Option pricing formulas in a new uncertain stock model with floating interest rate

1 Introduction

An option is a contract, which gives its holder the right, but not the obligation, to buy or sell a certain amount of an underlying asset (for example, stock), at a certain price (called the strike price), at a certain date (called the expiration date) written in the contract. There exist many options in financial market that are divided into different categories. This paper considers the pricing for the European, American and Asian options. The European option is low cost and high profit, so most of the international trading options are European options. The feature of an American option is its exercise privilege, that is, the holder can exercise the option prior to the date of expiration. Since the addition right should not be worthless, we expect an American option to be worth more than its European counterpart. Asian options are averaging options whose terminal payoff depends on some form of averaging of the price of the underlying asset over the whole of the option’s life.

With the development of financial market and the need of the hedging financial risk, option pricing has been a key issue of mathematical research in finance since the publication of Black-Scholes formula [1] in 1973. From then on, many researchers have investigated the prices of many kinds of options based on the Black-Scholes model and they obtain some useful results. As well known, the Black-Scholes model is based on probability theory in which the underlying asset price process follows the Brownian motion. However, there is a property of the path of Brownian motion which has infinite variation on any interval, no matter how small the interval is. It leads to the stock price having infinite changes in a very short period of time. This case is almost impossible in practical financial market. It makes the Black-Scholes model be inconsistent with the financial market. To avoid the problem, Liu [9] designed Liu process and suggested that uncertain differential equation may be a potential mathematical foundation of finance theory. Liu process can be regarded as a counterpart of Brownian motion. Different from Brownian motion, almost all sample path of Liu process are Lipschitz continuous, so using Liu process to describe the stock price process is more reasonable than using Brownian motion [12].

Since 2008, Liu [9] introduced the uncertain differential equation theory into financial field for the first time. And he proposed an uncertain stock model and derived European option pricing formulas in 2009 [10]. Many researchers have investigated the option pricing problems based on uncertainty theory. In 2011, Chen [2] gave the American option pricing formulas and Yao [6] investigated a new uncertain stock model that is now called Peng-Yao’s model. Yao [15], in 2012, gave no-arbitrage determinant theorems on mean-reverting stock model in uncertain market. Chen et al. [3] proposed an uncertain stock model with periodic dividends and obtained some options pricing formulas for this type of model in 2013. Zhang and Liu [20] obtained geometric average Asian option pricing formulas in 2014. Furthermore, in 2015, Ji and Zhou [5] discussed the option pricing formulas based on the uncertain stock model with jumps and Yao [19] derived a sufficient and necessary condition for the stock model being no-arbitrage, in the same year, Yao [18] also investigated uncertain contour process and applied it to stock model with floating interest rate. Zhang et al. [21] presented the power option pricing formulas in uncertain financial market.

In this paper, a new uncertain stock model with floating interest rate will be introduced and some option pricing formulas will be derived for this stock model. This paper is structured as follows. Section 2 will review some basic definitions and useful theorems on uncertain calculus and uncertain processes. A new uncertain stock model with floating interest rate will be introduced in Section 3. Sections 4, 5 and 6 will derive the pricing formulas for European option, American option and Asian option, respectively. Finally, a brief summary will be given in Section 7.

2 Preliminaries

In this section, we recall some basic concepts and properties of uncertainty theory and uncertain differential equation, which are used throughout this paper.

Definition 2.1. (Liu [8]) Let Γ be a nonempty set and ℒ be a σ-algebra on Γ. The set function M is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality Axiom) ℳ {Γ} =1 for the universal set Γ.

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ ^c } =1 for any Λ∈ ℒ.

Axiom 3. (Subadditivity Axiom) For every countable sequence Λ₁, Λ₂, ⋯ in ℒ, we have ℳ { ⋃ i = 1 ∞ Λ i } ≤ ∑ i = 1 ∞ ℳ { Λ i } .

The triple (Γ, ℒ , ℳ) is called an uncertainty space, and each element Λ in ℒ is called an event. In 2009, (Liu [10]) defined product uncertain measure by the following axiom:

Axiom 4. (Product Axiom) Let (Γ _k , ℒ  _k , _ℳk) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure ℳ is an uncertain measure satisfying ℳ { ∏ k = 1 ∞ Λ k } = ⋀ k = 1 ∞ ℳ k { Λ k } , where Λ _k are arbitrarily chosen events from ℒ _k , k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [8]) An uncertain variable is a function ξ from an uncertainty space (Γ, ℒ , ℳ) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers.

Definition 2.3. (Liu [8]) An an uncertain variable ξ is called normal if it has a normal uncertainty distribution Φ ( x ) = ( 1 + exp ( π ( e - x ) 3 σ ) ) - 1 , x ∈ R denoted by N ( e , σ ) , where e and σ are real numbers with σ > 0. The inverse uncertainty distribution of normal uncertain variable N ( e , σ ) is Φ - 1 ( α ) = e + σ 3 π ln α 1 - α .

Definition 2.4. An uncertainty variable ξ is called lognormal if it has a lognormal uncertainty distribution Φ ( x ) = ( 1 + exp ( π ( e - ln x ) 3 σ ) ) - 1 , x ≥ 0 denoted by ℒ𝒪𝒢𝒩 (e, σ), where e and σ are real numbers with σ > 0. The inverse uncertainty distribution of lognormal uncertain variable ℒ𝒪𝒢𝒩 (e, σ) is Φ - 1 ( α ) = exp ( e + σ 3 π ln α 1 - α ) .

Definition 2.5. (Liu [8]) The uncertain varibles ξ₁, ξ₂, ⋯ , ξ _n are said to be independent if ℳ { ⋂ i = 1 n ( ξ i ∈ B i ) } = ⋀ i = 1 n ℳ { ξ i ∈ B i } for any Borel sets B₁, B₂, ⋯ , B _n of real numbers.

Theorem 2.1. (Liu [11]) Let ξ₁, ξ₂, ⋯ , ξ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f (ξ₁, ξ₂, ⋯ , ξ _n ) is a strictly increasing function, then ξ = f ( ξ 1 , ξ 2 , ⋯ , ξ n ) has an inverse uncertainty distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , Φ 2 - 1 ( α ) , ⋯ , Φ n - 1 ( α ) ) .

Theorem 2.2. (Liu [11]) Let ξ₁, ξ₂, ⋯ , ξ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f (ξ₁, ξ₂, ⋯ , ξ _n ) is a strictly decreasing function, then ξ = f ( ξ 1 , ξ 2 , ⋯ , ξ n ) has an inverse uncertainty distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( 1 - α ) , Φ 2 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) .

Definition 2.6. (Liu [8]) The expected value of an uncertain variable ξ is defined by E [ ξ ] = ∫ 0 + ∞ ℳ { ξ ≥ x } d x - ∫ - ∞ 0 ℳ { ξ ≤ x } d x provided that at least one of the two integrals exists.

Definition 2.7. (Liu [11]) Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then E [ ξ ] = ∫ 0 1 Φ - 1 ( α ) d α .

Theorem 2.3. (Liu and Ha [14]) Assume ξ₁, ξ₂, ⋯ , ξ _n are independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If f (ξ₁, ξ₂, ⋯ , ξ _n ) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ _m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ _n , then ξ = f ( ξ 1 , ξ 2 , ⋯ , ξ n ) has an expected value E ( ξ ) = ∫ 0 1 f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ Φ n - 1 ( 1 - α ) ) d α .

An uncertain process is essentially a sequence of uncertain variables indexed by time. Liu process is the most important uncertain process.

Definition 2.8. (Liu [9]) Let T be a totally ordered set (e.g. time) and let (Γ, ℒ, ℳ) be an uncertainty space. An uncertain process is a function X _t  (γ) from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X _t  ∈ B} is an event for any Borel set B of real numbers at each time t.

Definition 2.9. (Liu [13]) Uncertain processes X_1t, X_2t, ⋯ , X _nt are said to be independent if for any positive integer k and any times t₁, t₂, ⋯ , t _k , the uncertain vectors ξ i = ( X it 1 , X it 2 , ⋯ , X it k ) , i = 1 , 2 , ⋯ , n are independent, i.e., for any Borel sets B₁, B₂, ⋯ , B _n of k-dimensional real vectors, we have ℳ { ⋂ i = 1 n ( ξ i ∈ B i ) } = ⋀ i = 1 n ℳ { ξ i ∈ B i } .

Definition 2.10. (Liu [8]) An uncertain process C _t is said to be a Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous,

Definition 2.11. (Liu [10]) Let X _t be an uncertain process and C _t be a Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as Δ = max 1 ≤ i ≤ k | t i + 1 - t i | .

Then Liu integral of X _t with respect to C _t is defined by ∫ a b X t d C t = lim Δ → 0 ∑ i = 1 k X t i ( C t i + 1 - C t i ) provided that the limit exists almost surely and is finite. In this case, the uncertain process X _t is said to be integrable.

The result of Liu integral is another uncertain process.

Definition 2.12. (Liu [9]) Suppose C _t is a Liu process, and f and g are two given functions. Then d X t = f ( t , X t ) d t + g ( t , X t ) d C t(1) is called an uncertain differential equation. A solution is an uncertain process that satisfies (1) identically in t.

Definition 2.13. (Yao and Chen [16]) Let α be a number with 0 < α < 1. An uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t is said to have an α-path X t α if it solves the corresponding ordinary differential equation d X t α = f ( t , X t α ) d t + | g ( t , X t α ) | Φ - 1 ( α ) d t where Φ^-1 (α) is the inverse uncertainty distribution of standard normal uncertain variable, i.e., Φ - 1 ( α ) = 3 π ln α 1 - α .

Theorem 2.4. (Yao-Chen Formula [16]) Let X _t and X t α be the solution and α-path of the uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t . Then ℳ { X t ≤ X t α , ∀ t } = α , ℳ { X t > X t α , ∀ t } = 1 - α .

Theorem 2.5. (Yao [16]) Let X _t and X t α be the solution and α-path of the uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t , respectively. Then the solution X _t has an inverse uncertainty distribution Φ t - 1 ( α ) = X t α .

Theorem 2.6.(Yao [17]) Let X _t and X t α be the solution and α-path of the uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t , respectively. Then for any time s > 0 and strictly increasing function J (x), the supremum sup 0 ≤ t ≤ s J ( X t ) has an inverse uncertainty distribution Ψ s - 1 ( α ) = sup 0 ≤ t ≤ s J ( X t α ) ; and the infimum inf 0 ≤ t ≤ s J ( X t ) has an inverse uncertainty distribution Ψ s - 1 ( α ) = inf 0 ≤ t ≤ s J ( X t α ) .

Theorem 2.7.(Yao [17]) Let X _t and X t α be the solution and α-path of the uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t , respectively. Then for any time s > 0 and strictly decreasing function J (x), the infimum sup 0 ≤ t ≤ s J ( X t ) has an inverse uncertainty distribution Ψ s - 1 ( α ) = sup 0 ≤ t ≤ s J ( X t 1 - α ) ; and the infimum inf 0 ≤ t ≤ s J ( X t ) has an inverse uncertainty distribution Ψ s - 1 ( α ) = inf 0 ≤ t ≤ s J ( X t 1 - α ) .

Theorem 2.8.(Yao [17]) Let X _t and X t α be the solution and α-path of the uncertain differential equation d X t = f ( t , X t ) d t + g ( t , X t ) d C t , respectively. Then for any time s > 0 and strictly increasing function J (x), the time integral ∫ 0 s J ( X t ) d t has an inverse uncertainty distribution Ψ s - 1 ( α ) = ∫ 0 s J ( X t α ) d t .

In order to get financial derivatives price, the scholars usually suppose the underlying asset prices follow differential equations. In 2009 Liu [9] established a stock model { d X t = rX t d t d S t = μ S t d t + σ S t d C t(2) where X _t is the bond price, S _t is the stock price, r is the riskless interest rate, μ is the log-drift, σ is the log-diffusion, and C _t is a Liu process. Based on the model (2), many researchers investigated the pricing formulas of many kinds of options.

In 2017, Dai et al. [4] gave a nonlinear model which is called the exponential Ornstein-Uhlenbeck model as follows, { d X t = rX t d t d S t = μ ( 1 - c ln S t ) S t d t + σ S t d C t(3) where X _t is the bond price, S _t is the stock price, r > 0, c > 0, σ > 0 and μ are all constants and C _t is a Liu process. Furthermore, Sun et al. [7] gave the analysis solution of this model.

However, in the practical market, interest rate is an important economic indicator and it is always influenced by some uncertain factors. In order to meet the need of the actual financial market, this paper introduces a new uncertain stock model with floating interest rate. The stock price S _t and the short-term uncertain interest rate r _t are defined by { dr t = μ 1 t d t + σ 2 t d C 1 t dS t = μ 2 S t d t + σ 2 S t d C 2 t(4) where μ_1t and σ_2t are the functions with respect to t, μ₂ and σ₂ are two constants and C_1t and C_2t are two independent Liu processes.

In this section, we give European option pricing formulas for model (4). European options are divided into the European call option and the European put option. The European call option allows the holder to buy this stock for strike price at expiration date while the European put option allows the holder to sell this stock for strike price at expiration date. Assume a European option has a strike K and an expiration T. Then the payoff of European call option at expiration T is ( S T - K ) + and the payoff of European put option at expiration T is ( K - S T ) + .

Since the money resulted from the bond has the time value, the option price should be the expected value of the terminal payoff discounted to the present value. Then the European call option has the price f c = E [ exp ( - ∫ 0 T r t d t ) ( S T - K ) + ](5) and the European put option has the price f p = E [ exp ( - ∫ 0 T r t d t ) ( K - S T ) + ] .(6)

Theorem 4.1. Assume a European call option for the model (4) has a strike K and an expiration T. Then the European call option price is f c = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( ϒ T - 1 ( α ) - K ) + d α where Φ T - 1 ( α ) = ∫ 0 T ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , ϒ T - 1 ( α ) = S 0 ( μ 2 T + 3 σ 2 T π ln α 1 - α ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s .

It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ T - 1 ( α ) = ∫ 0 T r t α d t = ∫ 0 T ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s d t .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 T r s d s ) has an inverse uncertainty distribution Φ 1 T - 1 ( α ) = exp ( - Φ T - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

It follows from Theorem 2.5 that the uncertain process S _T has an inverse uncertainty distribution ϒ T - 1 ( α ) = S 0 exp ( μ 2 T + 3 σ 2 T π ln α 1 - α ) .

Since (S _T  - K) ⁺ is increasing with respect to S _T , it has an inverse uncertainty distribution Φ 2 T - 1 ( α ) = ( ϒ T - 1 ( α ) - K ) + .

It follows from Theorem 2.1 that exp ( - ∫ 0 T r s d s ) ( S T - K ) + has an inverse uncertainty distribution Ψ 2 T - 1 ( α ) = Φ 1 T - 1 ( α ) ( ϒ T - 1 ( α ) - K ) + .

From the Equation (5), we have the price of the European call option f c = E [ exp ( - ∫ 0 T r s d s ) ( S T - K ) + ] = ∫ 0 1 Φ 1 T - 1 ( α ) ( ϒ T - 1 ( α ) - K ) + d α = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( ϒ T - 1 ( α ) - K ) + d α .

We finish the proof.

Example 4.1. Assume that the initial price of the stock is S₀ = 40, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 45. Then the price of a European call option at expiration T = 1 is f _c  = 5.3304.

Theorem 4.2.Assume a European put option for the model (4) has a strike K and an expiration T. Then the European put option price is f p = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( K - ϒ T - 1 ( 1 - α ) ) + d α where Φ T - 1 ( α ) = ∫ 0 T ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , ϒ T - 1 ( α ) = S 0 ( μ 2 T + 3 σ 2 T π ln α 1 - α ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s .

It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ T - 1 ( α ) = ∫ 0 T r t α d t = ∫ 0 T ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s d t .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 T r s d s ) has an inverse uncertainty distribution Φ 1 T - 1 ( α ) = exp ( - Φ T - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

It follows from Theorem 2.5 that the uncertain process S _T has an inverse uncertainty distribution ϒ T - 1 ( α ) = S 0 exp ( μ 2 T + 3 σ 2 T π ln α 1 - α ) .

Since (K - S _T ) ⁺ is decreasing with respect to S _T , it has an inverse uncertainty distribution Φ 2 T - 1 ( α ) = ( K - ϒ T - 1 ( 1 - α ) ) + .

It follows from Theorem 2.1 that exp ( - ∫ 0 T r s d s ) ( K - S T ) + has an inverse uncertainty distribution Ψ 2 T - 1 ( α ) = Φ 1 T - 1 ( α ) ( K - ϒ T - 1 ( 1 - α ) ) + .

From the Equation (6), we have the price of the European put option f p = E [ exp ( - ∫ 0 T r s d s ) ( K - S T ) + ] = ∫ 0 1 Φ 1 T - 1 ( α ) ( K - ϒ T - 1 ( 1 - α ) ) + d α = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( K - ϒ T - 1 ( 1 - α ) ) + d α .

The theorem is completed.

Example 4.2. Assume that the initial price of the stock is S₀ = 40, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 38. Then the price of a European put option at expiration T = 1 is f _p  = 4.7096.

An American call or put option is a contract that gives the holder the right to buy or, respectively, to sell the underlying asset for the strike price at any time between now and the expiration date. In other words, an American option can be exercised at any time up to and including expiry.

In this section, we give the American option pricing formulas based on model (4). Assume the American option has a strike K and an expiration T. Considering the time value of the money resulted from the bond, if the price of stock is S _t , the American call option has the price f c a = E [ sup 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( S t - K ) + ](7) and the American put option has the price f p a = E [ inf 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( K - S t ) + ] .(8)

Theorem 5.1. Assume a American call option for the model (4) has a strike K and an expiration T. Then the American call option price is f c a = ∫ 0 1 sup 0 ≤ t ≤ T exp ( - Φ t - 1 ( 1 - α ) ) ( ϒ t - 1 ( α ) - K ) + d α where Φ t - 1 ( α ) = ∫ 0 t ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , ϒ t - 1 ( α ) = S 0 ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s .

It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ t - 1 ( α ) = ∫ 0 t r s α d s = ∫ 0 t ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 t r s d s ) has an inverse uncertainty distribution Φ 1 t - 1 ( α ) = exp ( - Φ t - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

It follows from Theorem 2.5 that the uncertain process S _t has an inverse uncertainty distribution ϒ t - 1 ( α ) = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Since (S _t  - K) ⁺ is increasing with respect to S _t , it has an inverse uncertainty distribution Φ 2 t - 1 ( α ) = ( ϒ t - 1 ( α ) - K ) + .

Since C_1t and C_2t are independent, exp ( - ∫ 0 t r s d s ) ( S t - K ) + has an inverse uncertainty distribution Ψ 2 t - 1 ( α ) = Φ 1 t - 1 ( α ) ( ϒ t - 1 ( α ) - K ) + .

Since J (x) = x is a strictly increasing function, it follows from Theorem 2.6 [17], the supremum sup 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( S t - K ) + has an inverse uncertainty distribution Ψ 2 T - 1 ( α ) = sup 0 ≤ t ≤ T Ψ 2 t - 1 ( α ) = sup 0 ≤ t ≤ T Φ 1 t - 1 ( α ) ( ϒ t - 1 ( α ) - K ) + .

From the Equation (7), we have the price of the American call option f c a = E [ sup 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( S t - K ) + ] = ∫ 0 1 sup 0 ≤ t ≤ T Φ 1 t - 1 ( α ) ( ϒ t - 1 ( α ) - K ) + d α = ∫ 0 1 sup 0 ≤ t ≤ T exp ( - Φ t - 1 ( 1 - α ) ) ( ϒ t - 1 ( α ) - K ) + d α .

We finish the proof.

Example 5.1. Assume that the initial price of the stock is S₀ = 40, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 45. Then the price of the American call option at expiration T = 1 is f _{c a}  = 4.7628.

Theorem 5.2. Assume a American put option for the model (4) has a strike K and an expiration T. Then the American put option price is f p a = ∫ 0 1 sup 0 ≤ t ≤ T exp ( - Φ t - 1 ( 1 - α ) ) ( K - ϒ t - 1 ( 1 - α ) ) + d α where Φ t - 1 ( α ) = ∫ 0 t ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , ϒ t - 1 ( α ) = S 0 ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s .

It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ t - 1 ( α ) = ∫ 0 t r s α d s = ∫ 0 t ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 t r s d s ) has an inverse uncertainty distribution Φ 1 t - 1 ( α ) = exp ( - Φ t - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

It follows from Theorem 2.5 that the uncertain process S _t has an inverse uncertainty distribution ϒ t - 1 ( α ) = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Since (K - S _t ) ⁺ is decreasing with respect to S _t , it has an inverse uncertainty distribution Φ 2 t - 1 ( α ) = ( K - ϒ t - 1 ( 1 - α ) ) + .

Since C_1t and C_2t are independent, exp ( - ∫ 0 t r s d s ) ( S t - K ) + has an inverse uncertainty distribution Ψ 2 t - 1 ( α ) = Φ 1 t - 1 ( α ) ( K - ϒ t - 1 ( 1 - α ) ) + .

Since J (x) = x is a strictly increasing function, it follows from Theorem 2.6 [17], the supremum sup 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( K - S t ) + has an inverse uncertainty distribution Ψ 2 T - 1 ( α ) = sup 0 ≤ t ≤ T Ψ 2 t - 1 ( α ) = sup 0 ≤ t ≤ T Φ 1 t - 1 ( α ) ( K - ϒ t - 1 ( 1 - α ) ) + .

From the Equation (8), we have the price of the American put option f p a = E [ sup 0 ≤ t ≤ T exp ( - ∫ 0 t r s d s ) ( K - S t ) + ] = ∫ 0 1 sup 0 ≤ t ≤ T Φ 1 t - 1 ( α ) ( K - ϒ t - 1 ( 1 - α ) ) + d α = ∫ 0 1 sup 0 ≤ t ≤ T exp ( - Φ t - 1 ( 1 - α ) ) ( 1 - ϒ t - 1 ( 1 - α ) ) + d α . We finish the proof.

Example 5.2. Assume that the initial price of the stock is S₀ = 40, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 35. Then the price of the American put option at expiration T = 1 is f _{p a}  = 3.2543.

Asian option is a kind of path dependent option. The terminal payoff of an Asian option depends on the average of the underlying asset price S _t over period within the life of the option. Assume an Asian option has a fixed strike K and an expiration T. We define the continuous arithmetic average of S _t over the time period [0, T] by A T = 1 T ∫ 0 T S t d t .

In the view of the time value of the money resulted from the bond, the option price should be the expected value of the terminal payoff discounted to the present value. Then the Asian call option with the continuous arithmetic average has the price

f c 1 = E [ exp ( - ∫ 0 T r t d t ) ( A T - K ) + ] = E [ exp ( - ∫ 0 T r t d t ) ( 1 T ∫ 0 T S t d t - K ) + ] .(9)

The continuous geometric average of S _t over the time period [0, T] is defined by G T = exp ( 1 T ∫ 0 T ln S t d t ) .

Similarly, the Asian call option with the continuous geometric average has the price

f c 2 = E [ exp ( - ∫ 0 T r t d t ) ( G T - K ) + ] = E [ exp ( - ∫ 0 T r t d t ) ( exp ( 1 T ∫ 0 T ln S t d t ) - K ) + ] .(10)

Theorem 6.1.Assume an Asian call option for the model (4) has a strike K and an expiration T. Then the Asian call option based on continuous arithmetic average price is f c 1 = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( Φ 2 T - 1 ( α ) - K ) + d α where Φ T - 1 ( α ) = ∫ 0 T ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , Φ 2 T - 1 ( α ) = S 0 ( μ 2 T + 3 σ 2 T π ln α 1 - α ) - 1 ( exp ( μ 2 T + 3 σ 2 T π ln α 1 - α ) - 1 ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s . It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ T - 1 ( α ) = ∫ 0 T r t α d t = ∫ 0 T ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s d t .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 T r s d s ) has an inverse uncertainty distribution Φ 1 T - 1 ( α ) = exp ( - Φ T - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Since J ( x ) = x T is increasing with respected to x, by applying Theorem 2.8, the arithmetic average A T = 1 T ∫ 0 T S t d t has an inverse uncertainty distribution Φ 2 T - 1 ( α ) = S 0 T ∫ 0 T exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) d t , = S 0 ( μ 2 T + 3 σ 2 T π ln α 1 - α ) - 1 ( exp ( μ 2 T + 3 σ 2 T π ln α 1 - α ) - 1 ) .

Since [A _T  - K] ⁺ is increasing with respect to A _T , it has an inverse uncertainty distribution Ψ T - 1 ( α ) = ( Φ 2 T - 1 ( α ) - K ) + .

By using the definition of expected value of uncertain variable and the Equation (9), we have the price of Asian call option f c 1 = E [ exp ( - ∫ 0 T r s d s ) ( A T - K ) + ] = ∫ 0 1 Φ 1 T - 1 ( α ) ( Φ 2 T - 1 ( α ) - K ) + d α = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( Φ 2 T - 1 ( α ) - K ) + d α .

Thus the theorem is proved.

Example 6.1. Assume that the initial price of the stock is S₀ = 20, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 25. Then the price of the Asian call option based on continuous arithmetic average price at expiration T = 1 is f _c  = 1.37096.

Theorem 6.2.Assume an Asian call option for the model (4) has a strike K and an expiration T. Then the Asian call option based on continuous geometric average price is f c 2 = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( Ψ 2 T - 1 ( α ) - K ) ) + d α where Φ T - 1 ( α ) = ∫ 0 T ∫ 0 s ( μ 1 u + 3 | σ 1 u | π ln α 1 - α ) d u d s , Ψ 2 T - 1 ( α ) = S 0 exp ( μ 2 T 2 + 3 σ 2 T 2 π ln α 1 - α ) .

Proof. By applying Definition 2.13, the uncertain differential equation d r t = μ 1 t d t + σ 1 t d C t has an α-path r t α = ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s .

It follows from Theorems 2.5 and 2.8 that the uncertain process ∫ 0 T r t d t has an inverse uncertainty distribution Φ T - 1 ( α ) = ∫ 0 T r t α d t = ∫ 0 T ∫ 0 t ( μ 1 s + 3 | σ 1 s | π ln α 1 - α ) d s d t .

Since y = exp(- x) is strictly decreasing with respect to x, from Theorem 2.3, exp ( - ∫ 0 T r s d s ) has an inverse uncertainty distribution Φ 1 T - 1 ( α ) = exp ( - Φ T - 1 ( 1 - α ) ) .

Similarly, the uncertain differential equation d S t = μ 2 S t d t + σ 2 S t d C t has also an α-path S t α = S 0 exp ( μ 2 t + 3 σ 2 t π ln α 1 - α ) .

Since J ( x ) = ln x T is increasing with respected to x, by applying Theorem 2.8, the uncertain process 1 T ∫ 0 T ln S t d t has an inverse uncertainty distribution Φ 2 T - 1 ( α ) = 1 T ∫ 0 T ln S t α d t = 1 T ∫ 0 T ( ln S 0 + μ 2 t + 3 σ 2 t π ln α 1 - α ) d t = ln S 0 + μ 2 T 2 + 3 σ 2 T 2 π ln α 1 - α .

Since y = exp(x) is a strictly function, from Theorem 2.3, the geometric average on the interval [0, T] G T = exp ( 1 T ∫ 0 T ln S t d t ) has an inverse uncertainty distribution Ψ 2 T - 1 ( α ) = exp ( Φ 2 T - 1 ( α ) ) = S 0 exp ( μ 2 T 2 + 3 σ 2 T 2 π ln α 1 - α )

Since (G _T  - K) ⁺ is increasing with respect to G _T , it has an inverse uncertainty distribution Ψ T - 1 ( α ) = ( Ψ 2 T - 1 ( α ) - K ) + .

By using the definition of expected value of uncertain variable and the Equation (10), we have the price of the Asian call option f c 2 = E [ exp ( - ∫ 0 T r s d s ) ( A T - K ) + ] = ∫ 0 1 Φ 1 T - 1 ( α ) ( Ψ 2 T - 1 ( α ) - K ) + d α = ∫ 0 1 exp ( - Φ T - 1 ( 1 - α ) ) ( Ψ 2 T - 1 ( α ) - K ) + d α .

We complete the theorem.

Example 6.2. Assume that the initial price of the stock is S₀ = 40, the other parameters are σ_1t = 0.3t², μ_1t = 0.2t, σ₂ = 0.2, μ₂ = 0.16 and the strike price K = 45. Then the price of the Asian call option based on continuous geometric average price at expiration T = 1 is f _c  = 1.1524.

This paper introduced a new uncertain stock model with floating interest rate which generalized Liu’s stock model. Furthermore we derived the pricing formulas for the European, American and Asian options when the stock price follows the new uncertain stock model with floating interest rate. The future study may consider pricing lookback options and barrier options and we also use granular computing techniques [22–27] to develop new option pricing formulas in this uncertain stock models.