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oplib

Option Volatility and Pricing Models.

Installation

You can install the development version of oplib like so:

if (!require(devtools)) install.packages("devtools")
devtools::install_github("wilsonfreitas/oplib")

Examples

Pricing Equity Options with Black-Scholes-Merton Model

Pricing options with Black-Scholes-Merton Model (BSM Model).

library(oplib)

bsmprice(
  type = "put", spot = 20, strike = 30, time = 1:4 / 2,
  rate = 0.15, yield = 0, sigma = 0.25
)
#> [1] 7.882056 6.272936 5.098353 4.203993
bsmprice(
  type = c("call", "put"), spot = c(50, 49, 48, 47),
  strike = 45, time = 0.25, rate = 0.13, yield = 0.01, sigma = 0.2
)
#> [1] 6.5031266 0.2854164 4.7405597 0.6079959
bsmprice(
  type = "call", spot = 50, strike = 45, time = 0.25, rate = 0.13, yield = 0.01,
  sigma = seq(0.1, 0.5, 0.1)
)
#> [1] 6.316558 6.503127 7.012037 7.695351 8.464750

Works nicely with {tidyverse}.

library(tidyverse)

tibble(
  type = "call",
  spot = 100,
  strike = seq(90, 110, 5),
  time = 0.5,
  rate = 0.02,
  yield = 0,
  sigma = 0.2
) |> mutate(
  price = bsmprice(type, spot, strike, time, rate, yield, sigma),
  delta = bsmdelta(type, spot, strike, time, rate, yield, sigma),
  gamma = bsmgamma(type, spot, strike, time, rate, yield, sigma),
  rho = bsmrho(type, spot, strike, time, rate, yield, sigma),
  vega = bsmvega(type, spot, strike, time, rate, yield, sigma),
  theta = bsmtheta(type, spot, strike, time, rate, yield, sigma),
)
#> # A tibble: 5 x 13
#>   type   spot strike  time  rate yield sigma price delta  gamma   rho  vega theta
#>   <chr> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl>
#> 1 call    100     90   0.5  0.02     0   0.2 12.5  0.812 0.0190  34.4  19.0 -5.18
#> 2 call    100     95   0.5  0.02     0   0.2  8.95 0.693 0.0248  30.2  24.8 -6.18
#> 3 call    100    100   0.5  0.02     0   0.2  6.12 0.556 0.0279  24.8  27.9 -6.58
#> 4 call    100    105   0.5  0.02     0   0.2  3.99 0.419 0.0276  19.0  27.6 -6.29
#> 5 call    100    110   0.5  0.02     0   0.2  2.47 0.297 0.0245  13.6  24.5 -5.44

Pricing Equity Options with Corrado-Su Model

Corrado-Su model introduces skewness and kurtosis to option pricing formula.

tibble(
  type = "call",
  spot = seq(70, 130, 1),
  strike = 100,
  time = 0.5,
  rate = 0.02,
  yield = 0,
  sigma = 0.2
) |>
  mutate(
    bsm_price = bsmprice(type, spot, strike, time, rate, yield, sigma),
    csm_price = csmprice(type, spot, strike, time, rate, yield, sigma, -0.5, 6)
  ) |>
  tidyr::pivot_longer(
    cols = ends_with("price"),
    names_to = "model",
    values_to = "price"
  ) |>
  ggplot(aes(x = spot, y = price, color = model)) +
  geom_line()

See that Corrado-Su modelo is cheaper than Black-Scholes for ATM options and expensive for OTM and ITM, as a consequence of the kurtosis.

Options Implied Volatility

We can get option prices with {rb3} and compute their implied volatility. Let’s start getting all PETR4 optoins.

library(rb3)
library(bizdays)
library(tidyverse)

refdate <- preceding(Sys.Date() - 3, "Brazil/B3")
ch <- cotahist_get(refdate, "daily")
yc <- yc_get(refdate)
op <- cotahist_equity_options_superset(ch, yc)

symbol_ <- "PETR4"
op1 <- op |>
  filter(
    symbol.underlying == symbol_
  )

Filter the first two available maturities.

maturities <- unique(op1$maturity_date) |> sort()

op1 <- op1 |>
  filter(maturity_date %in% maturities[1:2])

Compute implied volatility and delta of the options. Note that it is necessary to:

  • compute business days biz_days to get time_to_maturity in years
  • convert interest rate (rate column) to continuous rate
op_vol <- op1 |>
  mutate(
    biz_days = bizdays(
      refdate, following(maturity_date, "Brazil/B3"), "Brazil/B3"
    ),
    time_to_maturity = biz_days / 252,
    rate = log(1 + r_252),
    impvol = bsmimpvol(
      close, type, close.underlying, strike, time_to_maturity, rate, 0
    ),
    delta = bsmdelta(
      type, close.underlying, strike, time_to_maturity, rate, 0, impvol
    )
  )

Visualize with {ggplot2}

op_vol |>
  filter(!is.na(impvol)) |>
  ggplot(aes(x = delta, y = impvol, size = volume)) +
  geom_point(alpha = 0.5) +
  facet_wrap(type ~ biz_days, scales = "free_x") +
  theme(legend.position = "bottom") +
  labs(
    x = "Delta", y = "Implied Volatility",
    title = str_glue("Equity Options Volatility - {symbol_} {format(refdate)}")
  )

Fit Corrado-Su Model to Options Prices

Use options prices to find the parameters of Corrado-Su model. The function csm_fit_min_price executes the fit by minimizing the difference between the given prices and prices returned by the formula.

Execute the fit only for the first maturity call options and see the result (fitted parameters) in the end.

params <- with(op_vol |> filter(type == "Call", maturity_date == maturities[1]), {
  csm_fit_min_price(
    par = c(0.1, 0, 3),
    type, close.underlying, strike, rate, 0, time_to_maturity, close, 1,
    control = list(trace = 3)
  )
})
#> N = 3, M = 5 machine precision = 2.22045e-16
#> This problem is unconstrained.
#> At X0, 0 variables are exactly at the bounds
#> At iterate     0  f=       3.9264  |proj g|=       18.082
#> At iterate    10  f =      0.97023  |proj g|=        1.3768
#> At iterate    20  f =      0.88965  |proj g|=      0.028646
#> 
#> iterations 25
#> function evaluations 47
#> segments explored during Cauchy searches 1
#> BFGS updates skipped 0
#> active bounds at final generalized Cauchy point 0
#> norm of the final projected gradient 1.88873e-05
#> final function value 0.889647
#> 
#> F = 0.889647
#> final  value 0.889647 
#> converged

params
#>      sigma        mu3        mu4 
#>  0.3075525 -0.0836883  4.4146539

Compute the theoretical price with Corrado-Su model.

op_csm_call <- op_vol |>
  filter(type == "Call", maturity_date == maturities[1]) |>
  mutate(
    theo_price = csmprice(
      type, close.underlying, strike, time_to_maturity, rate, 0,
      as.numeric(params[1]), as.numeric(params[2]), as.numeric(params[3])
    )
  )

Visualize the given prices and model together in a plot.

op_csm_call |>
  ggplot(aes(x = strike, y = close)) +
  geom_point(alpha = 0.75) +
  geom_line(aes(y = theo_price), colour = "red") +
  labs(
    y = "Price",
    title = str_glue("{symbol_}/{format(maturities[1])} Calls Corrado-Su Model Fit"),
    caption = str_glue("Reference Date: {format(refdate)}")
  ) +
  theme(legend.position = "none")

An interesting view is the relative error.

op_csm_call |>
  ggplot(aes(x = delta, y = (close - theo_price) / close), size = volume) +
  geom_point(alpha = 0.75) +
  labs(
    y = "Error",
    title = str_glue("{symbol_}/{format(maturities[1])} Calls Corrado-Su Model Fit Error"),
    caption = str_glue("Reference Date: {format(refdate)}")
  ) +
  scale_y_continuous(labels = scales::percent)
#> Warning: Removed 26 rows containing missing values (geom_point).

It is clear that OTM options have the greater error, mainly due to their small prices.