European option pricing model based on uncertain fractional differential equation

Abstract

In this paper, we investigate a new version of stock model under uncertain circumstances for uncertain stock markets. Firstly, solutions to some uncertain fractional differential equations are presented by employing the Mittag-Leffler function. Then, a new uncertain stock model with mean-reverting process is formulated on the basis of uncertain fractional differential equations. Finally, European option pricing formulas based on the proposed model are investigated as well as some numerical examples.

Appendix

In this appendix, some special functions and their properties will be recalled, and two types of fractional order derivatives of a function will be reviewed. More details can be seen in Podlubny (1999) and Kilbas et al. (2006). The gamma function defined by

$$\begin{aligned} \varGamma (p)=\int _{0}^{+\infty }t^{p-1}e^{-t}dt, p>0 \end{aligned}$$

has the properties that $\varGamma (p+1)=p\varGamma (p), p>0$; $\varGamma (1)=1$, $\varGamma (\frac{1}{2}) =\sqrt{\pi },\varGamma (n+1)=n!$. The Mittag-Leffler function is defined by

$$\begin{aligned} E_{\eta ,\nu }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\varGamma (k\eta +\nu )},(\eta>0,\nu >0). \end{aligned}$$

We usually denote $E_{\eta }(z)=E_{\eta ,1}(z)$ and $E_{1,1}(z)=\sum _{k=0}^{\infty }\frac{z^k}{\varGamma (k+1)}=e^z$.

Definition 7

For a function f(t) given on the interval [a, b], and $p>0$. Then the p-th fractional order integral of f is defined as

$$\begin{aligned} I_{a+}^pf(t)=\frac{1}{\varGamma (p)}\int _{a}^{t}(t-s)^{p-1}f(s)ds. \end{aligned}$$

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Definition 8

For a function f given on the interval [a, b], and $0\le n-1\le p<n$. Then the p-th Riemann–Liouville fractional derivative of f is defined by

$$\begin{aligned} D_{a+}^pf(t)=\frac{1}{\varGamma (n-p)}\frac{d^n}{dt^{n}}\int _{a}^{t}(t-s)^{n-p-1}f(s)ds. \end{aligned}$$

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Remark 5

For an arbitrary positive number p with $0\le n-1\le p<n$, the fractional derivative and the fractional integral have the relations:

• (i) $D_{a+}^p$$I_{a+}^pf(t)=f(t)$;

• (ii) $I^{p}_{a+}$ $D_{a+}^{p}f(t)=f(t)-\sum _{j=1}^{n}[D_{a+}^{p-j}f(t)]_{t=a}\frac{(t-a)^{p-j}}{\varGamma (p-j+1)}.$

Definition 9

Let $f:[a,b]\rightarrow R$ be a differentiable function at least n-order, and $0\le n-1< p\le n$. Then the p-th Caputo fractional derivative of f is defined by

$$\begin{aligned} ^cD_{a+}^pf(t)=\frac{1}{\varGamma (n-p)}\int _{a}^{t}(t-s)^{n-p-1}f^{(n)}(s)ds, \end{aligned}$$

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where $f^{(n)}(t)$ represents the n-order derivatives of f(t).

Remark 6

For an arbitrary positive number p with $0\le n-1< p\le n$, the fractional derivative and the fractional integral have the relations:

• (i) $^cD_{a+}^p$$I_{a+}^pf(t)=f(t)$;

• (ii) $I^{p}_{a+}$ $^cD_{a+}^{p}f(t)=f(t)-\sum _{k=0}^{n-1}\frac{(t-a)^{k}}{\varGamma (k+1)}f^{(k)}(a).$

Remark 7

For $0\le n-1<p\le n$ and $t>0$, the relationship between these two type of fractional derivatives satisfies

$$\begin{aligned} ^cD^p_{a+}f(t)=D^p_{a+}f(t)-\sum _{k=0}^{n-1}\frac{(t-a)^{k-p}}{\varGamma (k-p+1)}f^{(k)}(a). \end{aligned}$$

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Remark 8

Denote the $I_{0+}^p, D_{0+}^p$ and $^cD_{0+}^p$ by the abbreviations $I^p, D^p$ and $^cD^p$ respectively.