1. Introduction
Financial options are securities that convey rights to conduct specific trades in the future. For example, an S&P 500 call option with strike price 4500 and expiration date December 31, 2021, provides the right to buy one share of S&P 500 at on that day.^{1}^{1}1For simplicity, examples are given without the typical 100x contract multiplier. A standard financial option is a derivative instrument of a single underlying asset or index, and its payoff is a function of the underlying variable. The example option contract pays , where is the value of the S&P 500 index on the expiration date.^{2}^{2}2 Throughout this paper, we restrict to European options which can be exercised only at expiration. The settlement value is calculated as the opening value of the index on the expiration date or the last business day (usually a Friday) before the expiration date. In cashsettled markets, instead of actual physical delivery of the underlying asset, the option holder gets a cash payment that is equivalent to the value of the asset.
Investors trade options to hedge risks and achieve certain return patterns, or to speculate about the movement of the underlying asset price, buying an option when its price falls below their estimate of its expected value. Thus, option prices reveal the collective riskneutral belief distribution of the underlying asset’s future value.
Despite the significant volume of trade and interest in option contracts, financial exchanges that support trading of standard options have two limitations that compromise its expressiveness and efficiency. First, the exchange offers only a selective set of markets for options written on a specific underlying asset (i.e., option chain), in which each market features a predetermined strike price and expiration date. For example, as of this writing, the Chicago Board Options Exchange (CBOE) offers 120 distinct strike prices, ranging from 1000 to 6000 at intervals of 25, 50, or 100, for S&P 500 options expiring by the end of 2021. While traders can engineer custom contracts (e.g., a S&P 500 call option with a strike price of $4520) by simultaneously purchasing multiple available options in appropriate proportions, they must monitor several markets to ensure that a bundle can be constructed at a desired price. Often, execution risk and transaction costs prevent individual traders from carrying out such strategies. As the prescriptive markets prevent traders from conveniently expressing custom strike values, the exchange may fail to aggregate supply and demand requests of greater detail, leading to a loss of economic efficiency.
Second, each market within an option chain independently aggregates and matches orders in regard to a single contract with a specified option type, strike price, and expiration date, despite its interconnectedness to other option markets and their common dependency on the underlying asset. Such independent market design fails to notify traders of strictly better, cheaper options and generically allows arbitrage opportunities. Moreover, investments get diluted across independent markets even when participants are interested in the same underlying asset. This can lead to the problem of thin markets, where few trades happen and bidask spreads become wide. Even for some of the most actively traded option families, empirical evidence has shown that liquidity can vary much across option types and strikes. Cao and Wei studied eight years of options trading data and find consistently lower liquidity in puts and deep inthemoney options (Cao and Wei, 2010).
1.1. An Exchange for Standard Financial Options
The paper first focuses on the design of an exchange for standard financial options to address limitations discussed above. We propose a mechanism to aggregate and match orders on options across different strike prices that are logically related to the same underlying asset. As a result, the exchange operates a consolidated market for each underlying security and expiration, where traders can specify custom options of arbitrary strike value. The mechanism is an exchange that generalizes the double auction; it is not a market maker or any individual arbitrageur. It works by finding a match that maximizes net profit subject to zero loss even in the worst case, and thus poses no risk to the exchange regardless of the value of the underlying security at expiration. We show that the mechanism is computationally efficient, and key market operations (i.e., match and price quotes) can be computed in time polynomial in the number of orders. We conduct experiments on realmarket options data, using our mechanism to consolidate outstanding orders from independentlytraded options markets. Empirical results demonstrate that the proposed mechanism can match options that the current exchange cannot and provide more competitive bid and ask prices. The improved efficiency and expressiveness may help to aggregate more finegrained information and recover a complete and fully general probability distribution of the underlying asset’s future value at expiration.
1.2. An Exchange for Combinatorial Financial Options
In the second part of the work, we generalize standard financial options to define combinatorial financial options—derivatives that give contract holders the right to buy or sell any linear combination of underlying assets at any strike price. For example, a call option written on “AAPL MSFT” with strike price 300 specifies the right to buy one share of Apple and one share of Microsoft at total price on the expiration date, whereas a call on “AAPL MSFT” with strike price 50 confers the ability to buy two shares of Apple and sell one share of Microsoft at expiration for a net cost of $50. Combinatorial options allow traders to conveniently and precisely hedge their exact portfolio, replicating any payoff functions that standard options can achieve and exponentially more. Some of the standard options on mutual funds and stock indices, like the S&P 500 and Dow Jones Industrial Average (DJI), are indeed combinatorial options written on the respectively predefined portfolios. The CBOE recently launched options on eleven Select Sector Indices,^{3}^{3}3https://markets.cboe.com/tradable_products/sp_500/cboe_select_sectors_index_options/ each of which can be considered as a combinatorial option of a prespecified linear combination of stocks representing one economic sector. The goal of this work is to allow traders to create options on their own custom indices, representing any linear combination of stocks and sectors that they want.
We design an exchange for combinatorial financial options. Such a market enables the elicitation and recovery of future correlations (e.g., complimentary or substitution effects) among assets. However, as traders are offered the expressiveness to specify weights (or shares) for multiple underlying assets, new challenges arise and the thin market problem exacerbates. Opening a new, separate exchange for each new combination of stocks and weights would rapidly grow intractable. Naively matching buy and sell orders for only the exact same portfolio may yield few or no trades, despite plenty of acceptable trades among orders.
Extending the mechanism that consolidates standard options on a single underlying security, we propose an optimization formulation that can match combinatorial options written on different linear combinations of underlying assets. The mechanism maximizes net profit subject to no risk to the exchange, regardless of (any combination of) values of all assets at expiration. We show that the proposed matching mechanism with increased expressiveness, however, comes at the cost of higher computational complexity: the optimal clearing of a combinatorial options market is coNPhard.
We demonstrate that the proposed mechanism can be solved relatively efficiently by exploiting constraint generation techniques. We propose an algorithm that finds the exact optimalmatching solution by satisfying an increasing set of constraints iteratively generated from different future values of the underlying assets. In experiments on synthetic combinatorial orders generated from realmarket standard options prices, we show that the matching algorithm terminates quickly, with its running time growing linearly with the number of orders and the size of underlying assets.
1.3. Roadmap
Section 2 discusses related work on option pricing and combinatorial markets. Section 3 introduces notations and background on the current options market. We first focus on the market design for standard financial options. Section 4 introduces the formal setting of a consolidated options market with limit orders, and proposes a matching mechanism for options across types and strikes (Mechanism M.1). Section 5, our main contribution, defines a combinatorial financial options exchange (Mechanism M.2), analyzes the computational complexity of optimally clearing such a market (Theorem 4 and Theorem 5), and proposes a constraint generation algorithm to find the exact optimal match (Algorithm 1). Section 6 and Section 7 evaluate the two proposed mechanisms on realmarket standard options data and synthetically generated combinatorial options data, respectively. Section 8 concludes and discusses possible future directions.
2. Related Work
2.1. Rational Option Pricing
Our proposed market designs relate to arbitrage conditions that have been studied extensively in financial economics (Modigliani and Miller, 1958; Varian, 1987). In short, an arbitrage describes the scenario of “free lunches”—configurations of prices such that one can get something for nothing. The matching operation of an exchange (i.e., the auctioneer function) can be considered as arbitrage elimination: matching orders in effect works by identifying combinations of orders that reflect a riskfree surplus (i.e., gains from trade). In the classic double auction, a match accepting the highest buy order at, say, $12 and the lowest sell order at $10 in effect yields an arbitrage surplus of $2 that can go to the buyer, the seller, the exchange, or split among them in any way.
Merton first investigated the noarbitrage pricing for options, stating the necessity of convexity in option prices (Merton, 1973). Other relevant works examine noarbitrage conditions for options under different scenarios, such as modeling the stochastic behavior of the underlying asset (e.g., the BlackScholes model) (Black and Scholes, 1973; Perrakis and Ryan, 1984) and considering the presence of other types of securities (e.g., bonds and futures) (Garman, 1976; Ritchken, 1985). The most relevant work to our proposed approach is by Herzel, who makes little assumption on the underlying process of the asset price and other financial instruments (Herzel, 2005)
. The paper proposes a linear program to check the convexity between every strikeprice pair; it finds arbitrage opportunities that yield positive cash flows now and no liabilities in the future on European options written on the same underlying security.
Our first contribution on a market design that consolidates standard options generalizes Herzel’s by adding an extra degree of freedom to incorporate potential future payoff gain or loss into the matching. To our knowledge, no prior work has defined general combinatorial financial options or investigated the matching mechanism design and complexity for such a market.
2.2. Combinatorial Market Design
Much prior work examines the design of combinatorial markets, both exchanges and market makers (Chen et al., 2008a; Hanson, 2003), for different applications including prediction markets with Boolean combinations (Fortnow et al., 2005), permutations (Chen et al., 2007), hierarchical structures (Guo and Pennock, 2009), tournaments (Chen et al., 2008b), and electronic sourcing (Sandholm, 2007). Dudík et al. show how to employ constraint generation in linear programming to keep complex related prices consistent (Dudík et al., 2013). Kroer et al. generalize this approach using integer programming (Kroer et al., 2016). Rothschild and Pennock investigate misaligned prices for logically related binarypayoff contracts in prediction market, and uncover persistent arbitrage opportunities for riskneutral investors across different exchanges (Rothschild and Pennock, 2014).
Designing combinatorial markets faces the tradeoff between expressiveness and computational complexity: giving participants greater flexibility to express preferences can help to elicit better information and increase economic efficiency, but in the meantime leads to a more intricate mechanism that is computationally harder. Several works have formally described and quantified such tradeoff (Benisch et al., 2008; Golovin, 2007), and studied how to balance it by exploiting the outcome space structure and limiting expressivity (Chen et al., 2008a; Xia and Pennock, 2011; Laskey et al., 2018; Dudík et al., 2020). Our work here contributes to this rich literature by designing a vastly more expressive version of the popular financial options market. The state space and payoff function for combinatorial financial options differ from the previously studied combinatorial structures in both meaning and computational complexity. The associated matching problem is novel (with its new form of payoff function), and is not a specialization of previously studied combinatorial prediction market.
3. Background and Notations
There are two types of options, referred to as call and put options. We denote a call option as and a put option as , which respectively gives the option buyer the right to buy and sell an underlying asset at a specified strike price on the expiration date . In the rest of this paper, we omit from the tuples for simplicity, as the mechanism aggregates options within the same expiration.
The option buyer decides whether to exercise an option. Suppose that a buyer spends $8 and purchases a call option, . If the S&P 500 index is $4700 at expiration, the buyer will pay the agreed strike $4500, receive the index, and get a payoff of $200 and a net profit of $192 (assuming no time value). If the S&P 500 price is $4200, the buyer will walk away without exercising the option. Therefore, the payoff of a purchased option is
where is the value of underlying asset at expiration and equals 1 for calls and for puts. As the payoff for a buyer is always nonnegative, the seller receives a premium now (e.g., $8) to compensate for future obligations.
Option contracts written on the same underlying asset, type, strike, and expiration are referred to as an option series. Consider options of a single security offering both calls and puts, ten expiration dates and fifty strike prices. All option series render a total of a thousand markets, with each maintaining a separate limit order book. In such a market, deciding the existence of a transaction takes time by comparing the best bid and ask prices, and matching an incoming order can take up to time depending on its quantity, where is the number of orders on the opposite side of the order book.
4. Consolidating Standard Financial Options
This section proposes a mechanism to consolidate buy and sell orders on standard financial options written on the same underlying asset across different types and strike prices. The model does not make any assumption on the option’s pricing model or the stochastic behavior of the underlying security.
4.1. Matching Orders on Standard Financial Options
Consider an option market in regard to a single underlying asset with an expiration date . Traders can specify orders on such options with any positive strike value. The market has a set of buy orders, indexed by , and a set of sell orders, indexed by
. Buy orders are represented by a type vector
with each entry specifying a put or a call option, a strike price vector , and bid prices . Sell orders are denoted by a separate type vector , a strike vector , and ask prices .The exchange aims to match buy and sell orders submitted by traders. Specifically, it decides the fraction to sell to buy orders and the fraction to buy from sell orders. We start with a formulation that matches orders by finding a common form of arbitrage in which the exchange maximizes net profit at the time of order transaction, subject to zero payoff loss in the future for all possible states of :
s.t. 
Here, the first term in the constraint calculates the total payoff of sold options as a function of , which is the obligation or liability of the exchange at the time of option expiration. The second term computes the total payoff of bought options, by which the exchange has the right to exercise. The constraint guarantees that the liability of the exchange does not exceed its payoff, regardless of the value of on the expiration date. We denote options bought by the exchange as Portfolio and options sold as Portfolio , and the constraint enforces that Portfolio (weakly) dominates Portfolio .
We further extend the above formulation. Instead of restricting the exchange to zero loss at expiration, we allow it to take into account the potential (worstcase) deficit or gain in the future. We denote this extra degree of freedom as a decision variable , and have the following matching mechanism:
(M.1)  
(1)  s.t. 
The constraint now guarantees that the difference between the liability and the payoff of the exchange will be bounded by , regardless of the value of on the expiration date. Following the definition below, we say that the constraint enforces Portfolio to (weakly) dominate Portfolio with some constant offset .
Definition 0 (Payoff dominance with an offset. Adapted from Merton (Merton, 1973)).
Portfolio (weakly) dominates Portfolio with an offset , if the payoff of Portfolio plus a constant is greater than or equal to that of Portfolio for all possible states of the underlying variable at expiration. Portfolio is said to be (weakly) dominated by Portfolio with an offset .
Since this potential gain or loss is further incorporated in the objective at the time of matching, the mechanism M.1 guarantees no overall loss for each match, and enjoys an extra degree of freedom to trade. We give two motivating examples based on realmarket options data to illustrate the economic meaning and the usefulness of a flexible . Each of the matches shown below would not have been found if we restrict a zero payoff loss for the exchange. Example 2 showcases the scenario where allows the exchange to take a (worstcase) deficit at the time of option expiration, if it is preemptively covered by a surplus (i.e., revenue) at the time of order transaction.
Example 0 (A match with a positive ).
We use M.1 to consolidate options of Walt Disney Co. (DIS) that are priced on January 23, 2019 and expire on June 21, 2019. We find the following match, where each order would not transact in its corresponding independent market. The exchange can

sell to the buy order on at bid $7.2,

sell to the buy order on at bid $38.75,

buy from the sell order on at ask $0.05,

buy from the sell order on at ask $5.1.
Therefore, the exchange gets an immediate gain of $40.8 (). Figure 0(a) plots the payoffs of bought and sold options as a function of DIS, showing that the exchange will have a net liability of (i.e., ), regardless of the DIS value at expiration. The exchange makes a net profit of $0.80 from the match at no risk. ∎
Example 3 below demonstrates a different scenario where the decision variable allows the exchange to have a temporary deficit (i.e., expense) at the time of order transaction, if it is guaranteed to earn it back later at the time of option expiration.
Example 0 (A match with a negative ).
We consolidate options of Apple Inc. (AAPL) that are priced on January 23, 2019 and expire on January 17, 2020. We find the following match, where the exchange can

sell to the buy order on at bid $14.1,

sell to the buy order on at bid $0.62,

buy from the sell order on at ask $74.2,

buy from the sell order on at ask $19.1.
The match incurs an expense of $78.58 now (), and yields a guaranteed payoff of $80 (i.e., ) at expiration. Figure 0(b) depicts the respective payoffs of bought and sold options. From the match, we can infer an interest rate of 1.82%, calculated by . ∎
We now analyze the complexity of running Mechanism M.1. The lefthand side of constraint (1) is a linear combination of max functions, and thus is a piecewise linear function of . Therefore, it suffices to solve M.1 by satisfying constraints defined by at each breakpoint. In our case, breakpoints of the constraint (1) are the defined strike values in the market, plus two endpoints: . Let denotes the number of distinct strike values in the market, which is bounded above by , the total number of orders in the market. Therefore, M.1 is a linear program that has payoff constraints, and requires time polynomial in the size of the problem instance to solve. We defer the complete proof to Appendix A.1.
Theorem 4.
Mechanism M.1 matches financial options written on the same underlying asset and expiration date across all types and strike prices in time polynomial in the number of orders.
Remarks.
Several extensions can be directly applied to the mechanism:

We can incorporate the time value of investments by multiplying by a (discount) rate in the objective of M.1.

We note that matching solutions returned by M.1 may involve fractional shares of stocks. Ideally, an exchange (e.g., cashsettled markets) allows it, and if an exchange (e.g., physicalsettled markets) does not, we would need to normalize or round.

Mechanism M.1 identifies a maximal bundle of orders that can be accepted without risk to the exchange, but does not specify what to do with the surplus if any. The surplus can be split arbitrarily among the involved traders and the exchange. We note that while the surplus is guaranteed to be nonnegative, it may have an uncontingent (cash) component and a statecontingent component.
4.2. Quoting Prices for Standard Financial Options
A standard exchange maintains the best quotes (i.e., the highest bid and lowest ask) for each independent options market. This section extends Mechanism M.1 to quote the most competitive prices for any custom option of a specified type and strike by considering all other options related to the same underlying security. We describe the price quote procedure in an arbitragefree options market^{4}^{4}4That is, no match will be returned by Mechanism M.1. in regard to a single underlying asset with an expiration date , represented by . We defer the detailed proof of correctness to Appendix A.2.

The best bid for an option is the maximum gain of selling a portfolio of options that is weakly dominated by with some constant offset .
We derive by adding the option to the sell side of the market indexed (as the exchange buys from sell orders), initializing its price to 0, and solving for M.1. The best bid is then the returned objective.

The best ask for an option is the minimum cost of buying a portfolio of options that weakly dominates with some constant offset .
We derive by adding the option to the buy side of the market indexed , initializing its price to a large number (e.g., ), and solving for M.1. The best ask is then minus the returned objective.
In the case of matching orders with multiple units, it is necessary to consider all orders in the market. For deciding the existence of a match or quoting (instantaneous) prices, however, we only need to consider a set of orders that have the most competitive prices. We define a set with such orders as a frontier set .
Definition 0 (A Frontier Set of Options Orders).
An option order is in the frontier set if its bid price is greater than or equal to the maximum gain of selling a weakly dominated portfolio of options with some offset , or if its ask price is less than or equal to the minimum cost of buying a weakly dominant portfolio of options for some offset .
Corollary 6.0.
Mechanism M.1 determines the existence of a match and returns the instantaneous price quote in time polynomial in .
The proof of Corollary 6 is deferred to Appendix A.3, which shows that in order to determine the existence of a match or quote the most competitive prices (i.e., the highest bid and the lowest ask) for any target option , it suffices to consider options orders in and run Mechanism M.1. The runtime complexity follows immediately from Theorem 4.
5. Combinatorial Financial Options
This section introduces combinatorial financial options and designs a market to trade such options. Combinatorial options extend standard options to more general derivative contracts that can be written on any linear combination of underlying assets. We formally define a combinatorial financial option and its specifications.
Definition 0 (Combinatorial Financial Options).
Consider a set of underlying assets. Combinatorial financial options are derivative contracts that specify the right to buy or sell a linear combination of the underlying assets at a strike price on an expiration date. Each contract specifies a call or put type , a weight vector , a strike price , and an expiration date . It has a payoff of , where is a vector of the underlying assets’ values at .
Consider a combinatorial option that has weight for MSFT, weight for AAPL, and a strike price of zero. An investor who buys the option bets on the event that Microsoft outperforms Apple Inc., and will exercise it if . Thus, unlike standard options that will pay off due to price changes of a single security, combinatorial options bet on relative movements between assets or groups of assets, thus enabling the expression of future correlations among different underlying assets.
We note several distinctions and interpretations in regard to the definition of combinatorial financial options:

Distinction between a combinatorial option and a combination of standard options.
Consider the above combinatorial option and a combination of two standard options and . One would exercise the combinatorial option and possess the portfolio, i.e., long a share of MSFT and short sell a share of AAPL, as long as , whereas in the combination of standard options, one would exercise both and possess the same portfolio, if and .^{5}^{5}5In terms of possessing the portfolio MSFTAAPL, captures all combination of two standard options of the form, and for all positive . Thus, the exercise of a combinatorial option does not imply a simultaneous exercise of all standard options in the combination, and vice versa, the simultaneous buy and sell of several standard financial options with different underlying securities and strike prices (e.g., multileg options orders) cannot replicate the payoff of a combinatorial option.

Distinction between a call and a put combinatorial option.
We note that the distinction between a call and a put for combinatorial options depends on the strike price and coefficients that one specifies in a contract. For instance, in the above example, is identical to , as they have the same payoff function . Despite the different interpretations and expressions, we follow the convention of standard financial options, and have the strike price always be nonnegative.
5.1. Matching Orders on Combinatorial Financial Options
The increased expressiveness in combinatorial options brings new challenges in market design: only matching buy and sell orders on options related to the same assets and weights may yield few or no trades, despite plenty of profitable trades among options written on different portfolios. We start by giving the following motivating examples to illustrate such scenarios.
Example 0 (Matching combinatorial option orders).
Consider a combinatorial options market with four orders

: buy one at bid $110;

: buy one at bid $70;

: sell one at ask $160;

: sell one at ask $5.
The exchange returns no match if it only considers options related to the same combination of assets. However, a profitable match does exist. The exchange can sell to and and simultaneously buy from and to get an immediate gain of $15 (110+701605). Figure 2 plots the overall payoff (which is always nonnegative), as a function of and on the expiration date:
Therefore, the exercises of options cannot subtract from the $15 immediate gain, but could add to it, depending on the future prices of the two stocks. ∎
In the above example, the exchange can consider matching each individual buy order sequentially to some combination of sell orders. For example, the exchange can first sell to buy order and buy from sell orders and , and then sell to buy order and buy from sell orders and . Both are profitable trades subject to no loss, leading to the same ultimate match as in Example 2. The next example shows that such sequential matching of each individual buy order to a combination of sell orders may fail to find valid trades.
Example 0 (Matching combinatorial option orders in batch).
Consider the following four combinatorial options orders

: buy one at bid $6;

: buy one at bid $6;

: sell one at ask $10;

: sell one at ask $2.
No match will be found if we consider each buy order individually: covering a sold combinatorial option in or requires buying the same fraction of , which is at a higher price and will incur a net loss. However, a valid match does exist by selling to and and buying from and . It costs the exchange $0, and can yield a positive payoff in the future. The exchange has
meaning the liability will always be no larger than the payoff of bought options, for all nonnegative .^{6}^{6}6To see this, we can add on both sides of the inequality, and have and by Jensen’s inequality. ∎
We now introduce the formal setting of a combinatorial financial options market. A combinatorial options market is a twosided market with a set of buy limit orders, indexed by , and a set of sell limit orders, indexed by . Buy orders are represented by a type vector with each entry specifying a put or a call combinatorial option, a weight matrix specifying the linear combinations, a strike vector , and a bidprice vector . Sell orders are defined by a separate type vector , a weight matrix , a strike vector , and an askprice vector . Similar to a standard options market, the exchange decides the fraction to sell to buy orders and the fraction to buy from sell orders to maximize net profit. We generalize the proposed mechanism M.1 for standard options to facilitate trading combinatorial options:
(M.2)  
(2)  s.t. 
However, unlike M.1, it is no longer feasible to solve the optimization problem M.2 by enumerating the constraint at each breakpoint, which requires iterating over every combination of underlying asset values. The number of constraints can grow exponentially as or by Sauer’s Lemma.^{7}^{7}7In fact, we can write M.2 as an exponentialsized linear program.
We analyze the complexity of finding the optimal match in a combinatorial options market, i.e., solving for M.2. We first show in the following theorem that given a market instance, it is NPcomplete to decide if a certain matching assignment, and , violates the constraint (2) for a fixed .
Theorem 4.
Proof.
The decision problem is in NP. Given a certificate which is a value vector , we plug into the constraint of M.2 to compute the payoff and check whether it is less than . This takes time .
For the NPhardness, we prove by reducing from the Vertex Cover problem — given an undirected graph and an integer , decide if there is a subset of vertices of size such that each edge has at least one vertex in . Given a Vertex Cover instance , we construct an instance of the combinatorial options matching problem. Let the set of underlying assets correspond to vertices in , i.e., . For each vertex indexed , we associate four options with it, one on the buy side and three on the sell side. They have payoff functions as follows:
where we choose and for some large numbers with . For example, we choose and . For each edge , we define two options that involve its two endpoints and , one on the buy side and one on the sell side. They have payoff functions:
Finally, we include one sell order on an option with payoff . Since is fixed in advance, we assume without loss of generality. Thus, we have buy orders and sell orders. The construction takes time polynomial in the size of the Vertex Cover instance.
Suppose the Vertex Cover instance is a Yes instance. We assume that is a vertex cover. We show that assigning to 1 for the selected underlying assets (i.e., vertices) and 0 for the rest unselected ones gives an that violates the constraint (2). The lefthand side of the constraint (2) is
(3) 
For , it is easy to see that . Thus, we have by our assignment. Since at least one of is for any edge , we have and . Therefore, we have
Suppose the Vertex Cover instance is a No instance. We aim to show that for the given and , there does not exist an that violates the constraint. We prove by maximizing and demonstrating . We start by proving the following claim.
Claim 1.
For an optimal , we have .
We prove by contradiction. First assume for some . We rewrite Eq. (3) and have
We first analyze and have
(by assumption of )  
Recall that we choose , e.g., and . Thus, we have increase with rate , as decreases uniformly. Since decreases at most and the rest two terms, and , do not depend on , decreasing increases . It is suboptimal to have for some .
Next, we assume for some and have
As is large, by a similar argument analyzing the growth rate of each term, we show that (and also ) increases by assigning to if and by assigning to if .
We finish proving Claim 1, and now have . Following Eq. (3), we have
The goal is to maximize and show . As is an integer, to show , it suffices to show that . We prove by contradiction, assuming . Recall that to have for , we need at least one of to be . We consider the following two possible cases, and aim to refute them:

and .
This means we cover all edges with at most vertices assigned to 1.

and .
For any with , we can assign 1 to one of its endpoints, and have without changing . This leads back to (a).
Both cases contradict to the fact that the Vertex Cover instance is a No instance. We have as desired, and thus for all . ∎
Using a slightly stronger version of Theorem 4 (see Theorem 5 in Appendix B.1), we show that optimal clearing of a combinatorial options market is coNPhard. We defer the proof of Theorem 5 to Appendix B.1.
Theorem 5.
Optimal clearing of a combinatorial options market is coNPhard, even assuming that each combinatorial option is written on at most two underlying assets.
5.2. A Constraint Generation Algorithm to Match Combinatorial Financial Options
Since it is no longer practical to solve M.2 by identifying all constraints defined by different combinations of underlying asset values, we propose an algorithm (Algorithm 1) that finds the exact optimal match through iterative constraint generation. We first explain how Algorithm 1 works, and then prove that it is an equivalent formulation of M.2.
At the core of Algorithm 1 is a constraint generation process, where a new optimization problem is defined per iteration to find a violated constraint. The constraint set includes the value vectors that define different constraints. We can plug each back into the constraint to define and , which are the realized payoffs of bought and sold options with respect to the . We start with the constraint set of a zero vector, meaning that all underlying assets have price zero at expiration. In each iteration, the upperlevel optimization problem (M.3U) computes the optimal match (i.e., , , and ) that satisfies this restrictive set of generated constraints. Thus, it is a linear program with constraints. We then use the lowerlevel optimization problem (M.3L) to find the value vector that violates the returned matching solution the most. That is, given , , and , it generates the most adversarial realization of that yields the worstcase payoff loss for the exchange at expiration. We show that this lowerlevel optimization can be formulated as a mixedinteger linear program. We then include this generated vector into the constraint set.
The exact optimal match is returned when the lowerlevel M.3L gives an objective value of zero, meaning there exists no such that violates the upperlevel matching assignment. Therefore, the final number of constraints in is the number of iterations that takes Algorithm 1 to terminate. The algorithm trivially terminates finitely, but similar to the simplex method, it has no guarantee on the rate of convergence. We later evaluate its performance on synthetic combinatorial options markets of different scales, and demonstrate that Algorithm 1 converges relatively fast, with the number of iterations increasing linearly in the size of a market instance.
We next prove Algorithm 1 returns the same optimal clearing solution as M.2. We first show in the Lemma that M.3L finds the value vector that violates the constraint (2) in M.2 the most.
(M.3U)  
(M.3L)  
Lemma 0.
Proof.
First, it is easy to see that the formulation below returns the that violates constraint (2) the most, since we will have the largest feasible and the smallest feasible at the optimum. That is, and .
It remains to show the set of constraints related to any buy order in M.3L is equivalent to . In short, the set of constraints in M.3L linearize the max functions by using the big trick on each binary decision variable , where is a large constant, say .
Consider each case of . The first two constraints, and , guarantee that . Then, following the third and fourth constraints, i.e., and , we have
Therefore, at the optimum, we have
which is equivalent to . ∎
6. Experiments: RealMarket Standard Financial Options
We first evaluate the proposed mechanism M.1 that matches orders on standard options across types and strike values. We conduct empirical analysis on the OptionMetrics dataset provided by the Wharton Research Data Services (WRDS), which contains realmarket option prices, i.e., the best bid and ask prices, for each market defined by an underlying asset, an option type, a strike price, and an expiration date.^{8}^{8}8Our data includes American options that allow exercise before expiration. In practice, American options are almost always more profitable to sell than to exercise early (Singh, 2019). In experiments, we ignore early exercise and treat them as European options. We choose options data on 30 stocks that compose the DJI, as these stocks have actively traded options that cover a wide range of moneyness and maturity levels. There are a total of 25,502 distinct options markets for the 30 stocks in DJI on January 23, 2019, covering around 12 expiration dates for each stock.
We use M.1 to consolidate the outstanding buy orders and sell orders from independentlytraded options markets that associated with the same security and expiration date. This reduces the original 25,502 distinct markets to a total of 366 markets, which have standard options across different types and strikes. We run M.1 on these consolidated markets to

find trades that the current independentmarket design cannot match,

compute new price quotes implied by this consolidated market design, and

compare matches and price quotes to those of the case when we restrict to 0.
Out of the 366 consolidated options markets, we find profitable matches in 150 markets, which failed to transact under the independent market design. Among those trades, 94 have nonnegative (i.e., the case of Example 2), making an average net profit of $1.03, with a maximum of $9.64; the remaining 56 have negative (i.e., the case of Example 3), implying an average interest rate of 0.7%, with a maximum at 2.02%. When we restrict to 0 (i.e., is no longer a decision variable), we are able to find matches in 74 markets. Detailed statistics for options of each stock are available in the Appendix C.
For the arbitragefree consolidated markets (i.e., markets with no match returned by M.1), we find that approximately 49% of the orders belong to the frontier set. Using these orders to derive the most competitive bids and asks, we find that the bidask spreads can be reduced by 73%, from an average of 80 cents for each option series in the independent markets to 21 cents in consolidated options markets. For the case of set to 0, the bidask spreads can be reduced by 52%.
These results show that the exchange, by consolidating options markets across types and strike prices, can potentially achieve a higher economic efficiency, matching orders that the current independent design cannot and providing more competitive bid and ask prices.
7. Experiments: Synthetic Combinatorial Options Market
Since there is no combinatorial option traded in financial markets, we evaluate the proposed Algorithm 1 on synthetic combinatorial options, with prices calibrated using realmarket standard options written on each related underlying security.^{9}^{9}9We adopt the same dataset as Section 6, and use standard options that expire on Febuary 1, 2019, to calibrate order prices for generated combinatorial options. We implement Algorithm 1 using Gurobi 9.0 (Optimization, 2020). We are interested in quantifying the performance of Algorithm 1 in parametrically different markets that vary in the likelihood of matching, the number of orders, and the number of underlying assets. We first describe our synthetic dataset.
7.1. Generate Synthetic Orders
We generate combinatorial options markets of underlying assets. Each combinatorial option is written on a combination of two stocks, and , randomly selected from the underlying assets. This gives a total number of asset pairs. Weights for the selected assets, and , are picked uniformly randomly from and are preprocessed to be relatively prime.
We generate strikes and bid/ask prices using realmarket standard options data related to each individual asset to realistically capture the value of the synthetic portfolio. Let and respectively denote the set of strike prices offered by standard options on each selected asset. We generate the strike by first sampling two strike values, and , and scaling them by the associated weights to get . If is positive, we have a call option. Instead, if is negative, we generate a put option, and update the strike to and weights to and to comply with the representation and facilitate payoff computations.
We randomly assign each option to the buy or sell side of the market with equal probability, and generate a bid or ask price accordingly. Similar to the price quote procedure in Section
4, we derive the bid by calculating the maximum gain of selling a set of standard options whose payoff is dominated by the combinatorial option of interest, and compute the ask by calculating the minimum cost of buying a set of standard options whose payoff dominates the generated conbinatorial option. We add noises around the derived prices to control the likelihood of matching in a market, and set final prices to or , where and is a noise parameter.7.2. Evaluation
We explore a wide range of markets that vary in price noise , the number of orders , and the number of underlying assets . For each market, we measure (1) the number of iterations (i.e., the total number of constraints generated) that Algorithm 1
takes to find an exact optimal clearing and (2) the net profit made from the trade. For all experiments, we show results averaged over 40 simulated markets, with the error bars denoted one standard error around the means.
We first validate that as larger noise is added to the derived bids and asks, the likelihood of matching in our simulated combinatorial options market becomes higher. We generate markets with four underlying assets (arbitrarily selected from the 30 stocks in DJI) and 150 synthetic combinatorial options orders, and vary the noise level . Figure 2(a) plots the averaged results. As expected, the net profit made from the optimal match increases, as increases. Moreover, we find that as the matching probability increases, the number of iterations that takes to find the optimal solution consistently decreases. This makes sense as intuitively, in thin markets where few trades are likely to occur, the lowerlevel optimization program will keep coming up with candidate values to refute a large number of matching proposals until convergence.
Figure 2(b) further quantifies the change in iteration numbers and net profits, as we vary the number of combinatorial options orders. We fix these markets to have four underlying assets with a price noise of . As we see from Figure 2(b), as a market aggregates more orders, transactions are more likely to happen, leading to larger net profits. We also find that the number of generated constraints grows (sub)linear in the number of orders. Since different values are generated to define payoffs of distinct options and the number of distinct options increases sublinear in the number of total orders, the rate of increase in the number of iterations tends to decrease as a market aggregates more orders.
Finally, we evaluate how Algorithm 1 scales to markets with increasingly larger numbers of underlying assets. In this case, as the dimension of becomes large, the number of assetvalue combinations grows exponentially. Figure 2(c) (black lines) demonstrates a much faster increase in the number of iterations and a steady decrease in the net profit.^{10}^{10}10We report average runtimes to quantify the impact of increasing constraints. The average times (in seconds) that Algorithm 1 computes the optimal match are 4, 31, 47, 59, 68, and 72 for the respective markets with . It suggests that as the market provides a large set of underlying assets (e.g., all 30 stocks in DJI), the thin market problem may still arise even when the mechanism facilitates matching options written on different combinations of underlying assets. Here, in this set of experiments, we make the assumption that every asset pair in the is equally likely to be traded. In real markets, investors may be more interested in certain asset pairs, trading them more frequently than the others. Based on such observations, a market can specify a prescriptive set of asset pairs, , to alleviate the thin market problem. For the experiments, we choose and have underlying securities in those asset pairs cover the assets. Traders can choose from any asset pairs within this prescriptive set and specify custom weights for each security. Figure 2(c) (red lines) shows that such prescriptive design may indeed attenuate the thin market problem, leading to a faster convergence and higher profits.
8. Discussion
The paper proposes market designs to facilitate trading standard options and the more general combinatorial financial options. We start by examining standard financial exchanges that operate separate options markets and have each independently aggregate and match orders on options of a designated type, strike price, and expiration. When logically related financial markets run independently, traders may remove arbitrage and close bidask spreads themselves. Our OptionMetrics experiments show that they do so suboptimally. Execution risk and transaction cost often prevent arbitrage opportunities from being fully removed. Moreover, profits can flow to agents with better computational power and little information. Our consolidated market design, by putting computational power into the exchange, supports trading related options across different strikes and reduces bidask spreads, thus reducing the arms race among agents and rewarding informed traders only.
We extend standard options to define combinatorial financial options. We are interested in designing a fully expressive combinatorial options market that allows options to be specified on all linear combinations of assets. In such a market, traders are granted the convenience and expressiveness to speculate on correlated movements among stocks and more precisely hedge their risks (e.g., in one transaction, purchase a put option on their exact investment portfolio). Increasing expressiveness may also benefit the exchange: the market is able to incorporate information of greater detail to optimize outcome, improving economic efficiency, and obtain a surplus from trade to split between the exchange and traders. Like many other combinatorial markets, the combinatorial options market comes at the cost of a higher computational complexity: optimal clearing of such a market is coNPhard. We propose a constraint generation algorithm to solve for the exact optimal match, and demonstrate its viability on synthetically generated combinatorial options markets of different scales.
Two immediate questions arise from our work. First, can we design computationally efficient matching algorithms by limiting expressivity within a combinatorial options market or imposing certain assumptions on the underlying assets? One next step is to explore naturally structured markets where combinations are limited to components in a graph of underlying assets. One special case is a hierarchical graph (Guo and Pennock, 2009), for example, the S&P 500, sectors like travel and technology, subsectors like airlines and internet within those, etc. Another direction is to relax our current constraint that guarantees no loss for all possible states, assuming or learning a probability distribution over the security’s future price (e.g., recovering probability from option prices (Jackwerth and Rubinstein, 1996)).
Second, when operating such a combinatorial options market, how should an exchange set the market clearing rule? While the proposed mechanism (M.2) and matching algorithm (Algorithm 1) allow flexibility in the timing of market clear operations, and so can work in either continuous or periodic (batch) form, adopting an appropriate clearing rule is important and can affect both computational and economic efficiency. Our experiments on synthetic combinatorial options market (Section 7) provide some preliminary understanding. Results on markets with low noise suggest potential limitations of continuous clearing: as there is no match among existing orders and the matching probability is low with only one incoming order, the lowerlevel program may take a long time to refute a large number of matching proposals until convergence. Surplus can also be low in a continuous clearing market that looks for immediate trades.^{11}^{11}11We conduct preliminary experiments on synthetic data and find that the average net profit achieved in a market with continuous clearning is about 30% lower than that of a market with batch clearing of the same 100 orders. Batch clearing might be suitable for this combinatorial market, achieving a more practical convergence speed and higher surplus. The next question becomes how frequently should the market clear? Like discussed in prior works studying the proposal of having financial markets move from continuous to batch clearing (Budish et al., 2014, 2015; Brinkman and Wellman, 2017), choosing the appropriate clearing interval is an art. In our case, it depends on the market scale (e.g., the number of underlying assets and orders) and liquidity (e.g., matching probability), and a designer needs to balance price discovery and matching surplus (longer batches improve trading surplus, but can significantly slow down price discovery). Analysis based on empirical mechanism design (Vorobeychik et al., 2006; Brinkman and Wellman, 2017) may be useful to help a market designer set the optimal clearing interval and maximize efficiency.
We leave these questions open for future research.
Acknowledgements.
We thank Miroslav Dudík and Haoming Shen for helpful discussions. We are also grateful to the anonymous reviewers for constructive feedback.References

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Appendix A Deferred Proofs from Section 4
a.1. Proof of Theorem 4
As the lefthand side of the constraint in M.1 is a piecewise linear function of , it suffices to solve M.1 by including constraints defined by at breakpoint values. In our case, breakpoints of the constraint are the strike values specified in the market, and the number of distinct strikes grows sublinear in the number of orders . Therefore, we can specify the constraint in M.1 as payoff constraints for each stock value . Moreover, there are at most quantity constraints, if we consider the quantity specified in each order. Thus, M.1 is a linear program with decision variables and constraints. Mechanism M.1 matches options written on the same underlying asset and expiration across all types and strikes in time polynomial in the number of orders.
a.2. Proof of Price Quote Procedure
We reiterate the procedure of using mechanism M.1 to price a target options with existing options orders in the market defined by .

The best bid for an option is the maximum gain of selling a portfolio of options that is weakly dominated by for some constant .
We derive by adding to the sell side of the market indexed , initializing its price to 0, and solving for M.1. The best bid is the returned objective of M.1.
Proof.
We show that the above procedure finds the best bid, by returning the maximum gain of selling a weakly dominated portfolio of options. After adding to the market, we have the updated M.1 as the following:
s.t. Since we set , it is always optimal to buy option and have . Therefore, we have the following optimization problem extended from M.1:
(M.1bid) s.t. Following Definition 1, the lefthand side of the above inequality describes the payoff of a portfolio of existing options that is weakly dominated by the target options with an offset , and the objective maximizes the gain of selling such a portfolio. One is willing to buy at the highest price (i.e., the best bid) , as one can always sell the weakly dominated Portfolio () and get back without losing anything in the future. ∎

The best ask