Prices European options on an event-exposed asset whose terminal value, normalized by its forward, is the two-component lognormal event mixture of the accompanying working paper: with probability \(q\) (outcome 1) the gross event multiplier is lognormal with conditional mean \(m_1\) and total dispersion \(s_1\), with probability \(1-q\) (outcome 2) lognormal with mean \(m_2\) and dispersion \(s_2\). The outcome-conditional means are parametrized by the event exposure $$\theta = \log(m_1 / m_2)$$ and pinned down by the pricing-measure adding-up (scale) normalization \(q\,m_1 + (1-q)\,m_2 = 1\), so that the mixture has forward value one.
Usage
ec_mix_price(k, q, theta, s1, s2, type = c("call", "put"))
ec_mix_iv(k, q, theta, s1, s2)Arguments
- k
Log-forward moneyness \(\log(K/F)\) (vector).
- q
State-price-implied probability of outcome 1.
- theta
Event exposure \(\log(m_1/m_2)\);
theta = 0collapses to a single lognormal.- s1, s2
Outcome-conditional total dispersions (standard deviation of the conditional log multiplier over the remaining life; annualized volatility times \(\sqrt{\tau}\)).
- type
"call"or"put".
Value
ec_mix_price(): numeric vector of forward-normalized prices.
ec_mix_iv(): numeric vector of total implied dispersions (NA where
the price is at an arbitrage bound).
Details
ec_mix_price() returns undiscounted forward-normalized option prices
(multiply by the discount factor and the forward to obtain currency
prices). ec_mix_iv() returns the Black total implied dispersion of
the mixture price; divide by \(\sqrt{\tau}\) (time to expiry in years)
for an annualized implied volatility.
Sign identification. At \(q = 1/2\) the mixture is observationally
symmetric in \((\theta, s_1, s_2) \mapsto (-\theta, s_2, s_1)\): option
prices alone cannot tell which outcome the asset favors. Take the sign
from an external source — e.g. the returns-based exposure of
event_beta() — when q is close to one half.
References
Hanke, Schadner, Stöckl, and Weissensteiner (Working Paper), "Learning Before Scheduled Events: Prediction Markets, State Prices, and Option Valuation".
See also
ec_theta_fit() to estimate theta from observed surfaces,
ec_theta_hump() for the first-order variance-hump estimator.