1 Introduction

Early researches on option pricing were based on probability theory. As we all know, when using probability theory, a basic premise is that the estimated probability distribution is close enough to the long-run cumulative frequency. Otherwise, the law of large numbers is no longer valid and probability theory is no longer applicable. However, in many situations, there are not enough (or even no) historical data. For example, there are no historical data when new stocks are issued.

Furthermore, many assumptions on Black–Scholes model are too idealistic, many scholars later have made various amendments and promotions. Lots of studies have shown that the distribution of stock returns is not consistent with the assumption of the normal distribution. In many situations, researchers pointed that stochastic differential equation is inappropriate to model the actual financial market. Many empirical studies showed the price of underlying asset does not behave like randomness, and it is often affected by the belief degrees of investors in Kahneman and Tversky (1979) and Tversky and Kahneman (1992). Liu (2013b) also gave some paradoxes about stochastic finance theory and suggested that uncertain differential equation may be a potential mathematical foundation of finance theory.

In order to deal with this case, we have to invite some domain experts to evaluate their belief degree that each event will occur. For modeling the uncertainty of human beings, uncertainty theory was founded by Liu (2007) and refined by Liu (2009). Liu (2007) gave the definitions of uncertainty distribution, inverse uncertainty distribution, expected value and variance. Liu (2008) introduced the concept of uncertain process, which can describe dynamic uncertain systems, and presented a type of differential equations driven by canonical Liu process.

After that, uncertainty theory was used to investigate options pricing. Liu (2009) proposed an uncertain differential equation to describe the stock price, and obtained the European option pricing formula. Later, Chen (2011) gave the corresponding analytic solution of American option price. In the same year, Peng and Yao (2011) proposed an uncertain mean-reversion stock model and presented the pricing formulas of European and American options. Yu (2012) studied the option pricing formula under the uncertain stock model with jump. In the following year, Zhang and Liu (2014) and Sun and Chen (2015) derived the pricing formulas of geometric average Asian options and arithmetic average Asian options respectively. Under an uncertain Ornstein–Uhlenbeck model, Gao et al. (2018) obtained the pricing formulas of lookback options. Furthermore, see more details on other options pricing in Gao et al. (2019), Zhang and Sun (2020), and Zhang et al. (2020).

It is well known that a lookback option is a path-dependent option with payoff determined not only by the settlement but also by the maximum or minimum price of the asset within the life of the option. In this paper, we are concerned on lookback options with floating interest rate and uncertain strike price. We extend the lookback options pricing problems discussed in Gao et al. (2018) to an uncertain strike lookback options pricing problems with floating interest rate. Compared with existing models, our contributions are as follows. As is known to all, interest rate and strike price are also important indicators of economic measurement in options pricing and they are often fluctuated by uncertain factors such as the financial markets and economic policies. Thus, it is suitable to regard the interest rate and the strike price as uncertain variables when we study the options pricing problem. On one hand, interest rate is a changing quantity rather than a constant. On the other hand, the strike price is uncertain and the strike price is determined by another asset which is described by an uncertain differential equation. In addition, we also conducted empirical research and sensitivity analysis on the parameters in the model in order to better describe the true situation of the financial market.

The rest of this paper is organized as follows. In Sect. 2, the elementary concepts and theorems on uncertainty theory are introduced. In Sect. 3, we derive lookback call/put options pricing formulas with floating interest rate and uncertain strike price, and give the numerical algorithm to calculate the prices. Finally, a brief conclusion is given in Sect. 4.

2 Preliminaries

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms, which was founded by Liu (2007) and refined by Liu (2009). In this section, we will introduce some basic definitions and results on uncertainty theory. As for the detailed applications of uncertainty theory, please see the references Gao (2013) and Liu (2013a, (2014, (2015).

2.1 Uncertain variables

Definition 1

(Liu 2007, 2009) Let L be a \(\sigma \)-algebra on a nonempty set \(\varGamma \). A set function \(M:L\rightarrow [0,1]\) is called an uncertain measure if it satisfies the following axioms:

  • Axiom 1. (Normality Axiom) \(\mathrm{M}\left\{ \varGamma \right\} = 1\) for the universal set \(\varGamma \).

  • Axiom 2. (Duality Axiom) \(\mathrm{M}\left\{ \varLambda \right\} + \mathrm{M}\left\{ {{\varLambda ^c}} \right\} = 1\) for any event \(\varLambda \).

  • Axiom 3. (Subadditivity Axiom) For every countable sequence of events \(\varLambda _1\), \(\varLambda _2\),..., we have

    $$\begin{aligned} \mathrm{M}\left\{ {\bigcup \limits _{i = 1}^\infty {{\varLambda _i}} }\right\} \le \mathop \sum \limits _{i = 1}^\infty \mathrm{M}\left\{ {{\varLambda _i}} \right\} . \end{aligned}$$

  • Axiom 4. (Product Axiom) Let \(({\varGamma _k},{L_k},{\mathrm{M}_k})\) be uncertainty spaces for \(k = 1,2,\ldots \), the product uncertain measure \(\mathrm{M}\) is an uncertain measure satisfying

    $$\begin{aligned} \mathrm{M}\left\{ {\prod \limits _{k = 1}^\infty {{\varLambda _k}} } \right\} = \mathop \bigwedge \limits _{k = 1}^\infty {\mathrm{M}_k}\left\{ {{\varLambda _k}} \right\} \end{aligned}$$

    where \({\varLambda _k}\) are arbitrarily chosen events from \({L_k}\) for \(k = 1,2,\ldots \), respectively.

Definition 2

(Liu 2007) An uncertain variable is a function from an uncertainty space \((\varGamma , L, M)\) to the set of real numbers, such that, for any Borel set B of real numbers, the set \(\left\{ {\xi \in B} \right\} = \left\{ {\gamma \in \varGamma |\xi \left( \gamma \right) \in B} \right\} \) is an event.

Next, we will define the uncertainty distribution of an uncertain variable and the independence of multi-variables.

Definition 3

The uncertainty distribution \(\varPhi (x)\) of an uncertain variable \(\xi \) is defined by \(\varPhi (x) = \mathrm{M}\left\{ {\xi \le x} \right\} \) for any real number x. An uncertainty distribution \(\varPhi (x)\) is said to be regular if it is a continuous and strictly increasing function with respect to x at which \(0<\varPhi (x)<1\), and \(\mathop {\lim }\limits _{x \rightarrow -\infty } \varPhi (x) = 0,\mathop {\lim }\limits _{x \rightarrow +\infty }\varPhi (x)={\mathrm{1}}.\)

If \(\xi \) has a regular uncertainty distribution \(\varPhi (x)\), then the inverse function \({\varPhi ^{-1}}(\alpha )\) is called the inverse uncertainty distribution of \(\xi \).

Definition 4

(Liu 2009) The uncertain variables \({\xi _1},{\xi _2},\ldots ,{\xi _m}\) are said to be independent if

$$\begin{aligned} \mathrm{M}\left\{ {\bigcap \limits _{i = 1}^m {\left\{ {{\xi _i} \in {B_i}} \right\} } } \right\} = \mathop \bigwedge \limits _{i = 1}^m \mathrm{M}\left\{ {{\xi _i} \in {B_i}} \right\} \end{aligned}$$

for any Borel sets \({B_1},{B_2},\ldots ,{B_m}\) of real numbers.

By virtue of the operation law of uncertain variables, Liu (2010) gave the inverse uncertainty distribution of strictly monotone function.

Theorem 1

(Liu 2010) Let \({\xi _1},{\xi _2},\ldots ,{\xi _n}\) be independent uncertain variables with uncertainty distributions \({\varPhi _1},{\varPhi _2},\ldots ,{\varPhi _n}\). If the function \(f({x_1},{x_2},\ldots ,{x_n})\) is strictly increasing with respect to \({x_1},{x_2},\ldots ,{x_m}\) and strictly decreasing with \({x_{m + 1}},{x_{m + 2}},\ldots ,{x_n}\), then

$$\begin{aligned} \xi = f\left( {{\xi _1},{\xi _2},\ldots ,{\xi _m},{\xi _{m + 1}},{\xi _{m + 2}},\ldots ,{\xi _n}} \right) \end{aligned}$$

is an uncertain variable with inverse uncertainty distribution

$$\begin{aligned} \Psi ^{-1}(\alpha ) = f \left( \varPhi _1^{-1}(\alpha ),\ldots , \varPhi _m^{-1}(\alpha ),\varPhi _{m+1}^{-1}(1-\alpha ),\ldots ,\varPhi _n^{-1}(1-\alpha ) \right) . \end{aligned}$$

Definition 5

(Liu 2007) The expected value of an uncertain variable \(\xi \) is defined by

$$\begin{aligned} E[\xi ] = \int _0^{ + \infty } {\mathrm{M}\left\{ {\xi \ge x} \right\} } dx - \int _{ - \infty }^0 {\mathrm{M}\left\{ {\xi \le x} \right\} } dx \end{aligned}$$

provided that at least one of the two integrals exists.

Liu (2010) showed that the expected value of an uncertain variable with a regular uncertainty distribution was given by the following theorem.

Theorem 2

(Liu 2010) Assume the uncertain variable \(\xi \) has a regular uncertainty distribution \(\varPhi \), then

$$\begin{aligned} E[\xi ] = \int _0^{\mathrm{{1}}} {{\varPhi ^{ - 1}}(\alpha )} d\alpha . \end{aligned}$$

2.2 Uncertain differential equations

In this section, we will give the concepts and theorems on uncertain differential equation. For more details on them, please see the references Liu (2008, (2009) and Yao and Chen (2013).

Definition 6

(Liu 2008) Let T be an index set and let \((\varGamma ,L,\mathrm{M})\) be an uncertainty space. An uncertain process is a measurable function from \(T \times (\varGamma ,L,\mathrm{M})\) to the set of real numbers, i.e., for each \(t \in T\) and any Borel set B,

$$\begin{aligned} \left\{ {{X_t} \in B} \right\} = \left\{ {\gamma \in \varGamma |{X_t}\left( \gamma \right) \in B} \right\} \end{aligned}$$

is an event.

Definition 7

(Liu 2009) An uncertain process \({C_t}\) is said to be a Liu process if

  1. (i)

    \({C_t} = 0\) and almost all sample paths are Lipschitz continuous,

  2. (ii)

    \({C_t}\) has stationary and independent increments,

  3. (iii)

    every increment \({C_{s + t}} - {C_s}\) is a normal uncertain variable with expected value 0 and variance \({t^2}\), whose uncertainty distribution is

    $$\begin{aligned} \varPhi (x) = {\left( {1 + \exp \left( {\frac{{ - \pi x}}{{\sqrt{3} t}}} \right) } \right) ^{ - 1}},\quad x \in R. \end{aligned}$$

Definition 8

(Liu 2009) Let \({X_t}\) be an uncertain process and let \({C_t}\) be a Liu process. For any partition of closed interval [a, b] with \(a = {t_1}< {t_2}< \cdots < {t_{k + 1}} = b\), the mesh is written as

$$\begin{aligned} \varDelta = \mathop {\max }\limits _{1 \le i \le k} \left| {{t_{i + 1}} - {t_i}} \right| , \end{aligned}$$

then Liu integral of \({X_t}\) with respect to \({C_t}\) is defined as

$$\begin{aligned} \int _a^b {{X_t}} d{C_t} = \mathop {\lim }\limits _{\varDelta \rightarrow 0} \sum \limits _{i = 1}^k {{X_t} \cdot ({C_{t_{i + 1}}} - {C_{{t_i}}})} \end{aligned}$$

provided that the limit exists almost surely and is finite.

Definition 9

(Liu 2008) Suppose \({C_t}\) is a Liu process, and f and g are two functions. Then

$$\begin{aligned} d{X_t} = f(t,{X_t})dt + g(t,{X_t})d{C_t} \end{aligned}$$

is called an uncertain differential equation. A solution is an uncertain process \({X_t}\) that satisfies the equation identically in t.

Definition 10

(Yao and Chen 2013) Let \(\alpha \) be a number with \(0< \alpha < 1\). An uncertain differential equation

$$\begin{aligned} d{X_t} = f(t,{X_t})dt + g(t,{X_t})d{C_t} \end{aligned}$$

is said to have an \(\alpha \)-path \(X_t^\alpha \) if it solves the corresponding ordinary differential equation

$$\begin{aligned} dX_t^\alpha = f(t,X_t^\alpha )dt + \left| {g(t,X_t^\alpha )} \right| {\varPhi ^{ - 1}}(\alpha )dt, \end{aligned}$$

where \({\varPhi ^{ - 1}}(\alpha )\) is the inverse uncertainty distribution of standard normal uncertain variable,

$$\begin{aligned} {\varPhi ^{ - 1}}(\alpha ) = \frac{{\sqrt{3} }}{\pi }\ln \frac{\alpha }{{1 - \alpha }}. \end{aligned}$$

Theorem 3

(Yao and Chen 2013) Let \({X_t}\) and \(X_t^\alpha \) be the solution and \(\alpha \)-path of the uncertain differential equation

$$\begin{aligned} d{X_t} = f(t,{X_t})dt + g(t,{X_t})d{C_t} \end{aligned}$$

respectively. Then

$$\begin{aligned}&\mathrm{M}\left\{ {{X_t} \le X_t^\alpha ,\forall t} \right\} = \alpha ,\\&\mathrm{M}\left\{ {{X_t} > X_t^\alpha ,\forall t} \right\} = 1 - \alpha . \end{aligned}$$

Theorem 4

(Yao 2013) Let \({X_t}\) and \(X_t^\alpha \) be the solution and \(\alpha \)-path of the uncertain differential equation

$$\begin{aligned} d{X_t} = f(t,{X_t})dt + g(t,{X_t})d{C_t} \end{aligned}$$

respectively. Then for any time \(s>0\) and strictly increasing (decreasing) function J(x), the supremum \(\mathop {\sup }\limits _{0\le t\le s}J({x_t})\) has an inverse uncertainty distribution

$$\begin{aligned} \varPhi _s^{-1}(\alpha )=\mathop {\sup }\limits _{0 \le t \le s} J(x_t^\alpha )\left( {\varPhi _s^{ - 1}(\alpha ) = \mathop {\sup }\limits _{0 \le t \le s} J(x_t^{1 - \alpha })} \right) \end{aligned}$$

and the time integral \(\int _{\mathrm{{0}}}^s {J({x_t})} dt\) has an \(\alpha \)-path

$$\begin{aligned} Y_s^\alpha = \int _0^s {J(x_t^\alpha )} dt\left( {Y_s^\alpha = \int _0^s {J(x_t^{{\mathrm{1 - }}\alpha })} dt} \right) . \end{aligned}$$

2.3 Some results on uncertain Square–Root process and uncertain exponential Ornstein–Uhlenbeck process

In this section, we will introduce uncertain Square–Root process and uncertain exponential Ornstein–Uhlenbeck process, which are used to describe the interest rate and the stock price respectively and recall their inverse uncertainty distributions. The uncertain Square–Root process satisfies the following uncertain equation

$$\begin{aligned} dr_t =a\left( b-r_t\right) dt+\sigma _0\sqrt{r_t}dC_t, \end{aligned}$$

where \(a, b, \sigma _0\) are some positive real numbers and \({C_t}\) is an independent canonical Liu processe.

Theorem 5

(Jiao and Yao 2015) Suppose that an uncertain process satisfies the following uncertain equation

$$\begin{aligned} dr_t =a\left( b-r_t\right) dt+\sigma _0\sqrt{r_t}dC_t, \end{aligned}$$

then the inverse uncertainty distribution \({r_t^{-1}(\alpha )}\) of \({r_t}\) satisfies the following ordinary differential equation

$$\begin{aligned} dr_t^{-1}\left( \alpha \right) =a\left( b - r_t^{-1}\left( \alpha \right) \right) dt+\sigma _0\sqrt{r_t^{-1}\left( \alpha \right) }\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }dt. \end{aligned}$$

Dai et al. (2017) firstly gave the uncertain exponential Ornstein–Uhlenbeck process which was determined by the following uncertain differential equation

$$\begin{aligned} dY_t=\mu \left( 1 - c\ln {Y_t}\right) Y_t dt + \sigma Y_t d C_t, \end{aligned}$$

where \(c > 0\), \(\sigma > 0\) and \(\mu \) are constants. Subsequently, Sun et al. (2018) gave the analytic solution of this model and Dai et al. (2017) derived the inverse uncertainty distribution of the uncertain exponential Ornstein–Uhlenbeck process.

Theorem 6

(Sun et al. 2018) Suppose that an uncertain process satisfies the following uncertain equation

$$\begin{aligned} d Y_t = \mu \left( 1 - c\ln {Y_t}\right) Y_t dt + \sigma Y_t d C_t, \end{aligned}$$

then

$$\begin{aligned} Y_t = \exp \left( \frac{1}{c} + \left( {\ln {Y_0} - \frac{1}{c}} \right) \exp ({\mathrm{- }}\mu ct)+\sigma \int _0^t {\exp (c\mu (s - t))d{C_s} } \right) . \end{aligned}$$

Theorem 7

(Dai et al. 2017) Suppose an uncertain process satisfies the following uncertain equation

$$\begin{aligned} d{Y_t} = \mu (1 - c\ln {Y_t}){Y_t}dt + \sigma {Y_t}d{C_t}, \end{aligned}$$

then the inverse uncertainty distribution of \({Y_t}\) is

$$\begin{aligned} Y_t^{ - 1}(\alpha )= \exp \left( \exp \left( - \mu ct\right) \ln {Y_0} + {\frac{1}{c}(1 - \exp ({\mathrm{- }}\mu ct))\left( {1 + \frac{{\sqrt{3} \sigma }}{{\mu \pi }}\ln \frac{\alpha }{{1 - \alpha }}} \right) } \right) . \end{aligned}$$

3 Lookback options pricing model with floating interest and uncertain strike

In this paper, we assume the interest rate \(r_t\) satisfies uncertain Square–Root process, and uncertain strike price \(k_t\) and the stock price \(X_t\) follow uncertain exponential Ornstein–Uhlenbeck process, which are stated as below

$$\begin{aligned} \left\{ \begin{array}{ll} dr_t =a\left( b-r_t\right) dt+\sigma _0\sqrt{r_t}dC_t, \\ dX_t=\mu _1(1 - c_1\ln X_t)X_tdt + \sigma _1 X_t dC_{1t},\\ dk_t=\mu _2(1 - c_2\ln k_t) k_tdt + \sigma _2 k_t dC_{2t}, \end{array}\right. \end{aligned}$$
(1)

where a, b, \({\mu _1}\), \({\mu _{\mathrm{{2}}}}\), \({\sigma _{\mathrm{{0}}}}\), \({\sigma _1}\) and \({\sigma _{\mathrm{{2}}}}\) are positive real numbers and \({c_1}\), \({c_2} \ne 0\). Furthermore, \({C_t}\), \({C_{1t}}\) and \({C_{{\mathrm{2}}t}}\) are independent canonical Liu processes.

By virtue of Theorems 5 and 7, the inverse uncertainty distributions of the interest rate \(r_t\), the stock price \(X_t\) and uncertain strike price \(k_t\) are given by the following equation

$$\begin{aligned} dr_t^{-1}\left( \alpha \right)= & {} a\left( b-r_t^{-1}\left( \alpha \right) \right) dt+\sigma _0\sqrt{r_t^{-1}\left( \alpha \right) }\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }dt,\\ X_t^{ - 1}(\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{1}}}}{c_{\mathrm{{1}}}}t)\ln {X_0} +\frac{1}{c_1}(1 - \exp (- \mu _1 c_1 t))\left( {1 + \frac{\sqrt{3} \sigma _1}{\mu _1 \pi } }\ln \frac{\alpha }{1 - \alpha } \right) \right) ,\\ k_t^{ - 1}(\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{2}}}}{c_{\mathrm{{2}}}}t)\ln {k_0} +\frac{1}{c_2}(1 - \exp (- \mu _2 c_2 t))\left( {1 + \frac{\sqrt{3} \sigma _2}{\mu _2 \pi } }\ln \frac{\alpha }{1 - \alpha } \right) \right) . \end{aligned}$$

The following two subsections will give uncertain strike lookback call/put options pricing formulas with the floating interest rate.

3.1 Lookback call option pricing formula with the uncertain strike price

A lookback call option provides the holder with the right to sell a certain asset at the highest price during a certain period. Here, we suppose that a lookback call option has an expiration time T, then the payoff of selling a lookback call option is

$$\begin{aligned} {\left( {\mathop {\max }\limits _{0 \le t \le T} {X_t} - {k_T}} \right) ^ + }. \end{aligned}$$

Taking into account the time value of the stock return, the present value of the payoff is

$$\begin{aligned} \exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {\left( {\mathop {\max }\limits _{0 \le t \le T} {X_t} - {k_T}} \right) ^ + }. \end{aligned}$$

For the lookback call option is the expected present value of the payoff, then the lookback call option has a price

$$\begin{aligned} f_{call}=E\left[ {\exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {{\left( {\mathop {\max }\limits _{0 \le t \le T} {X_t}- {k_T}} \right) }^ + }}\right] . \end{aligned}$$

Theorem 8

Suppose that a lookback call option for the stock model (1) has an expiration time T, then the lookback call option pricing formula is

$$\begin{aligned} f_{call} = \int _0^1 \exp \left( - \int _0^T r_t^{ - 1}(1 - \alpha )dt \right) \mathop {\max }\limits _{0 \le t \le T} \left( X_t^{ - 1}(\alpha ) - k_T^{ - 1}(1 - \alpha )\right) ^ + d\alpha , \end{aligned}$$
(2)

where

$$\begin{aligned} dr_t^{-1}\left( 1-\alpha \right)= & {} a\left( b-r_t^{-1}\left( 1-\alpha \right) \right) dt+\sigma _0\sqrt{r_t^{-1}\left( 1-\alpha \right) }\frac{\sqrt{3}}{\pi }\ln \frac{1-\alpha }{\alpha }dt,\\ X_t^{ - 1}(\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{1}}}}{c_{\mathrm{{1}}}}t)\ln {X_0} +\frac{1}{c_1}(1 - \exp (- \mu _1 c_1 t))\left( {1 + \frac{\sqrt{3} \sigma _1}{\mu _1 \pi } }\ln \frac{\alpha }{1 - \alpha } \right) \right) ,\\ k_T^{ - 1}(1-\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{2}}}}{c_{\mathrm{{2}}}}T)\ln {k_0} +\frac{1}{c_2}(1 - \exp (- \mu _2 c_2 T))\left( {1 + \frac{\sqrt{3} \sigma _2}{\mu _2 \pi } }\ln \frac{1-\alpha }{ \alpha } \right) \right) . \end{aligned}$$

Proof

According to Theorem 4, the time integral \(\int _0^T {{r_t}} dt\) has an inverse uncertainty distribution

$$\begin{aligned} \varphi ^{ - 1}(\alpha ) = \int _0^T {r_t^{ - 1}(\alpha )}dt. \end{aligned}$$

Since \(\exp ({\mathrm{- }}x)\) is a strictly decreasing function, the uncertain variable

$$\begin{aligned} \exp \left( { - \int _0^T {{r_t}dt} } \right) \end{aligned}$$

has an inverse uncertainty distribution

$$\begin{aligned} \gamma ^{{\mathrm{- }}1}(\alpha )= \exp \left( {{\mathrm{- }}{\varphi ^{{\mathrm{- }}1}}(1 - \alpha )} \right) =\exp \left( {{\mathrm{- }}\int _0^T {r_t^{ - 1}(1 - \alpha )dt} } \right) \end{aligned}$$

Thus, the inverse distribution function of

$$\begin{aligned} \exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {\left( \mathop {\max }\limits _{0 \le t \le T} {X_t} - {k_T}\right) ^ + } \end{aligned}$$

is given by the following equation

$$\begin{aligned} \Psi ^{- 1}(\alpha )=\exp \left( - \int _0^T {r_t^{ - 1}(1{\mathrm{- }}\alpha )dt} \right) \mathop {\max }\limits _{0 \le t \le T} {\left( {X_t^{{\mathrm{- }}1}(\alpha ) - k_T^{{\mathrm{- }}1}({\mathrm{1}} - \alpha )} \right) ^ +}. \end{aligned}$$

So the price of lookback call option can be obtained

$$\begin{aligned} f_{call}= & {} E\left[ {\exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {{\left( {\mathop {\max }\limits _{0 \le t \le T} {X_t} - {k_T}} \right) }^ + }} \right] \\= & {} \int _0^1 {{\Psi ^{{\mathrm{- 1}}}}} (\alpha )d\alpha \\= & {} \int _0^1 \exp \left( - \int _0^T r_t^{-1}(1 - \alpha )dt \right) \mathop {\max }\limits _{0 \le t \le T} \left( X_t^{- 1}(\alpha ) - k_T^{- 1}(1 - \alpha ) \right) ^ + d\alpha . \end{aligned}$$

The pricing formula of lookback call option is derived. \(\square \)

From \(\left( \mathop {\max }\limits _{0 \le t \le T} X_t- k_T\right) ^+=\mathop {\max }\limits _{0 \le t \le T} {(X_t - k_T)^ + }\) and Theorem 8, the algorithm of calculating lookback call option price is given as follows.

  • Step 0: Set \({\alpha _i} = {i / N} \) and \({t_j} = {{jT} / M}\), \(i = 1,2,\ldots ,N - 1,\) \(j = 1,2,\ldots ,M,\) where N and M are two large numbers.

  • Step 1: Set \(i = {\mathrm{0}}\).

  • Step 2: Set \(i \leftarrow i + 1\).

  • Step 3: Set \(j = {\mathrm{0}}\).

  • Step 4: Set \(j \leftarrow j + 1\).

  • Step 5: Set \(G_{{t_0}}^{{\alpha _i}} = 0\).

  • Step 6: Calculate the interest rate, the difference between stock price and the terminal strike price at the time \({t_j}\) are

    $$\begin{aligned} r_{t_j}^{-1}\left( 1-\alpha _i\right)= & {} r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) + a\left( b-r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) \right) \varDelta t_j\\&+\,\sigma _0\sqrt{r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) }\frac{\sqrt{3}}{\pi }\ln \frac{1-\alpha _i}{\alpha _i}\varDelta t_j,\\ G_{{t_j}}^{{\alpha _i}}= & {} \exp \left( \exp ( - \mu _1 c_1 t_j)\ln X_0 + \frac{1}{c_1}(1 - \exp ( - \mu _1 c_1 t_j))\left( 1 +\frac{\sqrt{3} \sigma _1}{\mu _1 \pi }\ln \frac{\alpha _i}{1 - \alpha _i} \right) \right) \\&-\, \exp \left( \exp ( - \mu _2 c_2 T)\ln k_0 + \frac{1}{c_2}(1 - \exp ( - \mu _2 c_2 T))\left( 1 +\frac{\sqrt{3} \sigma _2}{\mu _2 \pi }\ln \frac{1-\alpha _i}{\alpha _i} \right) \right) . \end{aligned}$$

  • Step 7: Calculate the discount rate

    $$\begin{aligned} \exp \left( {{\mathrm{- }}\int _0^T {r_t^{ - 1}{\mathrm{(1 - }}{\alpha _i})dt} } \right) \leftarrow \exp \left( {{\mathrm{- }}\frac{T}{M}\sum \limits _{j = 1}^M {r_{{t_j}}^{{\mathrm{1 - }}{\alpha _i}}} } \right) \end{aligned}$$

    and the maximum difference between stock price and the terminal strike price over the time interval [0, T]

    $$\begin{aligned} \mathop {\max }\limits _{0 \le t \le T} {\left( {X_t^{{\mathrm{- 1}}}({\alpha _i}) - k_T^{{\mathrm{- 1}}}({\mathrm{1 - }}{\alpha _i})} \right) ^ + } \leftarrow max(G_{{t_1}}^{{\alpha _i}},\ldots ,G_{{t_M}}^{{\alpha _i}},0). \end{aligned}$$

  • Step 8: Set

    $$\begin{aligned} f^{\alpha _i}\leftarrow \exp \left( - \frac{T}{M}\sum \limits _{j = 1}^M r_{t_j}^{-1}(1-\alpha _i) \right) max(G_{t_1}^{\alpha _i},\ldots ,G_{t_M}^{\alpha _i},0). \end{aligned}$$

    If \(i < N- 1\), then return to Step 2.

  • Step 9: Calculate the lookback call option price

    $$\begin{aligned} {f_{call}} \leftarrow \frac{1}{{N - 1}}\sum \limits _{i = 1}^{N - 1} {{f^{{\alpha _i}}}}. \end{aligned}$$

Example 1

Assume that the parameters of the stock price are \(X_0=1, c_1= 1, \mu _1=\sqrt{3}, \sigma _1=\pi \), the parameters of the interest rate is \(r_0=0.03, a=1, b=0.5, \sigma _0=0.02 \), the parameters of the strike price is \(k_0=3, c_2=0.9, \mu _2=1.5, \sigma _2=3.1\) and \(M=999, N=9999\), then, the price of the lookback call option with a maturity date \(T=5\) is 3.4549.

Finally, we present a graph of the lookback call option pricing formula with different parameters as shown below. Figure 1a represents the price with respect to the floating interest rate; Fig. 1b represents the price with respect to the expiration time; Fig. 1c represents the price with respect to the starting value of stock price; Fig. 1d represents the price with respect to the original uncertain strike price.

Fig. 1
figure 1

Lookback call option price with respect to different parameters

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3.2 Lookback put option pricing formula with the uncertain strike price

Suppose that a lookback put option has an expiration time T, then the payoff of buying a lookback put option is

$$\begin{aligned} {\left( {{k_T} - \mathop {\min }\limits _{0 \le t \le T} {X_t}} \right) ^ + }. \end{aligned}$$

Taking into account the time value of money, the present value of this payoff is

$$\begin{aligned} \exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {\left( {{k_T} - \mathop {\min }\limits _{0 \le t \le T} {X_t}} \right) ^ + }. \end{aligned}$$

For the lookback put option is the expected present value of the payoff, then the lookback put option has a price

$$\begin{aligned} f_{put} = E\left[ {\exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {{\left( {{k_T} - \mathop {\min }\limits _{0 \le t \le T} {X_t}} \right) }^ + }} \right] . \end{aligned}$$

Theorem 9

Suppose that a lookback put option for the stock model has an expiration time T. Then the pricing formula lookback put option is

$$\begin{aligned} f_{put} = \int _0^1 \exp \left( - \int _0^T r_t^{ - 1}(1 - \alpha )dt \right) \mathop {\max }\limits _{0 \le t \le T} \left( k_T^{ - 1}( \alpha )- X_t^{ - 1}(1-\alpha ) \right) ^ + d\alpha , \end{aligned}$$
(3)

where

$$\begin{aligned} dr_t^{-1}\left( 1-\alpha \right)= & {} a\left( b-r_t^{-1}\left( 1-\alpha \right) \right) dt+\sigma _0\sqrt{r_t^{-1}\left( 1-\alpha \right) }\frac{\sqrt{3}}{\pi }\ln \frac{1-\alpha }{\alpha }dt,\\ X_t^{ - 1}(1-\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{1}}}}{c_{\mathrm{{1}}}}t)\ln {X_0} +\frac{1}{c_1}(1-\exp (- \mu _1 c_1 t))\left( {1 + \frac{\sqrt{3} \sigma _1}{\mu _1 \pi } }\ln \frac{1-\alpha }{ \alpha }\right) \right) ,\\ k_T^{ - 1}(\alpha )= & {} \exp \left( {\exp (\mathrm { - }}{\mu _{\mathrm{{2}}}}{c_{\mathrm{{2}}}}T)\ln {k_0} + \frac{1}{c_2}(1 - \exp (- \mu _2 c_2 T))\left( {1 + \frac{\sqrt{3} \sigma _2}{\mu _2 \pi } }\ln \frac{\alpha }{ 1-\alpha } \right) \right) . \end{aligned}$$

Proof

From Theorem 4, it is easily to derive the inverse distribution function of \(\exp \left( {-\int _0^T {{r_t}dt} } \right) {\left( {k_T} - \mathop {\min }\limits _{0 \le t \le T} {X_{t}}\right) ^ + }\) is

$$\begin{aligned} \Psi ^{ - 1}(\alpha )=\exp \left( {{\mathrm{- }}\int _0^T {r_t^{ - 1}(1 - \alpha )dt} } \right) {\left( {k_T^{{\mathrm{- }}1}(\alpha ) - \mathop {\min }\limits _{0 \le t \le T} X_t^{{\mathrm{- }}1}({\mathrm{1}} - \alpha )} \right) ^ + }, \end{aligned}$$

where \(r_t^{-1}(1-\alpha ), X_t^{-1}(1-\alpha ), K_T^{-1}(\alpha )\) are given in Theorem 9. Thus

$$\begin{aligned} f_{put}= & {} E\left[ {\exp \left( {{\mathrm{- }}\int _0^T {{r_t}dt} } \right) {{\left( { {k_T}- \mathop {\min }\limits _{0 \le t \le T} {X_t} } \right) }^ + }} \right] \\= & {} \int _0^1 {{\Psi ^{{\mathrm{- 1}}}}} (\alpha )d\alpha \\= & {} \int _0^1 \exp \left( - \int _0^T r_t^{-1}(1 - \alpha )dt \right) \left( k_T^{ - 1}(\alpha ) - \mathop {\min }\limits _{0 \le t \le T} X_t^{ - 1}(1 - \alpha ) \right) ^ + d\alpha . \end{aligned}$$

The pricing formula of the lookback put option is verified. \(\square \)

From\({\left( {{k_T}-\mathop {\min }\limits _{0 \le t \le T} {X_t}} \right) ^+}=\mathop {\max }\limits _{0 \le t \le T} {({k_T} - {X_t})^ + }\) and Theorem 9, the algorithm of calculating the lookback put option price is as follows.

  • Step 0: Set \({\alpha _i} = {i / N}\) and \({t_j} = {{jT} / M}\), \(i = 1,2,\ldots ,N - 1\), \(j = 1,2,\ldots ,M\), where N and M are two large numbers.

  • Step 1: Set \(i = {\mathrm{0}}\).

  • Step 2: Set \(i \leftarrow i + 1\).

  • Step 3: Set \(j = {\mathrm{0}}\).

  • Step 4: Set \(j \leftarrow j + 1\).

  • Step 5: Set \(H_{{t_0}}^{{\alpha _i}} = 0\).

  • Step 6: Calculate the interest rate, the difference between the terminal strike price and stock price at the time \({t_j}\) are

    $$\begin{aligned} r_{t_j}^{-1}\left( 1-\alpha _i\right)= & {} r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) + a\left( b-r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) \right) \varDelta t_j\\&+\,\sigma _0\sqrt{r_{t_{j-1}}^{-1}\left( 1-\alpha _i\right) }\frac{\sqrt{3}}{\pi }\ln \frac{1-\alpha _i}{\alpha _i}\varDelta t_j,\\ H_{{t_j}}^{{\alpha _i}}= & {} \exp \left( \exp ( - \mu _2 c_2 T)\ln k_0 + \frac{1}{c_2}(1 - \exp ( - \mu _2 c_2 T))\left( 1 +\frac{\sqrt{3} \sigma _2}{\mu _2 \pi }\ln \frac{\alpha _i}{1 - \alpha _i} \right) \right) \\&-\, \exp \left( \exp ( - \mu _1 c_1 t_j)\ln X_0 + \frac{1}{c_1}(1 - \exp ( - \mu _1 c_1 t_j))\left( 1 +\frac{\sqrt{3} \sigma _1}{\mu _1 \pi }\ln \frac{1-\alpha _i}{\alpha _i} \right) \right) . \end{aligned}$$

  • Step 7: Calculate the discount rate

    $$\begin{aligned} \exp \left( {{\mathrm{- }}\int _0^T {r_t^{ - 1}{\mathrm{(1 - }}{\alpha _i})dt} } \right) \leftarrow \exp \left( {{\mathrm{- }}\frac{T}{M}\sum \limits _{j = 1}^M {r_{{t_j}}^{{\mathrm{1 - }}{\alpha _i}}} } \right) \end{aligned}$$

    and the maximum difference between stock price and the terminal strike price over the time interval [0, T]

    $$\begin{aligned} \mathop {\max }\limits _{0 \le t \le T} {\left( {k_T^{{\mathrm{- 1}}}({\alpha _i}) - X_t^{{\mathrm{- 1}}}({\mathrm{1 - }}{\alpha _i})} \right) ^ + } \leftarrow max(H_{{t_1}}^{{\alpha _i}},\ldots ,H_{{t_M}}^{{\alpha _i}},0). \end{aligned}$$

  • Step 8: Set

    $$\begin{aligned} {f^{{\alpha _i}}} \leftarrow \exp \left( - \frac{T}{M}\sum \limits _{j = 1}^M r_{t_j}^{-1}(1 -\alpha _i ) \right) max(H_{{t_1}}^{{\alpha _i}},\ldots ,H_{{t_M}}^{{\alpha _i}},0). \end{aligned}$$

    If \(i < N -1\), then return to Step 2.

  • Step 9: Calculate the lookback put option price is

    $$\begin{aligned} {f_{put}} \leftarrow \frac{1}{{N - 1}}\sum \limits _{i = 1}^{N - 1} {{f^{{\alpha _i}}}}. \end{aligned}$$

Example 2

Assume that the parameters of the stock price are \(X_0=1, c_1=1, \mu _1=\sqrt{3}, \sigma _1=\pi \), the parameters of the interest rate is \(r_0=0.03, a=1, b=0.5, \sigma _0 =0.02\), the parameters of the strike price is \(k_0=3, c_2=0.9, \mu _2=1.5, \sigma _2=3.1\), then, the price of lookback put option with a maturity date \(T = 5\) is 21.1753.

Figure 2 shows the relationship of the lookback put option pricing formula and the parameters. Figure 2a represents the price with respect to the floating interest rate; Fig. 2b represents the price with respect to the expiration time; Fig. 2c represents the price with respect to the original value; Fig. 2d represents the price with respect to the original uncertain strike price.

Fig. 2
figure 2

Lookback put option price with respect to different parameters

Full size image

4 Conclusion

This paper was concerned on lookback options pricing problem within the framework of uncertainty theory. We extend the pricing problem discussed by Gao et al. (2018) to the lookback option pricing with uncertain strike price in a floating interest rate environment. Based on the assumption that floating interest rate and uncertain strike price follow the uncertain differential equations respectively, the lookback call option and lookback put option pricing formulas were derived by the method of uncertain calculus. In addition, some numerical algorithms and examples are given to illustrate the pricing formulas. Besides, the relationship between the option price and the parameter were discussed. Further research could establish the uncertain-stochastic integral and investigate the integral formulas. Our future research is to investigate the multi-assets options pricing in random-uncertain environment.