fit_variance_cumulation function

Fit piecewise constant volatilities to a set of equity options

Given a set of equity options with increasing tenors, along with target prices for those options, and a set of equity-lined default SDE parameters, fit a vector of piecewise constant volatilities and an associated cumulative variance function to them.

fit_variance_cumulation(
  S0,
  eq_options,
  mid_prices,
  spreads = NULL,
  initial_vols_guess = 0.55 + 0 * mid_prices,
  use_impvol = TRUE,
  relative_spread_tolerance = 0.01,
  force_same_grid = FALSE,
  num_time_steps = 40,
  const_short_rate = 0,
  const_default_intensity = 0,
  discount_factor_fcn = function(T, t, ...) {     exp(-const_short_rate * (T - t)) },
  survival_probability_fcn = function(T, t, ...) {     exp(-const_default_intensity * (T
    - t)) },
  default_intensity_fcn = function(t, S, ...) {     const_default_intensity + 0 * S },
  dividends = NULL,
  borrow_cost = 0,
  dividend_rate = 0,
  ...
)

Arguments

• S0: Current stock price

• eq_options: A list of options to find prices for. Each must have fields callput, maturity, and strike. This list must be in strictly increasing order of maturity.

• mid_prices: Prices to match

• spreads: Spreads within which any match is tolerable

• initial_vols_guess: Initial set of volatilities to try in the root finder

• use_impvol: Judge fit quality on implied vol distance rather than price distance

• relative_spread_tolerance: Tolerance multiplier on bid-ask spreads taken from vol normalization

• force_same_grid: Price all options on the same grid, rather than having smaller timestep sizes for earlier maturities

• num_time_steps: Minimum number of time steps in the grid

• 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

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

• survival_probability_fcn: A function for probability of survival, with arguments T, t and T>t. E.g. with a constant volatility $s$ this takes the form $(T-t)s^2$. This argument is only used in normalization of prices to vols for root finder tolerance, and is therefore entirely optional

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

• dividends: A data.frame with columns time, fixed, and proportional. Dividend size at the given time is

• borrow_cost: Stock borrow cost, affecting the drift rate

• dividend_rate: Continuous dividend rate, affecting the drift rate

• ...: Futher arguments to find_present_value

Returns

A list with two elements, volatilities and cumulation_function. The cumulation_function will be a 2-parameter function giving cumulated variances, as created by codevariance_cumulation_from_vols

Details

By default, the fitting happens in implied Black-Scholes volatility space for better normalization. That is to say, the fitting does pricing using the full SDE and PDE solver via find_present_value, but judges fit quality on the basis of running resulting prices through a nonlinear transformation that just happens to come from the straight Black-Scholes model.

See Also

Other Implied Volatilities: american_implied_volatility(), equivalent_bs_vola_to_jump(), equivalent_jump_vola_to_bs(), implied_jump_process_volatility(), implied_volatilities_with_rates_struct(), implied_volatilities(), implied_volatility_with_term_struct(), implied_volatility()

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