On model robustness of the regime switching approach for pegged foreign exchange markets

Abstract

We test the robustness of the regime switching model for pegged markets introduced by Drapeau et al. [How rational are the option prices of the Hong Kong dollar exchange rate? J. Derivatives, 2021, 28(3), 140–161]. In particular, there are two disputable underlying assumptions: (1) a Black and Scholes model with low volatility for the pre-depegging regime and (2) a thin tail distribution—Exponential type—for the time of the depegging. For the pre-depegging regime, we consider a bounded model within the peg—from Ingersoll and Rady. For the depegging time, we consider fat tail distributions more in line with catastrophic events—Pareto/Fréchet. We derive the option prices formula for each combination of these models. We then calibrate to option data from USD-HKD as well as EUR-CHF. In comparison to the benchmark model in Drapeau et al. [How rational are the option prices of the Hong Kong dollar exchange rate? J. Derivatives, 2021, 28(3), 140–161], it turns out that the relevant resulting characteristics—probability of a depegging before maturity, appreciation/depreciation at the depegging time as well as post-depegging volatility—are strongly robust in terms of model choice for this regime switching approach. However, from a term structure perspective, fat tail distributions fit the data significantly better and provide more rational depegging probabilities for short and long maturities.

Keywords:

• FX markets
• Pegged exchange rates
• Regime switching
• Option pricing
• Calibration

Disclosure statement

No potential conflict of interest was reported by the author(s).

Open Scholarship

 

This article has earned the Center for Open Science badges for Open Data and Open Materials through Open Practices Disclosure. The data and materials are openly accessible at https://doi.pangaea.de/10.1594/PANGAEA.915881.

Notes

1 Under some risk neutral probability measure assuming independence between Jump and Brownian motion, see Appendix.

2 As confirmed is an extended study (Zhang and Drapeau 2021) in terms of hedging.

3 We refer to Drapeau et al. (2021) and Zhang and Drapeau (2021) for extensive discussion of which.

4 See the drop of the peg by the Swiss franc on January 15th 2015.

5 $\underset{\_}{S}$ is actually the discounted exchange rate in the model for arbitrage reasons, see Ingersoll (1997).

6 Further extensions are scarce and rely on technical PDE approaches with boundary conditions, see Carr and Kakushadze (2017) and Carr (2017).

7 Unlike (Zhang and Drapeau 2021), tractable efficient Fourier approach is not available for bounded diffusion model (2(2) $\mathrm{d}\underset{\_}{S}=\underset{\_}{\sigma }(\underset{\_}{S}-l)(1-\frac{\underset{\_}{S}}{u})\mathrm{d}W$(2) ) due to the lack of a moment generating function. For the numerical implementation and calibration procedure, we therefore make use of quasi Monte-Carlo methods that prove to be efficient.

8 Note that generalized hyperbolic discounting functions are from the class of fat tail distributions of the Pareto kind.

9 Stopping time provided by the first Jump of a Poisson process.

10 In the case of Exponential distribution $s=1/\lambda$ where λ is the classical intensity.

11 See Appendix 3.

12 In other terms $\left(\underset{\_}{X}\right(t)1_{\{t<\tau \}}+\overline{X}(t\left)1_{\{t\ge \tau \}}\right)e^{-\kappa h(t\wedge \tau )}(1+\kappa 1_{\{t\ge \tau \}})$ is a martingale under P and τ is independent of the Brownian motion W. See appendix A.2.

13 Calibration of models to option data is particularly exotic for FX markets, see Clark (2011). The generic procedure is thoroughly explained in Drapeau et al. (2021) and Zhang and Drapeau (2021). Calibration process and details are presented in appendix 1.

14 The mean error in calibration procedure is defined in (A1(A1) $\mathrm{M}\mathrm{E}=\frac{1}{5}\sum _{i\in \mathcal{I}}\left|\frac{{\sigma }_{model}(K_i;{\theta }^{\ast })-{\sigma }_i}{{\sigma }_i}\right|\times 100\mathrm{\%}$(A1) ) in appendix 1.

15 In appendix 2, there are overall errors for all maturities.

16 In appendix 2, there are the relative errors for all the other maturities. The results are similar.

17 See appendix 2 for all the other maturities. Results do not differ.

18 See appendix 2 for all maturities. Similar conclusions.

19 The Swiss Franc was considered as a safe and stable currency due to its historical commitment to gold.

20 OTC option quotations in this case are not available to us.

21 Here is one-side bounded IR to BS model.

22 In appendix 2, the complete descriptive statistics can be found for all maturities, the results are in line with the $6M$.

23 See Appendix 2 for the comprehensive error for all maturities.

24 The average calibration error of $3M$, $6M$, $9M$, $1Y$, $18M$ and $2Y$

25 The double integral for IR/BS in theorem 2.1 is ${\int }_0^T\overline{\pi }(\tau =s)\mathrm{d}{F}_{\tau }\left(s\right)=e^{-r_{f}T-\kappa h\left(s\right)}(1+\kappa ){\int }_0^T{\int }_0^{1}BS(y,s)F_{\frac{x-l}{u-l},\underset{\_}{\sigma }\sqrt{s}}\left(\mathrm{d}y\right)\mathrm{d}{F}_{\tau }\left(s\right)$ where $BS(y,s):=BS\left(\right(u-l)y,\frac{K\left(s\right)}{1+\kappa }-l,\overline{\sigma }\sqrt{T-s})$.

Additional information

Funding

This work was supported by National Science Foundation of China, Grant Numbers: 11971310 and 11671257.