README

ragtop prices equity derivatives using variants of the famous Black-Scholes model, with special attention paid to the case of American and European exercise options and to convertible bonds.

Installation

install.packages("ragtop")

# install.packages("pak")
pak::pak("brianboonstra/ragtop")

Usage

Basic Usage

You can price american and european exercise options, either individually, or in groups. In the simplest case that looks like this for European exercise

blackscholes(c(CALL, PUT), S0 = 100, K = c(100, 110), time = 0.77, r = 0.06, vola = 0.20)
#> $Price
#> [1] 9.326839 9.963285
#> 
#> $Delta
#> [1]  0.6372053 -0.5761608
#> 
#> $Vega
#> [1] 32.91568 34.36717

american(PUT, S0 = 100, K = c(100, 110), time = 0.77, const_short_rate = 0.06, const_volatility = 0.20)
#> A100_281_0 A110_281_0 
#>    5.24386   11.27715

Including Term Structures

There are zillions of implementations of the Black-Scholes formula out there, and quite a few simple trees as well. One thing that makes ragtop a bit more useful than most other packages is that it treats dividends and term structures without too much pain. Assume we have some nontrivial term structures and dividends

## Dividends
divs = data.frame(time = seq(from = 0.11, to = 2, by = 0.25),
                  fixed = seq(1.5, 1, length.out = 8),
                  proportional = seq(1, 1.5, length.out = 8))

## Interest rates
disct_fcn = ragtop::spot_to_df_fcn(data.frame(time = c(1, 5, 10),
                                              rate = c(0.01, 0.02, 0.035)))

## Default intensity
disc_factor_fcn = function(T, t, ...) {
  exp(-0.03 * (T - t)) }
surv_prob_fcn = function(T, t, ...) {
  exp(-0.07 * (T - t)) }

## Variance cumulation / volatility term structure
vc = variance_cumulation_from_vols(
   data.frame(time = c(0.1, 2, 3),
              volatility = c(0.2, 0.5, 1.2)))
paste0("Cumulated variance to 18 months is ", vc(1.5, 0))
#> [1] "Cumulated variance to 18 months is 0.369473684210526"

black_scholes_on_term_structures(
   callput = TSLAMarket$options[500, 'callput'],
   S0 = TSLAMarket$S0,
   K = TSLAMarket$options[500, 'K'],
   discount_factor_fcn = disct_fcn,
   time = TSLAMarket$options[500,'time'],
   variance_cumulation_fcn = vc,
   dividends = divs)
#> $Price
#> [1] 62.55998
#> 
#> $Delta
#> [1] 0.7977684
#> 
#> $Vega
#> [1] 52.21925

american(
    callput = TSLAMarket$options[400, 'callput'],
    S0 = TSLAMarket$S0,
    K = TSLAMarket$options[400, 'K'],
    discount_factor_fcn = disct_fcn,
    time = TSLAMarket$options[400, 'time'],
    survival_probability_fcn = surv_prob_fcn,
    variance_cumulation_fcn = vc,
    dividends = divs)
#> A360_137_2 
#>   2.894296

implied_volatility_with_term_struct(
    option_price = 19, callput = PUT,
    S0 = 185.17, K = 182.50,
    discount_factor_fcn = disct_fcn,
    time = 1.12,
    survival_probability_fcn = surv_prob_fcn,
    dividends = divs)
#> [1] 0.1133976

american_implied_volatility(
    option_price = 19, callput = PUT,
    S0 = 185.17, K = 182.50,
    discount_factor_fcn = disct_fcn,
    time = 1.12,
    survival_probability_fcn = surv_prob_fcn,
    dividends = divs)
#> [1] 0.113407

Pricing Convertible Bonds

In addition to standard options, you can price more sophisticated instruments, including in particular convertible bonds

S0=241.80
varc = ragtop::variance_cumulation_from_vols(
  data.frame(
    time=c(0.13, 0.38, 0.63, 0.72, 1.72), 
    volatility=c(0.48, 0.46, 0.46, 0.45, 0.45)
    ))
disct_fcn = ragtop::spot_to_df_fcn(
  data.frame(time=c(1, 5, 10, 15),rate=c(0.01, 0.02, 0.03, 0.05))
  )

cb = ragtop::ConvertibleBond(
  maturity=2.87, conversion_ratio=2.7788, notional=1000,
  coupons=data.frame(
    payment_time=seq(2.8, 0, by=-0.25),
    payment_size=1000*0.0025/4),
  discount_factor_fcn = disct_fcn
  )
price_and_greeks = ragtop::find_greeks(
  greeks=c("delta", "vega", "credit_dv01"),
  S0=S0,
  instruments=list(CB=cb),
  num_time_steps=250,
  default_intensity_fcn = function(t, S, ...){0.03 + 0.01 * (S0/S)^1.5},
  discount_factor_fcn = disct_fcn,
  variance_cumulation_fcn=varc)
round(unlist(price_and_greeks),4)
#>       price.CB       delta.CB       gamma.CB        vega.CB credit_dv01.CB 
#>       989.2065         1.6682         0.0040       371.9968     -1717.6481

More Sophisticated Calibration

You can also find more complete calibration routines in ragtop. See the vignette or the documentation for fit_variance_cumulation and fit_to_option_market.

Technical Documentation

The source for the technical paper is in this repository. You can also find the pdf here