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Lookback option pricing models based on the uncertain fractional-order differential equation with Caputo type

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Abstract

Owing to the ability to discern the future and the past in the over-the-counter market, the lookback option is regarded as one of the well-known path-dependent financial derivatives. Firstly, because the historical actual financial data is usually unreliable, scarce and unavailable, the underlying asset of the lookback option is regarded as an uncertain variable. Fractional-order derivative p rather than the integer derivative is applied and a more fine-grained portrayal of the real economic market is obtained based upon the uncertain fractional-order differential equation (UFDE). Next, through the existing extreme value theorems of the UFDE, European lookback (containing call and put cases) option pricing formulas are obtained for the uncertain fractional-order stock model (UFSM) and uncertain fractional-order mean-reverting models (UFMM), respectively. At last, the predictor-corrector method (PCM) is used to design numerical algorithms for calculating European lookback option price. Moveover, some numerical example are experienced to demonstrate the reasonability of our models.

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Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Abbreviations

IUD:

Inverse uncertain distribution

ODE:

Ordinary differential equation

UDE:

Uncertain differential equation

UFDE:

Uncertain fractional-order differential equation

\(\alpha\) :

Belief degree

\(\Gamma\) :

Gamma function

\(\Phi\) :

Uncertain normal distribution

\(C_t\) :

Canonical Liu process

F :

Continuous function

G :

Continuous function

k :

Stock drift

K :

Strike price

n :

Positive integer

p :

Caputo fractional-order derivative

\(\rho\) :

Stock diffusion

r :

no-risk rate

T :

Expiration time

\(X_t\) :

Stock price

\(Y_t\) :

Bond price

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos.12071219), Natural Science Foundation of Jiangsu Province (No. BK20210605) and supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD), and Student Innovation Training Program (2021NFUSPITP0717).

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Authors and Affiliations

  1. School of Science, Nanjing Forestry University, Nanjing, 210037, Jiangsu, China

    Ting Jin

  2. College of International Education, Nanjing Forestry University, Nanjing, 210037, Jiangsu, China

    Hongxuan Xia

  3. Faculty of Forestry, University of British Columbia, 2424 Main Mall, Vancouver, BC V6T 1Z4, Canada

    Hongxuan Xia

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  1. Ting Jin
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  2. Hongxuan Xia
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Correspondence to Ting Jin.

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Jin, T., Xia, H. Lookback option pricing models based on the uncertain fractional-order differential equation with Caputo type. J Ambient Intell Human Comput 14, 6435–6448 (2023). https://doi.org/10.1007/s12652-021-03516-y

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