We investigate upper and lower hedging prices of multivariate contingent claims from the viewpoint of game-theoretic probability and submodularity. By considering a game between “Market” and “Investor” in discrete time, the pricing problem is reduced to a backward induction of an optimization over simplexes. For European options with payoff functions satisfying a combinatorial property called submodularity or supermodularity, this optimization is solved in closed form by using the Lovász extension and the upper and lower hedging prices can be calculated efficiently. This class includes the options on the maximum or the minimum of several assets. We also study the asymptotic behavior as the number of game rounds goes to infinity. The upper and lower hedging prices of European options converge to the solutions of the Black–Scholes–Barenblatt equations. For European options with submodular or supermodular payoff functions, the Black–Scholes–Barenblatt equation is reduced to the linear Black–Scholes equation and it is solved in closed form. Numerical results show the validity of the theoretical results.

中文翻译：

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多元或有债权和子模性上限对冲价格的博弈论推导

我们从博弈论概率和子模性的角度研究多元或有债权的上限和下限对冲价格。通过考虑离散时间“市场”和“投资者”之间的博弈，定价问题被简化为对单纯形优化的向后归纳。对于具有满足称为子模或超模的组合属性的支付函数的欧式期权，该优化通过使用 Lovász 扩展以封闭形式求解，并且可以有效地计算上限和下限对冲价格。此类包括几个资产的最大值或最小值的选项。我们还研究了随着游戏轮数趋于无穷大的渐近行为。欧式期权的上限和下限对冲价格收敛于 Black-Scholes-Barenblatt 方程的解。对于具有次模或超模收益函数的欧式期权，Black-Scholes-Barenblatt 方程简化为线性 Black-Scholes 方程，并以封闭形式求解。数值结果表明了理论结果的有效性。