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

## 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 availability

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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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 (2021). https://doi.org/10.1007/s12652-021-03516-y