Barrier option pricing with floating interest rate based on uncertain exponential Ornstein–Uhlenbeck model

1.   Introduction

Barrier options are categorized as knock-in and knock-out options and have a reward that is contingent on whether the underlying asset price reaches a specified barrier level within the contract period. Barrier options are an essential tool for risk management in the financial market and are extensively applied to different domains, such as risk control, asset management, and so on. The option holders could control the potential returns and losses by setting different barriers, and the barrier options are usually cheaper than the standard options, which makes them popular with many investors.

In a standard Black–Scholes model, barrier options were first analytically valued by Merton ^[1], using classical results about the first passage time of Brownian motion to a point that can be traced back to Lévy ^[2]. Heynen and Kat ^[3] and Carr ^[4] pioneered partial and outside barrier options, while Kunitomo and Ikeda ^[5] first tackled double barrier options, Armstrong ^[6] first dealt with window barrier options, and Guillaume ^[7] provided the first closed form formulae for step barrier options. Other seminal contributions that cannot all be cited here tackle non-constant boundaries, either deterministic or stochastic, or have been devoted to numerical approximations of barrier option values in more general models, especially those featuring stochastic volatility or jumps. What all these papers have in common is a stochastic approach based on the martingale method of pricing.

The above studies assumed that stock prices follow a Wiener process. In fact, Liu ^[8] provided a paradox: the actual stock price is impossible to follow any Ito's stochastic differential equation. Additionally, the application of probability theory requires that the real frequency closely resemble the probability distribution, which means that it must be possible to acquire sufficient samples. However, it is sometimes difficult to obtain enough or no samples for an uncertain event. Therefore, Liu ^[9] proposed the Liu process and developed the uncertainty theory, which was refined by Liu ^[10].

Liu ^[11] applied the uncertainty theory to the field of finance, and he first proposed the uncertain stock model. Afterward, many scholars started investigating the issue of option pricing with uncertainty theory. The uncertain stock model, including a mean-reverting process, was presented by Peng and Yao ^[12]. Chen and Liu ^[13] established a new uncertain stock model that has periodic dividends. Liu et al. ^[14] proposed the uncertain currency model, and Deng and Qin ^[15] examined the pricing issue of the Parisian option within this framework. Liu et al. ^[16] discussed the pricing problem of the European option using the Caputo–Hadamard UFDEs to simulate the change in the stock price. Pan et al. ^[17] investigated the pricing problem of Bermudan options. Yao and Qin ^[18] investigated the European barrier option using Liu's stock model. Similarly, Yang et al. ^[19] and Gao et al. ^[20] examined American and Asian barrier options using the same model. Furthermore, Dai et al. ^[21] explored a non-linear stock model that is named the uncertain exponential Ornstein–Uhlenbeck model. Based on this model, Liu et al. ^[22] and Gao et al. ^[23] investigated the power option and the lookback option, respectively.

We further investigate the price functions of the barrier option with a floating interest rate under the uncertain exponential Ornstein–Uhlenbeck model. Section 2 introduces some definitions and theorems used in the paper. The price formulas of two knock-in options and two knock-out option prices are investigated in Section 3. We design the numerical algorithms to calculate the option prices and provide several numerical examples in Section 4. In Section 5, the values of the knock-in and knock-out options are calculated by utilizing the Shanghai Interbank Offered Rate (SHIBOR) and the closing price for Haitian food. Section 6 gives a concise conclusion.

2.   Preliminaries

Definition 2.1. (Liu ^[11]) An uncertain process $C_t$ is called the Liu process if the following three conditions are satisfed:

(1) $C_0 = 0$, and almost all sample paths are Lipschitz continuous,

(2) $C_t$ has stationary and independent increments,

(3) Every increment $C_{s+t}-C_s$ is a normal uncertain variable with an expected value $0$ and variance $t^2$.

Theorem 2.2. (Liu ^[9]) Suppose $M$ is an uncertain measure, and for events $\Lambda_{1}$ and $\Lambda_{2}$ with $\Lambda_{1}\subset \Lambda_{2}$, we can obtain

Theorem 2.3. (Liu ^[10]) Let $\xi$ be an uncertain variable with a regular uncertainty distribution $\Phi$. Then

Definition 2.4. (Yao and Chen ^[24]) Assume that $f(t, x)$ and $g(t, x)$ are two continuous functions, respectively, and $C_{t}$ is a Liu process. Then

is called an uncertain differential equation.

Theorem 2.5. (Yao and Chen ^[24]) Suppose $X_{t}$ and $X_{t}^{\alpha}$ be the solution and $\alpha$-path of the uncertain differential equation

respectively. Then

and

where $\Phi^{-1}(\alpha)$ is the inverse uncertainty distribution of the uncertain variable $X_{t}$.

3.   Uncertain exponential Ornstein–Uhlenbeck model

Liu ^[11] first presented the uncertain stock model

where $Y_t$ represents the stock price, $\mu$ and $\sigma$ are the drift item and the diffusion item of $Y_t$. $X_t$ represents the bond price, the interest rate $r$ is a constant, and $C_{t}$ is a Liu process. Let $B$ be the strike price and $T$ be the expiration date. Liu ^[11] studied the European call option pricing formula

and the European put option pricing formula

In the model (3.1), which considers stock price movements in the short term, it is assumed that the interest rate is a fixed constant. However, the stock price varies around a constant rather than rising or falling constantly in the long term. In order to improve the model (3.1) to reflect the real financial markets, it is vital to take into account the volatility of interest rates. Then, Sun and Su ^[25] presented the following model:

where $r_{t}$ denotes the floating interest rate, $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$, $\sigma_{1}$ and $\sigma_{2}$ are positive constants, and $b_{1}\neq 0$, $b_{2}\neq 0$, $C_{1t}$ and $C_{2t}$ are two mutually independent Liu processes.

The above uncertain stock models are both linear. Liu ^[26] explored a nonlinear model with a floating interest rate that can better reflect the financial markets compared to the linear models

where $m$, $a$, $c$, $\sigma_{1}$ and $\sigma_{2}$ are positive constants with $a\neq 0$, $\mu$ is a constant.

The model (3.3) is an uncertain exponential Ornstein–Uhlenbeck stock model that takes into account a floating interest rate. It ensures that the stock price is non-negative and does not fluctuate dramatically in a short period of time. Hence, we investigate the knock-in and knock-out options and derive the price functions under the model (3.3).

3.1.   Knock-in options

This part primarily investigates the European up-and-in call option and the down-and-in put option. Suppose a barrier option has a barrier level $D$, a maturity date $T$, and an exercise price $B$. We define an indicator function to easily describe the barrier option

where $D$ is a specified constant.

For an up-and-in call option, the initial asset price is below the barrier level, and the option is activated only when the price moves up to the barrier level before the expiration date.

Let $C_{ui}$ be the option price. The investor buys the option with $C_{ui}$ at the initial time and has a payoff

at the expiration time $T$. The present value of the return is

Since money has a time value. At the initial moment, the net income of the investor is

The seller receives $C_{ui}$ for selling the option and pays the investor

Similarly, the seller has a net income

at the initial moment.

The fair option price should ensure the investor and the seller get the same expected return. Hence, the option price is

Theorem 3.1. Assume that a European up-and-in call option for the uncertain exponential Ornstein–Uhlenbeck model (3.3) has a barrier level $D$, a maturity date $T$, and a strike price $B$. Then the option price is

where

and

is the $\alpha$-path of $Y_t$.

Proof. First, we prove that

is the inverse uncertainty distribution of

Define two events

and

where $r^{1-\alpha}_t$ is the $\alpha$-path of $r_t$.

Since

we obtain

Similarly, because of

we obtain

According to the duality axiom, we obtain $M\{\Lambda_1 \}+M\{\Lambda_2 \} = 1,$ which means that $M\{ \Lambda_1\} = \alpha.$ Hence, we obtain

has an inverse uncertainty distribution

By using the calculation formula for the expected value, we have

Note that $r^{\alpha}_t$ satisfies the following ordinary differential equation:

So it is easily verified that

From Theorem 6 in ^[21], we obtain

Additionally, the equation

is equivalent to

Note that $Y_{0} < D$ in the up-and-in option and $Y^{\alpha }_{t}$ is a monotone function of $t$. $Y^{\alpha }_{t}$ increases with $t$, we can obtain

which implies that

and

Conversely, if $Y^{\alpha }_{t}$ decreases with $t$, we obtain

It means that

which contradicts with

Therefore, $Y^{\alpha }_{t}$ is an increasing function of $t$. We derive that

which implies that

Consequently, we can rewrite the option price as

□

For a down-and-in put option, the initial asset price is above the barrier level, and the option is not activated until the price decreases to the barrier level before the expiration date.

Let $P_{di}$ be the option price. The investor purchases the option at the initial time for $P_{di}$ and has a payoff of

The present value of the return is

Then, the investor's net return is

Similarly, the seller's net return is

Hence, the option price is

Corollary 3.1. Assume that a European down-and-in put option for the uncertain exponential Ornstein–Uhlenbeck model (3.3) has a barrier level $D$, a maturity date $T$, and a strike price $B$. Then the price of this option is

where

and

is the $\alpha$-path of $Y_t$.

3.2.   Knock-out options

This part investigates two European knock-out options under the uncertain exponential Ornstein–Uhlenbeck model (3.3), including the up-and-out option and the down-and-out option.

The down-and-out call option is a contract whose price of the asset is above the barrier level at the beginning of the transaction. The option is void until the price decreases to the barrier level before the expiration date.

Let $C_{do}$ be the option price. The investor buys an option with $C_{do}$ at the initial time and has a payoff

Then the investor's net return is

And the seller receives $C_{do}$ for selling the option at the initial time and has a net return

Therefore, the option price is

Theorem 3.2. Assume that a European down-and-out call option for the uncertain exponential Ornstein–Uhlenbeck model (3.3) has a barrier level $D$, a maturity date $T$, and a strike price $B$. Then the option price is

where

and

is the $\alpha$-path of $Y_t$.

Proof. First, we prove that

is the inverse uncertainty distribution of

Define two events

and

where $r^{1-\alpha}_t$ is the $\alpha$-path of $r_t$.

Since

we obtain

Similarly, due to

we obtain

According to the duality axiom, we obtain

which indicates that

Thus, we obtain

has an inverse uncertainty distribution

From the calculation formula for the expected value, we obtain

Additionally, note that

is equivalent to

Note that $Y_{0} > D$, and $Y^{\alpha }_{t}$ is a monotonic function of time $t$. If $Y^{\alpha }_{t}$ increases with $t$, we can obtain

which implies

Conversely, if $Y^{\alpha }_{t}$ decreases with $t$, we obtain

which means that

and $Y_T^{\alpha} \geq D.$

Therefore, we deduce that

which indicates that

Consequently, we can rewrite the option price as

□

The up-and-out put option is a contract whose price of the asset is below the barrier level at the beginning of the transaction. The option is invalid until the price exceeds the barrier level before the expiration date.

Let $P_{uo}$ be the option price. The investor buys an option with $P_{uo}$ at the initial moment and has a payoff

Then, the investor's net return at the initial time is

And the seller receives $P_{uo}$ for selling the option at the initial time and has a net return

Therefore, the option price is

Corollary 3.2. Assume that a European up-and-out put option for the uncertain exponential Ornstein–Uhlenbeck model (3.3) has a barrier level $D$, a maturity date $T$, and a strike price $B$. Then the price of this option is

where

and

is the $\alpha$-path of $Y_t$.

4.   Numerical experiments

This section focuses on developing numerical methods to compute the prices of knock-in options and analyzing the effects of different parameters on the option values. The numerical algorithms for calculating the knock-out option prices are similar to those presented in this section. Furthermore, the effects of the parameters on the knock-out option prices can be analyzed in the same way.

4.1.   Up-and-in call option

The algorithm for calculating the price $C_{ui}$ is designed according to Theorem 3.1.

Step 0: Set the values of $r_0$, $m$, $a$, $\sigma_1$, $Y_0$, $\mu$, $c$, $\sigma_2$, $B$, $T$, and $D$.

Step 1: Calculate

Step 2: Set $\alpha_j = \theta+j(1-\theta)/N$, $j = 1, 2, ..., N-1$, where $N$ is a large positive integer.

Step 3: Set $j = 0$.

Step 4: Set $j \leftarrow j+1$.

Step 5: Compute the positive deviation

Step 6: Calculate

and $W_j = Z_j\times G_j.$ Return to Step 4 if $j < N-1$.

Step 7: The option price function is

Example 4.1. Assume the initial interest rate $r_0 = 0.03$, the initial stock price $Y_0 = 16$, and other parameters of model (3.3) are $m = 0.01, a = 0.8, \sigma_1 = 0.01, \mu = 0.9, c = 0.35$, and $\sigma_2 = 0.1$, and the parameters of the option are $B = 18, T = 5$, and $D = 20$, respectively. Then the price $C_{ui}$ is $1.3657$.

It is noted that there are many parameters in the pricing formula of $C_{ui}$ in Theorem 3.1. Next, we investigate the influence of the parameters on the price $C_{ui}$ through numerical experiments. Several examples are given to illustrate the change of $C_{ui}$ on one parameter, in which the other parameters are consistent with Example 4.1.

First, the strike price $B$ and the parameter $m$ are discussed.

Example 4.2. Let the strike price $B$ change from 18 to 23 with step 0.01, and the other parameters remain unchanged. Figure 1 displays the results.

It illustrates a negative correlation between the price $C_{ui}$ and $B$ in Figure 1. The result can be explained intuitively from the option pricing formula in Theorem 3.1. The strike price $B$ appears only in the positive deviation, which demonstrates that $C_{ui}$ has a monotonically decreasing relationship with $B$. Similarly, it can be shown that the price $C_{ui}$ demonstrates a monotonically decreasing relationship with $m$.

Then, we study the parameter $\sigma_{2}$, the maturity time $T$, and the barrier level $D$.

Example 4.3. Let the parameter $\sigma_2$ change from 0.01 to 0.2 with step 0.01, and the other parameters remain unchanged. Figure 2 displays the results.

It displays that the price $C_{ui}$ ascends with the parameter $\sigma_2$ in Figure 2. The asset price is more likely to move up to the barrier level if $\sigma_2$ increases. Thus, the option price $C_{ui}$ increases with $\sigma_2$.

Example 4.4. Let the maturity time $T$ change from 1 to 6 with step 0.01, and the other parameters remain unchanged. Figure 3 displays the results.

As it is illustrated in Figure 3, the price $C_{ui}$ grows when the maturity time $T$ gets longer. The investor possibly gets more profits, and the seller takes more risks when $T$ ascends. Therefore, the option price $C_{ui}$ increases.

Example 4.5. Let the barrier level $D$ change from 18 to 25 with step 0.01, and the other parameters remain unchanged. Figure 4 displays the results.

Figure 4 demonstrates that the price $C_{ui}$ decreases with $D$. This result may be attributed to the fact that the price of stock is less likely to move up to the barrier level as $D$ increases, and the option is less likely to get activated.

Finally, we consider the parameters $\sigma_1$, $a$, $\mu$, and $c$.

Example 4.6. Let the parameter $\sigma_1$ change from 0 to 0.1 with step 0.01. Figure 5 displays the results.

Figure 5 illustrates that the price $C_{ui}$ exhibits a positive correlation with the parameter $\sigma_1$. It is difficult to investigate the relationship between the price $C_{ui}$ and $\sigma_1$ from the pricing formula. Next, we investigate the impact of $\sigma_1$ on $C_{ui}$ with different values of $\sigma_2$, $\mu$, $c$, $Y_0$, $D$, and $T$. The curves are illustrated in Figure 6.

In all cases, it reveals that the price $C_{ui}$ increases with the parameter $\sigma_1$. Thus, the price $C_{ui}$ is a monotonically increasing function of $\sigma_1$ as the remaining parameters remain within an acceptable range. Analogously, the changes in the price $C_{ui}$ on the parameters $a$, $\mu$, and $c$ can be analyzed in the same way.

4.2.   Down-and-in put option

The algorithm for calculating the price $P_{di}$ is designed according to Corollary 1.

Step 0: Set the values of $r_0$, $m$, $a$, $\sigma_1$, $Y_0$, $\mu$, $c$, $\sigma_2$, $B$, $T$, and $D$.

Step 1: Calculate

Step 2: Set $\alpha_j = j\theta/N$, $j = 1, 2, ..., N-1$, where $N$ is a large positive integer.

Step 3: Set $j = 0$.

Step 4: Set $j \leftarrow j+1$.

Step 5: Compute the positive deviation

Step 6: Calculate

and $W_j = Z_j\times G_j.$ Return to Step 4 if $j < N-1$.

Step 7: The option price function is

Example 4.7. Assume the initial interest rate $r_0 = 0.03$, the initial stock price $Y_0 = 16$, and other parameters of model (3.3) are $m = 0.01, a = 0.8, \sigma_1 = 0.01, \mu = 0.9, c = 0.35$, and $\sigma_2 = 0.1$, and the parameters of the option are $B = 15, T = 5$, and $D = 14$, respectively. Then the price $P_{di}$ is $0.5425$.

Considering the price function of the up-and-in call option is similar to that of the down-and-in put option, we only investigate the influence of $B$ and $D$ on the option price $P_{di}$.

Example 4.8. Let the strike price $B$ change from 10 to 15 with step 0.01, and the other parameters remain unchanged. Figure 7 displays the results.

Figure 7 illustrates that the price $P_{di}$ increases with $B$. The result can be deduced immediately from Theorem 1. Since $B$ appears only in the positive deviation, the price $P_{di}$ is a monotonically increasing function of $B$.

Example 4.9. Let the barrier level $D$ change from 9 to 14 with step 0.01, and the other parameters remain unchanged. Figure 8 displays the results.

Figure 8 demonstrates that the price $P_{di}$ increases with $D$. This result may be attributed to the fact that the price of stock is more likely to decrease to the barrier level as $D$ increases and the option is more likely to be activated.

5.   Real data analysis

In this section, real financial data are used to illustrate the performances of the four option pricing formulas given in Sections 3.1 and 3.2. Moreover, the method of moments is chosen for estimating the unknown parameters in the model (3.3). The uncertain hypothesis test is utilized in the following example to assess the reasonableness of the estimations.

5.1.   Parameter estimation

We choose the Shanghai Interbank Offered Rate (SHIBOR) and the closing price of Haitian food stock for the period of October 20, 2023, to December 27, 2023, which are displayed in Tables 1 and 2.

According to the method of moments for uncertain differential equations ^[27], the estimations of the parameters in the model (3.3) are $m = 0.0122, a = 0.7139, \sigma_{1} = 0.0011$, and $\mu = 0.8669, c = 0.2774, \sigma_{2} = 0.0166.$ Thus, the model (3.3) can be expressed as

As we can see from Figures 9 and 10, all the observations of interest rate and stock price fall between the $0.05$-path and the $0.95$-path, which implies that the estimates are acceptable.

5.2.   Hypothesis test

In this part, we employ the uncertain hypothesis testing proposed by Zhang et al. ^[28] to assess how well the uncertain model (5.1) fits the observed data.

For the first differential equation in the model (3.3),

by using the Euler difference, we obtained

Since

it can be obtained that

Similarly, for the second differential equation, we have

The sample values of $\omega_j$ and $z_j$ can be obtained from the observed data of ${r_{t_j}}$ and ${Y_{t_j}}$, where $j = 1, 2, ..., 48$.

The issue of determining whether the model (5.1) fits the data well is converted into a test to verify whether $\omega_j$ and $z_j$ obey the standard normal uncertain distribution $\mathcal N(0, 1)$. Let the significance level $\alpha$ take the value of 0.05, and the two rejection domains are

and

We can see that only $\omega_9 = -2.1621 \notin [-2.0198, 2.0198]$ in Figure 11, thus $(\omega_1, \omega_2, ..., \omega_{48}) \notin W_1$. It can also be found that $z_{9} > 2.0198$ and $z_{42} < -2.0198$ in Figure 12, so we have $(z_1, z_2, ..., z_{48}) \notin W_2$. In summary, we can conclude that the model (5.1) is able to fit the observed data well.

5.3.   Numerical results

We use the option price formulas given in Sections 3.1 and 3.2 to calculate the option prices with Haitian food stock as the underlying asset. Suppose that $Y_0$ is $37.33$, $r_0$ is $1.6260\%$, and the expiration date $T$ is $8$. {According to these initial conditions, we calculate the option prices under three different models, including the Black–Scholes model

where $r$ is a constant, the stochastic exponential Ornstein–Uhlenbeck model with stochastic interest rates

and the model (3.3) used in this paper. The four barrier option prices on the three models are outlined in Table 3. According to Table 3, it can be observed that the price estimates under the model (3.3) are higher than those of the two stochastic models, with the exception of the down-and-out call option.

Finally, based on the three different models, some numerical results for the up-and-in call option are given to visually demonstrate the sensitivity of the up-and-in call option price to the parameters, including barrier level, strike price and expiration date. Tables 4 and 5 illustrate that the price $C_{ui}$ decreases with the barrier level and the strike price, respectively. Table 6 indicates an increasing trend in the price $C_{ui}$ by varying $T$ from $6$ to $10$. As seen in Tables 4–6, for the comparison of the up-and-in call option prices with different parameters, the uncertain $O-U$ model's price estimates are closer to the stochastic $B-S$ model's price estimates.

6.   Conclusions

This paper primarily focused on the pricing issue of the European barrier option in the uncertain exponential Ornstein–Uhlenbeck model with a floating interest rate. The price functions of knock-in and knock-out barrier options were given. Then we developed numerical algorithms to compute the option prices and provided several numerical examples to show the effect of parameters on the option prices. In the end, we chose Haitian food stock as the underlying asset to demonstrate how to obtain the option prices and compare the option prices under different models.

Author contributions

Shaoling Zhou: Responsible for the review and editing of the manuscript, as well as project administration. Huixin Chai: Responsible for visualization and preparation of the original draft of the manuscript. Xiaosheng Wang: Responsible for the supervision of the research project. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61873084).

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Appendix

Suppose the stock price of Haitian food follows the stochastic differential equation

where $W_t$ is a Wiener process, $\mu$, $c$ and $\sigma_2$ are unknown parameters. Based on the stock price data shown in Table 2, using Maximum Likelihood Estimation (MLE), we obtain the estimates of the parameters in the above stochastic differential equation as

Hence, we get a stochastic stock model

By using the Euler difference, we obtain

Therefore, the sample value of $\varepsilon_j$ can be regarded as a sample of the standard normal distribution $N(0, 1)$.

Next we test whether the stochastic stock model fits the stock price of Haitian-food by the "Shapiro-Wilk" test. That is, we should determine whether the sample values of $\varepsilon_j$ are derived from the standard normal distribution $N(0, 1)$. The Shapiro function is used in the test with a significance level of $0.05$. The result indicates that the value of $P$ is $0.0172$, which implies that the sample values of $\varepsilon_j$ don't come from the standard normal distribution $N(0, 1)$. Therefore, the stochastic stock model mentioned in this part is not suitable for the observed stock data well.