options.

66. The method of estimating expected volatility in financial markets of claim 65 further including selecting out-of-the-money call options that have a strike price greater than the forward index level.

67. The method of estimating expected volatility in financial markets of claim 65 further including selecting out-of-the-money put options that have a strike price less than the forward index level.

68. The method of estimating expected volatility in financial markets of claim 65 further including adding both put and call options with strike prices equal to a strike price immediately below the forward index level.

28. A method of estimating expected volatility in financial markets comprising:

selecting out-of-the money options on a financial instrument; and

averaging weighted prices of the out-of-the money options.

29. The method of estimating expected volatility in financial markets of claim 28 further including selecting put and call options.

30. The method of estimating expected volatility in financial markets of claim 29 further including determining a forward index level based on at-the-money option prices and selecting out-of-the-money call options that have a strike price greater than the forward index level.

31. The method of estimating expected volatility in financial markets of claim 29 further including determining a forward index level based on at-the-money option prices and selecting out-of-the-money put options that have a strike price less than the forward index level.

32. The method of estimating expected volatility in financial markets of claim 29 further including determining a forward index level based on at-the-money option prices and adding both put and call options with strike prices equal to a strike price immediately below the forward index level.

33. The method of estimating expected volatility in financial markets of claim 28 further including using options that have non-zero bid prices.

34. The method of estimating expected volatility in financial markets of claim 33 further including determining a forward index level based on at-the-money option prices and selecting options that have a strike price greater than the forward index level.

35. The method of estimating expected volatility in financial markets of claim 33 further including determining a forward index level based on at-the-money option prices and selecting options that have a strike price less than the forward index level.

36. The method of estimating expected volatility in financial markets of claim 33 further including determining a forward index level based on at-the-money option prices and adding options with strike prices equal to a strike price immediately below the forward index level.

37. The method of estimating expected volatility in financial markets of claim 28 further including determining a forward index level based on at-the-money option prices and centering the options around a strike price immediately below the forward index level.

38. The method of estimating expected volatility in financial markets of claim 37 further wherein the centering comprises selecting two options at a strike price immediately below the forward index level.

39. The method of estimating expected volatility in financial markets of claim 38 further including averaging the put and call prices at a strike price immediately below the forward index level to arrive at a single value.

40. The method of estimating expected volatility in financial markets of claim 37 further wherein the centering comprises selecting a single option, either a put or a call, for every other strike price.

41. The method of estimating expected volatility in financial markets of claim 40 further including averaging the put and call prices at a strike price immediately below the forward index level to arrive at a single value.

42. The method of estimating expected volatility in financial markets of claim 28 further including determining a forward index level based on at-the-money option prices and selecting out-of-the-money put options with a strike price less than a strike price immediately below the forward index level.

43. The method of estimating expected volatility in financial markets of claim 28 further including determining a forward index level based on at-the-money option prices and selecting out-of-the money call options with a strike price greater than a strike price immediately below the forward index level.

44. The method of estimating expected volatility in financial markets of claim 28 further including selecting options in the two nearest-term expiration months in order to bracket a calendar period selected from the group consisting of 30 to 365 days.

45. The method of estimating expected volatility in financial markets of claim 28 further including rolling the options to subsequent contract months in order to minimize pricing anomalies that might occur close to expiration.

46. The method of estimating expected volatility in financial markets of claim 45 further wherein the options used have between and including 8 to 68 days to expiration.

47. The method of estimating expected volatility in financial markets of claim 46 further wherein the options used have 16 days and 44 days to expiration.

48. The method of estimating expected volatility in financial markets of claim 28 further wherein the same number of options is used for each contract month.

49. The method of estimating expected volatility in financial markets of claim 28 further wherein the interval between strike prices is uniform.

50. The method of estimating expected volatility in financial markets of claim 28 further including determining the volatility (σ

) from the variance (σ

²) in accordance with;

${\mathrm{\sigma <br><br>}}^2=\frac{2}{T}<br><br>\underset{i}{\overset{\text{\hspace{1em}<br><br>}}{\sum <br><br>}}<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_i}{K_i^2}<br><br>{\mathrm{e<br><br>}}^{\mathrm{RT}}<br><br>Q<br><br>\left(K_i\right)-{\frac{1}{T}<br><br>\left[\frac{F}{K_0}-1\right]}^2$

where;

T is the time to expiration;

F is the forward index level;

K_i is the strike price of i^th out-of-the-money option—

a call if K_i>

F and a put if K_i<

F;

Δ

K_i is the interval between strike prices;

K₀ is the first strike below the forward index level, F;

R is the risk-free interest rate to expiration; and

Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i.

51. The method of estimating expected volatility in financial markets of claim 50 further wherein the time to expiration is calculated in minutes.

52. The method of estimating expected volatility in financial markets of claim 51 further wherein the time to expiration T is calculated in accordance with the following:

T={M_{Current day}+M_{Settlement day}+M_{Other days}}/Minutes in a year;

where;

M_{Current day} is the number of minutes remaining until midnight of the current day;

M_{Settlement day} is the number of minutes from midnight until the target time on the settlement day; and

M_{Other days} is the Total number of minutes in the days between current day and settlement day.

53. The method of estimating expected volatility in financial markets of claim 28 further wherein the contribution of a single option is proportional to the price of that option and inversely proportional to the square root of a strike price of that option.

54. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is a security.

55. The method of estimating expected volatility in financial markets of claim 54 further wherein the security is a stock.

56. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is a stock index.

57. The method of estimating expected volatility in financial markets of claim 56 further wherein the stock index is the S&

P 500®

index.

58. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is a bond.

59. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is a basket of stocks.

60. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is an exchange-traded fund.

61. The method of estimating expected volatility in financial markets of claim 28 further wherein the financial instrument is a commodity.

62. The method of estimating expected volatility in financial markets of claim 28 further including interpolating near and future term options volatility to arrive at a single value.

63. A method of estimating expected volatility in financial markets comprising:

selecting a series of options with different expiration dates;

for a time period, determining a forward index level based on at-the-money option prices;

determining the forward index level for the near and future term options;

determining a strike price immediately below the forward index level;

averaging quoted bid-ask prices for each option;

calculating volatility of the near and future term options; and

interpolating the near and future term options volatility to arrive at a single value.

69. The method of estimating expected volatility in financial markets of claim 63 further including using options that have non-zero bid prices.

70. The method of estimating expected volatility in financial markets of claim 69 further including selecting options that have a strike price greater than the forward index level.

71. The method of estimating expected volatility in financial markets of claim 69 further including selecting options that have a strike price less than the forward index level.

72. The method of estimating expected volatility in financial markets of claim 69 further including adding options with strike prices equal to a strike price immediately below the forward index level.

73. The method of estimating expected volatility in financial markets of claim 63 further including centering the options around a strike price immediately below the forward index level.

74. The method of estimating expected volatility in financial markets of claim 73 further wherein the centering comprises selecting two options at the strike price immediately below the forward index level.

75. The method of estimating expected volatility in financial markets of claim 74 further including averaging the put and call prices at the strike price immediately below the forward index level to arrive at a single value.

76. The method of estimating expected volatility in financial markets of claim 73 further wherein the centering comprises selecting a single option, either a put or a call, for every other strike price.

77. The method of estimating expected volatility in financial markets of claim 76 further including averaging the put and call prices at the strike price immediately below the forward index level to arrive at a single value.

78. The method of estimating expected volatility in financial markets of claim 63 further including selecting out-of-the-money put options with a strike price less than a strike price immediately below the forward index level.

79. The method of estimating expected volatility in financial markets of claim 63 further including selecting out-of-the money call options with a strike price greater than a strike price immediately below the forward index level.

80. The method of estimating expected volatility in financial markets of claim 63 further including rolling the put and call options to subsequent contract months in order to minimize pricing anomalies that might occur close to expiration.

81. The method of estimating expected volatility in financial markets of claim 80 further wherein the options used have between and including 8 to 68 days to expiration.

82. The method of estimating expected volatility in financial markets of claim 81 further wherein the options used have 16 days and 44 days to expiration.

83. The method of estimating expected volatility in financial markets of claim 63 further wherein the same number of options is used for each contract month.

84. The method of estimating expected volatility in financial markets of claim 63 further wherein the interval between strike prices is uniform.

85. The method of estimating expected volatility in financial markets of claim 63 further including determining the forward index level (F) in accordance with:

F=Strike Price+e^RT×

(Call Price−

Put Price),

where

R is the risk-free interest rate to expiration; and

T is the time to expiration.

86. The method of estimating expected volatility in financial markets of claim 63 further including determining the volatility by averaging weighted prices of out-of-the money put and call options.

87. The method of estimating expected volatility in financial markets of claim 63 further including determining the volatility (σ

) from the variance (σ

²) in accordance with;

${\mathrm{\sigma <br><br>}}^2=\frac{2}{T}<br><br>\underset{i}{\overset{\text{\hspace{1em}<br><br>}}{\sum <br><br>}}<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_i}{K_i^2}<br><br>{\mathrm{e<br><br>}}^{\mathrm{RT}}<br><br>Q<br><br>\left(K_i\right)-{\frac{1}{T}<br><br>\left[\frac{F}{K_0}-1\right]}^2$

where;

T is the time to expiration;

F is the forward index level;

K_i is the strike price of i^th out-of-the-money option—

a call if K_i>

F and a put if K_i<

F;

Δ

K_i is the interval between strike prices;

K₀ is the first strike below the forward index level, F;

R is the risk-free interest rate to expiration; and

Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i.

88. The method of estimating expected volatility in financial markets of claim 87 further wherein the time to expiration is calculated in minutes.

89. The method of estimating expected volatility in financial markets of claim 88 further wherein the time to expiration T is calculated in accordance with the following:

T={M_{Current day}+M_{Settlement day}+M_{Other days}}/Minutes in a year;

where;

M_{Current day} is the number of minutes remaining until midnight of the current day;

M_{Settlement day} is the number of minutes from midnight until the target time on the settlement day; and

M_{Other days} is the Total number of minutes in the days between current day and settlement day.

90. The method of estimating expected volatility in financial markets of claim 63 further wherein the contribution of a single option is proportional to the price of that option and inversely proportional to the square root of a strike price of that option.

91. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is a security.

92. The method of estimating expected volatility in financial markets of claim 91 further wherein the security is a stock.

93. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is a stock index.

94. The method of estimating expected volatility in financial markets of claim 93 further wherein the stock index is the S&

P 500®

index.

95. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is a bond.

96. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is a basket of stocks.

97. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is an exchange-traded fund.

98. The method of estimating expected volatility in financial markets of claim 63 further wherein the financial instrument is a commodity.

99. A derivative contract comprising:

basing the derivative contract on an underlying index that estimates expected volatility in financial markets.

100. The derivative contract of claim 99 further wherein the estimated expected volatility is estimated with average weighted prices of out-of-the money options from a financial instrument.

101. The derivative contract of claim 100 further including determining a forward index level based on at-the-money option prices and selecting for the underlying index out-of-the-money call options that have a strike price greater than the forward index level.

102. The derivative contract of claim 100 further including determining a forward index level based on at-the-money option prices and selecting for the underlying index out-of-the-money put options that have a strike price less than the forward index level.

103. The derivative contract of claim 100 further including determining a forward index level based on at-the-money option prices and adding to the underlying index both put and call options with strike prices equal to a strike price immediately below the forward index level.

104. The derivative contract of claim 100 further including using for the underlying index options that have non-zero bid prices.

105. The derivative contract of claim 104 further including determining a forward index level based on at-the-money option prices and selecting for the underlying index options that have a strike price greater than the forward index level.

106. The derivative contract of claim 104 further including determining a forward index level based on at-the-money option prices and selecting for the underlying index options that have a strike price less than the forward index level.

107. The derivative contract of claim 104 further including determining a forward index level based on at-the-money option prices and adding to the underlying index options with strike prices equal to a strike price immediately below the forward index level.

108. The derivative contract of claim 100 further including selecting for the underlying index put and call options in the two nearest-term expiration months in order to bracket a calendar period selected from the group consisting of 30 to 365 days.

109. The derivative contract of claim 100 further including rolling the options in the underlying index to the subsequent contract months in order to minimize pricing anomalies that might occur close to expiration.

110. The derivative contract of claim 109 further wherein the options used have between and including 8 to 68 days to expiration.

111. The derivative contract of claim 110 further wherein the options used have 16 days and 44 days to expiration.

112. The derivative contract of claim 100 further wherein the same number of options in the underlying index is used for each contract month and the interval between strike prices is uniform.

113. The derivative contract of claim 100 further wherein the contribution of a single option to the underlying index is proportional to the price of that option and inversely proportional to the square of a strike price of that option.

114. The derivative contract of claim 100 further wherein the financial instrument is a security.

115. The derivative contract of claim 114 further wherein the further wherein the security is a stock.

116. The derivative contract of claim 114 further wherein the financial instrument is a stock index.

117. The derivative contract of claim 116 further wherein the stock index is the S&

P 500®

index.

118. The derivative contract of claim 100 further wherein the financial instrument is a bond.

119. The derivative contract of claim 100 further wherein the financial instrument is a basket of stocks.

120. The derivative contract of claim 100 further wherein the financial instrument is an exchange-traded fund.

121. The derivative contract of claim 100 further wherein the financial instrument is a commodity.

122. The derivative contract of claim 99 further wherein the volatility (σ

) is determined from the variance (σ

²) in accordance with;

${\mathrm{\sigma <br><br>}}^2=\frac{2}{T}<br><br>\underset{i}{\overset{\text{\hspace{1em}<br><br>}}{\sum <br><br>}}<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_i}{K_i^2}<br><br>{\mathrm{e<br><br>}}^{\mathrm{RT}}<br><br>Q<br><br>\left(K_i\right)-{\frac{1}{T}<br><br>\left[\frac{F}{K_0}-1\right]}^2$

where;

T is the time to expiration;

F is the forward index level;

K_i is the strike price of i^th out-of-the-money option—

a call if K_i>

F and a put if K_i<

F;

Δ

K_i is the interval between strike prices;

K₀ is the first strike below the forward index level, F;

R is the risk-free interest rate to expiration; and

Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i.

123. The derivative contract of claim 122 further wherein the time to expiration is calculated in minutes.

124. The derivative contract of claim 123 further wherein the time to expiration T is calculated in accordance with the following:

T={M_{Current day}+M_{Settlement day}+M_{Other days}}/Minutes in a year;

where;

M_{Current day} is the number of minutes remaining until midnight of the current day;

M_{Settlement day} is the number of minutes from midnight until the target time on the settlement day; and

M_{Other days} is the Total number of minutes in the days between current day and settlement day.

125. The derivative contract of claim 99 further wherein near and future term options volatility is interpolated to arrive at a single value.

126. The derivative contract of claim 99 further wherein the derivative contract is an options contract.

127. The derivative contract of claim 99 further wherein the derivative contract is a futures contract.

128. A method of creating a derivative contract from an underlying financial instrument comprising:

selecting options on a financial instrument;

determining a forward index level based on at-the-money option prices;

determining the forward index level for the options;

determining a strike price immediately below the forward index level;

averaging quoted bid-ask prices for each option; and

calculating volatility of the options.

129. The method of claim 128 further including selecting put and call options.

130. The method of claim 129 further including selecting out-of-the-money call options that have a strike price greater than the forward index level.

131. The method of claim 129 further including selecting out-of-the-money put options that have a strike price less than the forward index level.

132. The method of claim 129 further including adding both put and call options with strike prices equal to a strike price immediately below the forward index level.

133. The method of claim 128 further including using options that have non-zero bid prices.

134. The method of claim 133 further including selecting options that have a strike price greater than the forward index level.

135. The method of claim 133 further including selecting options that have a strike price less than the forward index level.

136. The method of claim 133 further including adding options with strike prices equal to a strike price immediately below the forward index level.

137. The method of claim 128 further including centering the options around a strike price immediately below the forward index level.

138. The method of claim 137 further wherein the centering comprises selecting two options at the strike price immediately below the forward index level.

139. The method of claim 138 further including averaging the put and call prices at the strike price immediately below the forward index level to arrive at a single value.

140. The method of claim 137 further wherein the centering comprises selecting a single option, either a put or a call, for every other strike price.

141. The method of claim 140 further including averaging the put and call prices at the strike price immediately below the forward index level to arrive at a single value.

142. The method of claim 128 further including selecting out-of-the-money put options with a strike price less than a strike price immediately below the forward index level.

143. The method of claim 128 further including selecting out-of-the money call options with a strike price greater than a strike price immediately below the forward index level.

144. The method of claim 128 further including selecting put and call options in the two nearest-term expiration months in order to bracket a calendar period selected from the group consisting of 30 to 365 days.

145. The method of claim 128 further including rolling the put and call options to subsequent contract months in order to minimize pricing anomalies that might occur close to expiration.

146. The method of claim 145 further wherein further wherein the options used have between and including 8 to 68 days to expiration.

147. The method of claim 146 further wherein the options used have 16 days and 44 days to expiration.

148. The method of claim 128 further wherein the same number of options is used for each contract month.

149. The method of claim 128 further wherein the interval between strike prices is uniform.

150. The method of claim 128 further wherein the forward index level (F) is calculated:

F=Strike Price+e^RT×

(Call Price−

Put Price),

where

R is the risk-free interest rate to expiration; and

T is the time to expiration.

151. The method of claim 128 further wherein the volatility is calculated by averaging weighted prices of out-of-the money put and call options.

152. The method of claim 128 further including determining the volatility (σ

) from the variance (σ

²) in accordance with;

${\mathrm{\sigma <br><br>}}^2=\frac{2}{T}<br><br>\underset{i}{\overset{\text{\hspace{1em}<br><br>}}{\sum <br><br>}}<br><br>\frac{\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>K_i}{K_i^2}<br><br>{\mathrm{e<br><br>}}^{\mathrm{RT}}<br><br>Q<br><br>\left(K_i\right)-{\frac{1}{T}<br><br>\left[\frac{F}{K_0}-1\right]}^2$

where;

T is the time to expiration;

F is the forward index level;

K_i is the strike price of i^th out-of-the-money option—

a call if K_i>

F and a put if K_i<

F;

Δ

K_i is the interval between strike prices;

K₀ is the first strike below the forward index level, F;

R is the risk-free interest rate to expiration; and

Q(K_i) is the midpoint of the bid-ask spread for each option with strike K_i.

153. The method of claim 152 further wherein the time to expiration is calculated in minutes.

154. The method of claim 153 further wherein the time to expiration T is calculated in accordance with the following:

T={M_{Current day}+M_{Settlement day}+M_{Other days}}/Minutes in a year;

where;

M_{Current day} is the number of minutes remaining until midnight of the current day;

M_{Settlement day} is the number of minutes from midnight until the target time on the settlement day; and

M_{Other days} is the Total number of minutes in the days between current day and settlement day.

155. The method of making a derivative contract of claim 128 further wherein the contribution of a single option is proportional to the price of that option and inversely proportional to the square of a strike price of that option.

156. The method of making a derivative contract of claim 128 further wherein the financial instrument is a security.

157. The method of making a derivative contract of claim 156 further wherein the security is a stock.

158. The method of making a derivative contract of claim 128 further wherein the financial instrument is a stock index.

159. The method of making a derivative contract of claim 158 further wherein the stock index is the S&

P 500®

index.

160. The method of making a derivative contract of claim 128 further wherein the financial instrument is a bond.

161. The method of making a derivative contract of claim 128 further wherein the financial instrument is a basket of stocks.

162. The method of making a derivative contract of claim 128 further wherein the financial instrument is an exchange-traded fund.

163. The method of making a derivative contract of claim 128 further wherein the financial instrument is a commodity.

164. The method of making a derivative contract of claim 128 further including interpolating near and future term options volatility to arrive at a single value.

165. The method of making a derivative contract of claim 128 further wherein the derivative contract is an options contract.

166. The method of making a derivative contract of claim 128 further wherein derivative contract is a futures contract.

167. A method of settling a derivative contract comprising:

collecting the opening traded price, if any, and the first bid/ask quote for each eligible option series;

determining the forward index level for each eligible contract month based on at-the-money option prices;

determining the strike price immediately below the forward index level for each eligible contract month;

sorting the options in ascending order by strike price;

selecting call options that have strike prices greater than the strike price immediately below the forward index level and a non-zero bid price, beginning with the strike price closest to the strike price immediately below the forward index level and moving to the next higher strike prices in succession;

selecting put options that have strike prices less than the strike price immediately below the forward index level and a non-zero bid price, beginning with the strike price closest to the strike price immediately below the forward index level and then moving to the next lower strike prices in succession;

calculating a special opening quotation using the options selected;

determining the settlement price from the special opening quotation.

168. The method of settling a derivative contract of claim 167 further wherein the price of each option used in the calculation is the opening traded price of that option.

169. The method of settling a derivative contract of claim 168 further wherein in the event that there is no opening traded price for an option, the price used in the calculation is the average of the first bid/ask quote for that option.

170. The method of settling a derivative contract of claim 167 further wherein after two consecutive calls with a bid price of zero are encountered, selecting no other calls.

171. The method of settling a derivative contract of claim 167 further wherein after encountering two consecutive puts with a bid price of zero, selecting no other puts.

172. The method of settling a derivative contract of claim 167 further including selecting both the put and call with the strike price immediately below the forward index level.

173. The method of settling a derivative contract of claim 167 further including multiplying the special opening quotation by 10 in order to determine the final settlement price.

174. The method of settling a derivative contract of claim 167 further wherein the derivative contract comprises a futures contract.

175. The method of settling a derivative contract of claim 167 further wherein the derivative contract comprises an options contract.