Quantile hedging for contingent claims in an uncertain financial environment

Jun Zhao; Peibiao Zhao; Jun Zhao; Peibiao Zhao

doi:10.3934/math.2023799

AIMS Mathematics

2023, Volume 8, Issue 7: 15651-15669. doi: 10.3934/math.2023799

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Research article

Quantile hedging for contingent claims in an uncertain financial environment

Jun Zhao ^{1
,
,},
Peibiao Zhao ²

1.
School of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi, China
2.
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, Jiangsu, China

Received: 08 February 2023 Revised: 27 March 2023 Accepted: 31 March 2023 Published: 27 April 2023
MSC : 90C70, 91G20, 91G80

This paper first studies the quantile hedging problem of contingent claims in an uncertain market model. A special kind of no-arbitrage, that is, the absence of immediate profit, is characterized. Instead of the traditional no-arbitrage targeting the whole market, the absence of immediate profit depends on the confidence level of the portfolio manager for hedging risk. We prove that the condition of absence of immediate profit holds if and only if the initial price of each risky asset lies between the $ \alpha $-optimistic value and $ \alpha $-pessimistic value of its discounted price at the end of the period. The bounds of the minimal quantile hedging price are derived under the criterion of no-arbitrage in this paper, that is, the absence of immediate profit. Moreover, numerical experiments are implemented to verify that the condition of absence of immediate profit can be a good substitute for the traditional no-arbitrage, since the latter is difficult to achieve. Thus, it may provide a better principle of pricing due to the flexibility from the optional confidence level for the market participants in the increasingly complex financial market.
- quantile hedging,
- uncertain market,
- no-arbitrage,
- absence of immediate profit,
- hedging risk
Citation: Jun Zhao, Peibiao Zhao. Quantile hedging for contingent claims in an uncertain financial environment[J]. AIMS Mathematics, 2023, 8(7): 15651-15669. doi: 10.3934/math.2023799

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Abstract

This paper first studies the quantile hedging problem of contingent claims in an uncertain market model. A special kind of no-arbitrage, that is, the absence of immediate profit, is characterized. Instead of the traditional no-arbitrage targeting the whole market, the absence of immediate profit depends on the confidence level of the portfolio manager for hedging risk. We prove that the condition of absence of immediate profit holds if and only if the initial price of each risky asset lies between the $ \alpha $-optimistic value and $ \alpha $-pessimistic value of its discounted price at the end of the period. The bounds of the minimal quantile hedging price are derived under the criterion of no-arbitrage in this paper, that is, the absence of immediate profit. Moreover, numerical experiments are implemented to verify that the condition of absence of immediate profit can be a good substitute for the traditional no-arbitrage, since the latter is difficult to achieve. Thus, it may provide a better principle of pricing due to the flexibility from the optional confidence level for the market participants in the increasingly complex financial market.

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