Multi-assets Asian rainbow options pricing with stochastic interest rates obeying the Vasicek model

1.   Introduction

Asian options are path-dependent options, meaning that the payoffs are based on the average price of the underlying asset over a specific period of time ^[1]. Asian options can lessen their sensitivity to asset price volatility thanks to this averaging feature, which also lowers trading risk ^[2]. As a result, it is frequently utilized for risk management. Multi-asset options, such as rainbow options, can be used in multi-asset allocation to lower the risks posed by multiple assets. In incentive contract design and risk management, Asian rainbow options, a hybrid of Asian and rainbow options, are frequently used ^[3]. Asian rainbow options combine the characteristics of Asian and rainbow options. They are a multi-asset, path-dependent option. Therefore, the maximum or minimum average price of the underlying assets determines whether Asian rainbow options will pay off ^[4]. However, the price of multi-asset options is rarely mentioned in the existing literature, and two- or three-asset options are frequently employed as a substitute for multi-asset options ^[5]. To derive analytical pricing formulas for Asian rainbow call options on two geometric mean assets, the authors of ^[6] expanded the methodology for pricing Asian and Rainbow options using PDE. Before and after taking into account fractional and mixed fractional Brownian motion, Wang et al. ^[4] and Ahmadian et al. ^[7,8] analyze the pricing of Asian rainbow options, respectively. Despite the fact that some researchers ^[5,7,8] attempted to increase the number of assets to $n$-dimensional, they weren't taking into account the randomness of real interest rates.

According to the efficient market hypothesis, the price of the underlying asset is currently determined by a widely understood stochastic process ^[9], and this stochastic process can be summarized as:

where according to different random processes, the interest rate ${{\bf{r}}}$ and volatility rate ${{{\boldsymbol{\sigma}}}}$ can be constants or specific functions. To reflect realism, the effects of stochastic volatility and stochastic interest rates have been incorporated into the model in several ways to reflect reality ^{[10,11,12,13,14,15,16]}. For example, Stein et al. ^[12] investigated the distributions of stock prices that result from diffusion processes with stochastically varying volatility parameters. In ^[13], the authors used a hybrid model that combines the CIR stochastic interest rate model and the Heston stochastic volatility model to get a closed-form pricing formula for European options. He and Lin ^[14] formed a rough Heston-CIR model to capture both the rough behavior of the volatility and the stochastic nature of the interest rate and present a semi-analytical pricing formula for European options. The pricing of volatility and variance swaps using a hybrid model with Markov-modulated jump-diffusion with discrete sampling times is the subject of ^[16].

Furthermore, given that some short-term government bonds have negative interest rates, some researchers have demonstrated that models like the Vasicek model that incorporate negative interest rates can improve option pricing and implied volatility forecasting ^{[17,18,19,20,21,22,23,24]}. Guo ^[20] used an assumption to derive the pricing formula for European call options: the price of the underlying asset will follow the Heston stochastic volatility model, while the interest will follow the Vasicek stochastic interest model. In ^[21], the issue of Bermuda option pricing on zero-coupon bonds was examined. In this case, the model's interest rate dynamics are based on the mixed fractional Vasicek model. Mehrdoust and Najafi ^[22] obtained the value of the European option on the zero-coupon bond with transaction cost by using the fractional Vasicek model to predict the interest rates in France and Australia. Under the Vasicek interest rate model, Zhao and Xu ^[23] calibrated the time-dependent volatility function for European options. Kharrat and Arfaoui ^[24] disentangled the Vasicek model for European options to verify the stability and reliability of the model with option theory.

Based on the above, in this paper, we adopt the Vasicek model as a stochastic interest rate and expand the number of assets of Asian rainbow options to $n$-dimensional cases to aim to present the pricing formula of geometric mean Asian rainbow options. The rest of the paper is organized as follows. In Section 2, we present the Vasicek and underlying asset price models and introduce the payoff of the Geometric Asian Rainbow options. Four explicit formulas for the geometric Asian rainbow options are derived in Section 3. In Section 4, the call option of the two and three assets are obtained by applying the Monte Carlo simulation method and different values of the model's parameters. Section 5 carries out a sensitivity analysis of the parameters in the pricing formula. In Section 6, conclusions are drawn.

2.   Asian rainbow options under the Vasicek interest rate model

In this paper, we use the Vasicek model ^[19] as the stochastic interest rate model, which enables negative interest rates and follows the actual requirements of the financial market. Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{0\leq t\leq T}, {{\bf{Q}}})$ be a complete probability space equipped with a filtration $\{\mathcal{F}_t\}$ satisfying the usual conditions (i.e., it is increasing and right-continuous when $\mathcal{F}_0$ contains all ${{\bf{Q}}}$-null sets), where ${{\bf{Q}}}$ is a risk-neutral probability measure. For the risk-neutral measure ${{\bf{Q}}}$, consider the change process of the risk-free interest rate satisfying the Vasicek model; that is, $\{r(t)\colon t \ge 0\}$ is the dynamic process of the risk-free interest rate change at time $t$ that follows the model

where $\alpha$ and $\beta \neq 0$ are positive constants, the parameter $\sigma_r > 0$ is the volatility of the risk-free interest rate, and $W_r(t)$ is a standard Wiener process. Next, in the risk-neutral measure ${{\bf{Q}}}$, we introduce the Vasicek stochastic interest rate model to the process of change in the underlying asset and investigate the dynamic process of underlying asset price change $\{S_i(t)\colon t \ge 0\}$ at time $t$, which follows

where the constant $\sigma_i > 0$ is the volatility of the underlying asset. $W_i(t)$ is a standard Wiener process and any two dependent standard Brownian motions have a correlation coefficient $\rho^{ij}\in [-1, 1]$, i.e., ${{\mathrm{d}}}W_i(t){{\mathrm{d}}}W_j(t) = \rho^{ij}{{\mathrm{d}}}t > (j\neq i)$. In addition, we assume that the Brownian motion $W_i(t)$ and $W_r(t)$ are independent of each other, i.e., ${{\mathrm{d}}}W_i(t){{\mathrm{d}}}W_r(t) = 0$, and present the following assumptions:

(A1) The underlying asset price dynamics follow a log-normal distribution, and price volatility is constant.

(A2) The risk-free interest rate is not constant, and its change follows the Vasicek model.

(A3) Securities trading is continuous; there are no riskless arbitrage opportunities; no dividends will be paid during the option's validity period; there are no transaction costs or taxes; all securities are perfectly divisible; investors can borrow or lend funds at the same interest rate.

(A4) The option can only be exercised at maturity.

Asian rainbow options have the same characteristics as Asian options because they are a combination of the two, i.e., the option price is determined by the average price of the underlying asset from the start date to the expiration date ^[7]. Considering that the price of the underlying asset (2.2) is followed by a geometric Brownian motion, we let

be the geometric mean of the price $S_i\, (i = 1, 2, \cdots, n)$ in $[0, T]$. The four geometric Asian rainbow options contract that we can take into consideration are call on max, call on min, put on max, and put on min ^[3]. The payoffs of geometric Asian rainbow call on max and min options with the strike price $K$ and maturity $T$ on $n$ underlying assets are given by

and

respectively. The payoffs of the geometric Asian rainbow put on max and min options ^[7] are given by

and

respectively.

Combining the above models and assumptions, the pricing of an Asian rainbow option ${{\bf{op}}}(T, r, S_i, V)$ under the ${{\bf{Q}}}$-measure can be written as follows:

where $V(T, S_i)$ is the payoff of the option and is a function of $T$ and $S_i(T)$. The conditional expectation under the $\mathcal{F}_0$ condition refers to valuing the expected payoff of the option when the initial time $t = 0$.

The stochastic interest rate function and the logarithm of the underlying asset price functions have definite integrals from time $0$ to $T$, according to Eqs (2.3) and (2.8). The stochastic interest rate function $r(t)$ is integrated as follows:

Let

which are the constant and integral terms of $\int_{0}^{T}r(t){\rm d}t$, respectively. The logarithm of the underlying asset price function $\ln S_i(t)$ is integrated as follows:

And let

which are a constant term and two integral terms of $\frac{1}{T}\int_{0}^{T}\ln S_i(t){\rm d}t$, respectively. For the sake of concise writing, we abbreviate the above functions respectively as $C_0$, $m$, $C_i$, $f$, and $g_i$.

Based on the above model, we will analyze the pricing formulas of geometric Asian rainbow options for multiple assets.

3.   Pricing formula of multi-asset

In this section, we consider geometric Asian rainbow option prices for multiple assets ($n$), including call options, put options, and the parity relationship.

3.1.   Call options

Referring to ^[25], we derive the following lemma first to determine the precise expression for this option's price.

Lemma 3.1. Let ${{\boldsymbol{Z}}} = (Z_1, Z_2, Z_3)^{\rm T}$ be standard normal random variables, follow the three-dimensional standard normal distribution $N({{\boldsymbol{0}}}, {{\boldsymbol{\Lambda}}})$ with the covariance matrix ${{\boldsymbol{\Lambda}}} = \begin{pmatrix}1 & \rho_{12} & \rho_{13}\\\rho_{12} & 1 & \rho_{23}\\\rho_{13} & \rho_{23} & 1\end{pmatrix}$. Then, for arbitrary constants $a$, $b$, $c$, $k$ and $l$,

where ${{\bf{I}}}(\cdot)$ is the indicator function that takes the value $1$ if the expression in parentheses is true and $0$ otherwise, and ${{\bf{N}}}(\cdot)$ is the two-dimensional cumulative standard normal distribution.

Proof. Let ${{\boldsymbol{z}}} = (z_1, z_2, z_3)^{\rm T}$, $t = \frac{1}{\sqrt{|{{\boldsymbol{\Lambda}}}|}}\left(\sqrt{1-\rho_{23}^2}z_1+\frac{(\rho_{13}\rho_{23}-\rho_{12})z_2+(\rho_{12}\rho_{23}-\rho_{13})z_3-a|{{\boldsymbol{\Lambda}}}|}{\sqrt{1-\rho_{23}^2}}\right)$, $\xi = z_2-\rho_{12}a$, $\eta = z_3-\rho_{13}a$, we then obtain

This completes the proof of Lemma 3.1.

From Lemma 3.1, we extend to the $n$-dimensional case.

Corollary 3.1. Let ${{\boldsymbol{Z}}} = (Z_0, Z_1, \cdots, Z_n)^{\rm T}$ be standard normal random variables, follow the $n+1$ dimensional standard normal distribution $N({{\boldsymbol{0}}}, {{\boldsymbol{\Lambda}}})$ with the covariance matrix ${{\boldsymbol{\Lambda}}} = \begin{pmatrix}1 & \rho_{01} & \cdots & \rho_{0n}\\\rho_{01} & 1 & \cdots & \rho_{1n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{0n} & \rho_{1n} & \cdots & 1\end{pmatrix}$. Then, for arbitrary constants $a_0$, $a_1$, $\cdots$, $a_n$, $k_1$, $\cdots$, $k_n$,

where ${{\bf{I}}}(\cdot)$ is the indicator function that takes the value $1$ if the expression in parentheses is true and $0$ otherwise, and ${{\bf{N}}}(\cdot)$ is the $n$-dimensional cumulative standard normal distribution.

Set the initial time and expiration time as $0$ and $T$, respectively, we then calculate the geometric mean of the time period from $0$ to $T$ for $S_1$, $S_2$, $\cdots$, $S_n$. The strike price of the option is set to $K$, and the payoff $V^c_{\max}(T, S_1, S_2, \cdots, S_n)$ for this option is:

which is the $n$-dimensional case of Eq (2.4). It denotes that the price of a call on max option is determined by the geometric mean of $n$ underlying assets at their maximum. We use Eqs (2.8) and (3.4) to calculate the price of a call on max option $C_{\max}(T, r, S_1, S_2, \cdots, S_n)$, that is,

Through direct calculation, the following theorem provides the expression for the price of a call on the max option.

Theorem 3.1. (Call on max option) Assume that the underlying assets $S_i > \, (i = 1, 2, \cdots,$ $n)$ obey generalized geometric Brownian motion, the stochastic interest rate follows the Vasicek model (2.1), and the initial to expiration time is $0$ to $T$, then the price of the geometric Asian rainbow call on max option $C_{\max}(T, r, S_1, S_2, \cdots, S_n)$ is given as follows:

where $i, j = 1, 2, \cdots, n$, $i\neq j$, $C_0, \, m, \, C_i, \, f$ and $g_i$ are defined in Eqs (2.10) and (2.12). ${{\bf{N}}}_i(\cdot)$ is the $n$-dimensional cumulative standard normal distribution with the covariance matrix ${{\boldsymbol{\Lambda_i}}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$ as follows:

and

Proof. From Eq (3.4), the payoff of the call on max option $V_{\max}^c(T, S_1, S_2, \cdots, S_n)$ can be expressed by the following equation

Set $A_i = \{G_i(T, S_i)\ge K, \cdots, G_i(T, S_i)\ge G_j(T, S_j), \cdots\}$ ($i, j = 1, 2, \cdots, n$, $i\neq j$), based on Eqs (2.3) and (2.11), we can obtain

It follows from Eq (3.5), we derive the call on the max option price as

From Eqs (2.9), (2.11) and (3.8), it is easy to see that

where $i, j = 1, 2, \cdots, n$ and $i\neq j$. Let

Since ${\rm d}W(u) = \varepsilon{\rm d}\sqrt{u}$ and $\varepsilon\sim N(0, 1)$ is a standard normal random variable, $Z_0$ can be expressed as

then the variance of the random variable $Z_0$ is

Similarly, the variance of $Z_1$ and $Z_a$ can be obtained as

and

respectively. The correlation coefficients between the $n+1$ random variables $Z_0$, $Z_1$, $Z_a$ ($a = 2, \cdots, n$) with respect to

and

Given the above, $(Z_0, Z_1, \cdots, Z_n)^{\rm T}\sim N({{\boldsymbol{0}}}$, $\tilde{{{\boldsymbol{\Lambda}}}}_i)$ with $\tilde{{{\boldsymbol{\Lambda}}}}_i = \begin{pmatrix}\sigma_{0}^2 & \rho_{01}\sigma_{0}\sigma_{1} & \cdots & \rho_{0n}\sigma_{0}\sigma_{n}\\\rho_{01}\sigma_{0}\sigma_{1} & \sigma_{1}^2 & \cdots & \rho_{1n}\sigma_{1}\sigma_{n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{0n}\sigma_{0}\sigma_{n} & \rho_{1n}\sigma_{1}\sigma_{n} & \cdots & \sigma_{n}^2\end{pmatrix}$, $i = 1, 2, \cdots, n$. According to Corollary 3.1, we derive the following result

where $i, j = 1, 2, \cdots, n$, $i\neq j$, $C_0, \, m, \, C_i, \, f$ and $g_i$ are defined in Eqs (2.10) and (2.12). ${{\bf{N}}}_i(\cdot)$ is the $n$-dimensional cumulative standard normal distribution with the covariance matrix ${{\boldsymbol{\Lambda_i}}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$, where the correlation coefficients $\rho_{1a}$ and $\rho_{cd}$ are defined in Eqs (3.18) and (3.19).

Moreover, $I_{n+i}\, (i, j = 1, 2, \cdots, n, > i\neq j)$ follows

Hence, the proof of Theorem 3.1 is confirmed.

Remark 3.1. Equation (3.6) displays the geometric Asian rainbow call on max option, which supports the investor in choosing multiple ($n$) underlying assets to diversify risk. Suppose that an investor purchases an option at time $0$ and chooses whether to trade it at time $T$ based on market conditions. During this period, the price of the underlying asset is fluctuating randomly. In the absence of market-based arbitrage opportunities, an option purchased at $t = 0$ takes into account the expectation of the future asset price. Due to the path dependence of Asian rainbow options, it is necessary to determine the mean value of the $n$ assets being chosen for the period between purchase and execution. We employ a geometric mean approach, which enables the option price to depend on both the asset price at expiration and the average price over the time between purchase and execution. As a result, the risk of market conditions changes can be reduced. The option price given in Theorem 3.1 supports both of these risk reduction options.

The following Theorem 3.2 provides an analytical solution for the call option with the minimum geometric mean of $n$ underlying assets, where the option has the payoff $V^c_{\min}(T, S_1, S_2, \cdots, S_n)$ as:

which is the $n$-dimensional case of Eq (2.5). That means that the price of a call on min option is determined by the minimum of the geometric mean of the $n$ underlying assets. It implies by (2.8) and (3.22) that the price of a call on min option is

Theorem 3.2. (Call on min option) Assume that the underlying assets $S_i > (i = 1, 2, \cdots, n)$ obey generalized geometric Brownian motion, the stochastic interest rate follows the Vasicek model (2.1), and the initial to expiration time range is $0$ to $T$, then the price of the geometric Asian rainbow call on min option $C_{\min}(T, r, S_1, S_2, \cdots, S_n)$ is determined by the following equation

where $i, \, j\in \{1, 2, \cdots, n\}$ ($i\neq j$), $C_0, \, m, \, C_i, \, f$ and $g_i$ are given in Eqs (2.10) and (2.12). ${{\bf{N}}}_i(\cdot)$ is the $n$-dimensional cumulative standard normal distribution with the covariance matrix ${{{\boldsymbol{\Lambda_i}}}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$ and

The proof of Theorem 3.2 is attached in A.

3.2.   Put options

In order to price put options, the following Theorem 3.3 provides an analytical solution for the put option with the maximum geometric mean of $n$ underlying assets, where the option has the payoff $V^p_{\max}(T, S_1, S_2, \cdots, S_n)$ as

This means that the price of a put on max option is determined by the maximum of the geometric mean of the $n$ underlying assets. It follows from Eqs (2.8) and (3.25) that the price of a put on max option

Through direct calculation, the following theorem provides the expression for the price of a put on max option.

Theorem 3.3. (Put on max option) Assume that the underlying assets $S_i > (i = 1, 2, \cdots, n)$ obey generalized geometric Brownian motion, the stochastic interest rate follows the Vasicek model (2.1), and the initial to expiration time is $0$ to $T$, then the price of the geometric Asian rainbow put on max option $P_{\max}(T, r, S_1, S_2, \cdots, S_n)$ is described by

where $i, \, j\in \{1, \dots, n\}$ $(i\neq j)$, $C_0, \, m, \, C_i, \, f$ and $g_i$ are taken as (2.10) and (2.12). ${{\bf{N}}}_i(\cdot)$ is the $n$-dimensional cumulative standard normal distribution with the covariance matrix ${{{\boldsymbol{\Lambda_i}}}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$ and

The proof of Theorem 3.3 is similar to the proof shown in Theorem 3.1, thus it is omitted here.

The put option with the minimum geometric mean of $n$ underlying assets is summarized in Theorem 3.4, where the option has the payoff

Combining (2.8) and (3.28), we have the price of a put on min option

Straight forward computation yields that the following theorem provides the expression for the price of a put on min option.

Theorem 3.4. (Put on min option) Assume that the underlying assets $S_i > (i = 1, 2, \cdots, n)$ obey generalized geometric Brownian motion, the stochastic interest rate follows the Vasicek model (2.1), and the initial to expiration time is $0$ to $T$, then the price of the geometric Asian rainbow put on min option $P_{\min}(T, r, S_1, S_2, \cdots, S_n)$ is given by

where $i, \, j\in \{1, \dots, n\}$ $(i\neq j)$, $C_0, \, m, \, C_i, \, f$ and $g_i$ are stated in (2.10) and (2.12). ${{\bf{N}}}_i(\cdot)$ is the $n$ dimensional cumulative standard normal distribution with the covariance matrix ${{{\boldsymbol{\Lambda_i}}}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$ and

Since the proof of this Theorem is similar to Theorem 3.2, we omit the proof here.

3.3.   Parity relationship

For multi-asset options, put-call and min-max combined parity describe the relationship between the prices of call on max, call on min, put on max, and put on min options with the same underlying portfolio, strike price, and expiration date. This section presents this relationship for Asian rainbow options based on stochastic interest rate dynamics equations obeying the Vasicek model (2.1). Here, we fix $n = 2$ and summarize the relationship as the following theorem.

Theorem 3.5. The put-call and min-max combined parity relationship for two-asset geometric Asian rainbow options under the Vasicek interest rate model (2.1) with a fixed strike price of $K$ at maturity $T$ can be given by

where $C_0, \, m, \, C_1, \, C_2, \, f, \, g_1$ and $g_2$ are respectively taken as (2.10) and (2.12).

Proof. For the sake of simplicity, we set

Then, by direct calculating, the sum of the call on max and min option prices is given by

According to the properties of two-dimensional normal distribution, we have

Therefore, we can get

Similarly, we sum the put on max and min option prices

In view of (3.33), we obtain that

As a result, we deduce that

The proof is hence complete.

Remark 3.2. For the geometric Asian rainbow option using the stochastic interest rate obeying the Vasicek model, the put-call and min-max combined parity formulas are provided in Eq (3.31). It means that the sum of call on max, call on min, put on max, and put on min option prices is a fixed value. This fixed value ensures that there is no arbitrage opportunity in the pricing formula of Theorems 3.1–3.4 and that the pricing formula is realistic and consistent with market conditions. Investors can construct a portfolio of long and short positions to obtain additional returns from the portfolio.

4.   Monte Carlo simulations

This section discusses the simulated values from the Monte Carlo simulations and analytical values from (3.6) for the multi-asset Asian rainbow call option.

4.1.   Two-asset case

For convenience, we choose $n = 2$ and assume that the initial prices of two assets $S_1$ and $S_2$ are both $40$, and the option expiry period is 6 months, i.e., $T = 0.5$. The stochastic volatility values $\sigma_r = 0.1$, $\sigma_1 = 0.1$ and $\sigma_2 = 0.2$ are adopted, and other parameters are shown in Table 1.

Assume that the initial time is $0$ and the expiration time is $T$, we divide the option period $[0, T]$ into $m$ equal intervals, namely, $0 = t_0 < t_1 < \cdots < t_m = T$ and the time interval $\Delta t = \frac{T}{m}$, where $m$ can be understood as the step size of the Monte Carlo simulation under a path. $\frac{S_i(t_{j+1})}{S_i(t_j)}\, (i = 1, 2;\, j = 0, 1, \cdots, m-1)$ follows the log-normal distribution with a mean and variance of $\left[r(t_{j-1}){\mathrm{e}}^{-\beta\Delta t}+\frac{\alpha}{\beta}\left(1-{\mathrm{e}}^{-\beta\Delta t}\right)-\frac{1}{2}\sigma_i ^2\right]\Delta t$ and $\left(\sigma_r{\mathrm{e}}^{\beta t_{j-1}}\Delta t+\sigma_i\right)^2\Delta t$, respectively. Then we can classify the simulation procedure in the following four steps.

Step 1. Simulating the path of the underlying asset price and stochastic risk-free interest rate changes. Combining time division and Brownian motion ${\rm d}W(t) = \varepsilon{\rm d}\sqrt{\Delta t}$, where $\varepsilon\sim N(0, 1)$ is a standard normal random variable, we derive

Step 2. Iterating $\kappa$ times in the first step, we obtain the underlying asset price and the risk-free rate under the $\kappa$ paths, namely,

Step 3. Calculate the $\kappa$ samples of call on max option payoff for the given $\kappa$ paths

Step 4. Calculate the sample mean to obtain the Monte Carlo simulation of the two-asset geometric Asian rainbow call on max option price

Using the above four steps, we calculate the Monte Carlo simulation values (see S-value in Table 1). The strike price is set at $K = 35, \, 40, \, 45$, the initial risk-free rate is set at $r(0) = 0.03, \, 0.05, \, 0.07$, the drift term is set at $\alpha = 0.005, \, 0.01$, and the diffusion term is set at $\beta = 0.1, \, 0.2$, and the correlation coefficient is set at $\rho^{12} = -0.3, \, 0.1, \, 0.5$, respectively. We separately combine each parameter. As an example, when $K = 35, \, r(0) = 0.03, \, \alpha = 0.005$, and $\beta = 0.1$, $\rho$ is equal to $-0.3, \, 0.1$, and $0.5$, respectively. The pricing results of Eq (3.6) are calculated using the same inputs; for details, see A-values in Table 1. R-error in Table 1 shows the results of our comparison of the errors between the simulated and analyzed values.

Table 1 demonstrates how the impact of changing parameters on the option price can have a significant economic impact when the strike price $K$ is allowed to rise. Take $r(0) = 0.03$, $\alpha = 0.005$, $\beta = 0.1$, and $\rho^{12} = -0.3$ as an example, the simulated prices for $K$ equal to $35$, $40$, and $45$ are $6.6460$, $1.9041$, and $0.1435$, respectively. The option price may decrease significantly with an increase in the strike price of $K$. The same phenomenon is noted for the analytical values, $6.7868$, $1.9943$, and $0.1380$, respectively. When $K = 35$, the analytical value is greater than the simulated value, and the relative error between the two values is $2.0745\%$. The simulated value, analytical value, and relative errors are $6.7325, \, 6.9223$, and $2.7418\%$, respectively, when the other parameters are held constant and the initial risk-free rate, $r(0)$, is raised to $0.05$. The errors do not significantly rise as $r(0)$ increases. This indicates that our pricing results are reasonable. The simulated, analyzed, and relative errors are, respectively, $6.5672, \, 6.5641$, and $-0.0483\%$ when all other parameters remain the same and the correlation coefficient $\rho$ is increased to $0.1$. There is no discernible difference in the errors after changing $\rho$. Similar results are discovered for the other parameters as well. As can be seen, changing one of the parameters does not affect the relative errors but changes the simulated or analytical values. In other words, the relative errors do not significantly change when a particular parameter is changed. The results of the pricing formula are accurate and reasonable following the verification of the Monte Carlo simulation.

4.2.   Three-asset case

Taking $n = 3$, we set the initial prices of three assets $S_1$, $S_2$ and $S_3$ are all $40$, and the option expiry period is still 6 months, i.e., $T = 0.5$. The stochastic volatility values are respectively $\sigma_r = 0.1$, $\sigma_1 = 0.1$, $\sigma_2 = 0.2$ and $\sigma_3 = 0.3$, and other parameters are shown in Table 2. Since the steps of the simulation procedure are comparable to those in the case of the two assets, we omit the details.

5.   Sensitivity analysis

To assess the effectiveness of the model, we look at the effects of a few key parameters on the analysis findings in this section. The strike price $K$, the correlation of the underlying assets $\rho^{ij} > (i = 1, 2; j = 2, 3; i < j)$, and the initial value of the risk-free interest rate $r(0)$ were the main factors we took into account. As a result, we can see how these parameters influence how much the option price changes.

5.1.   Sensitivity analysis of interest rate randomness

The effects of stochastic and constant interest rates on option prices are contrasted in this subsection. In Figure 1, we depict the changes in Asian rainbow call on max option prices at two assets ($n = 2$) and three assets ($n = 3$) with the stochastic or constant interest rates, respectively. The initial risk-free interest rate is $r(0)$, which ranges from $-0.1$ to $0.1$. When the interest rate is stochastic, the drift term, diffusion term, and volatility are all set to $\alpha = 0.01, \, \beta = 0.2$, and $\sigma_r = 0.2$, respectively. The drift term, diffusion term, and volatility are defined by $\alpha = 0, \, \beta = -1$, and $\sigma_r = 0$ when the interest rate is constant. From Figure (1a) and (1b), we can see that an increase in the risk-free interest rate causes the option's price to rise, but the option price increases more slowly at a stochastic rate than it does at a constant interest rate. This is a result of the stochastic interest rate's volatility, which affects a portion of the option price and makes it less sensitive than it would be if the interest rate were constant. In other words, under the stochastic interest rate, the option price is less influenced by the risk-free interest rate's size, reducing the dependence of the option price on the interest rate.

Figure 2 shows how the risk-free interest rate changes as expiration $T$ increases, where the range of the option's expiration period, $T$, is $0$ to $2$. The other parameters of interest rates are the same as in Figure 1. Figure (2a and (2b) illustrate that an extension of the expiration period causes the option's price to rise. The impact of the expiration period on the option price will, however, be diminished and will rise logarithmically at the stochastic interest rate. The price of options rises more quickly over time with a constant interest rate. The stochastic interest rate will not produce an absolute return and can be viewed as a risk if the interest rate changes at random. Options will not experience the same absolute return as a constant rate when this risk is present. Therefore, with a stochastic interest rate, the option's price must be reduced in order to offset the risk that the interest rate poses, preventing the price from rising too quickly as the time period is extended.

5.2.   Sensitivity analysis of the initial risk-free interest rate

Figure (3a) and (3b) respectively depict the changes in Asian rainbow call on max option prices at two-assets ($n = 2$) and three-assets ($n = 3$) with the strike price $K$ ranging from $35$ to $45$ for the initial risk-free interest rate $r(0)$ is equal to $0.03$, $0.05$, and $0.07$. For two assets $n = 2$ and three assets $n = 3$, we set the initial prices $S_i > (i = 1, 2)$ and $S_j > (j = 1, 2, 3)$ to be all $40$ respectively, and the option expiry period is still $6$ months, i.e., $T = 0.5$. The stochastic volatility values are $\sigma_r = 0.1$, $\sigma_1 = 0.1$, $\sigma_2 = 0.2$, and $\sigma_3 = 0.3$, respectively. The correlation of {the two and three underlying assets} are respectively $\rho^{12} = 0.5$ and $\rho^{12} = -0.3$, $\rho^{13} = 0.1$, $\rho^{23} = 0.5$.

The option price rises with the initial risk-free interest rate $r(0)$ at the same strike price $K$, as shown in Figure 3. The expected future return of the underlying asset will rise along with the rise in the risk-free interest rate when the price of the underlying asset remains constant. When discounted time prices are taken into account, a rise in the return on asset value appreciation, however, results in a fall in the value of the discount. Investors who purchase call options will profit when the price of options increases due to the higher risk-free rate. Furthermore, investing in stocks will cost more than purchasing the same number of call options. In the end, when interest rates are high, investors prefer to purchase call options because it is more expensive to buy stocks and hold them until they mature. Figure 3 also demonstrates that as the strike price $K$ rises, the option price gradually converges to $0$. It is more challenging for investors to profit from the options trade at expiration when the strike price of $K$ is very high. As a result of discounting to a very short initial time, the option price converges to zero.

Additionally, as can be seen in Figure 4, the option price of the two/three-asset Asian rainbow increases as the time to maturity $T$ increases and decreases as the strike price $K$ increases. For call options, the underlying asset price continues to rise over time. Therefore, extending the strike time is beneficial for investors to meet their desired expectations. The longer the term of the option, the greater the option's time value, which in turn increases the option price.

5.3.   Sensitivity analysis of the correlation coefficient of underlying assets

Asian rainbow call option prices at $n = 2$ and $n = 3$ are displayed in Figure (5a) and (5b), respectively, with strike prices $K$ ranging from $35$ to $45$ and various correlation coefficients of the underlying asset $\rho^{ij} > (i = 1, \, 2; j = 2, \, 3; > i < j)$.

As can be seen in Figure 5, the higher correlation coefficient of the underlying assets, $\rho^{ij}$, increases the likelihood that the underlying assets $S_i$ and $S_j$ will rise or fall together. The implied volatility of the option is higher because the Asian Rainbow call on max option is based on the higher price of the underlying asset. The price of options has increased as implied volatility has increased. Implied volatility, however, only has a modest impact on option prices. As a result, the variations in option prices under the various correlation coefficients are not significant when the strike price $K$ is high.

6.   Discussion

In this study, we take into account the pricing of geometric Asian rainbow options for multiple assets ($n$) when the stochastic interest rate followed the Vasicek model. Our study focuses not only on the use of stochastic interest rates but also more importantly on the expansion of the asset dimension. This is also the significance of our study on rainbow options. The relevance of multi-asset options in diversifying risk is obvious and is one of the purposes of this work. In detail, we propose an option payoff based on the geometric mean using the Vasicek model and the underlying asset price model combined. Four Asian rainbow option pricing formulas, including a call on max, call on min, put on max, and put on min option, are derived through rigorous derivation. These four options' pricing outcomes all satisfy the option parity relationship. Put-call and min-max combination parity theories are suggested based on this, and justifications are provided. Utilizing Monte Carlo simulations, the option prices were determined, and the validity of the pricing formulas was confirmed. In order to further demonstrate the compatibility of the pricing formulas with the financial derivatives market, a sensitivity analysis of variables including strike price, the initial value of risk-free rate, correlation coefficient, and strike time is carried out.

Certain limitations still exist in our model (2.2). To define time-varying interest rates, some stochastic processes are generally used: the Vasicek model, the CIR model, and the Hull-White model, which is an expansion of the Vasicek and CIR interest-rate models. In this study, we are only concerned with the former. However, it is reasonable to speculate that in some situations, the market's predictions about future interest rates include time-dependent parameters. The results in the Vasicek model might not be the same as those of the CIR model and the Hull-White model. Additionally, it might be valuable to search for a more inclusive and liberal result when choosing a stochastic interest rate model. We shall deploy these in future work.

Acknowledgements

This research is supported by Qing Lan Project of Jiangsu Province, NSFC $(11601226)$, Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2021SJB0081), Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX22_0406), Postgraduate Education Reform Project of Nanjing Tech University (YJG2212), and Postgraduate Excellent Teaching Case Project of Nanjing Tech University (2022-23).

Conflict of interest

The authors declare that they have no conflicts of interest.

A.   Proof of Theorem 3.2

Proof. From Eq (3.22), the payoff of the call on min option $V_{\min}^c(T, S_1, S_2, \cdots, S_n)$ can be expressed by the following equation

Set $A_i = \{G_i(T, S_i)\ge K, \cdots, G_j(T, S_j)\ge G_i(T, S_i), \cdots\}$ ($i, j = 1, 2, \cdots, n$, $i\neq j$), based on Eqs (2.3) and (2.11), we can obtain

It follows from Eq (3.23), we derive the call on min option price as

It follows from (2.9), (2.11) and (A.2) that we compute

where $i, j = 1, 2, \cdots, n$ and $i\neq j$. Taking

due to ${\rm d}W(u) = \varepsilon{\rm d}\sqrt{u}$ and $\varepsilon\sim N(0, 1)$ is a standard normal random variable, then $Z_0$ is rewritten as

and the variance of the random variable $Z_0$ is

Similarly, the variances of $Z_1$ and $Z_a$ take the form of

and

respectively. The correlation coefficients between the $n+1$ random variables $Z_0$, $Z_1$, $Z_a$ ($a = 2, \cdots, n$) satisfy

and

Summing up the above, $(Z_0, Z_1, \cdots, Z_n)^{\rm T}\sim N({\boldsymbol{0}}$, $\tilde{{\boldsymbol{\Lambda}}}_i)$ with $\tilde{{\boldsymbol{\Lambda}}}_i = \begin{pmatrix}\sigma_{0}^2 & \rho_{01}\sigma_{0}\sigma_{1} & \cdots & \rho_{0n}\sigma_{0}\sigma_{n}\\\rho_{01}\sigma_{0}\sigma_{1} & \sigma_{1}^2 & \cdots & \rho_{1n}\sigma_{1}\sigma_{n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{0n}\sigma_{0}\sigma_{n} & \rho_{1n}\sigma_{1}\sigma_{n} & \cdots & \sigma_{n}^2\end{pmatrix}$, $i = 1, 2, \cdots, n$. In view of Corollary 1, we can obtain

where $i, j = 1, 2, \cdots, n$, $i\neq j$, $C_0, \, m, \, C_i, \, f$ and $g_i$ are defined in Eqs (2.10) and (2.12). ${\bf{N}}_i(\cdot)$ is the $n$-dimensional cumulative standard normal distribution with the covariance matrix ${\boldsymbol{\Lambda_i}} = \begin{pmatrix}1 & \rho_{12} & \cdots & \rho_{1n}\\\rho_{12} & 1 & \cdots & \rho_{2n}\\\vdots & \vdots & \ddots & \vdots\\\rho_{1n} & \rho_{2n} & \cdots & 1\end{pmatrix}$, where the correlation coefficients $\rho_{1a}$ and $\rho_{cd}$ are defined in Eqs (9.12) and (9.13).

In addition, $I_{n+i}\, (i, j = 1, 2, \cdots, n, > i\neq j)$ satisfies

This completes the proof of Theorem 3.2.