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European Call Option Price and the Inverse

Description

Computes the European call option and the inverse. All vectors with length greater than one needs to have the same length.

Usage

BS_call(V, D, T., r, vol)

get_underlying(S, D, T., r, vol, tol = 1e-12)

Arguments

Value

Numeric vector or scalar with price of the underlying asset or equity price.

See Also

[BS_fit](#topic+BS%5Ffit)

Examples

library(DtD)
set.seed(58661382)
sims <- BS_sim(
  vol = .2, mu = .03, dt = .1, V_0 = 100, T. = 1, D = rep(80, 20), r = .01)

stopifnot(with(
  sims, isTRUE(all.equal(V, get_underlying(S, D, T, r, vol)))))
stopifnot(with(
  sims, isTRUE(all.equal(S, BS_call(V, D, T, r, vol)))))

Fit Black-Scholes Parameters

Description

Function to estimate the volatility, \sigma, and drift, \mu. Seevignette("Distance-to-default", package = "DtD") for details. All vectors with length greater than one needs to have the same length. The Nelder-Mead method from [optim](../../../../doc/manuals/r-patched/packages/stats/refman/stats.html#topic+optim) is used whenmethod = "mle". Either time or dt should be passed.

Usage

BS_fit(
  S,
  D,
  T.,
  r,
  time,
  dt,
  vol_start,
  method = c("iterative", "mle"),
  tol = 1e-12,
  eps = 1e-08
)

Arguments

Value

A list with the following components

Warning

Choosing tol >= eps or roughly equal may make the method alternate between two solutions for some data sets.

Examples

library(DtD)
set.seed(83486778)
sims <- BS_sim(
  vol = .1, mu = .05, dt = .1, V_0 = 100, T. = 1, D = rep(80, 20), r = .01)

with(sims,
     BS_fit(S = S, D = D, T. = T, r = r, time = time, method = "mle"))

Fit Black-Scholes Parameters Over Rolling Window

Description

Function to estimate the volatility, \sigma, and drift, \mu. E.g., the window can be over a given number of months. Seevignette("Distance-to-default", package = "DtD") for details.

Usage

BS_fit_rolling(
  S,
  D,
  T.,
  r,
  time,
  dt,
  vol_start,
  method = c("iterative", "mle"),
  tol = 1e-12,
  eps = 1e-08,
  grp,
  width,
  min_obs
)

Arguments

Value

Matrix with the grp, number of observation in the window, parameter estimates, and 'n_iter' as in [BS_fit](#topic+BS%5Ffit), and whether the estimation method was successful.

An error attribute is added in case other code than[optim](../../../../doc/manuals/r-patched/packages/stats/refman/stats.html#topic+optim) fails. It is a list of lists with the grp index where the method failed and the output from [try](../../../../doc/manuals/r-patched/packages/base/refman/base.html#topic+try).

See Also

[BS_fit](#topic+BS%5Ffit)

Examples

# Simulate data
set.seed(55770945)
n <- 21L * 3L * 12L # 21 trading days for 3 years w/ 12 months
sims <- BS_sim(
  vol = .1, mu = .05, dt = .1, V_0 = 100, T. = 1,
  D = runif(n, 80, 90), r = runif(n, 0, .01))
sims$month <- (1:nrow(sims) - 1L) %/% 21L + 1L

# throw out some months
sims <- subset(sims, !month %in% 15:24)

# assign parameters
grp <- sims$month
width <- 12L        # window w/ 12 month width
min_obs <- 21L * 3L # require 3 months of data

# estimate results with R loop which is slightly simpler then the
# implementation
grps <- unique(grp)
out <- matrix(
  NA_real_, nrow = length(grps), ncol = 6,
  dimnames = list(NULL, c("mu", "vol", "n_iter", "success", "n_obs", "grp")))
for(g in grps){
  idx <- which(grps == g)
  keep <- which(grp %in% (g - width + 1L):g)
  out[idx, c("n_obs", "grp")] <- c(length(keep), g)
  if(length(keep) < min_obs)
    next
  res <- with(
    sims[keep, ],
    BS_fit(S = S, D = D, T. = T, r = r, time = time, method = "iterative",
           vol_start = 1))
  out[idx, c("mu", "vol", "n_iter", "success")] <- rep(
    do.call(c, res[c("ests", "n_iter", "success")]), each = length(idx))
}

# we get the same with the R function
out_func <- with(sims, BS_fit_rolling(
  S = S, D = D, T. = T, r = r, time = time, method = "iterative",
  grp = month, width = width, min_obs = min_obs))

all.equal(out[, names(out) != "n_iter"],
          out_func[, names(out_func) != "n_iter"])

Simulate Stock Price and Price of Underlying Asset

Description

At least one of D, r, or T. needs to have the desired length of the simulated series. All vectors with length greater than one needs to have the same length.

Usage

BS_sim(vol, mu, dt, V_0, D, r, T.)

Arguments

See Also

[BS_fit](#topic+BS%5Ffit)

Examples

library(DtD)
set.seed(79156879)
sims <- BS_sim(
  vol = .1, mu = .05, dt = .2, V_0 = 100, T. = 1, D = rep(80, 20), r = .01)

# plot underlying
plot(sims$V)

# plot stock
plot(sims$S)

Compute Log-Likelihood of Merton Model

Description

Computes the log-likelihood for a given values of \mu and\sigma.

Usage

merton_ll(S, D, T., r, time, dt, vol, mu, tol = 1e-12)

Arguments

See Also

[BS_fit](#topic+BS%5Ffit)

Examples

# we get the same if we call `optim` as follows. The former is faster and is
# recommended
set.seed(4648394)
sims <- BS_sim(
  vol = .1, mu = .05, dt = .1, V_0 = 100, T. = 1, D = rep(80, 20), r = .01)

r1 <- with(
  sims, BS_fit(S = S, D = D, T. = T, r = r, time = time, method = "mle",
               eps = 1e-8, vol_start = .2))

r2 <- optim(c(mu = 0, log_vol = log(.2)), function(par)
  -with(
    sims, merton_ll(S = S, D = D, T. = T, r = r, time = time,
                    mu = par["mu"], vol = exp(par["log_vol"]))))

all.equal(r1$n_iter, unname(r2$counts[1]))
all.equal(r1$ests[1], r2$par[1])
all.equal(r1$ests[2], exp(r2$par[2]), check.attributes = FALSE)

# the log-likelihood integrates to one as it should though likely not the
# most stable way to test this
ll <- integrate(
  function(x) sapply(x, function(S)
    exp(merton_ll(
      S = c(1, S), D = .8, T. = 3, r = .01, dt = 1/250, vol = .2,
      mu = .05))),
  lower = 1e-4, upper = 6)
stopifnot(isTRUE(all.equal(ll$value, 1, tolerance = 1e-5)))