---
title: "European options and Greeks"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{European options and Greeks}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(greeks)
```

European options can be exercised only at maturity. In the Black-Scholes model,
the package computes their prices and sensitivities with closed formulas through
`BS_European_Greeks()`. The generic `Greeks()` wrapper dispatches to the same
implementation when `option_type = "European"` and `model = "Black_Scholes"`.

The examples below are compact versions of the checks used in the test suite:
they compute a few Greeks directly, check one Greek with a finite difference,
and compare exact Black-Scholes values with the Malliavin Monte Carlo estimator.

## Exact Black-Scholes values

The `greek` argument can contain more than one quantity. The result is a named
numeric vector.

```{r}
european_put <- BS_European_Greeks(
  initial_price = 120,
  exercise_price = 100,
  r = 0.02,
  time_to_maturity = 4.5,
  dividend_yield = 0.015,
  volatility = 0.22,
  payoff = "put",
  greek = c("fair_value", "delta", "gamma", "vega", "theta", "rho")
)

round(european_put, 4)
```

The same calculation can be written with the wrapper:

```{r}
round(
  Greeks(
    initial_price = 120,
    exercise_price = 100,
    r = 0.02,
    time_to_maturity = 4.5,
    dividend_yield = 0.015,
    volatility = 0.22,
    payoff = "put",
    greek = c("fair_value", "delta", "gamma")
  ),
  4
)
```

Digital payoffs are supported by the European Black-Scholes implementation. A
cash-or-nothing call pays one unit of cash at maturity if the option finishes in
the money.

```{r}
digital_call <- BS_European_Greeks(
  initial_price = 100,
  exercise_price = 105,
  r = 0.03,
  time_to_maturity = 1,
  volatility = 0.25,
  payoff = "cash_or_nothing_call",
  greek = c("fair_value", "delta", "vega")
)

round(digital_call, 4)
```

## A finite-difference check

Delta is the derivative of the option value with respect to the initial price
of the underlying asset. The tests verify this over many random inputs. For a
single option, the same idea can be seen with a central finite difference.

```{r}
base_args <- list(
  exercise_price = 100,
  r = 0.02,
  time_to_maturity = 1.5,
  dividend_yield = 0,
  volatility = 0.3,
  payoff = "call"
)

fair_value_at <- function(initial_price) {
  do.call(
    BS_European_Greeks,
    c(base_args, list(initial_price = initial_price, greek = "fair_value"))
  )
}

step_size <- 1e-4
finite_difference_delta <-
  (fair_value_at(100 + step_size) - fair_value_at(100 - step_size)) /
  (2 * step_size)

exact_delta <- do.call(
  BS_European_Greeks,
  c(base_args, list(initial_price = 100, greek = "delta"))
)

round(
  c(
    exact_delta = exact_delta,
    finite_difference_delta = finite_difference_delta,
    absolute_error = abs(exact_delta - finite_difference_delta)
  ),
  8
)
```

## Malliavin Monte Carlo comparison

`Malliavin_European_Greeks()` estimates the same Greeks by simulation. This is
less efficient than closed formulas for plain European options, but it is useful
as a bridge to the Malliavin methods used for path-dependent options.

```{r}
greeks_to_compare <- c("fair_value", "delta", "vega", "theta", "rho", "gamma")

exact <- BS_European_Greeks(
  initial_price = 110,
  exercise_price = 100,
  r = 0.02,
  time_to_maturity = 1,
  volatility = 0.25,
  payoff = "call",
  greek = greeks_to_compare
)

monte_carlo <- Malliavin_European_Greeks(
  initial_price = 110,
  exercise_price = 100,
  r = 0.02,
  time_to_maturity = 1,
  volatility = 0.25,
  payoff = "call",
  greek = greeks_to_compare,
  paths = 50000,
  seed = 42,
  antithetic = TRUE
)

round(rbind(exact = exact, malliavin_monte_carlo = monte_carlo), 4)
```

## References

The closed-form European formulas are standard Black-Scholes results; see Hull
(2022). The Malliavin estimators are described in Hudde and Rueschendorf (2023).