penalty_with_intensity_link function

Helper function (volatility-normalized pricing error) for calibration of equity-linked default intensity

Given a set SDE parameters, form a volatility term structure that fairly precisely matches the supplied prices of the variance_instruments. Then use that term structure and the default intensity to price all the fit_instruments, and compare them to the fit_instrument_prices.

penalty_with_intensity_link(
  p,
  s,
  h,
  variance_instruments,
  variance_instrument_prices,
  variance_instrument_spreads,
  fit_instruments,
  fit_instrument_prices,
  fit_instrument_spreads,
  fit_instrument_weights,
  S0,
  num_time_steps = 30,
  const_short_rate = 0,
  discount_factor_fcn = function(T, t) {     exp(-const_short_rate * (T - t)) },
  ...,
  relative_spread_tolerance = 0.15,
  num_variance_time_steps = 30
)

Arguments

• p: Power of default intensity

• s: Proportion of constant default intensity

• h: Base default intensity

• variance_instruments: A list of instruments in strictly increasing order of maturity, from which the volatility term structure will be inferred. Once the calibration is finished, the chosen parameters will reproduce the prices of these instruments with fairly high precision.

• variance_instrument_prices: Central price targets for the variance instruments

• variance_instrument_spreads: Bid-offer spreads used to normalize errors in variance instrument prices during term structure fitting

• fit_instruments: A list of instruments in any order, from which the mispricing penalties used for judging fit quality will be computed

• fit_instrument_prices: Central price targets for the variance instruments

• fit_instrument_spreads: Bid-offer spreads used to normalize errors in fit instrument prices during default intensity

• fit_instrument_weights: Weights applied to relative errors in fit instrument prices before summing to form the penalty

• S0: Current underlying price

• num_time_steps: Time step count passed on to find_present_value

  while fitting instrument values

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

  is not given

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

• ...: Further arguments passed to price_with_intensity_link

• relative_spread_tolerance: Tolerance to apply in calling fit_variance_cumulation

• num_variance_time_steps: Number of time steps to use in calling fit_variance_cumulation

Details

Forms implied Black-Scholes volatilities from all supplied mid prices, and their implied bid and offer prices, as well as from the prices computed by the grid solver. Each instrument is then assigned an error term component in proportion to its weight and the pricing error (in implied vol terms) divided by the spread (also in implied vol terms).

See Also

price_with_intensity_link for the pricing function