README.md
In m-g-h/R.MFIV: Calculate Model Free Implied Volatility, i.e. the CBOE VIX from Option Quotes
R.MFIV 
R.MFIV can be installed via:
remotes::install_github("m-g-h/R.MFIV")
Package Functionality
The VIX is defined as:
where the term in brackets under the square root is a linear
interpolation of two Implied Variances, which are given by the second
formula. The CBOE approach of the VIX calculation is explained in the
CBOE VIX Whitepaper.
In order to calculate the VIX and its individual variables, this package
provides the following functionality as explained in the following
sections
1. Short Walkthrough
- Calculate the Risk-Free-Rate R
interpolate_rfr()
- Calculate the At-The-Money Forward Price F0
CBOE_F_O()
- Calculate the At-The-Money Strike Price K0
CBOE_K_0()
- Select the Option Quotes Q(Ki) according to the CBOE
Rule
CBOE_option_selection()
- Calculate the Implied Variance σ2
CBOE_sigma_sq()
- Do all of the above in one function
CBOE_VIX_variables()
- Interpolate the VIX using two Implied Variances
CBOE_VIX_index()
- Calculate Option Descriptives
option_descriptives()
2. Processing a whole dataset (using data.table)
- Automatically determine the Expiration Terms (“near-” and
“next-term”)
CBOE_interpolation_terms()
- Create the implied Variance (and VIX variables) for a whole dataset
- Interpolate the weekly or monthly VIX index
3. Visualise the VIX-Index
- As single plot
plot_VIX()
- Browse through plots of multiple tickers with an interactive Shiny
App
result_browser()
4. Analyse the Option-Data Quality
- Calculate Option Descriptives for the whole
dataset
option_descriptives()
- Visualise the option data quality [to be done]
5. Improve Option-Data Quality using Smoothing Methods
- Intra- and Extrapolate option quotes
JandT_2007_smoothing_method()
- Calculate the Jiang & Tian MFIV
JandT_2007_sigma_sq()
1. Short Walkthrough
Load exemplary package data
library(R.MFIV)
data(option_dataset)
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-06-16 147.390 <data.table[96x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 5: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> ---
#> 3117: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3121: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
We select an exemplary entry for the calculation. This observation will
be our “near-term” contract. Also note that option_quotes is a
“nested” data.table itself
t <- option_dataset$t[2]
exp <- option_dataset$exp[2]
option_quotes <- option_dataset$option_quotes[[2]]
option_quotes
#> K c p
#> 1: 105.0 42.550 NA
#> 2: 110.0 37.550 NA
#> 3: 115.0 32.525 NA
#> 4: 120.0 27.600 NA
#> 5: 123.0 24.600 0.105
#> 6: 124.0 23.600 0.115
#> 7: 125.0 22.675 0.135
#> 8: 126.0 21.650 0.155
#> 9: 127.0 20.725 0.170
#> 10: 128.0 19.725 0.195
#> 11: 129.0 18.725 0.220
#> 12: 130.0 17.825 0.245
#> 13: 131.0 16.775 0.285
#> 14: 132.0 15.825 0.325
#> 15: 133.0 14.875 0.370
#> 16: 134.0 13.925 0.425
#> 17: 135.0 12.975 0.485
#> 18: 136.0 11.975 0.560
#> 19: 137.0 11.150 0.640
#> 20: 138.0 10.325 0.750
#> 21: 139.0 9.450 0.870
#> 22: 140.0 8.550 1.020
#> 23: 141.0 7.625 1.185
#> 24: 142.0 6.825 1.390
#> 25: 143.0 6.150 1.630
#> 26: 144.0 5.425 1.895
#> 27: 145.0 4.675 2.235
#> 28: 146.0 4.125 2.590
#> 29: 147.0 3.525 3.000
#> 30: 148.0 3.045 3.475
#> 31: 149.0 2.545 4.000
#> 32: 150.0 2.130 4.575
#> 33: 152.5 1.280 6.375
#> 34: 155.0 0.745 8.275
#> 35: 157.5 0.430 10.450
#> 36: 160.0 0.265 12.800
#> 37: 162.5 0.165 15.175
#> 38: 165.0 0.115 17.650
#> 39: 167.5 0.085 20.100
#> 40: 170.0 NA 22.600
#> 41: 172.5 NA 25.075
#> 42: 175.0 NA 27.575
#> 43: 177.5 NA 30.075
#> 44: 180.0 NA 32.575
#> 45: 182.5 NA 35.075
#> 46: 185.0 NA 37.575
#> 47: 187.5 NA 40.075
#> 48: 190.0 NA 42.575
#> 49: 195.0 NA 47.575
#> 50: 200.0 NA 52.575
#> 51: 210.0 NA 62.575
#> K c p
Next we calculate the Risk-Free-Rate R
library(lubridate)
#>
#> Attache Paket: 'lubridate'
#> Die folgenden Objekte sind maskiert von 'package:base':
#>
#> date, intersect, setdiff, union
R <- interpolate_rfr(date = as_date(t),
exp = exp,
ret_table = F)
R
#> [1] 0.008769736
Calculate At-The-Money Forward Price F0
## Set expiration time to 4 PM
exp <- exp + hours(16)
## Calculate maturity in years
maturity <- time_length(exp-t,
unit = "years")
## Calculate ATM Forward
F_0 <- CBOE_F_0(option_quotes = option_quotes,
R = R,
maturity = maturity)
F_0
#> [1] 147.5697
Calculate At-The-Money Strike price K0
K_0 <- CBOE_K_0(option_quotes = option_quotes,
F_0 = F_0)
K_0
#> [1] 147
Select option quotes per CBOE rule
option_quotes <- CBOE_option_selection(option_quotes = option_quotes,
K_0 = K_0)
option_quotes
#> K c p
#> 1: 123.0 24.600 0.105
#> 2: 124.0 23.600 0.115
#> 3: 125.0 22.675 0.135
#> 4: 126.0 21.650 0.155
#> 5: 127.0 20.725 0.170
#> 6: 128.0 19.725 0.195
#> 7: 129.0 18.725 0.220
#> 8: 130.0 17.825 0.245
#> 9: 131.0 16.775 0.285
#> 10: 132.0 15.825 0.325
#> 11: 133.0 14.875 0.370
#> 12: 134.0 13.925 0.425
#> 13: 135.0 12.975 0.485
#> 14: 136.0 11.975 0.560
#> 15: 137.0 11.150 0.640
#> 16: 138.0 10.325 0.750
#> 17: 139.0 9.450 0.870
#> 18: 140.0 8.550 1.020
#> 19: 141.0 7.625 1.185
#> 20: 142.0 6.825 1.390
#> 21: 143.0 6.150 1.630
#> 22: 144.0 5.425 1.895
#> 23: 145.0 4.675 2.235
#> 24: 146.0 4.125 2.590
#> 25: 147.0 3.525 3.000
#> 26: 148.0 3.045 3.475
#> 27: 149.0 2.545 4.000
#> 28: 150.0 2.130 4.575
#> 29: 152.5 1.280 6.375
#> 30: 155.0 0.745 8.275
#> 31: 157.5 0.430 10.450
#> 32: 160.0 0.265 12.800
#> 33: 162.5 0.165 15.175
#> 34: 165.0 0.115 17.650
#> 35: 167.5 0.085 20.100
#> K c p
Calculate the Implied Variance σ2
sigma_sq <- CBOE_sigma_sq(sel_option_quotes = option_quotes,
K_0 = K_0,
F_0 = F_0,
maturity = maturity,
R = R)
sigma_sq
#> [1] 0.05416683
Do the same for a second maturity, this time all at once. This
observation will be our “next-term” contract.
## Select observation 5
t2 <- option_dataset$t[3]
exp2 <- option_dataset$exp[3]
option_quotes2 <- option_dataset$option_quotes[[3]]
## Risk free rate
R2 <- interpolate_rfr(date = as_date(t2),
exp = exp2,
ret_table = F)
## Set expiration time to 4 PM
exp2 <- exp2 + hours(16)
## Calculate maturity in years
maturity2 <- time_length(exp2-t2,unit = "years")
## Calculate all VIX vars at once
VIX_vars <- CBOE_VIX_vars(option_quotes = option_quotes2,
R = R2,
maturity = maturity2,
ret_vars = T)
VIX_vars
#> $F_0
#> [1] 147.5497
#>
#> $K_0
#> [1] 147
#>
#> $n_put_raw
#> [1] 15
#>
#> $n_call_raw
#> [1] 14
#>
#> $n_put
#> [1] 15
#>
#> $n_call
#> [1] 14
#>
#> $sigma_sq
#> [1] 0.05222205
Calculate and interpolate the VIX index from both maturities
CBOE_VIX_index(maturity = c(maturity, maturity2),
sigma_sq = c(sigma_sq,VIX_vars$sigma_sq))
#> [1] 22.91347
Optionally calculate Option Descriptives, i.e. the strike price range
and spacing in standard-deviation (SD) units of the used option quotes:
price <- option_dataset$price[2]
option_descriptives(option_quotes = option_quotes,
K_0 = K_0,
R = R,
price = price,
maturity = maturity)
#> $SD
#> [1] 0.2218317
#>
#> $max_K
#> [1] 167.5
#>
#> $min_K
#> [1] 123
#>
#> $mean_delta_K
#> [1] 1.308824
#>
#> $n_put
#> [1] 24
#>
#> $n_call
#> [1] 10
2. Processing a whole dataset (using data.table and future.apply)
To speed up the calculations, we employ parallelised versions of the
apply family. However, you can still use the standard ones.
library(future.apply)
#> Lade nötiges Paket: future
plan(multisession, workers = 4) ## Parallelize using four cores
Determine the expiration terms for the interpolation. 1 indicates the
near-term, 2 the next-term option. We only keep those observations which
are relevant for the weekly and monthly VIX calculation.
## Determine maturity
option_dataset[, maturity := time_length((exp + hours(16) - t),unit = "years")]
## Weekly and monthly expiration terms
option_dataset[, `:=`(term_wk = future_sapply(maturity, CBOE_interpolation_terms,
method = "weekly",
future.seed = 1337),
term_mn = future_mapply(CBOE_interpolation_terms,
maturity, as_date(t), as_date(exp),
MoreArgs = list(method = "monthly"),
future.seed = 1337)
)]
## Select only options needed for the weekly and monthly VIX
option_dataset <- option_dataset[!is.na(term_wk)
| !is.na(term_mn)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn
#> 1: 0.06644802 1 NA
#> 2: 0.08561297 2 NA
#> 3: 0.10477793 NA 1
#> 4: 0.18143775 NA 2
#> 5: 0.06644612 1 NA
#> ---
#> 3116: 0.18070005 NA 2
#> 3117: 0.06570842 1 NA
#> 3118: 0.08487337 2 NA
#> 3119: 0.10403833 NA 1
#> 3120: 0.18069815 NA 2
Calculate the Implied Variances for a complete dataset
## Calculate risk-free-rate and maturity
option_dataset[, `:=`(R = interpolate_rfr(date = as_date(t),
exp = exp))]
## Calculate CBOE MFIV for the whole dataset
option_dataset[, sigma_sq := future_mapply(CBOE_VIX_vars, option_quotes, R, maturity, future.seed = 1337)
]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq
#> 1: 0.06644802 1 NA 0.008769736 0.05416683
#> 2: 0.08561297 2 NA 0.008911253 0.05222205
#> 3: 0.10477793 NA 1 0.009049549 0.05213150
#> 4: 0.18143775 NA 2 0.009571069 0.06150820
#> 5: 0.06644612 1 NA 0.008769736 0.05362052
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823
Optionally, if you want to keep the intermediate variables:
## This function converts the mapply results below to a data.table
multicols <- function(matrix){
data.table::as.data.table(t(matrix))[, lapply(.SD, unlist)]
}
## Calculate VIX for the whole dataset, including intermediate variables
option_dataset[, c("F_0", "K_0", "n_put_raw", "n_call_raw", "n_put", "n_call", "sigma_sq") :=
multicols(future_mapply(CBOE_VIX_vars,
option_quotes, R, maturity,
MoreArgs = list(ret_vars = T),
future.seed = 1337)
)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call
#> 1: 10 24 10
#> 2: 14 15 14
#> 3: 9 8 9
#> 4: 11 9 11
#> 5: 10 24 10
#> ---
#> 3116: 38 52 38
#> 3117: 52 43 52
#> 3118: 34 25 34
#> 3119: 22 47 22
#> 3120: 38 52 38
Calculate the VIX indices
## Weekly VIX
weekly <- option_dataset[!is.na(term_wk)][, .(VIX_wk = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq)),
by = .(ticker, t)]
## Monthly VIX
monthly <- option_dataset[!is.na(term_mn)][, .(VIX_mn = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq)),
by = .(ticker, t)]
VIX_data <- weekly[monthly, on = .(ticker, t)]
VIX_data
#> ticker t VIX_wk VIX_mn
#> 1: AAAA 2017-06-13 09:31:00 22.91347 21.45530
#> 2: AAAA 2017-06-13 09:32:00 22.84365 21.17519
#> 3: AAAA 2017-06-13 09:33:00 22.78820 21.13499
#> 4: AAAA 2017-06-13 09:34:00 22.75515 21.11951
#> 5: AAAA 2017-06-13 09:35:00 22.75988 21.05913
#> ---
#> 776: BBBB 2017-06-13 15:56:00 21.46799 18.35661
#> 777: BBBB 2017-06-13 15:57:00 21.45854 18.36722
#> 778: BBBB 2017-06-13 15:58:00 21.44498 18.38045
#> 779: BBBB 2017-06-13 15:59:00 21.50976 18.45417
#> 780: BBBB 2017-06-13 16:00:00 21.60340 18.07499
3. Visualise the VIX-Index
Display the data
plot_VIX(VIX_data)
You can use result_browser(VIX_data) to display an interactive Shiny
App that allows to browse through the results
4. Analyse the Option-Data Quality
Calculate descriptive variables for the option-data quality
## Calculate the option descriptives using the `multicols()` function from above
option_dataset[, c("SD", "max_K", "min_K", "mean_delta_K", "n_put", "n_call") :=
multicols(future_mapply(option_descriptives,
option_quotes, K_0, R, price, maturity)
)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call SD max_K min_K mean_delta_K
#> 1: 10 24 10 0.2218317 167.5 123 1.308824
#> 2: 14 15 14 0.2189827 177.5 115 2.155172
#> 3: 9 8 9 0.2196033 190.0 100 5.294118
#> 4: 11 9 11 0.2395117 200.0 100 5.000000
#> 5: 10 24 10 0.2208317 167.5 123 1.308824
#> ---
#> 3116: 38 52 38 0.2695806 1400.0 720 7.555556
#> 3117: 52 43 52 0.2037625 1140.0 790 3.684211
#> 3118: 34 25 34 0.2050042 1130.0 800 5.593220
#> 3119: 22 50 22 0.2152011 1200.0 650 7.638889
#> 3120: 38 52 38 0.2670286 1400.0 720 7.555556
5. Improve Option-Data Quality using Smoothing Methods
## Use the Jiang & Tian (2007) smoothing method to fill in and extrapolate the option quotes.
option_dataset[, option_quotes_smooth := future_mapply(JandT_2007_smoothing_method,
option_quotes, K_0, price, R, maturity, F_0,
SIMPLIFY = F)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call SD max_K min_K mean_delta_K
#> 1: 10 24 10 0.2218317 167.5 123 1.308824
#> 2: 14 15 14 0.2189827 177.5 115 2.155172
#> 3: 9 8 9 0.2196033 190.0 100 5.294118
#> 4: 11 9 11 0.2395117 200.0 100 5.000000
#> 5: 10 24 10 0.2208317 167.5 123 1.308824
#> ---
#> 3116: 38 52 38 0.2695806 1400.0 720 7.555556
#> 3117: 52 43 52 0.2037625 1140.0 790 3.684211
#> 3118: 34 25 34 0.2050042 1130.0 800 5.593220
#> 3119: 22 50 22 0.2152011 1200.0 650 7.638889
#> 3120: 38 52 38 0.2670286 1400.0 720 7.555556
#> option_quotes_smooth
#> 1: <data.table[244x2]>
#> 2: <data.table[268x2]>
#> 3: <data.table[99x2]>
#> 4: <data.table[148x2]>
#> 5: <data.table[242x2]>
#> ---
#> 3116: <data.table[511x2]>
#> 3117: <data.table[595x2]>
#> 3118: <data.table[687x2]>
#> 3119: <data.table[383x2]>
#> 3120: <data.table[512x2]>
Let’s look at one nest of smoothed option_quotes
## Calculate the option descriptives using the `multicols()` function from above
smooth_quotes <- option_dataset$option_quotes_smooth[[1]]
min(smooth_quotes$K)
#> [1] 26
max(smooth_quotes$K)
#> [1] 269
length(smooth_quotes$K)
#> [1] 244
## Average spacing in price units
mean(smooth_quotes$K - data.table::shift(smooth_quotes$K), na.rm = T)
#> [1] 1
## Spacing in SD units
3 / (option_dataset$SD[[1]] * option_dataset$price[[1]] * sqrt(option_dataset$maturity[[1]]))
#> [1] 0.3559497
Now calculate the MFIV using the Jiang & Tian (2007) method:
option_dataset[, sigma_sq_smooth := future_mapply(JandT_2007_sigma_sq,
option_quotes_smooth, K_0, maturity, R)]
Calculate the respective VIX indices
## Weekly VIX
weekly_smooth <- option_dataset[!is.na(term_wk)][, .(VIX_wk_smooth = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq_smooth)),
by = .(ticker, t)]
## Monthly VIX
monthly_smooth <- option_dataset[!is.na(term_mn)][, .(VIX_mn_smooth = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq_smooth)),
by = .(ticker, t)]
VIX_data_smooth <- weekly_smooth[monthly_smooth, on = .(ticker, t)]
VIX_data_2 <- VIX_data[VIX_data_smooth, on = .(ticker, t)]
plot_VIX(data = VIX_data_2,
VIX_vars = c("VIX_wk", "VIX_mn",
"VIX_wk_smooth", "VIX_mn_smooth"))
m-g-h/R.MFIV documentation built on July 4, 2022, 3:35 a.m.
R Package Documentation
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R.MFIV
R.MFIV can be installed via:
remotes::install_github("m-g-h/R.MFIV")
Package Functionality
The VIX is defined as:
where the term in brackets under the square root is a linear interpolation of two Implied Variances, which are given by the second formula. The CBOE approach of the VIX calculation is explained in the CBOE VIX Whitepaper.
In order to calculate the VIX and its individual variables, this package provides the following functionality as explained in the following sections
1. Short Walkthrough
- Calculate the Risk-Free-Rate R
interpolate_rfr() - Calculate the At-The-Money Forward Price F0
CBOE_F_O() - Calculate the At-The-Money Strike Price K0
CBOE_K_0() - Select the Option Quotes Q(Ki) according to the CBOE
Rule
CBOE_option_selection() - Calculate the Implied Variance σ2
CBOE_sigma_sq() - Do all of the above in one function
CBOE_VIX_variables() - Interpolate the VIX using two Implied Variances
CBOE_VIX_index() - Calculate Option Descriptives
option_descriptives()
2. Processing a whole dataset (using data.table)
- Automatically determine the Expiration Terms (“near-” and
“next-term”)
CBOE_interpolation_terms() - Create the implied Variance (and VIX variables) for a whole dataset
- Interpolate the weekly or monthly VIX index
3. Visualise the VIX-Index
- As single plot
plot_VIX() - Browse through plots of multiple tickers with an interactive Shiny
App
result_browser()
4. Analyse the Option-Data Quality
- Calculate Option Descriptives for the whole
dataset
option_descriptives() - Visualise the option data quality [to be done]
5. Improve Option-Data Quality using Smoothing Methods
- Intra- and Extrapolate option quotes
JandT_2007_smoothing_method() - Calculate the Jiang & Tian MFIV
JandT_2007_sigma_sq()
1. Short Walkthrough
Load exemplary package data
library(R.MFIV)
data(option_dataset)
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-06-16 147.390 <data.table[96x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 5: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> ---
#> 3117: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3121: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
We select an exemplary entry for the calculation. This observation will
be our “near-term” contract. Also note that option_quotes is a
“nested” data.table itself
t <- option_dataset$t[2]
exp <- option_dataset$exp[2]
option_quotes <- option_dataset$option_quotes[[2]]
option_quotes
#> K c p
#> 1: 105.0 42.550 NA
#> 2: 110.0 37.550 NA
#> 3: 115.0 32.525 NA
#> 4: 120.0 27.600 NA
#> 5: 123.0 24.600 0.105
#> 6: 124.0 23.600 0.115
#> 7: 125.0 22.675 0.135
#> 8: 126.0 21.650 0.155
#> 9: 127.0 20.725 0.170
#> 10: 128.0 19.725 0.195
#> 11: 129.0 18.725 0.220
#> 12: 130.0 17.825 0.245
#> 13: 131.0 16.775 0.285
#> 14: 132.0 15.825 0.325
#> 15: 133.0 14.875 0.370
#> 16: 134.0 13.925 0.425
#> 17: 135.0 12.975 0.485
#> 18: 136.0 11.975 0.560
#> 19: 137.0 11.150 0.640
#> 20: 138.0 10.325 0.750
#> 21: 139.0 9.450 0.870
#> 22: 140.0 8.550 1.020
#> 23: 141.0 7.625 1.185
#> 24: 142.0 6.825 1.390
#> 25: 143.0 6.150 1.630
#> 26: 144.0 5.425 1.895
#> 27: 145.0 4.675 2.235
#> 28: 146.0 4.125 2.590
#> 29: 147.0 3.525 3.000
#> 30: 148.0 3.045 3.475
#> 31: 149.0 2.545 4.000
#> 32: 150.0 2.130 4.575
#> 33: 152.5 1.280 6.375
#> 34: 155.0 0.745 8.275
#> 35: 157.5 0.430 10.450
#> 36: 160.0 0.265 12.800
#> 37: 162.5 0.165 15.175
#> 38: 165.0 0.115 17.650
#> 39: 167.5 0.085 20.100
#> 40: 170.0 NA 22.600
#> 41: 172.5 NA 25.075
#> 42: 175.0 NA 27.575
#> 43: 177.5 NA 30.075
#> 44: 180.0 NA 32.575
#> 45: 182.5 NA 35.075
#> 46: 185.0 NA 37.575
#> 47: 187.5 NA 40.075
#> 48: 190.0 NA 42.575
#> 49: 195.0 NA 47.575
#> 50: 200.0 NA 52.575
#> 51: 210.0 NA 62.575
#> K c p
Next we calculate the Risk-Free-Rate R
library(lubridate)
#>
#> Attache Paket: 'lubridate'
#> Die folgenden Objekte sind maskiert von 'package:base':
#>
#> date, intersect, setdiff, union
R <- interpolate_rfr(date = as_date(t),
exp = exp,
ret_table = F)
R
#> [1] 0.008769736
Calculate At-The-Money Forward Price F0
## Set expiration time to 4 PM
exp <- exp + hours(16)
## Calculate maturity in years
maturity <- time_length(exp-t,
unit = "years")
## Calculate ATM Forward
F_0 <- CBOE_F_0(option_quotes = option_quotes,
R = R,
maturity = maturity)
F_0
#> [1] 147.5697
Calculate At-The-Money Strike price K0
K_0 <- CBOE_K_0(option_quotes = option_quotes,
F_0 = F_0)
K_0
#> [1] 147
Select option quotes per CBOE rule
option_quotes <- CBOE_option_selection(option_quotes = option_quotes,
K_0 = K_0)
option_quotes
#> K c p
#> 1: 123.0 24.600 0.105
#> 2: 124.0 23.600 0.115
#> 3: 125.0 22.675 0.135
#> 4: 126.0 21.650 0.155
#> 5: 127.0 20.725 0.170
#> 6: 128.0 19.725 0.195
#> 7: 129.0 18.725 0.220
#> 8: 130.0 17.825 0.245
#> 9: 131.0 16.775 0.285
#> 10: 132.0 15.825 0.325
#> 11: 133.0 14.875 0.370
#> 12: 134.0 13.925 0.425
#> 13: 135.0 12.975 0.485
#> 14: 136.0 11.975 0.560
#> 15: 137.0 11.150 0.640
#> 16: 138.0 10.325 0.750
#> 17: 139.0 9.450 0.870
#> 18: 140.0 8.550 1.020
#> 19: 141.0 7.625 1.185
#> 20: 142.0 6.825 1.390
#> 21: 143.0 6.150 1.630
#> 22: 144.0 5.425 1.895
#> 23: 145.0 4.675 2.235
#> 24: 146.0 4.125 2.590
#> 25: 147.0 3.525 3.000
#> 26: 148.0 3.045 3.475
#> 27: 149.0 2.545 4.000
#> 28: 150.0 2.130 4.575
#> 29: 152.5 1.280 6.375
#> 30: 155.0 0.745 8.275
#> 31: 157.5 0.430 10.450
#> 32: 160.0 0.265 12.800
#> 33: 162.5 0.165 15.175
#> 34: 165.0 0.115 17.650
#> 35: 167.5 0.085 20.100
#> K c p
Calculate the Implied Variance σ2
sigma_sq <- CBOE_sigma_sq(sel_option_quotes = option_quotes,
K_0 = K_0,
F_0 = F_0,
maturity = maturity,
R = R)
sigma_sq
#> [1] 0.05416683
Do the same for a second maturity, this time all at once. This observation will be our “next-term” contract.
## Select observation 5
t2 <- option_dataset$t[3]
exp2 <- option_dataset$exp[3]
option_quotes2 <- option_dataset$option_quotes[[3]]
## Risk free rate
R2 <- interpolate_rfr(date = as_date(t2),
exp = exp2,
ret_table = F)
## Set expiration time to 4 PM
exp2 <- exp2 + hours(16)
## Calculate maturity in years
maturity2 <- time_length(exp2-t2,unit = "years")
## Calculate all VIX vars at once
VIX_vars <- CBOE_VIX_vars(option_quotes = option_quotes2,
R = R2,
maturity = maturity2,
ret_vars = T)
VIX_vars
#> $F_0
#> [1] 147.5497
#>
#> $K_0
#> [1] 147
#>
#> $n_put_raw
#> [1] 15
#>
#> $n_call_raw
#> [1] 14
#>
#> $n_put
#> [1] 15
#>
#> $n_call
#> [1] 14
#>
#> $sigma_sq
#> [1] 0.05222205
Calculate and interpolate the VIX index from both maturities
CBOE_VIX_index(maturity = c(maturity, maturity2),
sigma_sq = c(sigma_sq,VIX_vars$sigma_sq))
#> [1] 22.91347
Optionally calculate Option Descriptives, i.e. the strike price range and spacing in standard-deviation (SD) units of the used option quotes:
price <- option_dataset$price[2]
option_descriptives(option_quotes = option_quotes,
K_0 = K_0,
R = R,
price = price,
maturity = maturity)
#> $SD
#> [1] 0.2218317
#>
#> $max_K
#> [1] 167.5
#>
#> $min_K
#> [1] 123
#>
#> $mean_delta_K
#> [1] 1.308824
#>
#> $n_put
#> [1] 24
#>
#> $n_call
#> [1] 10
2. Processing a whole dataset (using data.table and future.apply)
To speed up the calculations, we employ parallelised versions of the
apply family. However, you can still use the standard ones.
library(future.apply)
#> Lade nötiges Paket: future
plan(multisession, workers = 4) ## Parallelize using four cores
Determine the expiration terms for the interpolation. 1 indicates the near-term, 2 the next-term option. We only keep those observations which are relevant for the weekly and monthly VIX calculation.
## Determine maturity
option_dataset[, maturity := time_length((exp + hours(16) - t),unit = "years")]
## Weekly and monthly expiration terms
option_dataset[, `:=`(term_wk = future_sapply(maturity, CBOE_interpolation_terms,
method = "weekly",
future.seed = 1337),
term_mn = future_mapply(CBOE_interpolation_terms,
maturity, as_date(t), as_date(exp),
MoreArgs = list(method = "monthly"),
future.seed = 1337)
)]
## Select only options needed for the weekly and monthly VIX
option_dataset <- option_dataset[!is.na(term_wk)
| !is.na(term_mn)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn
#> 1: 0.06644802 1 NA
#> 2: 0.08561297 2 NA
#> 3: 0.10477793 NA 1
#> 4: 0.18143775 NA 2
#> 5: 0.06644612 1 NA
#> ---
#> 3116: 0.18070005 NA 2
#> 3117: 0.06570842 1 NA
#> 3118: 0.08487337 2 NA
#> 3119: 0.10403833 NA 1
#> 3120: 0.18069815 NA 2
Calculate the Implied Variances for a complete dataset
## Calculate risk-free-rate and maturity
option_dataset[, `:=`(R = interpolate_rfr(date = as_date(t),
exp = exp))]
## Calculate CBOE MFIV for the whole dataset
option_dataset[, sigma_sq := future_mapply(CBOE_VIX_vars, option_quotes, R, maturity, future.seed = 1337)
]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq
#> 1: 0.06644802 1 NA 0.008769736 0.05416683
#> 2: 0.08561297 2 NA 0.008911253 0.05222205
#> 3: 0.10477793 NA 1 0.009049549 0.05213150
#> 4: 0.18143775 NA 2 0.009571069 0.06150820
#> 5: 0.06644612 1 NA 0.008769736 0.05362052
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823
Optionally, if you want to keep the intermediate variables:
## This function converts the mapply results below to a data.table
multicols <- function(matrix){
data.table::as.data.table(t(matrix))[, lapply(.SD, unlist)]
}
## Calculate VIX for the whole dataset, including intermediate variables
option_dataset[, c("F_0", "K_0", "n_put_raw", "n_call_raw", "n_put", "n_call", "sigma_sq") :=
multicols(future_mapply(CBOE_VIX_vars,
option_quotes, R, maturity,
MoreArgs = list(ret_vars = T),
future.seed = 1337)
)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call
#> 1: 10 24 10
#> 2: 14 15 14
#> 3: 9 8 9
#> 4: 11 9 11
#> 5: 10 24 10
#> ---
#> 3116: 38 52 38
#> 3117: 52 43 52
#> 3118: 34 25 34
#> 3119: 22 47 22
#> 3120: 38 52 38
Calculate the VIX indices
## Weekly VIX
weekly <- option_dataset[!is.na(term_wk)][, .(VIX_wk = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq)),
by = .(ticker, t)]
## Monthly VIX
monthly <- option_dataset[!is.na(term_mn)][, .(VIX_mn = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq)),
by = .(ticker, t)]
VIX_data <- weekly[monthly, on = .(ticker, t)]
VIX_data
#> ticker t VIX_wk VIX_mn
#> 1: AAAA 2017-06-13 09:31:00 22.91347 21.45530
#> 2: AAAA 2017-06-13 09:32:00 22.84365 21.17519
#> 3: AAAA 2017-06-13 09:33:00 22.78820 21.13499
#> 4: AAAA 2017-06-13 09:34:00 22.75515 21.11951
#> 5: AAAA 2017-06-13 09:35:00 22.75988 21.05913
#> ---
#> 776: BBBB 2017-06-13 15:56:00 21.46799 18.35661
#> 777: BBBB 2017-06-13 15:57:00 21.45854 18.36722
#> 778: BBBB 2017-06-13 15:58:00 21.44498 18.38045
#> 779: BBBB 2017-06-13 15:59:00 21.50976 18.45417
#> 780: BBBB 2017-06-13 16:00:00 21.60340 18.07499
3. Visualise the VIX-Index
Display the data
plot_VIX(VIX_data)
You can use result_browser(VIX_data) to display an interactive Shiny
App that allows to browse through the results
4. Analyse the Option-Data Quality
Calculate descriptive variables for the option-data quality
## Calculate the option descriptives using the `multicols()` function from above
option_dataset[, c("SD", "max_K", "min_K", "mean_delta_K", "n_put", "n_call") :=
multicols(future_mapply(option_descriptives,
option_quotes, K_0, R, price, maturity)
)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call SD max_K min_K mean_delta_K
#> 1: 10 24 10 0.2218317 167.5 123 1.308824
#> 2: 14 15 14 0.2189827 177.5 115 2.155172
#> 3: 9 8 9 0.2196033 190.0 100 5.294118
#> 4: 11 9 11 0.2395117 200.0 100 5.000000
#> 5: 10 24 10 0.2208317 167.5 123 1.308824
#> ---
#> 3116: 38 52 38 0.2695806 1400.0 720 7.555556
#> 3117: 52 43 52 0.2037625 1140.0 790 3.684211
#> 3118: 34 25 34 0.2050042 1130.0 800 5.593220
#> 3119: 22 50 22 0.2152011 1200.0 650 7.638889
#> 3120: 38 52 38 0.2670286 1400.0 720 7.555556
5. Improve Option-Data Quality using Smoothing Methods
## Use the Jiang & Tian (2007) smoothing method to fill in and extrapolate the option quotes.
option_dataset[, option_quotes_smooth := future_mapply(JandT_2007_smoothing_method,
option_quotes, K_0, price, R, maturity, F_0,
SIMPLIFY = F)]
option_dataset
#> ticker t exp price option_quotes
#> 1: AAAA 2017-06-13 09:31:00 2017-07-07 147.390 <data.table[51x3]>
#> 2: AAAA 2017-06-13 09:31:00 2017-07-14 147.390 <data.table[40x3]>
#> 3: AAAA 2017-06-13 09:31:00 2017-07-21 147.390 <data.table[42x3]>
#> 4: AAAA 2017-06-13 09:31:00 2017-08-18 147.390 <data.table[61x3]>
#> 5: AAAA 2017-06-13 09:32:00 2017-07-07 147.130 <data.table[51x3]>
#> ---
#> 3116: BBBB 2017-06-13 15:59:00 2017-08-18 979.755 <data.table[92x3]>
#> 3117: BBBB 2017-06-13 16:00:00 2017-07-07 980.805 <data.table[99x3]>
#> 3118: BBBB 2017-06-13 16:00:00 2017-07-14 980.805 <data.table[65x3]>
#> 3119: BBBB 2017-06-13 16:00:00 2017-07-21 980.805 <data.table[121x3]>
#> 3120: BBBB 2017-06-13 16:00:00 2017-08-18 980.805 <data.table[92x3]>
#> maturity term_wk term_mn R sigma_sq F_0 K_0 n_put_raw
#> 1: 0.06644802 1 NA 0.008769736 0.05416683 147.5697 147 24
#> 2: 0.08561297 2 NA 0.008911253 0.05222205 147.5497 147 15
#> 3: 0.10477793 NA 1 0.009049549 0.05213150 147.5927 145 8
#> 4: 0.18143775 NA 2 0.009571069 0.06150820 147.4042 145 9
#> 5: 0.06644612 1 NA 0.008769736 0.05362052 147.2551 147 24
#> ---
#> 3116: 0.18070005 NA 2 0.009571069 0.07185948 982.6045 980 52
#> 3117: 0.06570842 1 NA 0.008769736 0.04598859 981.3243 980 43
#> 3118: 0.08487337 2 NA 0.008911253 0.04675662 982.0747 980 25
#> 3119: 0.10403833 NA 1 0.009049549 0.04784626 982.4523 980 50
#> 3120: 0.18069815 NA 2 0.009571069 0.07206823 983.2470 980 52
#> n_call_raw n_put n_call SD max_K min_K mean_delta_K
#> 1: 10 24 10 0.2218317 167.5 123 1.308824
#> 2: 14 15 14 0.2189827 177.5 115 2.155172
#> 3: 9 8 9 0.2196033 190.0 100 5.294118
#> 4: 11 9 11 0.2395117 200.0 100 5.000000
#> 5: 10 24 10 0.2208317 167.5 123 1.308824
#> ---
#> 3116: 38 52 38 0.2695806 1400.0 720 7.555556
#> 3117: 52 43 52 0.2037625 1140.0 790 3.684211
#> 3118: 34 25 34 0.2050042 1130.0 800 5.593220
#> 3119: 22 50 22 0.2152011 1200.0 650 7.638889
#> 3120: 38 52 38 0.2670286 1400.0 720 7.555556
#> option_quotes_smooth
#> 1: <data.table[244x2]>
#> 2: <data.table[268x2]>
#> 3: <data.table[99x2]>
#> 4: <data.table[148x2]>
#> 5: <data.table[242x2]>
#> ---
#> 3116: <data.table[511x2]>
#> 3117: <data.table[595x2]>
#> 3118: <data.table[687x2]>
#> 3119: <data.table[383x2]>
#> 3120: <data.table[512x2]>
Let’s look at one nest of smoothed option_quotes
## Calculate the option descriptives using the `multicols()` function from above
smooth_quotes <- option_dataset$option_quotes_smooth[[1]]
min(smooth_quotes$K)
#> [1] 26
max(smooth_quotes$K)
#> [1] 269
length(smooth_quotes$K)
#> [1] 244
## Average spacing in price units
mean(smooth_quotes$K - data.table::shift(smooth_quotes$K), na.rm = T)
#> [1] 1
## Spacing in SD units
3 / (option_dataset$SD[[1]] * option_dataset$price[[1]] * sqrt(option_dataset$maturity[[1]]))
#> [1] 0.3559497
Now calculate the MFIV using the Jiang & Tian (2007) method:
option_dataset[, sigma_sq_smooth := future_mapply(JandT_2007_sigma_sq,
option_quotes_smooth, K_0, maturity, R)]
Calculate the respective VIX indices
## Weekly VIX
weekly_smooth <- option_dataset[!is.na(term_wk)][, .(VIX_wk_smooth = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq_smooth)),
by = .(ticker, t)]
## Monthly VIX
monthly_smooth <- option_dataset[!is.na(term_mn)][, .(VIX_mn_smooth = CBOE_VIX_index(maturity = maturity,
sigma_sq = sigma_sq_smooth)),
by = .(ticker, t)]
VIX_data_smooth <- weekly_smooth[monthly_smooth, on = .(ticker, t)]
VIX_data_2 <- VIX_data[VIX_data_smooth, on = .(ticker, t)]
plot_VIX(data = VIX_data_2,
VIX_vars = c("VIX_wk", "VIX_mn",
"VIX_wk_smooth", "VIX_mn_smooth"))
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