1 Introduction
The pricing of contingent claims is a central problem in mathematical finance (Karatzas and Shreve, 1998). The fundamental models of financial markets are the binomial model (Shreve, 2003) and the geometric Brownian motion model (Shreve, 2005) in discretetime and continuoustime setting, respectively. These models describe complete markets and therefore the price of any contingent claim is obtained by arbitrage argument. Specifically, the CoxRossRubinstein formula (Cox et al., 1979) and the BlackScholes formula (Black and Scholes, 1973) provide the exact price in the binomial model and the geometric Brownian motion model, respectively. These formulas are derived by constructing a hedging portfolio for the seller to replicate the contingent claim.
In general, the market is incomplete and the above formulas are not applicable. Even in incomplete markets, we can define the upper and lower hedging prices of a contingent claim by considering superreplication (Karatzas and Shreve, 1998). Musiela (1997) and El Karoui and Quenez (1995) provide fundamental results on the upper hedging price in discretetime models and continuoustime models, respectively. As a special case, for discretetime models with bounded martingale differences, Ruschendorf (2002) pointed out the upper hedging price of a convex contingent claim is obtained by the extremal binomial model.
Although the above studies focused on contingent claims depending on a single asset, there are contingent claims for which the payoff depends on two or more assets (Stapleton, 1984), which are called multivariate contingent claims. For example, Stulz (1982) and Johnson (1987) investigated the pricing of options on the maximum or the minimum of several assets. Boyle et al. (1989) developed a numerical method for pricing multivariate contingent claims in discretetime models. Although their method is based on a lattice binomial model that is originally incomplete, they change the model to make it complete by specifying correlation coefficients between all the pairs of assets. Thus, their method does not compute the upper hedging price. On the other hand, Romagnoli and Vargiolu (2000) considered superreplication in continuoustime models and derived the pricing formula based on the BlackScholesBarenblatt equation. They also provided some sufficient conditions on payoff functions for reduction of the BlackScholesBarenblatt equation to the linear BlackScholes equation.
Whereas existing studies on the upper and lower hedging price are based on stochastic models of financial markets, Nakajima et al. (2012) investigated this problem from the viewpoint of the gametheoretic probability (Shafer and Vovk, 2001)
, in which only the protocol of a game between “Investor” and “Market” is formulated without specification of a probability measure. They showed that the upper hedging price in the discretetime multinomial model is obtained by a backward induction of linear programs, and that the upper hedging price of a European option converges to the solution of the onedimensional BlackScholesBarenblatt equation as the number of game rounds goes to infinity.
In this paper, we investigate the upper hedging price of multivariate contingent claims by extending the gametheoretic probability approach of Nakajima et al. (2012). We consider a discretetime multinomial model with several assets and show that the upper hedging price of multivariate contingent claims is given by a backward induction of a maximization problem whose domain is a set of simplexes, which becomes intractable in general as the number of assets increases. However, we find that this maximization is solved in closed form if the contingent claim satisfies a combinatorial property called submodularity or supermodularity (Fujishige, 2005). Specifically, the maximizing simplex is determined by using the Lovász extension (Lovász, 1983) for European options with supermodular payoff function on two or more assets and also European options with submodular payoff function on two assets. As realistic examples, we prove that options on the maximum and the minimum of several assets are submodular and supermodular, respectively. Then, by considering the asymptotics of the number of game rounds, we show that the upper hedging price of a European option converges to the solution of the BlackScholesBarenblatt equation. In particular, for European options with supermodular payoff function on two or more assets and also European options with submodular payoff function on two assets, the BlackScholesBarenblatt equation reduces to the linear BlackScholes equation, which is solved in closed form. Finally, we confirm the validity of the theoretical results by numerical experiments.
As in Nakajima et al. (2012), we consider the price processes in an additive form and the BlackScholesBarenblatt equation in section 4 is actually an additive form of the BlackScholesBarenblatt equation in the standard finance literature. Similarly, the linear BlackScholes equation is given in the form of a heat equation. However, the results of this paper holds for the usual multiplicative model with simple exponential transformations. Except for a few places we do not specifically indicate that our equations are in the additive form.
In section 2, we provide a formulation of the upper hedging price based on the gametheoretic probability. In section 3, we derive results for the special case of European options with submodular or supermodular payoff function, which include the option on the maximum or the minimum. In section 4, we derive PDE for asymptotic upper hedging prices. In section 5, we confirm the theoretical results by numerical experiments. In section 6, we give some concluding remarks and discuss future works.
2 Gametheoretic derivation of upper hedging prices of multivariate contingent claims
In this section, we present a formulation of the upper hedging price based on the gametheoretic probability (Shafer and Vovk, 2001). The pricing problem is reduced to a backward induction of linear programs. As special cases, we consider convex or separable payoff functions.
2.1 Definitions and notation
Let , , be a finite set, which we call a move set. Let denote the convex hull of . We assume that the dimension of is and contains the origin in its interior. The protocol of the round multinomial game with assets is defined as follows:
FOR
Investor announces
Market announces
END FOR
Here, denotes the initial capital of Investor,
corresponds to the vector of amounts Investor buys the assets,
corresponds to the vector of price changes of the assets and corresponds to Investor’s capital at the end of round . When , this game reduces to the game analyzed in Nakajima et al. (2012). One natural candidate for is a product set(1) 
where . Such with was adopted by the lattice binomial model (Boyle et al., 1989).
We call the sample space and a path of Market’s moves. For , is a partial path. The initial empty path is denoted as . We call a strategy. When Investor adopts the strategy , his capital at the end of round is given by , where
We call a function a payoff function or a contingent claim. If depends only on , then is called a European option. The upper hedging price (or simply upper price) of is defined as
(2) 
and the lower hedging price is defined as
A strategy attaining the infimum in (2) is called a superreplicating strategy for .
A market is called complete if the upper hedging price and the lower hedging price coincide for any payoff function. For example, the binomial model, which corresponds to and , is complete (Shreve, 2003).
As discussed in section 1 we consider an additive form of the game where the prices changes ’s are added rather than multiplied in .
2.2 Formulation with linear programming
Following Nakajima et al. (2012), we formulate the pricing problem as a recursive linear programming.
First, we consider the singleround game (). Note that the payoff function is . Let be the set of simplexes of dimension containing the origin. For each , define
where the probability vector is defined as the unique solution of the linear equations
(3) 
By extending Proposition 2.1 of Nakajima et al. (2012), we obtain the following.
Proposition 2.1.
For a singleround game, the upper hedging price of a payoff function is given by
(4) 
Proof.
From (2), the upper hedging price of is the optimal value of the following linear program:
The dual of this linear program is
From the complementary condition (Boyd and Vandenberghe, 2004), there exists an optimal solution of the dual problem that has at most nonzero variables. Then, from the constraint of the dual problem, we have for some . Therefore, we obtain (4). ∎
Similarly, the lower hedging price of for a singleround game is
The relation (4) is interpreted as follows. A probability vector is called a risk neutral measure on if the expectation under is zero:
Let be the set of risk neutral measures on . Note that is closed and convex. Then, the set coincides with the set of extreme points (vertices) of . Since the maximum of a linear function on a closed convex set is attained at extreme points, the maximization in (4) is interpreted as searching over all risk neutral measures:
where denotes the expectation with respect to .
When and , the maximization in (4) involves two candidates of . Specifically, the possible is ABC or ABD in Figure 1. When and , the maximization in (4) becomes more difficult since the number of possible becomes large. For example, when , the number of possible can be as large as 14, as shown in Appendix A. In general, the number of possible is at least , as shown in Appendix B. In the next section, we will provide some sufficient conditions on for the maximization in (4) to be solved explicitly.
Now, we consider the round game. As discussed by Nakajima et al. (2012) for , the upper hedging price is calculated by solving linear programs recursively. Specifically, let , be given by
(5) 
with the initial condition for . Here, denotes a function defined by . Then, we have the following result.
Proposition 2.2.
The upper hedging price of is given by
Therefore, the upper hedging price is calculated by a backward induction of (5). In particular, for a European option, depends only on and so the required number of calculations (5) grows only polynomially with .
The lower hedging price is also calculated by a backward induction. In general, a market with is complete since , irrespective of .
2.3 Convex payoff functions
Here, we consider the case where the payoff function is a European option with convex function . When , the calculation of the upper hedging price of a convex payoff function is reduced to the binomial model (Ruschendorf, 2002), since the maximization in (4) is always uniquely attained by the extreme pair . For general , we have the following result.
Proposition 2.3.
Suppose that is a European option with convex function . Let be the set of vertices of . Then, the upper hedging price of with move set coincides with that with move set :
(6) 
Proof.
We provide a proof for the singleround game. By applying it to each induction step, the proof for the multiround game is obtained.
Since , we obtain
(7) 
Let . From Caratheodory’s Theorem (Rockafeller, 1997), for each , there exists a subset of and a probability vector such that
Thus, from the convexity of ,
Therefore,
for any . By summing up the right hand side for each point of , we obtain
where since
and
Thus, for every , there exists such that
By taking the maximum in the left hand side, we obtain
(8) 
Suppose and the move set is a product set (1). Then, Proposition 2.3 implies that we can redefine as the move set. However, unlike the case , even with this reduction to , the maximizing in (4) is not unique in general. One example is the option on the maximum with , as we will see in the next section.
2.4 Separable payoff functions
We call a European option separable if it can be decomposed as
(9) 
where is a partition of :
We assume that the move set is a direct product of move sets for each subset . For example, when and , the move set is a product set as in (3). Then, the calculation of the upper hedging price of a separable European option is reduced to that for each component European option as follows.
Proposition 2.4.
Suppose is separable (9) and is a direct product of move sets for each subset . Then, the upper hedging price of coincides with the sum of the upper hedging prices of :
(10) 
Proof.
We provide a proof for the singleround game. By applying it to each round, the proof for the multistep game is obtained.
For simplicity of notation we consider the case and without essential loss of generality. Then, the move set is a product set and the payoff function is written as
(11) 
Let be an arbitrary risk neutral measure on and let be its marginals. Then, each is a risk neutral measure on . By taking expectations in (11),
Since ,
Therefore,
(12) 
Conversely, let be arbitrary risk neutral measures on , respectively. Define . Then, is a risk neutral measure on and it satisfies
Since ,
Therefore,
(13) 
3 Submodular and supermodular payoff functions
We have seen that the calculation of the upper hedging price reduces to backward induction of the maximization (4). In this section, we show that this maximization is solved in closed form if the payoff function satisfies a combinatorial property called submodularity or supermodularity. As a special case, we discuss the options on the maximum or the minimum. Throughout this section, we assume that the move set is a product set (1) with , i.e., the lattice binomial model. Note that for , because we are assuming that contains the origin in its interior.
3.1 Submodular and supermodular functions
The concept of submodularity is fundamental in combinatorial optimization theory
(Fujishige, 2005).Definition 3.1.

A set function is said to be submodular if it satisfies
for every .

A set function is said to be supermodular if it satisfies
for every .
It is well known (Theorem 44.1 of Schrijver (2003)) that in Definition 3.1 we only need to consider and such that
(14) 
We can extend the definition of submodularity and supermodularity to functions on the hypercube and . For vectors , we denote the vectors of componentwise minimum and maximum by and , respectively:
Then, submodular and supermodular functions on the hypercube and are defined as follows.
Definition 3.2.

A function or is said to be submodular if it satisfies
(15) for every or .

A function or is said to be supermodular if it satisfies
for every or .
We note that the concept of multivariate total positivity (MTP2) is closely related to submodularity (Karlin, 1968; Fallat et al., 2017). Namely, a positive function is MTP2 if and only if its logarithm is supermodular.
When or is twice continuously differentiable, the submodularity and supermodularity of are characterized by the signs of the mixed second order derivatives as follows.
Lemma 3.1.

A twice continuously differentiable function or is submodular if and only if
(16) for every .

A twice continuously differentiable function or is supermodular if and only if
for every .
3.2 Convex closure and Lovász extension
In considering the maximization (4), the concepts of convex closure and Lovász extension are useful. Dughmi (2009) presents a brief survey on these topics.
Let be a finite set. For a subset of , its characteristic vector is defined as
In the following, we identify with by the bijection . For a set function , a function or is said to be its extension if it satisfies .
Definition 3.3.
For a set function , its convex closure and concave closure are extensions of defined as
From the definition, and are convex and concave, respectively.
Lemma 3.2.
For a set function , its convex closure and concave closure are given by
(17) 
Proof.
See section 3.1.1 of Dughmi (2009). ∎
Another type of extension was introduced by Lovász (1983) and it plays an important role in submodular function optimization (Fujishige, 2005).
Definition 3.4.
Note that the Lovász extension can be viewed as taking a special value of in (17). Lovász (1983) showed that the submodularity of a set function is equivalent to the convexity of its Lovász extension . In fact, there is a stronger result as follows.
Lemma 3.3.

The convex closure of a submodular function is equal to the Lovász extension of : .

The concave closure of a supermodular function is equal to the Lovász extension of : .
Proof.
See section 3.1.3 of Dughmi (2009). ∎
3.3 Relation between upper hedging price and concave closure
Let be a bijective affine map defined by and be its restriction to . By using , we can identify the payoff function of the singleround game with a set function . Then, its concave closure is closely related to the maximization (4) as follows.
Proposition 3.1.
For a singleround game, the upper hedging price is given by
Proof.
From Lemma 3.2,
(18) 
Since is a bijective affine map, the first constraint on in (18) is equivalent to
(19) 
Here, for , each entry of is
Thus, (19) is rewritten as
Therefore, each satisfying the constraints in (18) is viewed as a risk neutral measure on . Since is the maximum over all risk neutral measures, we obtain .
3.4 Two assets case
Suppose and , where and .
For a singleround game, the maximization in (4) involves two candidates of . One of them () has positive correlation while the other () has negative correlation. For example, in Figure 1, ABD and BCD have positive correlation while ABC and ACD have negative correlation. Similarly to section 3.3, the payoff function is identified with a set function . If is submodular or supermodular, then the maximizer in (4) is determined as follows.
Proposition 3.2.
Proof.
From the definition of the Lovász extension,
Now, consider the round game and assume that the payoff function is a European option , where and . Recall that depends only on
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