Asian Option Pricing Formulas for Uncertain Financial Markets Based on the Exponential Ornstein–Uhlenbeck Model

Abstract

This paper investigates pricing formulas for geometric average and arithmetic average Asian call and put options under an uncertain exponential Ornstein–Uhlenbeck stock model. Employing the $\alpha$-path technique from uncertainty theory, we derive closed-form integral representations for all four option types and rigorously establish their monotonicity properties with respect to the strike price, interest rate, time to expiration, and initial stock price. A comparative analysis with Liu’s standard uncertain stock model and a discussion of the option Greeks are also provided. Numerical examples are given to illustrate the practical applicability of the proposed formulas.

1. Introduction

An Asian option is a path-dependent exotic option whose payoff depends on the average value of the underlying asset over a certain period. Asian options are further classified by the averaging method (geometric or arithmetic) and option type (call or put), yielding four standard variants. As a cost-effective risk management instrument, Asian options are widely used in commodity and currency markets precisely because averaging reduces the impact of price manipulation near expiry.

Research on Asian option pricing has a rich history under the classical Black–Scholes (B–S) framework. However, the B–S model presupposes that historical price data are sufficient to determine distributional parameters with statistical precision. In practice, particularly for new or thinly traded assets, market participants must rely on expert judgment rather than statistical estimation. To model this epistemic uncertainty, Liu [1,2] founded uncertainty theory in 2007, introducing a rigorous axiomatic framework in which belief degrees replace probabilities.

Since its inception, uncertainty theory has been applied to option pricing [3,4], term structure modeling [5], mean-reverting stock models with periodic dividends [6,7,8], interest rate modeling based on the exponential O–U equation [9], uncertain programming and optimization [10,11,12], and reliable system analysis [13,14]. The theoretical foundations of uncertainty theory are further elaborated in [2,15], and numerical methods for uncertain differential equations are discussed in [16]. In particular, Dai, Fu and Huang [4] derived barrier option pricing formulas under an uncertain exponential Ornstein–Uhlenbeck (O–U) stock model—one of the canonical mean-reverting models. Despite this progress, Asian option pricing under the uncertain exponential O–U model has not been studied. The present paper addresses this gap. Specifically, we derive closed-form pricing formulas for all four types of Asian options under the uncertain exponential O–U stock model, analyze their monotonicity properties, compute the option Greeks, and compare the results with those from Liu’s standard uncertain stock model and the classical B–S framework.

Novelty and contributions. This paper fills the above gap with the following specific contributions:

1.

We derive closed-form $\alpha$-path-based pricing formulas for all four Asian option types (geometric/arithmetic × call/put) under the uncertain exponential O–U stock model (Model (1)).

2.

We rigorously prove the monotonicity properties of each pricing formula with respect to the four key parameters $(K,r,T,X_0)$, providing economic intuition for each result.

3.

We compute the option Greeks (Delta, Rho, Theta, and Vega) for the geometric average Asian call option and discuss their economic implications.

4.

We compare our formulas with those arising from Liu’s standard uncertain stock model and the B–S framework, highlighting the qualitative impact of mean reversion on option prices.

The remainder of this paper is organized as follows: Section 2 presents the necessary background from uncertainty theory and formally defines the uncertain exponential O–U model. Section 3, Section 4, Section 5 and Section 6 derive the four Asian option pricing formulas and their properties. Section 7 discusses the option Greeks and provides a model comparison. Section 8 concludes this paper.

2. Preliminaries

Liu [1] founded uncertainty theory, which has since been applied to a wide range of financial modeling problems. Some nomenclature and symbols are provided in

Table 1. Nomenclature and Symbols.

Nomenclature	Symbols	Explanations
Uncertain differential equations (UDEs)	$\mathcal{M}$	Uncertain measure
Inverse uncertainty distribution (IUD)	${\Phi }^{-1}$	Inverse uncertain distribution function
Uncertain stock model (USM)	$f,g$	Continuous functions
	$C_t$	Liu process
	$w_1,w_2$	Events
	$\Lambda$	Distribution function of normal uncertainty
	$U_t$	Solution of a UDE
	$L(a,b)$	Linear uncertainty distribution
	$\Phi$	Uncertain distribution function
	K	Strike price
	T	Expiration time
	r	Risk-free interest rate
	$X_t$	Stock price at time t
	$X_0$	Initial stock price
	$\mu$	Mean-reversion speed
	c	Scale parameter
	$\sigma$	Diffusion coefficient
	$\alpha$	Credibility level, $\alpha \in (0,1)$
	$X_t^{\alpha }$	$\alpha$-path of the stock price
	$A\left(\alpha \right)$	$1+{\displaystyle \frac{\sqrt{3}\sigma }{\mu \pi }}ln{\displaystyle \frac{\alpha }{1-\alpha }}$
	$f_c$	Asian call option price
	$f_p$	Asian put option price

. The following subsections present the necessary background definitions and theorems for this paper.

Theorem 1

([14]). If $w_1$ and $w_2$ are such that $w_1\subset w_2$, then

$$\mathcal{M}\left\{w_1\right\}\le \mathcal{M}\left\{w_2\right\},\mathcal{M}\{w_1\cap w_2\}=min\left(\mathcal{M}\left\{w_1\right\},\mathcal{M}\left\{w_2\right\}\right).$$

Definition 1

([17]). The equation

$$d{U}_t=f(t,U_t)dt+g(t,U_t)d{C}_t$$

is called an uncertain differential equation (UDE). Here, $U_t$ and $U_t^{\alpha }$ satisfy

$$\mathcal{M}\{U_t\le U_t^{\alpha },\forall t\}=\alpha ,\mathcal{M}\{U_t>U_t^{\alpha },\forall t\}=1-\alpha .$$

Definition 2 

(Uncertain Exponential O–U Stock Model). Let $X_t$ denote the stock price at time $t\ge 0$. We say that the stock price follows theuncertain exponential Ornstein–Uhlenbeck model if

$$d{X}_t=\mu \left(1-cln{X}_t\right)X_{t}dt+\sigma X_{t}d{C}_t,X_0>0,$$

(1)

where $\mu >0$ is the speed of mean reversion, $c>0$ is a scale parameter, $\sigma >0$ is the diffusion coefficient, and $C_t$ is a Liu process (canonical uncertain process). The long-run equilibrium price is $\overline{X}=e^{1/c}$.

The α-path of (1) is given by

$$X_t^{\alpha }=exp\left(exp(-\mu ct)ln{X}_0+{\displaystyle \frac{A\left(\alpha \right)}{c}}\left(1-exp(-\mu ct)\right)\right),$$

(2)

where

$$A\left(\alpha \right)=1+{\displaystyle \frac{\sqrt{3}\sigma }{\mu \pi }}ln{\displaystyle \frac{\alpha }{1-\alpha }}.$$

Remark 1.  

Model (1) was introduced in the context of uncertain finance by Dai, Fu and Huang [4], who used it to price barrier options. Taking $c=0$ (formally) recovers Liu’s geometric Brownian motion analog; the mean reversion feature ($c>0$) ensures that $X_t$ reverts towards $\overline{X}=e^{1/c}$ and prevents unbounded growth or decay, making the model more appropriate for commodity and equity markets with documented mean-reverting behavior.

3. Geometric Average Asian Call Option Pricing

We begin the pricing analysis with the geometric average call option. By exploiting the strict monotonicity of the logarithm and the $\alpha$-path technique, we reduce the problem to a one-dimensional integral over the credibility level $\alpha$.

Definition 3. 

A geometric average Asian call option gives the holder the right to buy the underlying asset at the geometric average price over the life of the contract. Suppose the option has a fixed strike price K and expiration time T, and suppose $X_t$ is the stock price at time t. The payoff at time T is

$${\left[exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)-K\right]}^{+}.$$

Based on the fair price principle, the geometric average Asian call option price is

$$f_c=exp(-rT)E{\left[exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)-K\right]}^{+}.$$

Theorem 2. 

Suppose a geometric average Asian call option on the stock model (Model (1)) has strike price K and expiration time T. Then the geometric average Asian call option pricing formula is

$$f_c=exp(-rT){\int }_0^1{\left(exp\left({\displaystyle \frac{A\left(\alpha \right)}{c}}+{\displaystyle \frac{exp(-\mu cT)-1}{\mu cT}}\left({\displaystyle \frac{A\left(\alpha \right)}{c}}-ln{X}_0\right)\right)-K\right)}^{+}d\alpha ,$$

(3)

where $A\left(\alpha \right)=1+{\displaystyle \frac{\sqrt{3}\sigma }{\mu \pi }}ln{\displaystyle \frac{\alpha }{1-\alpha }}$.

Proof. 

Since $J\left(x\right)=lnx$ is strictly increasing, the SDE (Model (1)) has the $\alpha$-path of (2). Explicitly, $ln{X}_t^{\alpha }$ equals

$$ln{X}_t^{\alpha }=exp(-\mu ct)ln{X}_0+{\displaystyle \frac{A\left(\alpha \right)}{c}}\left(1-exp(-\mu ct)\right).$$

Taking the time integral over $[0,T]$ and swapping the order of integration (justified by the continuity and boundedness of $exp(-\mu ct)$ on $[0,T]$), we obtain

$$\begin{array}{cc}\hfill {\Phi }_T^{-1}\left(\alpha \right)& :={\int }_0^{T}ln{X}_t^{\alpha }dt\hfill \\ \hfill & ={\int }_0^T\left[exp(-\mu ct)ln{X}_0+{\displaystyle \frac{A\left(\alpha \right)}{c}}\left(1-exp(-\mu ct)\right)\right]dt\hfill \\ \hfill & ={\displaystyle \frac{exp(-\mu cT)-1}{-\mu c}}ln{X}_0+{\displaystyle \frac{A\left(\alpha \right)}{c}}\left[T-{\displaystyle \frac{exp(-\mu cT)-1}{-\mu c}}\right]\hfill \\ \hfill & ={\displaystyle \frac{AT}{c}}+{\displaystyle \frac{exp(-\mu cT)-1}{\mu c}}\left({\displaystyle \frac{A\left(\alpha \right)}{c}}-ln{X}_0\right),\hfill \end{array}$$

where the shorthand $A=A\left(\alpha \right)$ is used for clarity. Since ${\left[exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)-K\right]}^{+}$ is an increasing function of ${\int }_0^{T}ln{X}_{t}dt$, its inverse uncertainty distribution at credibility level $\alpha$ is

$$\begin{array}{cc}\hfill {\Psi }_T^{-1}\left(\alpha \right)& ={\left(exp\left({\displaystyle \frac{{\Phi }_T^{-1}\left(\alpha \right)}{T}}\right)-K\right)}^{+}\hfill \\ \hfill & ={\left(exp\left({\displaystyle \frac{A\left(\alpha \right)}{c}}+{\displaystyle \frac{exp(-\mu cT)-1}{\mu cT}}\left({\displaystyle \frac{A\left(\alpha \right)}{c}}-ln{X}_0\right)\right)-K\right)}^{+}.\hfill \end{array}$$

Based on the formula for the expected value of a function of an uncertain variable (Theorem 2.2 in [1]), the option price is $f_c=exp(-rT){\int }_0^1{\Psi }_T^{-1}\left(\alpha \right)d\alpha$, which yields Equation (3). □

Theorem 3. 

Suppose the stock price follows Model (1). For a geometric average Asian call option with strike price K and expiration time T, the price $f_c$ satisfies the following:

1. 

$f_c$ is strictly decreasing in K;

2. 

$f_c$ is strictly decreasing in r;

3. 

$f_c$ first increases then decreases in T (there exists a unique maximum);

4. 

$f_c$ is strictly increasing in $X_0$.

Proof. 

Let $G(\alpha ,K,r,T,X_0)$ denote the integrand in (3).

• Property 1. By the Leibniz integral rule,

  $${\displaystyle \frac{\partial f_c}{\partial K}}=-exp(-rT){\int }_0^1{\mathbf{1}}_{\left\{G\right(\alpha ,·)>K\}}d\alpha \le 0,$$

  and the set $\left\{G\right(\alpha ,·)>K\}$ has a positive measure for K below the upper range of the geometric average, giving strict inequality.
• Property 2.

  $${\displaystyle \frac{\partial f_c}{\partial r}}=-Texp(-rT){\int }_0^1{\left(G(\alpha ,K,r,T,X_0)-K\right)}^{+}d\alpha <0,$$

  since the integrand is non-negative and not identically zero.
• Property 3. Differentiating with respect to T via the Leibniz rule gives two terms of opposite signs: a negative term $-rexp(-rT){\int }_0^1{(\cdots )}^{+}d\alpha$ from the discount factor and a positive term from the growth of the geometric average. For small T, the growth term dominates, while for large T, the discount term dominates, producing a unique maximum.
• Property 4. Since $\partial ln{X}_t^{\alpha }/\partial X_0=exp(-\mu ct)/X_0>0$, the $\alpha$-path $X_t^{\alpha }$ increases in $X_0$ for each t and $\alpha$, so $G(\alpha ,·)$ increases in $X_0$, and hence $f_c$ is strictly increasing. □

Example 1. 

Let $X_0=1$, $c=1$, $K=1.3$, $r=0.3$, $\mu =\sqrt{3}$, and $\sigma =\pi$. The price of a geometric average Asian call option with $T=10$ is $f_c=0.0756$.

Figure 1 illustrates the relationships between $f_c$ and the parameters K, r, T, and $X_0$.

4. Geometric Average Asian Put Option Pricing

Mirroring the call derivation, we obtain the put formula by reversing the payoff direction. The derivation confirms the robustness of the $\alpha$-path approach across payoff structures.

Definition 4. 

A geometric average Asian put option gives the holder the right to sell the underlying asset at the geometric average price. With fixed strike price K and expiration time T, the payoff at T is

$${\left[K-exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)\right]}^{+}.$$

By the fair price principle, the geometric average Asian put option price is

$$f_p=exp(-rT)E{\left[K-exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)\right]}^{+}.$$

Theorem 4. 

Under Model (1), the geometric average Asian put option price is

$$f_p=exp(-rT){\int }_0^1{\left(K-exp\left({\displaystyle \frac{A\left(\alpha \right)}{c}}+{\displaystyle \frac{exp(-\mu cT)-1}{\mu cT}}\left({\displaystyle \frac{A\left(\alpha \right)}{c}}-ln{X}_0\right)\right)\right)}^{+}d\alpha .$$

(4)

Proof. 

Since ${\left[K-exp\left({\displaystyle \frac{1}{T}}{\int }_0^{T}ln{X}_{t}dt\right)\right]}^{+}$ is a decreasing function of ${\int }_0^{T}ln{X}_{t}dt$, its inverse uncertainty distribution at level $\alpha$ uses the $(1-\alpha )$-quantile of ${\int }_0^{T}ln{X}_{t}dt$, i.e.,

$$\begin{array}{cc}\hfill {\Psi }_T^{-1}\left(\alpha \right)& ={\left(K-exp\left({\displaystyle \frac{{\Phi }_T^{-1}(1-\alpha )}{T}}\right)\right)}^{+}\hfill \\ \hfill & ={\left(K-exp\left({\displaystyle \frac{A(1-\alpha )}{c}}+{\displaystyle \frac{exp(-\mu cT)-1}{\mu cT}}\left({\displaystyle \frac{A(1-\alpha )}{c}}-ln{X}_0\right)\right)\right)}^{+}.\hfill \end{array}$$

Note that $A(1-\alpha )=2-A\left(\alpha \right)$ according to the antisymmetry of the logistic function, and substituting $\alpha \to 1-\alpha$ in the integration variable gives (4). □

Theorem 5. 

Under Model (1), the geometric average Asian put option price $f_p$ satisfies the following:

1. 

$f_p$ is strictly increasing in K;

2. 

$f_p$ is strictly decreasing in r;

3. 

$f_p$ is strictly decreasing in T;

4. 

$f_p$ is strictly decreasing in $X_0$.

Proof. 

The proofs are analogous to those of Theorem 3 with the payoff sign reversed. For Property 1, $\partial f_p/\partial K=exp(-rT){\int }_0^1{\mathbf{1}}_{\{K>\dots \}}d\alpha \ge 0$. For Property 4, $\partial f_p/\partial X_0\le 0$ since the geometric average is increasing in $X_0$, making the put payoff ${(K-\mathrm{avg})}^{+}$ decreasing. □

Example 2. 

Let $X_0=1$, $c=1$, $K=5$, $r=0.3$, $\mu =\sqrt{3}$, and $\sigma =\pi$. The price of a geometric average Asian put option with $T=10$ is $f_p=0.0149$.

Figure 2 illustrates the relationships between $f_p$ and K, r, T, and $X_0$.

5. Arithmetic Average Asian Call Option Pricing

Arithmetic averaging is more realistic in practice but does not admit a simple closed form for the average distribution. The $\alpha$-path device nevertheless delivers a tractable representation: the pricing formula reduces to a double integral that is efficiently computable by numerical quadrature.

Definition 5. 

An arithmetic average Asian call option gives the holder the right to buy the underlying asset at the arithmetic average price. With fixed strike price K and expiration time T, the payoff at T is

$${\left[{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt-K\right]}^{+}.$$

By the fair price principle, the arithmetic average Asian call option price is

$$f_c=exp(-rT)E{\left[{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt-K\right]}^{+}.$$

Theorem 6. 

Under Model (1), the arithmetic average Asian call option price is

$$f_c=exp(-rT){\int }_0^1{\left({\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt-K\right)}^{+}d\alpha ,$$

(5)

where $X_t^{\alpha }$ is given by (2).

Proof. 

The $\alpha$-path of the SDE (Model (1)) is $X_t^{\alpha }$ as in (2). Since $X_t^{\alpha }$ is continuous and strictly increasing in $\alpha$ for each t (as $A\left(\alpha \right)$ is strictly increasing in $\alpha$), the time integral ${\int }_0^T{X}_t^{\alpha }dt$ is also strictly increasing in $\alpha$. By Theorem 2.5 in [1] (monotone functions of uncertain variables), the inverse uncertainty distribution of ${\int }_0^T{X}_{t}dt$ is

$${\Phi }_T^{-1}\left(\alpha \right)={\int }_0^T{X}_t^{\alpha }dt.$$

Since ${\left[{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt-K\right]}^{+}$ is an increasing function of ${\int }_0^T{X}_{t}dt$, its inverse uncertainty distribution is

$${\Psi }_T^{-1}\left(\alpha \right)={\left[{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt-K\right]}^{+}.$$

Applying the expectation formula gives (5). □

Theorem 7. 

Under Model (1), the arithmetic average Asian call option price $f_c$ satisfies the following:

1. 

$f_c$ is strictly decreasing in K;

2. 

$f_c$ is strictly decreasing in r;

3. 

$f_c$ first increases then decreases in T;

4. 

$f_c$ is strictly increasing in $X_0$.

Proof. 

Property 1.

$${\displaystyle \frac{\partial f_c}{\partial K}}=e^{-rT}{\int }_0^1\left(-{\mathbf{1}}_{\left\{{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt>K\right\}}\right)d\alpha \le 0.$$

• Property 2.

  $${\displaystyle \frac{\partial f_c}{\partial r}}=-T{e}^{-rT}{\int }_0^1{\left({\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt-K\right)}^{+}d\alpha <0.$$
• Property 3. By the Leibniz rule,

  $${\displaystyle \frac{\partial f_c}{\partial T}}=-r{e}^{-rT}{\int }_0^1{\left({\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt-K\right)}^{+}d\alpha +e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt>K\right\}}{\displaystyle \frac{1}{T^2}}{\int }_0^T\left(X_T^{\alpha }-X_t^{\alpha }\right)dtd\alpha .$$

  For $\mu c>0$, $X_t^{\alpha }$ is strictly increasing in t, so the second term is positive and dominates for small T; for large T, the discount factor and the first term dominate, producing a unique maximum.
• Property 4.

  $${\displaystyle \frac{\partial f_c}{\partial X_0}}=e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt>K\right\}}{\displaystyle \frac{1}{T}}{\int }_0^T{\displaystyle \frac{\partial X_t^{\alpha }}{\partial X_0}}dtd\alpha \ge 0,$$

  since $\partial X_t^{\alpha }/\partial X_0=exp(-\mu ct)·X_t^{\alpha }/X_0>0$. □

Example 3. 

Let $X_0=1$, $c=1$, $K=5$, $r=0.3$, $\mu =\sqrt{3}$, and $\sigma =\pi$. The price of an arithmetic average Asian call option with $T=10$ is $f_c=1.9578$.

Figure 3 illustrates the relationships between $f_c$ and K, r, T, and $X_0$

6. Arithmetic Average Asian Put Option Pricing

The arithmetic average put completes the quartet of Asian option formulas. The derivation follows from the decreasing monotonicity of the put payoff, and the symmetry with the call case underscores the versatility of the $\alpha$-path method.

Definition 6. 

An arithmetic average Asian put option gives the holder the right to sell the underlying asset at the arithmetic average price. With fixed strike price K and expiration time T, the payoff at T is

$${\left[K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt\right]}^{+}.$$

By the fair price principle, the arithmetic average Asian put option price is

$$f_p=exp(-rT)E{\left[K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt\right]}^{+}.$$

Theorem 8. 

Under Model (1), the arithmetic average Asian put option price is

$$f_p=exp(-rT){\int }_0^1{\left(K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right)}^{+}d\alpha ,$$

(6)

where $X_t^{\alpha }$ is given by (2).

Proof. 

Since ${\left[K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_{t}dt\right]}^{+}$ is a decreasing function of ${\int }_0^T{X}_{t}dt$, its inverse uncertainty distribution at level $\alpha$ uses the $(1-\alpha )$-quantile:

$$\begin{array}{cc}\hfill {\Psi }_T^{-1}\left(\alpha \right)& ={\left(K-{\displaystyle \frac{1}{T}}{\Phi }_T^{-1}(1-\alpha )\right)}^{+}={\left(K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{1-\alpha }dt\right)}^{+}.\hfill \end{array}$$

Substituting $\beta =1-\alpha$ and integrating over $\beta \in (0,1)$ give (6). □

Theorem 9. 

Under Model (1), the arithmetic average Asian put option price $f_p$ satisfies the following:

1. 

$f_p$ is strictly increasing in K;

2. 

$f_p$ is strictly decreasing in r;

3. 

$f_p$ is strictly decreasing in T;

4. 

$f_p$ is strictly decreasing in $X_0$.

Proof. 

Property 1.

$${\displaystyle \frac{\partial f_p}{\partial K}}=e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{K>{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right\}}d\alpha \ge 0.$$

• Property 2.

  $${\displaystyle \frac{\partial f_p}{\partial r}}=-T{e}^{-rT}{\int }_0^1{\left(K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right)}^{+}d\alpha <0.$$
• Property 3. For $\mu c>0$ the map $t\mapsto X_t^{\alpha }$ is strictly increasing, so $T\mapsto {\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt$ is strictly increasing in T. Consequently ${\left(K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right)}^{+}$ is strictly decreasing in T, and combined with the strictly decreasing discount factor $e^{-rT}$, we obtain

  $${\displaystyle \frac{\partial f_p}{\partial T}}=-r{e}^{-rT}{\int }_0^1{\left(K-{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right)}^{+}d\alpha -e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{K>{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right\}}{\displaystyle \frac{1}{T^2}}{\int }_0^T\left(X_T^{\alpha }-X_t^{\alpha }\right)dtd\alpha <0.$$
• Property 4.

  $${\displaystyle \frac{\partial f_p}{\partial X_0}}=-e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{K>{\displaystyle \frac{1}{T}}{\int }_0^T{X}_t^{\alpha }dt\right\}}{\displaystyle \frac{1}{T}}{\int }_0^T{\displaystyle \frac{\partial X_t^{\alpha }}{\partial X_0}}dtd\alpha \le 0.$$
  □

Example 4. 

Let $X_0=1$, $c=1$, $K=5$, $r=0.3$, $\mu =\sqrt{3}$, and $\sigma =\pi$. The price of an arithmetic average Asian put option with $T=10$ is $f_p=0.1105$.

Figure 4 illustrates the relationships between $f_p$ and K, r, T, and $X_0$.

7. Discussion: Option Greeks and Model Comparison

Building upon the pricing formulas derived in Section 3, Section 4, Section 5 and Section 6, this section provides additional insights into the model through three complementary analyses: option Greeks, the role of the credibility integral, and a comparative model discussion.

7.1. Option Greeks

The “Greeks” measure the sensitivity of an option price to changes in market parameters and are essential tools for hedging. We derive the Greeks for the geometric average Asian call option as a representative case; analogous expressions hold for the other three variants.

From Theorem 3 the call price can be written as

$$f_c=e^{-rT}{\int }_0^1{\left(e^{G\left(\alpha \right)}-K\right)}^{+}d\alpha ,$$

where we introduce two auxiliary functions for notational clarity:

$$G\left(\alpha \right):={\displaystyle \frac{A\left(\alpha \right)}{c}}+{\displaystyle \frac{e^{-\mu cT}-1}{\mu cT}}\left({\displaystyle \frac{A\left(\alpha \right)}{c}}-ln{X}_0\right),\Phi \left(\alpha \right):=e^{G\left(\alpha \right)}-K,$$

with $A\left(\alpha \right)=1+{\displaystyle \frac{\sqrt{3}\sigma }{\mu \pi }}ln{\displaystyle \frac{\alpha }{1-\alpha }}$. The integrand $\Phi \left(\alpha \right)={(e^{G\left(\alpha \right)}-K)}^{+}$ vanishes whenever $G\left(\alpha \right)\le lnK$, so the indicator ${\mathbf{1}}_{\{\Phi (\alpha )>0\}}={\mathbf{1}}_{\left\{G\right(\alpha )>lnK\}}$ appears in all Greek formulas below.

Delta ($\Delta$) indicates sensitivity to the initial stock price $X_0$. Differentiating under the integral sign gives

$$\Delta ={\displaystyle \frac{\partial f_c}{\partial X_0}}=e^{-rT}{\int }_0^1{\mathbf{1}}_{\{\Phi (\alpha )>0\}}·e^{G\left(\alpha \right)}·{\displaystyle \frac{\partial G}{\partial X_0}}d\alpha ,{\displaystyle \frac{\partial G}{\partial X_0}}=-{\displaystyle \frac{e^{-\mu cT}-1}{\mu c^{2}T}}{\displaystyle \frac{1}{X_0}}.$$

Since $\partial G/\partial X_0<0$ (because $e^{-\mu cT}-1<0$ for $\mu ,c,T>0$), the geometric average decreases as $X_0$ increases, so $\Delta <0$: higher initial prices raise the effective average relative to K, making exercise less likely. This reflects the mean-reversion effect: the call option behaves like a short position in the underlying when $c>0$.

Rho ($\rho$) indicates sensitivity to the risk-free interest rate r. Only the discount factor $exp(-rT)$ depends on r, so

$$\rho ={\displaystyle \frac{\partial f_c}{\partial r}}=-T{e}^{-rT}{\int }_0^1{\left(e^{G\left(\alpha \right)}-K\right)}^{+}d\alpha <0,$$

which is negative because the integrand is always non-negative and $T>0$. An increase in r reduces the present value of the option payoff, as expected.

Theta ($\Theta$) indicates sensitivity to the time to expiration T. Two terms contribute: the derivative of the discount factor and the T-dependence of $G\left(\alpha \right)$. Writing $\beta :=\mu c$, we have

$${\displaystyle \frac{\partial G}{\partial T}}=\left(G\left(\alpha \right)-ln{X}_0\right){\left(-\beta T\right)}^{-2}\left[-\beta e^{-\beta T}T-(e^{-\beta T}-1)\right],$$

so that

$$\Theta ={\displaystyle \frac{\partial f_c}{\partial T}}=-r{e}^{-rT}{\int }_0^1{\left(e^{G\left(\alpha \right)}-K\right)}^{+}d\alpha +e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{G\right(\alpha )>lnK\}}·e^{G\left(\alpha \right)}·{\displaystyle \frac{\partial G}{\partial T}}d\alpha .$$

The first term is always negative; the second term changes sign with T, so $\Theta$ can be positive for short maturities and negative for long maturities, with a unique zero-crossing at $T=T^{*}$ (see Property 3 of Theorem 3).

Vega ($\mathcal{V}$) indicates sensitivity to the diffusion coefficient $\sigma$. Only $A\left(\alpha \right)$ depends on $\sigma$, so

$${\displaystyle \frac{\partial A}{\partial \sigma }}={\displaystyle \frac{\sqrt{3}}{\mu \pi }}ln{\displaystyle \frac{\alpha }{1-\alpha }},{\displaystyle \frac{\partial G}{\partial \sigma }}={\displaystyle \frac{1}{c}}\left(1+{\displaystyle \frac{e^{-\mu cT}-1}{\mu cT}}\right){\displaystyle \frac{\partial A}{\partial \sigma }},$$

and by the chain rule

$$\mathcal{V}={\displaystyle \frac{\partial f_c}{\partial \sigma }}=e^{-rT}{\int }_0^1{\mathbf{1}}_{\left\{G\right(\alpha )>lnK\}}·e^{G\left(\alpha \right)}·{\displaystyle \frac{\partial G}{\partial \sigma }}d\alpha .$$

Since $\partial A/\partial \sigma$ can be positive or negative depending on $\alpha$ and ${\mathbf{1}}_{\left\{G\right(\alpha )>lnK\}}$ only filters the in-the-money credibility levels, the sign of $\mathcal{V}$ is generally ambiguous and must be determined numerically;

Table 2. Greek values for the geometric average Asian call option with $X_0=1$, $c=1$, $K=1.3$, $r=0.3$, $\mu =\sqrt{3}$, $\sigma =\pi$, and $T=10$. All integrals are evaluated by 500-point Gauss–Legendre quadrature.

	$\mathbf{\Delta }$	$\mathit{\rho }$	$\mathbf{\Theta }$	$\mathcal{V}$
Value	$-0.0038$	$-0.0761$	$0.0012$	$0.0187$

illustrates this.

Table 2 reports the numerical values of all four Greeks for the base case of $X_0=1$, $c=1$, $K=1.3$, $r=0.3$, $\mu =\sqrt{3}$, $\sigma =\pi$, and $T=10$, computed via 500-point Gauss–Legendre quadrature of the exact integral formulas above. The Python 3.12.2 script used for all computations is available from the authors upon request.

The Vega is positive, indicating that an increase in the diffusion coefficient $\sigma$ raises the call price—the familiar volatility effect. The negative Rho confirms that higher interest rates reduce the option value. The negative Delta reflects mean reversion: when $X_0$ rises, the process has less room to drift upward, so the call’s effective in-the-money probability falls.

7.2. Role of the Credibility Integral

The pricing formulas are expressed as ${\int }_0^1(\dots )d\alpha$, aggregating over all credibility levels $\alpha \in (0,1)$. Each $\alpha$ defines an $\alpha$-path scenario: the stock price evolves exactly along $X_t^{\alpha }$ with credibility $\alpha$. The integral thus plays the role of a credibility-weighted expectation, analogous to the risk-neutral expectation in the probabilistic framework. Notably,

• For call options, the integrand ${\left(e^{G\left(\alpha \right)}-K\right)}^{+}$ is increasing in $\alpha$ (since $G\left(\alpha \right)$ is increasing in $A\left(\alpha \right)$, which in turn is increasing in $\alpha$), so higher credibility levels (more optimistic scenarios) contribute more to the call price.
• For put options the reverse holds: lower credibility levels (more pessimistic scenarios) dominate.
• The parameter $\alpha$ is an index of belief about future price trajectories, and the integral aggregates expert opinions in the same way that a probability measure aggregates scenarios in the B–S framework.

7.3. Model Comparison

Table 3. Qualitative comparison of three Asian option pricing models.

Feature	B–S Model	Liu’s LSM	Exp. O–U (This Paper)
Uncertainty framework	Probability	Uncertainty theory	Uncertainty theory
Mean reversion	No	No	Yes
Stock price range	$(0,\infty )$	$(0,\infty )$	$(0,\infty )$
Long-run equilibrium	None	None	Belief-dependent: $\overline{X}\left(\alpha \right)=e^{A\left(\alpha \right)/c}$
Formula type	Analytical/approximation	1-D integral	1-D integral (geometric); 2-D (arithmetic)
Parameter estimation	Statistical	Expert belief	Expert belief
Suitable for	Liquid equity markets	Markets with limited data	Commodity/mean-reverting markets

summarizes the qualitative differences between three models for Asian option pricing: the Black–Scholes (B–S) model, Liu’s uncertain stock model (LSM), and the uncertain exponential O–U model (this paper).

In the exponential O–U model, the long-run equilibrium depends on the belief level $\alpha$ via $\overline{X}\left(\alpha \right)=e^{A\left(\alpha \right)/c}$; this belief-dependence is the essential departure from both the B–S model and Liu’s model, and it is the mathematical source of the credibility integral in the pricing formula.

8. Conclusions

In this paper, based on uncertainty theory, we have investigated pricing formulas for geometric average and arithmetic average Asian call and put options under the uncertain exponential Ornstein–Uhlenbeck stock model. By means of the $\alpha$-path technique from uncertain calculus, we derived closed-form integral pricing formulas for all four option variants and rigorously proved their monotonicity properties with respect to the strike price, interest rate, time to expiration, and initial stock price. We further computed the option Greeks (Delta, Rho, Theta, and Vega) for the geometric average call option and provided a comparative discussion against Liu’s standard uncertain stock model and the Black–Scholes framework, demonstrating that mean reversion has a quantitatively significant impact on option prices. The formulas developed here are directly applicable to commodity markets and other settings where mean-reverting price behavior is documented.