Educación

EWMA(0.95)

The RiskMetrics Volatility Model is a special case of the Exponential Weighted Moving Average Model (EWMA). The EWMA suggests that the variance of a financial asset can be calculated using the formula:

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Where 𝜎_𝑡−12 is the EWMA variance at time t=1, 𝑟_𝑡−12 the squared returns at t-1, and 𝜆 a weight between 0 and 1. Following IVS, a RiskMetrics Volatility Model with a weight of 0.95 was selected therefore calculating daily volatility as:

𝑅𝑖𝑠𝑘𝑀𝑒𝑡𝑟𝑖𝑐𝑠𝑉𝑜𝑙 = √0.95𝜎_𝑡−12 + 0.05𝑟_𝑡−12 (74)

Nikkei 225 Volatility Index

The Nikkei 225 VI indicates the expected degree of fluctuation of the Nikkei 225 Index in the future. The greater the index values are, the larger fluctuation investors expect in the market. Nikkei 225 VI Futures are based on the Nikkei 225 VI, which is an index,

calculated by Nikkei Inc., estimating the degree of expected fluctuation in the Nikkei 225. The Nikkei225 Volatility Index displays the following characteristics:

High Volatility. Nikkei 225 VI has a regular range of 22 ~ 28 points, however when the Nikkei 225 declines sharply, the Nikkei 225 VI may advance rapidly.

Mean Regression. The Nikkei 225 VI has a tendency to regress to its regular range of 22 ~ 28 points following a rapid increase without sticking in a high point range.

Calculation Method:

The Nikkei 225 Volatility Index is calculated using prices of Nikkei 225 futures and Nikkei 225 options on the Osaka Exchange (OSE). In the calculation, taking near-term future price as the basis of ATM, the volatility of near-term options and next-term options are calculated with OTM option prices of each delivery month. Next, the index value is calculated using linear interpolation or linear extrapolation between the volatilities of each delivery month to set the time to expiration as 30 days.

Base Date:

The commencement date of the calculation was November 19, 2010, which had been retroactively calculated in the past on the end-of-day basis, to June 12th, 1989. The index is currently calculated every 15 seconds during the day session of the Nikkei 225 options on the OSE.

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Relative Strength Index (Volatility)

𝑅𝑆𝐼(𝑉)_𝑡 = 𝑈𝑡 𝑈_𝑡+ 𝐷_𝑡

(75)

Where 𝑈_𝑡 denotes the cumulative advance i.e. the close-to-close increase on a day where the security under observation has closed higher than the previous day’s closing price over a certain period, and 𝐷_𝑡 denotes the cumulative decline i.e. the close-to-close decrease on a day where the security under observation has closed lower than the previous day’s closing price over the sample period x which by default is 14 trading days as prescribed by Wilder.

𝑈𝑡= ∑ 𝑙(𝑆𝑡−1− 𝑆𝑡−1−𝑖 > 0)(𝑆𝑡−1− 𝑆𝑡−1−𝑖) 𝑚 𝑖=1 (76) 𝐷_𝑡 = ∑ 𝑙(𝑆_𝑡−1− 𝑆_{𝑡−1−𝑖}< 0)|𝑆_𝑡−1− 𝑆_{𝑡−1−𝑖}| 𝑚 𝑖=1 (77)

Where S = Standard Deviation of the Close

Decreasing the value of x results in increased sensitivity whereas increasing the value of x

decreases sensitivity. The RSI is then normalised and presented on a vertical scale bound between 0 and 100 with values above 70 signalling overbought conditions and values below 30 signalling oversold conditions. A mid-range value of 50 signals equilibrium and acts as a boundary between bullish and bearish market conditions, the crossing of which generates a respective minor buy or sell signal.

Garman Klass Volatility

The Garman Klass Volatility estimator incorporates intraday information stored at daily frequency and is calculated as follows. Initially a scaled vector is determined equal to N, the number of trading days in a year, and n the chosen sample size. If n = N no scaling is required. Thereafter, the scaling factor is multiplied with the variance of the sample. The results obtained is the annualized Garman Klass variance. Lastly, taking the square root of this result produces the Garman Klass volatility estimator:

181 𝜎_𝑡 = √𝑁 𝑛 ∙ ∑ 1 2 𝑛 𝑖=1 ∙ (log (𝐻𝑖 𝐿_𝑖)) 2 _{− (2 ∙ log(2) − 1) ∙ 𝑙𝑜𝑔 (}𝐶𝑖 𝑂_𝑖) 2 (78) Where:

N = Number of trading days in 1 year

n = Sample Period H = High L = Low O = Open C = Close CV(L9)ROC12

Chaikin Volatility (CV), designed by Marc Chaikin, is a technical indicator employed to measure volatility. CV compares the spread between a asset’s high and low prices and quantifies volatility as a widening of the range between the high and the low price. CV is calculated by initially calculating an EMA of the difference between the periods’ high and low prices:

𝐻 − 𝐿 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 = 𝐸𝑀𝐴(𝐻 − 𝐿) (79)

Where:

H = High

L = Low

EMA = Exponential Moving Average

Lastly, the percent that the EMA has changed over a specified period is calculated as:

((𝐻 − 𝐿 𝐴𝑣𝑒𝑟𝑎𝑔𝑒) − (𝐻 − 𝐿 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛 − 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑎𝑔𝑜)

(𝐻 − 𝐿 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛 − 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑎𝑔𝑜) × 100

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Chapter 6: Input Variables

For Chapter 6: Macroeconomic Forecasting with Artificial Neural Networks: U.S.

Unemployment Forecasting, all variables of the candidate variable pool were sourced from the (FRED) Economic Data section of the Federal Bank of St Louis. Following 2-stage IVS, seven variables were selected:

Variable 1

Title: Civilian Unemployment Rate Series ID: UNRATE

Unit Measurement: Percent (%) Frequency: Monthly

Adjusted: Seasonally Adjusted

Variable 2

Title: Unemployment Rate: Ages 15-64, All Persons for the U.S. Series ID: LRUN64TTUSM156S

Unit Measurement: Percent (%) Frequency: Monthly

Adjusted: Seasonally Adjusted

Variable 3

Title: Unemployment Level: Men Series ID: LNS13000001

Unit Measurement: Thousands of Persons Frequency: Monthly

Adjusted: Seasonally Adjusted

Variable 4

Title: Unemployment Level: Looking for Full-Time Work Series ID: LNS13100000

Unit Measurement: Thousands of Persons Frequency: Monthly

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Variable 5

Title: Unemployment Level: White Series ID: LNS13000003

Unit Measurement: Thousands of Persons Frequency: Monthly

Adjusted: Seasonally Adjusted

Variable 6

Title: Unemployment Level: 20 Years + Series ID: LNS13000024

Unit Measurement: Thousands of Persons Frequency: Monthly

Adjusted: Seasonally Adjusted

Variable 7

Title: Unemployed Population: Ages 15-64, All Persons for the U.S. Series ID: LFUN64TTUSM647S

Unit Measurement: Persons Frequency: Monthly

Adjusted: Seasonally Adjusted

Chapter 7: Input Variables

For Chapter 7: Macroeconomic Forecasting with Artificial Neural Networks: U.S. GDP Forecasting, all variables of the candidate variable pool were sourced from the (FRED) Economic Data section of the Federal Bank of St Louis. Following 2-stage IVS, two variables were selected:

Variable 1

Title: GDP

Series ID: A191RP1Q027SBEA

Unit Measurement: % Change from Preceding Period Frequency: Monthly

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Variable 2

Title: GDP Per Capita

Series ID: A939RC0Q052SBEA Unit Measurement: USD ($) Frequency: Quarterly

Adjusted: Seasonally Adjusted