**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. To install the development version, use the command

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
```

and like this for American exercise

```
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
```

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"
```

then we can price vanilla options

```
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 exercise options

```
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
```

We can also find volatilities of European exercise options

```
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
```

as well as American exercise options

```
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
```

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*.

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