Estimate the event exposure theta from event-spanning option surfaces

Fits the lognormal event mixture of ec_mix_price() to a panel of event-spanning option surfaces: each surface (one valuation date and one expiry) gets its own outcome-conditional dispersions \((s_1, s_2)\), while the event exposure \(\theta = \log(m_1/m_2)\) is restricted to be common across all surfaces — the working paper's common conditional-mean-ratio design. The objective is the working paper's: price residuals divided by the Black vega at the observed node (the implied-volatility error to first order), averaged with node weights within each surface and surface-balanced across surfaces, so each surface carries equal weight regardless of how many quotes it has.

Usage

ec_theta_fit(surfaces, theta_range = c(-3, 3), common = TRUE, min_nodes = 4)

# S3 method for class 'ec_theta_fit'
print(x, ...)

Arguments

surfaces

A data.frame with one row per observed node and columns:

surface: surface identifier (valuation date \(\times\) expiry); anything coercible to a factor.
k: log-forward moneyness \(\log(K/F)\) (see ec_delta_to_k() for delta grids).
iv: observed implied volatility. Annualized if tau is supplied, otherwise interpreted as total dispersion.
q: state-price-implied probability of outcome 1 at the surface's valuation date (constant within surface).
tau: optional: time to expiry in years; converts between annualized and total units.
w: optional: node weight (e.g. quote count or inverse spread); defaults to 1.

theta_range

Search interval for the common exposure.

common

If FALSE, every surface gets its own \(\theta\) instead (the working paper's free-ratio diagnostic); the common estimate is then NA and the per-surface exposures are reported in the surfaces table.

min_nodes

Surfaces with fewer observed nodes are dropped with a warning (two free dispersions need cross-sectional information).

x, ...

Print method arguments.

Value

An object of class ec_theta_fit: a list with

theta, ratio: the common event exposure and \(m_1/m_2 = e^{\theta}\).
rmse: surface-balanced root mean squared vega-scaled pricing error — implied-volatility units to first order, in the units of the iv input (annualized if tau was supplied).
surfaces: per-surface tibble: q, nodes, fitted s1, s2 (total dispersions), per-surface rmse, and theta when common = FALSE.
n_surfaces, n_obs, common: bookkeeping.

Details

Sign at q near one half. See ec_mix_price(): with symmetric q the sign of \(\theta\) is not identified from option prices alone and the optimizer returns one of the two equivalent solutions. Fix the sign externally (e.g. event_beta()).

Methods (by generic)

print(ec_theta_fit): Print method.

References

Hanke, Schadner, Stöckl, and Weissensteiner (Working Paper), "Learning Before Scheduled Events: Prediction Markets, State Prices, and Option Valuation".

Examples

# recover a known exposure from two noiseless simulated surfaces
k <- seq(-0.5, 0.5, by = 0.25)
surf <- rbind(
  data.frame(surface = "d1", k = k, q = 0.60,
             iv = ec_mix_iv(k, 0.60, 0.8, 0.30, 0.35)),
  data.frame(surface = "d2", k = k, q = 0.65,
             iv = ec_mix_iv(k, 0.65, 0.8, 0.28, 0.32))
)
ec_theta_fit(surf)
#> -- Event-spanning mixture fit (2 surfaces, 10 nodes)
#> Common event exposure theta = 0.8000 (conditional-mean ratio m1/m2 = 2.226)
#> Surface-balanced RMSE = 0.0000 (IV units of the input)