First-order event exposure from the spanning variance hump

For the two-point event mixture, the event contributes exactly \(q(1-q)\,\theta_{\log}^2\) to the total variance of the terminal log multiplier, where \(\theta_{\log}\) is the difference of the conditional log means (equal to \(\theta\) when the conditional dispersions coincide). Reading an event-spanning option's total ATM variance in excess of its no-event baseline as that contribution gives the moment-based estimator $$|\hat\theta| = \sqrt{\Delta V / (q(1-q))}.$$

Usage

ec_theta_hump(dvar, q)

Arguments

dvar

Event-induced increment in total variance of the spanning expiry (total variance of the spanning maturity minus the no-event baseline, both as \(\sigma^2 \tau\)).

q

State-price-implied probability of outcome 1.

Value

\(|\hat\theta|\); NA (with a warning) where dvar is negative.

Details

This is the quick first-pass estimator: it needs only the total-variance hump of the spanning maturity (computed in variance units, not in annualized volatilities) and the event probability. It identifies the magnitude only; take the sign from event_beta() or from the smile asymmetry via the full fit in ec_theta_fit().

See also

ec_theta_fit() for the full surface fit.

Examples

# a 10-point IV hump at 40% baseline vol, one month to expiry, q = 0.5
tau <- 1 / 12
dvar <- (0.50^2 - 0.40^2) * tau
ec_theta_hump(dvar, q = 0.5)
#> [1] 0.1732051