  Financial options are contracts that specify the right to buy or sell an
underlying asset at a strike price by an expiration date. Standard exchanges
offer options of predetermined strike values and trade options of different
strikes independently, even for those written on the same underlying asset.
Such independent market design can introduce arbitrage opportunities and lead
to the thin market problem. The paper first proposes a mechanism that
consolidates and matches orders on standard options related to the same
underlying asset, while providing agents the flexibility to specify any custom
strike value. The mechanism generalizes the classic double auction, runs in
time polynomial to the number of orders, and poses no risk to the exchange,
regardless of the value of the underlying asset at expiration. Empirical
analysis on real-market options data shows that the mechanism can find new
matches for options of different strike prices and reduce bid-ask spreads.
  Extending standard options written on a single asset, we propose and define a
new derivative instrument -- combinatorial financial options that offer
contract holders the right to buy or sell any linear combination of multiple
underlying assets. We generalize our single-asset mechanism to match options
written on different combinations of assets, and prove that optimal clearing of
combinatorial financial options is coNP-hard. To facilitate market operations,
we propose an algorithm that finds the exact optimal match through iterative
constraint generation, and evaluate its performance on synthetically generated
combinatorial options markets of different scales. As option prices reveal the
market's collective belief of an underlying asset's future value, a
combinatorial options market enables the expression of aggregate belief about
future correlations among assets.
