# Game-theoretic derivation of upper hedging prices of multivariate contingent claims and submodularity

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## Abstract

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.

## Keywords

Game-theoretic probability Upper hedging price Multivariate contingent claim Submodular Lovász extension Black–Scholes–Barenblatt equation## Mathematics Subject Classification

91G20## Notes

### Acknowledgements

We thank the referee for constructive comments. We thank Naoki Marumo and Kengo Nakamura for helpful comments. This work was supported by JSPS KAKENHI Grant Numbers 16K12399, 18H04092 and 17H06569.

## Supplementary material

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