Convert traded state prices into risk-adjusted event probabilities

Traded event claims (prediction-market contracts, betting quotes, binary options) pay one unit at resolution. Their price is therefore a discounted probability: \(\psi_t = D(t,\tau)\, q_t\), where \(D(t,\tau)\) is the discount factor of the payout numeraire. This function undoes that discounting, or alternatively removes the bookmaker overround by normalizing with the sum of all outcome prices in the book.

Usage

q_from_price(
  price,
  discount = 1,
  book = NULL,
  method = c("discount", "overround")
)

Arguments

price: Numeric vector of raw claim prices \(\psi_t\) (on the 0-1 scale; rescale percent quotes first).
discount: Numeric scalar or vector, the discount factor \(D(t,\tau) \in (0, 1]\) applicable to the payout (default 1, i.e. no discounting).
book: Numeric scalar or vector, the sum of prices over all outcomes of the same market. Required for method = "overround".
method: Character, "discount" or "overround".

Value

A numeric vector of probabilities q. Values outside \([0,1]\) trigger a warning (they indicate a wrong discount/book), but are returned unchanged so that the caller can flag rather than drop them.

Details

Two normalizations are available:

method = "discount" (default): \(q = \psi / D\). Under deterministic interest rates the result is the risk-adjusted (risk-neutral) event probability; with stochastic rates it is a forward-measure probability. At short horizons the correction is a few basis points but it is systematic.
method = "overround": \(q_j = \psi_j / \sum_i \psi_i\), where book supplies \(\sum_i \psi_i\). Use this only when the raw quotes include a bookmaker margin that has not already been removed by the data provider. Note that series which are already margin-adjusted must not be normalized twice.

A level distortion that is (approximately) constant on the log-odds scale — a constant discount factor, a constant state-price tilt — shifts \(L_t = \mathrm{logit}(q_t)\) by a constant and therefore leaves the event clock unchanged: quadratic variation is invariant to level shifts. Getting the level of q exactly right matters for probability statements, not for event_clock().

Examples