form_present_value_grid function

Use a model to estimate the present value of financial derivatives on a grid of initial underlying values

Use a finite difference scheme to form estimates of present values for a variety of stock prices on a grid of initial underlying prices, determined by constructing a logarithmic equivalent conforming to the grid parameters structure_constant

and structure_constant

form_present_value_grid(
  S0,
  num_time_steps,
  instruments,
  const_volatility = 0.5,
  const_short_rate = 0,
  const_default_intensity = 0,
  override_Tmax = NA,
  discount_factor_fcn = function(T, t, ...) {     exp(-const_short_rate * (T - t)) },
  default_intensity_fcn = function(t, S, ...) {     const_default_intensity + 0 * S },
  variance_cumulation_fcn = function(T, t) {     const_volatility^2 * (T - t) },
  dividends = NULL,
  borrow_cost = 0,
  dividend_rate = 0,
  structure_constant = 2,
  std_devs_width = 3,
  grid_center = NA
)

Arguments

• S0: An initial stock price, for setting grid scale

• num_time_steps: Minimum number of time steps in the grid

• instruments: A list of instruments to be priced. Each one must have a strike and a optionality_fcn, as with GridPricedInstrument and its subclasses.

• const_volatility: A constant to use for volatility in case variance_cumulation_fcn

  is not given

• const_short_rate: A constant to use for the instantaneous interest rate in case discount_factor_fcn

  is not given

• const_default_intensity: A constant to use for the instantaneous default intensity in case default_intensity_fcn

  is not given

• override_Tmax: A different maximum time on the grid to enforce

• discount_factor_fcn: A function for computing present values to time t of various cashflows occurring during this timestep, with arguments T, t

• default_intensity_fcn: A function for computing default intensity occurring during this timestep, dependent on time and stock price, with arguments t, S.

• variance_cumulation_fcn: A function for computing total stock variance occurring during this timestep, with arguments T, t. E.g. with a constant volatility $s$ this takes the form $(T-t)s^2$.

• dividends: A data.frame with columns time, fixed, and proportional. Dividend size at the given time is then expected to be equal to fixed + proportional * S / S0

• borrow_cost: Stock borrow cost, affecting the drift rate

• dividend_rate: Continuous dividend rate, affecting the drift rate

• structure_constant: The maximum ratio between time intervals dt

  and the square of space intervals dz^2

• std_devs_width: The number of standard deviations, in sigma * sqrt(T)

  units, to incorporate into the grid

• grid_center: A reasonable central value for the grid, defaults to S0 or an instrument strike

Details

If any instrument in the instruments has a strike, then the grid will be normalized to the last such instrument's strike.

See Also

Other Equity Dependent Default Intensity: find_present_value(), fit_to_option_market_df(), fit_variance_cumulation(), implied_jump_process_volatility()

Other Implicit Grid Solver: construct_implicit_grid_structure(), find_present_value(), infer_conforming_time_grid(), integrate_pde(), iterate_grid_from_timestep(), take_implicit_timestep(), timestep_instruments()