Approximations and asymptotics of upper hedging prices in multinomial models

Nakajima, Ryuichi; Kumon, Masayuki; Takemura, Akimichi; Takeuchi, Kei

doi:10.1007/s13160-011-0047-8

Cited by 3 publications

(15 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When d = 1, the maximum in (22) only depends on the sign of the second derivative ofū and (22) reduces to the equation (13) in Nakajima et al (2012). This case is also discussed in Peng (2007) and Section 5 of Romagnoli and Vargiolu (2000).…”

Section: Derivation Of the Black-scholes-barenblatt Equationmentioning

confidence: 82%

“…Market announces x n ∈ χ K n := K n−1 + M n x n END FOR Here, α denotes the initial capital of Investor, M n corresponds to the vector of amounts Investor buys the assets, x n corresponds to the vector of price changes of the assets and K n corresponds to Investor's capital at the end of round n. When d = 1, this game reduces to the game analyzed in Nakajima et al (2012). One natural candidate for χ is a product set…”

Section: Definitions and Notationmentioning

confidence: 99%

“…Now, we consider the N -round game. As discussed by Nakajima et al (2012) for d = 1, the upper hedging price is calculated by solving linear programs recursively. Specifically, letf (·, N − n) : χ n → R, n = N, N − 1, · · · , 0 be given bȳ…”

Section: Formulation With Linear Programmingmentioning

confidence: 99%

“…In this paper, we investigate the upper hedging price of multivariate contingent claims by extending the game-theoretic probability approach of Nakajima et al (2012). We consider a discrete-time 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.…”

Section: Introductionmentioning

confidence: 99%

“…As in Nakajima et al (2012), we consider the price processes in an additive form and the Black-Scholes-Barenblatt equation in section 4 is actually an additive form of the Black-Scholes-Barenblatt equation in the standard finance literature. Similarly, the linear Black-Scholes equation is given in the form of a heat equation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Matsuda

Takemura

2019

Japan J. Indust. Appl. Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Derivation Of the Black-scholes-barenblatt Equationmentioning

confidence: 82%

Section: Definitions and Notationmentioning

confidence: 99%

Section: Formulation With Linear Programmingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Matsuda

Takemura

2019

Japan J. Indust. Appl. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Miyabe

Takemura

2012

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

We give a unified treatment of the convergence of random series and the rate of convergence of strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to existence of an absolute moment of order smaller than two in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in measure-theoretic framework, these results characterize convergence of a martingale in terms of convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality's strategies do not have any counterparts in measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in measure-theoretic framework.

show abstract

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Miyabe

Takemura

2011

Preprint

View full text Add to dashboard Cite

Approximations and asymptotics of upper hedging prices in multinomial models

Cited by 3 publications

References 24 publications

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

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

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability

Contact Info

Product

Resources

About