Abstract
In this paper, the pricing of foreign exchange options is considered under a modified Heston–Cox–Ingersoll–Ross hybrid model. This modified model reserves all the characteristics of the Heston–Cox–Ingersoll–Ross model and also additionally assumes regime switching in the key parameters of the volatility as well as the domestic and foreign interest rates. Even though complicated, we have derived a closed-form pricing formula for foreign exchange options after the affinity of this new model is verified. Various properties of the newly derived formula are also shown through numerical experiments. To show the performance of this newly proposed model, an empirical study is also conducted, the result of which suggests that our model is a good alternative to the Heston–Cox–Ingersoll–Ross model for practical purpose.
1. Introduction
Risk management forms an important part of modern management, particularly in portfolio management, as a result of the fast development of financial markets. Also, a more integrated global economy as a result of much better connectivity of global markets and financial institutions has posed new challenges to the effective management of financial risks. One of the most important types of financial risks is the foreign exchange risk, and it can be effectively managed through the trading of foreign exchange options, which give the holder the right but not the obligation to exchange money denominated in one currency into another at a pre-described exchange rate on a specific date. Therefore, the accurate and efficient determination of the price of foreign exchange options becomes an very important topic in the risk management area.
The pricing of foreign exchange options can be dated back to the early 1980s, when Biger & Hull (1983) and Garman & Kohlhagen (1983) independently expressed its price in an analytical form with the assumption that the underlying exchange rate follows a geometric Brownian motion, the so-called Black–Scholes model (Black & Scholes, 1973; Guardasoni et al., 2020; Lin & Zhu, 2020; Zhu & He, 2018). Even though simple and elegant, their model has inevitably led to some pricing biases as it is unable to appropriately describe what is reflected from real market data, and this has prompted the development of more sophisticated models to produce more accurate results.
A common practice is to add additional stochastic sources, providing further flexibility so that the fitness of the model to market data would be better. Among those modifications, stochastic volatility models are one of the most popular choices, which alleviate the so-called ‘volatility smile’ (Dumas et al., 1998), a phenomenon showing that the implied volatility extracted from market data actually displays a ‘smile’ curve instead of a flat line across different strike. However, in most cases, the introduction of stochastic volatility destroys the analytical achievability of the original model, and numerical methods should be adopted (Johnson & Shanno, 1987; Scott, 1987; Wiggins, 1987). However, these modifications are certainly not satisfactory because the lack of analytical pricing formula would make the calibration of the model very time consuming, posing a great obstacle in practical applications. In 1993, Heston (1993) made a great contribution to the literature by assuming that the volatility follows a CIR (Cox–Ingersoll–Ross) process and deriving a closed-form pricing formula for European-style options.
Although the Heston model enjoys great success in derivative pricing, one should still be aware that it sometimes fails to provide good fitness to real market data. For instance, Heston’s assumption of the constant interest rate is not appropriate (Bodurtha & Courtadon, 1987), because empirical evidence demonstrated that the incorporation of the stochastic interest rate would provide better model performance (Abudy & Izhakian, 2013), especially in modelling foreign exchange rate. As a result, the combination of stochastic volatility and stochastic interest rate has attracted a lot of attention. For example, the Hull–White interest rate model (Hull & White, 1993) was combined with the Heston model by Van Haastrecht et al. (2009), and combined with the correlated Stein-Stein model (Schöbel & Zhu, 1999) by Grzelak et al. (2012). Furthermore, the Heston model is also combined with the CIR interest rate model (Cox et al., 1985), formulating the Heston–CIR hybrid model, which gains popularity in pricing foreign exchange options Ahlip & Rutkowski (2013).
Recently, the existence of regime switching in real market data has been demonstrated from a lot of empirical evidence (Hamilton, 1990), making Markov-modulated models (Buffington & Elliott, 2002; He & Zhu, 2017, 2018) increasingly popular among researchers and practitioners. Some authors even went further by introducing regime switching parameters into stochastic volatility models Elliott & Lian (2013); He & Zhu (2016a) because regime-switching stochastic volatility models can not only enhance the forecasting power of the stochastic volatility models, but also better capture major events affecting the market (Vo, 2009). Being aware of this, in this paper, we introduce the regime-switching mechanics into the Heston–CIR hybrid model by allowing the key parameters of the volatility and the domestic and foreign interest rates jumping among different states instead of being constant. Although the introduction of regime switching makes the pricing model even more complicated, we have still managed to derive a closed-form pricing formula for foreign exchange options, by making use of the generalized moment generating function. Numerical experiments are also conducted to show various properties of the option prices under the regime switching Heston–CIR hybrid model. Finally, an empirical study is conducted to show the performance of our model when real market data is adopted.
The rest of the paper is organized as follows. In Section 2, the dynamics of the regime switching Heston-CIR hybrid model is introduced. In Section 3, a closed-from analytical solution is derived for the price of European-style foreign exchange options. In Section 4, numerical examples and discussions are presented. Concluding remarks are given in the last section.
2. The regime switching Heston–CIR hybrid model
In this section, the regime switching Heston–CIR hybrid model for the pricing of foreign exchange options is formally introduced, to take into account the presence of regime switching demonstrated by real market data. This model is constructed based on the well-known Heston–CIR hybrid model, under which the volatility and domestic and foreign interest rates follow the Heston model and CIR process, respectively, with the long-term means of the volatility and interest rates being allowed to vary among different different economic states of a Markov chain.
After the introduction of regime-switching mechanics into the Heston–CIR hybrid model, a natural question is whether the analytical tractability of foreign exchange options is still reserved, because this is an important property that is desired in practical applications. This issue will be discussed in the next section.
3. Pricing foreign exchange options
In this section, a closed-form analytical pricing formula will be derived for pricing foreign exchange options. In particular, after the affinity of the underlying dynamics is verified, a general form of the price of a foreign exchange call option is derived, with the generalized moment generating function to be further determined. The target function is then worked out after the derivation of the conditional generalized moment generating function with the assumption that the information of the Markov chain up to the expiry time is given.
3.1 The general pricing approach
One should notice that though the general pricing formula of foreign exchange options is now derived, as shown in (3.3), the determination of the option price requires further information of the generalized moment generating function |$f$|. Therefore, at this stage, our problem remains unsolved, and the key step becomes the derivation of |$f$|. In the next subsection, the technical details on the derivation of |$f$| will be provided.
3.2 The generalized moment generating function
After successfully deriving the generalized moment generating function in an analytical form, the pricing formula for foreign exchange options, as displayed in (3.3), is certainly exact and in a closed-form, which is very convenient to be implemented in practice. In the next section, some numerical examples are provided to show the properties of the prices of the foreign exchange options under our regime-switching Heston–CIR hybrid model.
4. Numerical experiments and discussions
In this section, the accuracy of the newly derived formula will be first checked, after which the influence of regime switching on foreign exchange options will be illustrated. In the following unless otherwise stated, we assume that the current state is 1, and the two transition rates, |$\lambda _{12}$| and |$\lambda _{21}$|, are both equal to 10. |$\bar{\theta }$| takes the value of |$(0.1,0.15)^T$|, while |$\bar{\beta }$| and |$\bar{\xi }$| are chosen to be |$(0.1,0.15)^T$| and |$(0.1,0.15)^T$|, respectively. Other parameters are set as |$\tau =1, k=10, \sigma =0.1, \alpha _1=5, \eta _1=0.1, \alpha _2=10, \eta _2=0.1, \rho =-0.8, r_t=0.1, v_t=0.1, \hat{r}_t=0.1, S_t=100, K=100$|. For comparison purposes, we use the same values for the corresponding parameters in the Heston–CIR hybrid model, with the constant long-term means, |$\theta $|, |$\beta $| and |$\xi $|, defaulted as the corresponding ones used for State 1 in our model.
Depicted in Fig. 1 is the comparison of foreign exchange call option computed through our newly derived formula and those obtained by directly simulating the model dynamics, (2.4), to demonstrate the accuracy of the formula. It is clear from Fig 1(a) that both prices are very close to each other, with the relative difference between the two prices being no greater than 0.8%. From this, one could certainly reach a conclusion that our formula produces accurate results, and it is safe to be implemented in practice.
The comparison of foreign exchange option prices calculated with our formula and those obtained through Monte Carlo simulation.
Figure 2 shows the pricing difference of our model and the Heston-CIR hybrid model when the regime switching occurs only in the long-term mean of the volatility. In this figure, we set |$\beta =\beta _1=\beta _2$| and |$\xi =\xi _1=\xi _2$| so that there are no actual regime switching in the interest rate directions. It can be observed from this figure the Heston–CIR price with |$\theta =\theta _1$| is lower than that with |$\theta =\theta _2$|, and the difference between the two increases when the time to expiry becomes larger. This is not surprising because a larger long-term mean of the volatility implies an overall higher volatility level and thus a greater option price. In addition, a larger time to expiry ensures more time for the volatility to reach the long-term mean, yielding further difference between the two. On the other hand, one can also observe from this figure that our price in State 1(2) is higher(lower) than the Heston–CIR price with |$\theta =\theta _1$|(|$\theta =\theta _2$|). This is mainly caused by the introduction of the regime switching, because the long-term mean of the volatility in our model could jump between |$\theta _1$| and |$\theta _2$| while it remains constant in the Heston–CIR model, yielding the average volatility level in State 1(2) being higher(lower) than the |$\theta _1$|(|$\theta _2$|) and thus also the corresponding option prices.
The Heston-CIR hybrid model VS our model with regime-switching long-term mean of volatility
Similar analysis can be conducted to investigate the impact of the introduction of the regime switching in the domestic and foreign interest rate directions, as shown in Fig. 3(a and b), respectively. From both figures, one can clearly observe that the prices produced from our model stay between the prices produced by the Heston–CIR model with two different long-term means. Moreover, if the regime switching is introduced only in the domestic(foreign) interest rate, our price in State 1 is lower(higher) than that of State 2. This is indeed reasonable. When only the long-term mean of the domestic interest rate is regime switching, the overall domestic interest rate lever in State 1 is lower than that in State 2. Financially, a lower domestic interest rate level implies a smaller price of the financial asset valued by domestic currency, resulting in a cheaper foreign exchange call option price, as shown in Fig. 3(a). Similarly, a higher foreign interest rate implies a relatively lower domestic interest rate, and thus the corresponding option price would become cheaper, as shown in Fig. 3(b).
The Heston-CIR hybrid model VS our model with regime-switching in the interest rate directions.
The effect of transition rates with respect to |$\theta _{X_t}$|.
The effect of transition rates with respect to different |$\beta _{X_t}$|.
In addition to |$\theta _{X_t}$|, |$\beta _{X_t}$| and |$\xi _{X_t}$|, the impact of the transition rates is also worth investigating when the regime-switching mechanics is introduced to model the economic status. Figure 4 displays the change of the foreign exchange call option prices w.r.t the transition rate of the long-term mean of the volatility. In this figure, we artificially let |$\beta _1=\beta _2$| and |$\xi _1=\xi _2$| so that only the long-term mean of the volatility is regime switching. In addition, for convenience, we assume that |$\lambda _{12}=\lambda _{21}$|. From this figure, it is clear that when the transition rates are both equal to zero, our price coincides with the Heston–CIR price, which can be understood by the fact that there is no actual regime switching in this case because of the zero transition rates. A further look at Fig. 4 reveals that the foreign exchange option price in State 1 is monotonically decreasing(increasing) w.r.t the transition rates if the long-term mean of the volatility of State 1 is higher(lower) than that of State 2. Financially, a larger transition rate implies that the long-term mean volatility would jump more often between |$\theta _1$| and |$\theta _2$|. Therefore, in this case, from an average point of view, the volatility in State 1 will in average have a lower lever if |$\theta _1>\theta _2$|, and the corresponding option price would become cheaper. A similar pattern can be observed when the long-term mean of the domestic interest rate in State 1 is smaller (larger) than that of State 2, i.e. |$\beta _1<\beta _2$| (|$\beta _1>\beta _2$|). In contrast, when the long-term mean of the foreign interest rate for State 1 is lower(higher) than that for State 2, i.e. |$\xi _1<\xi _2$| (|$\xi _1>\xi _2$|), the foreign exchange call option price increases (decreases) w.r.t the transition rate, as shown in Fig. 6. These two figures can both be explained from an average point of view as well as the positive impact of the domestic interest rate on the exchange option price, as discussed earlier.
The effect of transition rates with respect to different |$\xi _{X_t}$|.
Before leaving this section, it should be remarked that all the numerical experiments done in this section are based on artificial parameters, while in practice model parameters need to be extracted from real market data through a model calibration process. Therefore, to assess whether our model performs better the Heston–CIR model, an empirical study based on real market data is conducted, the details of which are presented in the next section.
5. Empirical studies
In this section, an empirical study is conducted to access the performance of our new model. This section is further divided into three subsections, according to three different stages of an empirical study. Some prior steps for model calibration will be introduced in the first two subsections, followed by a detailed analysis of empirical results in the last subsection.
5.1 Data description
This preliminary empirical study is based on Australian dollar/US dollar foreign exchange options, XDA, ranging from Jan to Mar 2016. However, this raw data set could not be directly used because sample noises should be eliminated first to avoid possible misleading conclusions, and the following filters are applied.
First of all, only the Wednesday options are adopted in the stage of parameter estimation because Wednesday is least likely to be a holiday in a week and less likely to be affected by the ‘day-of-the-week’ effect (Bakshi et al., 1997; Christoffersen et al., 2006). Secondly, options with the time to expiry being shorter than |$7$| days and longer than |$90$| days are both removed, since options that are very close to expiry have small time values and their prices could be very volatile, while the trading premiums of options that are far way from expiry are very large. These two situations imply that these kinds of options usually have liquidity problems (Le, 2015). Last but not the least, options with their bid prices1 being 0 are discarded since nobody is willing to buy these options at any cost.
With the filtered data set in hand, we are now ready to proceed to the parameter estimation step, which is presented in detail in the next subsection.
5.2 Parameter estimation
There are two different kinds of parameters in any option pricing formula, i.e. the parameters that are available when the contract is entered, and the parameters that are introduced by the adopted specific model. Thus, it is necessary to first figure out the parameters to be determined in the process of parameter estimation. The dynamics of the Heston–CIR model, as a benchmark, are specified in (2.1), from which one can clearly observe that there are actually thirteen model parameters, including four parameters associated with the volatility process, |$k,\theta ,\sigma ,v_0$|, four parameters associated with the domestic interest rate |$\alpha _1,\beta ,\eta _1,r_0$|, four parameters associated with the foreign interest rate |$\alpha _2,\xi ,\eta _2,\hat{r}_0$|, and the correlation factor |$\rho $|. After incorporating the effect of regime switching, our model has five additional parameters, i.e. the second mean-reversion level of volatility, domestic and foreign interest rate |$\theta _2$|, |$\beta _2$|, |$\xi _2$| and two transition rates |$\lambda _{12}$| and |$\lambda _{21}$|.
At this stage, it is quite clear that determining model parameters from real market data is equivalent to minimizing the MSE defined in (5.1). This can be solved by using optimization techniques. There are two main categories of optimization techniques, i.e. local optimizations and global optimizations. Although the approaches in the first category are easy and fast to implement, they depend heavily on the initial guess and are easily stuck in a local minimum. Considering the fact that our objective function (5.1) is not necessarily convex and there would be several local minima, the approaches in the latter category are preferred because they are able to skip local minima to ensure more possibility of attaining the global minimum.
Of all the global optimization approaches, the simulated annealing (SA) (Kirkpatrick et al., 1983) has received lots of attentions, because it not only has few parameters requiring tuning, but also theoretically guarantees the convergence to the global minimum. However, a noticeable shortcoming of this approach is its slow convergence, which makes it unsuitable in practice, especially in the recent trend of algorithmic trading. Therefore, a number of modifications have been proposed to address this issue. Among them, the adaptive simulated annealing (ASA), designed to find the best global fit of a non-linear constrained non-convex cost function over a D-dimensional space (Ingber et al., 2012), is one of the most popular variations (Ingber, 1989). The popularity of ASA results from the fact that this method is more efficient and less sensitive to user defined parameters than the SA does, while preserves all the advantages of the SA. Due to this unique feature, the ASA has been widely applied to different areas, including the model calibration Mikhailov & Nögel (2004); Poklewski-Koziell (2012). Moreover, the open-source code of the ASA (Ingber, 2018) makes it become more flexible and powerful, because the feedback from different users regularly assesses the algorithm. According to all the advantages of the ASA, we adopt this method in the current work to minimize the MSE, and the estimated daily-averaged parameters for both models are reported in Table 1.
Estimated parameters
| Parameters | Our model | Heston–CIR model |
|---|---|---|
| |$k$| | 14.9942 | 13.3853 |
| |$\theta _1(\theta )$| | 0.0029 | 0.0045 |
| |$\theta _2$| | 0.1791 | |
| |$\sigma $| | 0.0883 | 0.6510 |
| |$\alpha _1$| | 10.5919 | 11.3442 |
| |$\beta _1(\beta )$| | 0.1823 | 0.1502 |
| |$\beta _2$| | 0.3871 | |
| |$\eta _1$| | 1.1824 | 1.4673 |
| |$\alpha _2$| | 10.9166 | 11.5241 |
| |$\xi _1(\xi )$| | 0.1617 | 0.1375 |
| |$\xi _2$| | 0.3723 | |
| |$\eta _2$| | 1.1586 | 1.9407 |
| |$\rho $| | –0.6733 | –0.7446 |
| |$v_0$| | 0.0263 | 0.0292 |
| |$r_0$| | 0.1413 | 0.1915 |
| |$\bar{r}_0$| | 0.1654 | 0.2227 |
| |$\lambda _{12}$| | 2.05749 | |
| |$\lambda _{21}$| | 5.0562 |
| Parameters | Our model | Heston–CIR model |
|---|---|---|
| |$k$| | 14.9942 | 13.3853 |
| |$\theta _1(\theta )$| | 0.0029 | 0.0045 |
| |$\theta _2$| | 0.1791 | |
| |$\sigma $| | 0.0883 | 0.6510 |
| |$\alpha _1$| | 10.5919 | 11.3442 |
| |$\beta _1(\beta )$| | 0.1823 | 0.1502 |
| |$\beta _2$| | 0.3871 | |
| |$\eta _1$| | 1.1824 | 1.4673 |
| |$\alpha _2$| | 10.9166 | 11.5241 |
| |$\xi _1(\xi )$| | 0.1617 | 0.1375 |
| |$\xi _2$| | 0.3723 | |
| |$\eta _2$| | 1.1586 | 1.9407 |
| |$\rho $| | –0.6733 | –0.7446 |
| |$v_0$| | 0.0263 | 0.0292 |
| |$r_0$| | 0.1413 | 0.1915 |
| |$\bar{r}_0$| | 0.1654 | 0.2227 |
| |$\lambda _{12}$| | 2.05749 | |
| |$\lambda _{21}$| | 5.0562 |
Estimated parameters
| Parameters | Our model | Heston–CIR model |
|---|---|---|
| |$k$| | 14.9942 | 13.3853 |
| |$\theta _1(\theta )$| | 0.0029 | 0.0045 |
| |$\theta _2$| | 0.1791 | |
| |$\sigma $| | 0.0883 | 0.6510 |
| |$\alpha _1$| | 10.5919 | 11.3442 |
| |$\beta _1(\beta )$| | 0.1823 | 0.1502 |
| |$\beta _2$| | 0.3871 | |
| |$\eta _1$| | 1.1824 | 1.4673 |
| |$\alpha _2$| | 10.9166 | 11.5241 |
| |$\xi _1(\xi )$| | 0.1617 | 0.1375 |
| |$\xi _2$| | 0.3723 | |
| |$\eta _2$| | 1.1586 | 1.9407 |
| |$\rho $| | –0.6733 | –0.7446 |
| |$v_0$| | 0.0263 | 0.0292 |
| |$r_0$| | 0.1413 | 0.1915 |
| |$\bar{r}_0$| | 0.1654 | 0.2227 |
| |$\lambda _{12}$| | 2.05749 | |
| |$\lambda _{21}$| | 5.0562 |
| Parameters | Our model | Heston–CIR model |
|---|---|---|
| |$k$| | 14.9942 | 13.3853 |
| |$\theta _1(\theta )$| | 0.0029 | 0.0045 |
| |$\theta _2$| | 0.1791 | |
| |$\sigma $| | 0.0883 | 0.6510 |
| |$\alpha _1$| | 10.5919 | 11.3442 |
| |$\beta _1(\beta )$| | 0.1823 | 0.1502 |
| |$\beta _2$| | 0.3871 | |
| |$\eta _1$| | 1.1824 | 1.4673 |
| |$\alpha _2$| | 10.9166 | 11.5241 |
| |$\xi _1(\xi )$| | 0.1617 | 0.1375 |
| |$\xi _2$| | 0.3723 | |
| |$\eta _2$| | 1.1586 | 1.9407 |
| |$\rho $| | –0.6733 | –0.7446 |
| |$v_0$| | 0.0263 | 0.0292 |
| |$r_0$| | 0.1413 | 0.1915 |
| |$\bar{r}_0$| | 0.1654 | 0.2227 |
| |$\lambda _{12}$| | 2.05749 | |
| |$\lambda _{21}$| | 5.0562 |
Having successfully determined model parameters to minimize the distance between the model and market prices, the last stage of the empirical study is to analyse the model performance using the parameters reported in Table 1, the results of which are provided in the next subsection.
5.3 Empirical results
The model performance is reflected by the pricing errors, which can mainly be classified into two categories. The first one is the ‘in-sample error’, which is the remaining error between market and model prices after model calibration for the Wednesday data set used to determine parameter values, while the other is the ‘out-of-sample error’ which is another MSE calculated with market prices of another data set that is not used for the model calibration and the corresponding model prices with the estimated parameters. Thus, Table 2 below exhibits both of in- and out-of-sample errors of the two models produced by the Wednesday options data and Thursday options data, respectively.
In- and out-of-sample errors for the two models
| Error | In-sample | Out-of-sample |
|---|---|---|
| Our model | 1.88e-3 | 4.61e-2 |
| Heston–CIR model | 3.65e-3 | 6.25e-2 |
| Error | In-sample | Out-of-sample |
|---|---|---|
| Our model | 1.88e-3 | 4.61e-2 |
| Heston–CIR model | 3.65e-3 | 6.25e-2 |
In- and out-of-sample errors for the two models
| Error | In-sample | Out-of-sample |
|---|---|---|
| Our model | 1.88e-3 | 4.61e-2 |
| Heston–CIR model | 3.65e-3 | 6.25e-2 |
| Error | In-sample | Out-of-sample |
|---|---|---|
| Our model | 1.88e-3 | 4.61e-2 |
| Heston–CIR model | 3.65e-3 | 6.25e-2 |
Out-of-sample errors according to moneyness
| Moneyness | |$S/K<0.9$| | |$0.9\leq S/K\leq 1$| | |$1\leq S/K\leq 1.1$| | |$S/K>1.1$| |
|---|---|---|---|---|
| Our model | 3.78e-4 | 2.10e-3 | 3.54e-3 | 2.68e-3 |
| Heston–CIR model | 4.72e-3 | 2.34e-3 | 5.79e-3 | 9.90e-3 |
| Moneyness | |$S/K<0.9$| | |$0.9\leq S/K\leq 1$| | |$1\leq S/K\leq 1.1$| | |$S/K>1.1$| |
|---|---|---|---|---|
| Our model | 3.78e-4 | 2.10e-3 | 3.54e-3 | 2.68e-3 |
| Heston–CIR model | 4.72e-3 | 2.34e-3 | 5.79e-3 | 9.90e-3 |
Out-of-sample errors according to moneyness
| Moneyness | |$S/K<0.9$| | |$0.9\leq S/K\leq 1$| | |$1\leq S/K\leq 1.1$| | |$S/K>1.1$| |
|---|---|---|---|---|
| Our model | 3.78e-4 | 2.10e-3 | 3.54e-3 | 2.68e-3 |
| Heston–CIR model | 4.72e-3 | 2.34e-3 | 5.79e-3 | 9.90e-3 |
| Moneyness | |$S/K<0.9$| | |$0.9\leq S/K\leq 1$| | |$1\leq S/K\leq 1.1$| | |$S/K>1.1$| |
|---|---|---|---|---|
| Our model | 3.78e-4 | 2.10e-3 | 3.54e-3 | 2.68e-3 |
| Heston–CIR model | 4.72e-3 | 2.34e-3 | 5.79e-3 | 9.90e-3 |
Clearly, the introduction of regime switching has led to a significant improvement, in both in- and out-of-sample errors. In particular, the daily averaged in-sample MSE of our model is 1.88e-3, only around 50% of that of the Heston–CIR model, implying that the pricing biases can be reduced to half after the regime switching is introduced. On the other hand, although the improvement occurred in the out-of-sample MSE reduces a bit, compared with the in-sample one, it is still very significant with our out-of-sample error being less than 75% of that of the Heston–CIR model. The extent of improvement in both in- and out-of-sample errors indicates the importance of introducing regime switching into the pricing model, implying that our model serves as a better alternative than the Heston–CIR model, at least for the chosen data set.
Displayed in Table 3 are the out-of-sample errors of both models according to different moneyness, as it is always interesting to know how the model performs in ‘predicting’ prices of different options. In this table, the options are classified into four categories according to the magnitude of the underlying price |$S$| in relation to the strike price |$K$|, i.e. |$S/K<0.9$|, |$0.9\leq S/K\leq 1$|, |$1\leq S/K\leq 1.1$|, and |$S/K>1.1$|, representing the deep out-of-money, slightly out-of-money, slightly in-the-money and deep in-the-money options, respectively. Overall, our model outperforms the Heston–CIR model in all the four categories, with the deep out-of-money options enjoying the greatest improvement, in which our out-of-sample error is even less than one tenth of that of the Heston–CIR model. The categories of deep in-the-money and slightly in-the-money options also witness significant improvement, because our model produces respectively about 70% and 40% less errors than the Heston–CIR model. Although least improvement can be found in slightly out-of-money options, our model still reduces the error by more than 10%, in comparison to the error of the Heston–CIR model.
6. Conclusion
In this paper, a regime-switching Heston–CIR hybrid model is proposed for the pricing of foreign exchange options. This new model combines the effect of stochastic volatility, stochastic domestic/foreign interest rate and the regime switching mechanics. An exact and analytical formula for foreign exchange option prices is derived under this complicated model. Through numerical experiments, the accuracy of the formula is verified, and the impacts of regime switching on the foreign exchange options are also discussed quantitatively. To further assess the model performance, a comparison based on real market data is made between our model and the Heston-CIR model. The result suggests that our model greatly outperforms the Heston–CIR model in the test cases, implying that our model can be a good alternative to the Heston–CIR model in practice.
Acknowledgments
This work is supported by National Natural Science Foundation of China (11601189) and Research Base of Humanities and Social Sciences outside Jiangsu Universities “Research Center of Southern Jiangsu Capital Market” (2017ZSJD020).
Footnotes
As usual, the option price is deemed as the average of the bid and ask prices.
