A speculator, $i$ ( $i=1,2,\cdots ,N_a$), intends to drive the spot exchange rate up to her perceived fundamental value or the official intervention level (in case of the fixed exchange rate), s_ai, which is a random variable with perceived mean ${\bar{s}}_{ai}$ and perceived variance ${\sigma }_{sai}^2$. It is a “buy low and sell high” strategy with risk. As the N_a speculators hold the view of a potential appreciation of HKD, they will short the USD and long the HKD in the spot market. This is illustrated in Panel A of Table 1 . Having observed that the HKD is undervalued compared with their perceived value, i.e., $s>{\bar{s}}_{ai}$, speculators of group a expect the HKD to appreciate against USD soon. To make a speculative profit, suppose the speculation fund mobilized by speculator i to long domestic currency HKD is a_i and short foreign currency USD in the amount of a_i/s (column 1). This transaction incurs a cost of a_it in HKD (column 2), which is assumed to be proportional to the transaction volume a_i.

This table shows the transaction records of speculators. “+” indicates a purchase and “−” indicates a sale or a payment. s is the observed spot exchange rate, s_ai and s_bi are perceived spot rates in the future, which are random variables with mean of ${\bar{s}}_{ai}$ and ${\bar{s}}_{bj}$ by speculators of group a and b, respectively. t is the unit transaction cost, which is assumed to be paid in HKD. All profits are denominated in HKD for simplicity.

To close her position, the speculator i may reverse the transaction when the spot rate moves to her perceived level s_ai. This is illustrated in Panel A of Table 1 . The reverse transaction is to sell domestic currency HKD in the amount of a_is_ai/s and to buy foreign currency USD in the amount of a_i/s (column 3). This transaction incurs a cost of ta_is_ai/s in HKD (column 4). As s_ai is a random variable, the reverse transaction is associated with uncertainty.

Assume that there is a variable transaction cost, which includes the net financing cost to finance the speculation. The net financing cost may be negative, i.e., a net gain. This is because suppose the speculator initially borrowed USD to go short, got the HKD, which she would probably place as a deposit. So the LIBOR-HIBOR interest differential represents the net financing cost. In reversing the transaction, she withdraws the HKD deposit, and uses it to buy back the USD, repaying it to the initial lender. She may actually get a net gain if the amount of USD she buys in the spot market exceeds the principal plus interest of the original USD loan, with which she started speculating. To simplify the analysis, it is assumed that the transaction cost is charged at the domestic currency HKD. The speculator’s profit in HKD is given as:

${\omega }_{ai}=a_i\left[\left(1-t\right)-\left(s_{ai}/s\right)\left(1+t\right)\right],i=1,2,\cdots ,N_a.$ (1)

Profits made in the USD can also be modeled in a similar way.

Speculators are assumed to maximize their expected utility Eu_ai, which is defined as:

$E{u}_{ai}\equiv E{\omega }_{ai}-{\alpha }_i{\sigma }_{\omega ai}^2,$ (2)

where $E{\omega }_{ai}$ and ${\sigma }_{\omega ai}^2$ are the mean and variance of ${\omega }_{ai}$, respectively, and ${\alpha }_i$_i is the degree of risk-averse of the speculator i and a larger ${\alpha }_i$ indicates the speculator is more risk-averse.

The expected profit of the speculator i is given as:

$E{\omega }_{ai}=a_i\left[\left(1-t\right)-\left({\bar{s}}_{ai}/s\right)\left(1+t\right)\right]$(3)

and the variance of the profit is given as:

${\sigma }_{\omega ai}^2=a_i^2{\sigma }_{sai}^2{\left(1+t\right)}^2/s^2$ (4)

Strictly speaking, only speculators with expected non-negative profits will trade. This implies that the actual number of speculators ${N^{\prime }}_a\le N_a$. If the expected profits of the speculators are all negative, then there is no trade among speculators.

From the first order condition ∂Eu_ai/∂a_i = 0, the optimal speculation fund a_i(s) can be solved to maximize the expected utility Eu_ai:

$a_i\left(s\right)=\frac{\left(1-t\right)-\left({\bar{s}}_{ai}/s\right)\left(1+t\right)}{{\alpha }_i{\sigma }_{sai}^2{\left(1+t\right)}^2/s^2}$ (5)

The four key properties of the speculation fund a_i(s) are as follows:

(1) The higher the transaction cost (t) is, the less speculation funds are spent.

(2) The larger deviation of s from the perceived parity or fundamental value ${\bar{s}}_{ai}$ is, the more speculation activities are, i.e.

$\frac{\partial a_i\left(s\right)}{\partial s}=\frac{2s\left(1-t\right)-{\bar{s}}_{ai}\left(1+t\right)}{{\alpha }_i{\sigma }_{sai}^2{\left(1+t\right)}^2}\ge \frac{2s\left(1-t\right)-s\left(1+t\right)}{{\alpha }_i{\sigma }_{sai}^2{\left(1+t\right)}^2}\ge \frac{s\left(1-3t\right)}{{\alpha }_i{\sigma }_{sai}^2{\left(1+t\right)}^2}>0$ (6)

if t < 1/3. That is, the speculation fund a_i(s) is an upward sloping curve with s.

(3) The less uncertainty about the official intervention level or fundamental value of spot rate ( ${\sigma }_{sai}^2$) is, the more speculation activities are. In the limit, if ${\sigma }_{sai}^2=0$, the speculator will use up all her available resources to engage in speculation. In this case, the speculator becomes an arbitrageur. That is, $\partial a_i\left(s\right)/\partial {\sigma }_{sai}^2<0$, and $\underset{{\sigma }_{sai}^2\to 0}{\lim }{a}_i\left(s\right)={\bar{a}}_i$.

(4) The more risk-averse the speculator (α_i) is, the less speculation activities are.

In this study, it is focused on a devaluation game. This implies that the speculator group a is the potential arbitrageur group. However, if it is a revaluation game, then the potential arbitrageur group will be the group b.

The speculators of group b will do exactly the opposite of what the speculators of group a are doing at the foreign exchange spot market. They believe the HKD is overvalued compared with their perceived value, i.e., s < s_bj. Therefore, they decide to make a speculative profit to short domestic currency HKD and long foreign currency USD. This is illustrated in Panel B of Table 1 . It is a “sell high then buy low” strategy with risk.

Suppose the speculation fund utilized by the speculator, $j$ ( $j=N_a+1,N_a+2,\cdots ,K$), to short domestic currency HKD is b_j (see column 1 in Panel B of Table 1 ). As the spot rate is s, the amount he longs foreign currency USD is b_j/s (column 1). To close his position, the speculator j is expected to reverse his transaction when the market spot rate moves to his perceived level of s_bj. The reverse transaction is to long domestic currency HKD in the amount of b_js_bj/s and to short foreign currency USD in the amount of b_j/s (column 3). As s_bj is a random variable, the reverse transaction is associated with uncertainty. Subject to the transaction cost associated with each transaction, the speculators’ profit is given as (column 5):

${\omega }_{bj}=b_j\left[\left(s_{bj}/s\right)\left(1-t\right)-\left(1+t\right)\right],j=N_a+1,N_a+2,\cdots ,K$ (7)

The speculator is assumed to maximize his expected utility, Eu_bj, defined as:

$E{u}_{bj}\equiv E{\omega }_{bj}-{\beta }_j{\sigma }_{\omega bj}^2$ (8)

where ${\beta }_j$ is the degree of risk-averse of the speculator j.

The expected profit is:

$E{\omega }_{bj}=b_j\left[\left({\bar{s}}_{bj}/s\right)\left(1-t\right)-\left(1+t\right)\right]$ (9)

and the variance of the profit is:

${\sigma }_{\omega bj}^2=b_j^2{\sigma }_{sbj}^2{\left(1-t\right)}^2/s^2$ (10)

Similar to the previous discussion related to speculator group a, only speculators with expected non-negative profits will trade. This implies that the actual number of speculators ${N^{\prime }}_b\le N_b$. If the expected profits of the speculators are all negative, then there is no trade among speculators.

To maximize expected utility Eu_bj, the optimal speculation fund b_j(s) is:

$b_j\left(s\right)=\frac{\left({\bar{s}}_{bj}/s\right)\left(1-t\right)-\left(1+t\right)}{{\beta }_j{\sigma }_{sbj}^2{\left(1-t\right)}^2/s^2}$ (11)

The properties of speculation fund b_j(s) are similar to that of speculator group a, except that:

$\frac{\partial b_j\left(s\right)}{\partial s}=\frac{{\bar{s}}_{bj}\left(1-t\right)-2s\left(1+t\right)}{{\beta }_j{\sigma }_{sbj}^2{\left(1-t\right)}^2}\le \frac{2s\left(1-t\right)-2s\left(1+t\right)}{{\beta }_j{\sigma }_{sbj}^2{\left(1-t\right)}^2}\le \frac{-4st}{{\beta }_j{\sigma }_{sbj}^2{\left(1-t\right)}^2}<0$ (12)

if ${\bar{s}}_{bj}\le 2s$. That is, the speculation fund b_j(s) is a downward sloping curve with s.

Liquidity traders such as exporters/importers, consumers, and international investors may also engage foreign currency transactions at the foreign exchange market. These transactions are not of a speculative nature, i.e., they are not supposed to be profit-seeking in the foreign exchange spot market per se. Suppose that the net demand for USD is c_h and the net supply of HKD is sc_h for the liquidity traders, where c_h is assumed, for simplicity, to be independent of the observed spot rate s. And assume that the exogenous demand for HKD due to the government intervention is sq, where q = 0 means no intervention, and q > 0 and q < 0 indicate a revaluation and devaluation intervention, respectively.

Equilibrium of the spot market is attained when the aggregate demand for domestic currency HKD by speculators of group a and the government intervention (s q) is equal to the aggregate supply of domestic currency by speculators of group b plus the net supply of liquidity traders ( $s{c}_h$), i.e.,

$\underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}{a}_i\left(s\right)+sq=\underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}{b}_j\left(s\right)+s{c}_h$ (13)

Substituting Equations (5) and (11) into Equation (13), it has:

$\underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}\frac{\left(1-t\right)-\left({\bar{s}}_{ai}/s\right)\left(1+t\right)}{{\alpha }_i{\sigma }_{sai}^2{\left(1+t\right)}^2/s^2}+sq=\underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}\frac{\left({\bar{s}}_{bj}/s\right)\left(1-t\right)-\left(1+t\right)}{{\beta }_j{\sigma }_{sbj}^2{\left(1-t\right)}^2/s^2}+s{c}_h$ (14)

Hence,

$\begin{array}{l}\frac{1}{s}\left[\left(1+t\right)\underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}\frac{{\bar{s}}_{ai}}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^2}{1-t}\underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}\frac{{\bar{s}}_{bj}}{{\beta }_j{\sigma }_{sbj}^2}+{\left(1+t\right)}^2\left(c_h-q\right)\right]\\ =\left(1-t\right)\underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}\frac{1}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^3}{{\left(1-t\right)}^2}\underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}\frac{1}{{\beta }_j{\sigma }_{sbj}^2}\end{array}$ (15)

Solving for the equilibrium spot rate $s^{*}$:

$s^{*}=\frac{\left(1+t\right){{\displaystyle \sum }}_{i=1}^{N_a}\frac{{\bar{s}}_{ai}}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^2}{1-t}{{\displaystyle \sum }}_{j=N_a+1}^K\frac{{\bar{s}}_{bj}}{{\beta }_j{\sigma }_{sbj}^2}+{\left(1+t\right)}^2\left(c_h-q\right)}{\left(1-t\right){{\displaystyle \sum }}_{i=1}^{N_a}\frac{1}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^3}{{\left(1-t\right)}^2}{{\displaystyle \sum }}_{j=N_a+1}^K\frac{1}{{\beta }_j{\sigma }_{sbj}^2}}$ (16)

This suggests that an increase in government intervention—specifically, injecting USD into the market (q > 0) to boost demand for HKD—will lead to an appreciation of HKD and the equilibrium spot rate $s^{*}$ fall.

To establish the spot exchange rate market equilibrium, rewrite equation (13) without government intervention (q = 0) as follows:

$\underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}{a}_i\left(s\right)=\underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}{b}_j\left(s\right)+s{c}_h$ (17)

where a_i(s) and b_j(s) are defined in Equations (5) and (11) respectively, c_h is the net supply of HKD by liquidity traders.

Since speculators in group a anticipate that the HKD would revalue to their perceived level, they take long positions in HKD, becoming demanders of HKD in the spot market. Define their revaluation fund as their transaction volume in the left-hand-side of Equation (17):

$R\left(s\right)\equiv \underset{i=1}{\overset{N_a}{{\displaystyle \sum }}}{a}_i\left(s\right)$ (18)

Since ∂a_i/∂s > 0 from Equation (6), R(s) is an upward sloping curve in s as depicted in Figure 1 .

In contrast, since speculators in group b anticipate that the HKD may devalue to their perceived levels, they take short positions in HKD, becoming HKD suppliers in the spot market. Define their devaluation fund as their transaction volume, together with the net supply of the HKD of liquidity traders $s{c}_h$, in the right-hand-side of Equation (17):

$D\left(s\right)\equiv \underset{j=N_a+1}{\overset{K}{{\displaystyle \sum }}}{b}_j\left(s\right)+s{c}_h$ (19)

This figure shows that the equilibrium spot exchange rate $s^{*}$ may, or may not, equal to the true official intervention level or the fundamental value $\bar{s}$.

If c_h < 0, D(s) curve is depicted as downward sloping because ∂b_j/∂s < 0 from Equation (12). If c_h > 0 and c_h is not too large, then ∂D(s)/∂s < 0 still may hold and D(s) curve remains downward sloping. If c_h is excessively large and ∂D(s)/∂s > 0, then D(s) curve will slope upward; however, this does not alter the qualitative nature of the analysis. For simplicity, this paper focuses its analysis on cases where the D(s) curve slopes downward. The market equilibrium spot rate $s^{*}$ is defined at the level at which $R\left(s^{*}\right)=D\left(s^{*}\right)$, as depicted in Figure 1 .

Figure 1 shows an example that the market equilibrium rate $s^{*}$ is above the official parity or the fundamental value $\bar{s}$, indicating the market value of HKD is devalued at the spot market. This is a typical example that the market equilibrium may not be settled at its fundamental value if there are speculative activities betting on devaluation (Ma [8] ) or revaluation (Sun and Ma [32] ) of a domestic currency.

Rewrite the equilibrium spot rate $s^{*}$ Equation (16) as follows:

$s^{*}={\theta }_0+{\theta }_{1}q$(20)

where

${\theta }_0=\frac{\left(1+t\right){{\displaystyle \sum }}_{i=1}^{N_a}\frac{{\bar{s}}_{ai}}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^2}{1-t}{{\displaystyle \sum }}_{j=N_a+1}^K\frac{{\bar{s}}_{bj}}{{\beta }_j{\sigma }_{sbj}^2}+{\left(1+t\right)}^2{c}_h}{\left(1-t\right){{\displaystyle \sum }}_{i=1}^{N_a}\frac{1}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^3}{{\left(1-t\right)}^2}{{\displaystyle \sum }}_{j=N_a+1}^K\frac{1}{{\beta }_j{\sigma }_{sbj}^2}}$

${\theta }_1=\frac{-{\left(1+t\right)}^2}{\left(1-t\right){{\displaystyle \sum }}_{i=1}^{N_a}\frac{1}{{\alpha }_i{\sigma }_{sai}^2}+\frac{{\left(1+t\right)}^3}{{\left(1-t\right)}^2}{{\displaystyle \sum }}_{j=N_a+1}^K\frac{1}{{\beta }_j{\sigma }_{sbj}^2}}$

Both ${\theta }_0$ and ${\theta }_1$ are true parameter values of private information undisclosed to the government. Therefore, the government may not be able to identify the exact intervention $\bar{q}=\left(\bar{s}-{\theta }_0\right)/{\theta }_1$ such that $s^{*}=\bar{s}={\theta }_0+{\theta }_1\bar{q}$. However, the government may estimate ${\theta }_0$ and ${\theta }_1$ and derive q based on the estimated ${\hat{\theta }}_0$ and ${\hat{\theta }}_1$,i.e., $\hat{q}=\left(\bar{s}-{\hat{\theta }}_0\right)/{\hat{\theta }}_1$ such that

$\bar{s}={\hat{\theta }}_0+{\hat{\theta }}_1\hat{q}$(21)

Substituting $\hat{q}$ into Equation (20):

${\hat{s}}^{*}={\theta }_0+{\theta }_1\hat{q}$(22)

Subtracting Equation (22) by Equation (21), it has:

$E\left({\hat{s}}^{*}-\bar{s}|\hat{q}\right)=E\left({\theta }_0-{\hat{\theta }}_0\right)+\hat{q}E\left({\theta }_1-{\hat{\theta }}_1|\hat{q}\right)$(23)

If both ${\hat{\theta }}_0$ and ${\hat{\theta }}_1$ are unbiased estimation of ${\theta }_0$ and ${\theta }_1$, respectively, then it has:

$E\left({\hat{s}}^{*}-\bar{s}|\hat{q}\right)=0$

i.e.,

$E\left({\hat{s}}^{*}|\hat{q}\right)=\bar{s}$(24)

This shows that the government may be able to lock the spot rate s to $\bar{s}$ on average (provided that the government has sufficient foreign reserve), but not always. Indeed, from the spot exchange rate equilibrium Equation (16), it shows that there are so many factors that may affect the level of spot rate, rendering any government intervention a very difficult task to accomplish. For example, Taylor [33] highlights that central banks from Canada, France, Germany, Italy, Japan, Spain, Switzerland, the United Kingdom, and the United States incurred substantial losses, estimated at around $12 billion, while attempting to stabilize exchange markets during the 1970s. Furthermore, Naranjo and Nimalendran [34] find that bid-ask spreads diverged with government foreign exchange rate intervention during the period 1976-1994, which increased estimated $27.5 billion transaction costs on an annualized basis.

In the case of floating exchange rates where the fundamental value $\bar{s}$ is not easily observable. Hence, the perceived fundamental mean value of the spot rate ${\bar{s}}_k$ of agent k may be inconsistent with $\bar{s}$ and the perceived variance ${\sigma }_k^2$ may be non-zero. As a result, the equilibrium spot exchange rate may not be equal to $\bar{s}$, as Figure 1 illustrates. Over time, $\bar{s}$ may still not be observable and that may generate persistent deviations of the equilibrium spot exchange rate from $\bar{s}$. The lack of informational certainty about $\bar{s}$ may further create difficulties for arbitrage and may actually encourage speculations. As a result, the spot rate may drift further away from $\bar{s}$ because of noise.

In this case, the government establishes an official parity in a fully fixed exchange rate regime. Alternatively, the government sets up a central parity in a semi-fixed exchange rate regime. Examples of the former type include the situations in most developing countries with fixed exchange rate arrangements. As to the latter, the best example is Exchange Rate Mechanism (ERM) in Europe during 1980s and 1990s. Suppose $\bar{s}$ is the government-set central parity in such a case. This central parity may or may not equal the fundamental value of the exchange rate. As there is a clear target, the perceived mean value of the spot rate, ${\bar{s}}_k$, may be consistent with $\bar{s}$, at least for some agents k.

However, the historical evidence showed that the government interventions in these fixed rate regimes have not been transparent. Bhattacharya and Weller [36] and Fernholz [37] ), for example, argue that the “policy secrecy” or “constructive ambiguity” about the scale and the target of intervention might have advantage for the central bank as they have deterrent effects on the speculators. This result is consistent with the prediction of the model of this study. The non-transparency of policy generates uncertainty to speculators and the perceived variance ${\sigma }_k^2$ is non-zero. A higher degree of policy non-transparency will increase ${\sigma }_k^2$ and that will reduce speculative activities.

This should mean that spot rate stability would be enhanced. For example, Krugman’s [9] target-zone model shows that there should be an automatic stabilizing effect on the exchange rate. However, an evident empirical puzzle is that this has not been the case. The spot exchange rates in most of these regimes have never been fully locked as the authorities expected. Nor did they pass the empirical stability tests on target zones (Svensson [10] ).

The behavioral model of this paper may be useful to shed some light on this empirical puzzle. The above “constructive ambiguity” argument is only half of the story. In fact, non-transparency of policy generates uncertainty to both speculators and arbitrageurs. As a result, the perceived variance ${\sigma }_k^2$ for both arbitrageurs and speculators is non-zero and would rise if policy becomes less transparent. This deters both arbitrage as well as speculation activities. Overall, policy uncertainty violates the sufficient conditions to lock the spot rate to $\bar{s}$. In other words, in the absence of a market arbitrage mechanism, this type of fixed exchange rate regimes has not provided a firm anchor for arbitrageurs to restore the official parity of the exchange rate.

A gold standard requires the central bank to undertake two-way convertibility, i.e., from domestic to foreign currency and vice versa, at the fixed gold point to their fiat money. If two countries both adopt a gold standard, their bilateral exchange rate is effectively fixed via the gold points (Tsang [5] ). In a hypothetical case, if the Hong Kong Monetary Authority (HKMA) fixes one HKD to p_HK ounces of gold and the US Monetary Authority (USMA) fixes one USD to p_US ounces of gold. The bilateral exchange rate between HKD and USD is effectively fixed at

$\bar{s}=p_{US}/p_{HK}$ (HKD per USD)(35)

Any deviation of the spot rate from this fixed rate will generate a profitable opportunity for arbitrageurs via gold arbitrage, subject to transaction costs. As long as there is an adequate supply of gold, with the potential for recycling to support arbitrage activities, the need for government intervention in the foreign exchange market is reduced under the gold standard. This serves as a good example of a transparent and market-driven fixed exchange rate system.

To illustrate the arbitrage mechanism in the gold standard, suppose there is a group of gold arbitrageurs, G, who take the view that $s_{ai}={\bar{s}}_{ai}=\bar{s}$ without uncertainty, i.e., ${\sigma }_{sai}^2=0$, $\forall i\in \text{G}$. Table 3 presents the gold arbitrage mechanism in a devaluation game for the HKD with $\bar{s}<s$. In this example, the market spot rat s deviates from the official parity $\bar{s}$, indicating the HKD being devalued against the USD. The gold arbitrageur i seizes this arbitrage opportunity. She will short foreign currency USD in the amount of a_i/s and long domestic currency HKD in the amount of a_i at the spot exchange market (see column 1 in Table 3 ).

This table shows the transaction records of a gold arbitrageur i in a devaluation game with $\bar{s}<s$ under the gold standard. That is, the gold arbitrageur i seizes the arbitrage opportunity between the market spot exchange rate s and the official parity $\bar{s}$, where $\bar{s}=p_{US}/p_{HK}$ is the implicit official parity of bilateral spot exchange rate of HKD per USD, p_HK and p_US are the official gold parity per HKD and USD, respectively. “+” indicates a purchase and “−” indicates a sale or a payment. t is the unit transaction cost, which is assumed to be paid in HKD. All profits are denominated in HKD for simplicity.

To close her position, the gold arbitrageur would reverse the transaction by carrying out gold buying and selling with the HKMA and the USMA respectively. Firstly, she needs to sell the amount ( $a_i\bar{s}/s$) of HKD to HKMA and get $\left(a_i\bar{s}/s\right)p_{HK}$ amount of bullion in return (column 3). Secondly, as shown in column 5 of Table 3 , she has to transport this $\left(a_i\bar{s}/s\right)p_{HK}$ amount of bullion to the US and sell it to the USMA. In return, she will get USD $\left(a_i\bar{s}/s\right)p_{HK}/p_{US}=a_i/s$, which is exactly the same amount of USD she sold in column 1.

Transactions of USD in column 5 constitute the reverse transaction to column 1 of Table 3 . While transactions of gold in column 5 constitute the reverse transaction to column 3 of Table 3 . Effectively, gold arbitrage allows arbitrageurs to implement the reverse transactions at the fixed rate, $\bar{s}$, without uncertainty, as both HKMA and USMA fully commits to two-way convertibility undertaking at fixed gold points. Repeated arbitrage activities eventually will compete away the profits of gold arbitrageurs by appreciating the HKD against the USD, i.e.,

${\omega }_{ai}=a_i\left[\left(1-t_1-t_3\right)-\left(1+t_2\right)\bar{s}/s^{*}\right]=0$ (36)

Similar to equation (28), the arbitrage exchange rate model gives the equilibrium spot rate as follows:

$\underset{{\sigma }_{sai}^2\to 0}{lim}{s}^{*}=\bar{s}\left(1+t_2\right)/\left(1-t_1-t_3\right)$, (37)

if there is sufficient gold for arbitrage.

This implies that the spot exchange rate can be effectively locked subject to the transaction costs if gold bullion is in sufficient supply in countries adopting the gold standard. The collection of works by Officer [38] provides empirical evidence to support the prediction from the new model of this paper. Officer [38] shows that the gold standard was efficient if it takes account of the transaction costs, which included those on transportation and various forms of financing schemes.

Another type of fixed exchange rate regime without government intervention is the currency board system. Varieties of it were practiced mainly in former British colonies (Schwartz [39] ); and now in Hong Kong SAR, Macao SAR, Bulgaria, Brunei, amongst others (IMF [14] ). The classical “currency board” issues currency notes, i.e., cash, with 100 per cent foreign reserve backing at a fixed exchange rate under a two-way convertibility undertaking. This represents a strong commitment to economic discipline. Any holder of paper money therefore rests assured that he can exchange it into foreign currency at the fixed rate and vice versa.

Similar to the gold standard, the classical currency board arrangements (CBAs) depends on cash-based arbitrage to fix the spot exchange rate. Since the exchange rate of paper money is fixed, the rate of bank deposit has to follow suit. Any rate differential gives rise to profitable activity that closes the gap. As there is no government discretion, it eliminates the uncertainty for arbitrageurs. Suppose there is a group of arbitrageurs, G, take the view that $s_{ai}={\bar{s}}_{ai}=\bar{s}$ without uncertainty, i.e., ${\sigma }_{sai}^2=0$, $i\in \text{G}$. It may illustrate their cash arbitrage activities by Table 2 again. They long HKD and short USD at the spot market (see column 1 of Table 2 ). At the same time, they can also physically move their cash and sell it to the HKMA. They get USD in return and can sell it at spot market, reversing the transaction. This is illustrated in column 3. As HKMA fully commits to two-way convertibility at the fixed rate, these transactions face no uncertainty.

If arbitrage fund is abundant, it should be able to get the same equilibrium spot rate as that under the gold standard. In other words, the spot rate should be still be fixed subject to the transaction cost of cash arbitrage. For instance, in October 1983, the government of Hong Kong requested its Exchange Fund to have purchased US$100 million in cash, physically stored in a vault in Hong Kong, from the Federal Reserve Bank of New York (James [40] , p. 194). This action took place when the government adopted Greenwood’s [15] proposal to return to a classical currency board arrangements and resolved the currency crisis on 15th October 1983. At that time, the government anticipated that it would be a large shortage of USD cash arbitrage fund held by market arbitrageurs. Given the circumstances, government intervention by injecting USD cash to the market was inevitable, resembling the scenario previously discussed in Figure 4 . The purchase of USD cash in advance represented a significant political and economic commitment. The historical records of some small British colonies operating a sterling CB show that their exchange rates are fully fixed to the pounds (Williamson [16] ).

However, exceptions can also be found. For example, just before the 1997 Asian Financial Crisis, Tsang [5] [19] highlighted one of the weaknesses in Hong Kong’s currency board arrangements at that time. He argued that as Hong Kong developed into a modern financial economy and achieved the status of an international financial center, cash accounted for a very small—and increasingly shrinking—portion of the total money supply. There may be simply very little cash that can be employed for arbitrage fund. This violates the sufficient condition to lock the spot rate to the official parity $\bar{s}$ fully (see Figure 3 again for an illustration). In reality, since the inception of the currency board arrangements (CBAs) in Hong Kong in October 1983 and until September 1998, cash arbitrage had been never effectively functioned. As a result, the Hong Kong Monetary Authority (HKMA) had to depend on the management of interbank liquidity and interest rates, as well as outright intervention in the foreign exchange market to defend the Hong Kong dollar (HKMA [41] ; Tsang [5] ). Some would call that “conventional central banking dressed up as a currency board system” (Tsang [30] ). Due to imperfect information to the HKMA, significant deviations can be found in the spot rate from its official parity, although the CBAs in Hong Kong survived from the speculative attacks (Tsang [5] [19] ; Tsang and Ma [20] ). This is consistent with the theoretical prediction in Section 2.5 that government intervention is not effective to fix the spot rate under a classic currency board.

After completing a series of visits to the IMF, the World Bank, and relevant central banks to conduct his commissioned project by the Hong Kong Policy Research Institute in 1996, Tsang [5] [19] discovered that several former currency board countries, including Argentina, Estonia, and Lithuania, had adopted a modernized arrangement of convertibility undertakings by their monetary authorities, referred to as the “AEL model”. Convertibility at the fixed exchange rate $\bar{s}$ is extended to the whole monetary base, not just the cash base as in the case of the classical currency board arrangements (CBAs). The monetary base includes the cash base, as well as all the reserves and balances of banks with the monetary authority. This ingenious set-up bypasses the problem of moving cash around for arbitrage.

Under this “convertible reserves” system, arbitrage can be carried out by banks against each other and settled electronically through the inter-bank clearing system hosted by the monetary authority. That is, all transactions in Table 2 are carried out electronically and simultaneously, thereby reducing the transaction cost and risk to the minimal. A real-time gross settlement system (RTGS) hosted by the monetary authority, like what Hong Kong implemented in late 1996, provides such a kind of efficient arbitrage facility (Tsang [42] ).

This represents a breakthrough in arbitrage efficiency, surpassing that of both the old gold standard and the classical CBAs. Furthermore, “electronic arbitrage” should also increase substantially the available arbitrage fund, subject to the speed and cost of the settlement system. The mobilized arbitrage fund may never hit its upper limit $\bar{A}$. The mechanism of electronic settlement (that minimizes the transaction cost of arbitrage), the monetary authority’s two-way convertibility undertaking, and sufficient arbitrage fund fulfill the three sufficient conditions (i) to (iii) laid out in the theoretical model in Section 3 of this paper. A modern currency board system meeting these conditions can lock the spot rate firmly at the official parity $\bar{s}$.

This recommendation was initially put forward to the Hong Kong government and the Hong Kong Monetary Authority (HKMA) in 1996 (Tsang [5] [19] ; Law [43] ). Its objective was to tackle vulnerabilities in Hong Kong’s Linked Exchange Rate System and prepare for any potential speculative attacks on Hong Kong dollar during Hong Kong’s political transition in 1997. During and after the 1997 Asian Financial Crisis, this recommendation was revisited and debated again at the HKMA and the Legislative Council of Hong Kong (Legislative Council [44] ; Tsang [42] [45] ).

Subsequent institutional development in Hong Kong further supports this theoretical finding and Tsang’s [5] recommendations. In September 1998, the Hong Kong Monetary Authority (HKMA [46] ) adopted the seven technical measures to partially strengthen the Linked Exchange Rate System, the official term for the currency board arrangements. The HKMA introduced a weak-side Convertibility Undertaking to sell US dollars to licensed banks via an electronic system, without establishing a strong-side commitment. Consequently, from 1998 to 2003, the implicit strong-side threshold of HK$7.75 per US$ was breached on multiple occasions, although the weak-side edge remained intact throughout the period. In May 2005, the HKMA [47] further implemented a two-side Convertibility Undertaking to buy and sell US dollars with licensed banks at exchange rates of 7.75 and 7.85, respectively. This eliminated the uncertainty regarding the Hong Kong dollar’s appreciation or depreciation beyond the specified band and achieved symmetry around the Linked Exchange Rate of HK$7.8 per US dollar.

This is a big political and economic commitment. It marked the full implementation of modern currency board arrangements recommended by Tsang [5] [19] and satisfied the three sufficient conditions outlined in Section 3 of this paper, albeit the new measures have introduced a hybrid exchange rate system in Hong Kong. On the strong- and weak-side of the band, the banks may trigger convertibility undertakings with the HKMA. Within the band, the Hong Kong dollar is a managed floating rate arrangement with limited arbitrage uncertainty, as the HKMA [47] may choose to conduct market operations to smooth the operation of the exchange rate system.

Nevertheless, following these strengthening measures, the Hong Kong dollar has remained firmly pegged to the US dollar within the official committed band. Furthermore, the Hong Kong’s modern currency board system has withstood numerous crises, including the 9/11 terrorist attacks in 2001, the SARS outbreak in 2002, the global financial crisis of 2008, the Brexit event in 2016, Hong Kong’s social unrest in 2019, and the COVID-19 pandemic from 2019 to 2022. It has continued to function effectively during the ongoing tariff war that began in 2018.

To provide empirical data to substantiate the predictions of this paper’s new model, Figure 5 presents the monthly time series of the Hong Kong dollar’s interbank spot exchange rate against the US dollar over the period of September 1998 to March 2025. Figure 5 clearly shows that the Hong Kong dollar has been stayed within the official Convertibility Undertaking band of 7.75 to 7.85 per US dollar. This provides direct evidence to support the prediction of the newly developed behavioral model incorporating an arbitrage mechanism and Tsang’s [5] [19] foresight.

However, there are seven times before 2005 when the interbank spot rates appreciated against the US dollar that breached the 7.75 official band. It happened because the Hong Kong Monetary Authority had not yet implemented the strong-side Convertibility Undertaking to buy US dollars from licensed banks. As predicted by the behavioral model developed in this paper, when arbitrageurs encounter uncertainty regarding their arbitrage transactions, they are unable to fully commit to engaging in arbitrage. Consequently, the spot rate cannot be anchored within the official band. Only after May 2005, when the HKMA [47] implemented a two-sided Convertibility Undertaking, has the Hong Kong dollar spot rate been maintained within the official band, consistent with the predictions of the behavioral model.

To demonstrate that the Hong Kong dollar has endured multiple crises throughout the abovementioned sample period, Table 4 presents the results of OLS regressions using the Hong Kong dollar interbank spot exchange rate (HK$ per US$) as the dependent variable, incorporating dummy variables for six major historical events as explanatory variables. The sample period is from September 1998 to March 2025, a total of 319 monthly observations. The 9/11 Attacks dummy equals one for terrorist attacks in September 2001 and zero otherwise. The SARS2002 dummy equals one for the period of SARS outbreak from November 2002 to May 2004 and zero otherwise. The Crisis2008 dummy equals one over the period of global financial crisis from August 2007 to June 2009 and zero otherwise. The Brexit2016 dummy variable is assigned a value of one in June 2016 to represent the Brexit event—the UK-wide referendum held on 23 June 2016, in which 52% of voters supported leaving the EU—and zero in the remaining month-years. The Social unrest dummy equals one for the period of Hong Kong SAR’s social unrest during June 2019 to June 2020 and zero otherwise. The COVID-19 dummy equals one over the pandemic period of January 2020 to May 2023.

In Table 4 , columns 1 to 6 introduce each major event dummy variable

This table presents OLS regression results on the impact of six historical events on the Hong Kong dollar interbank spot exchange rate. The sample period is from September 1998 to March 2025. The dependent variable is the Kong dollar interbank spot rate HK$ per US$. A positive (negative) coefficient indicates a depreciation (appreciation) impact on the Kong dollar against the US dollar. P-values are reported in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively. The monthly time series of the interbank spot rate is downloaded from the Hong Kong Monetary Authority website: https://www.hkma.gov.hk/eng/data-publications-and-research/data-and-statistics/economic-financial-data-for-hong-kong .

individually in the regression, while column 7 incorporates all six event dummies within the same regression model. Additionally, each regression includes 11 month dummy variables (i.e., February to December) to account for seasonal effects, with January omitted as the reference month. Year fixed effects are also included in capturing the annual macroeconomic developments such as trade balance, GDP per capita, inflation, and employment.

Columns 1, 2, 4 and 5 of Table 4 show that these four historical events did not generate statistically significant impact on the Hong Kong dollar interbank spot exchange rate at the 10% level. But column 6 and 7 both show that the COVID-19 pandemic from 2019 to 2023 has a persistent devaluation impact on the Hong Kong dollar that is significant at the 5% level. Interestingly, the global financial crisis dummy shows an appreciation impact on Hong Kong dollar that is statistically significant at the 1% level. This was a result of the subprime crisis originating in the United States, which had spillover effects on Europe. These effects triggered capital flows into Hong Kong’s financial market, positioning it as a relatively safe international financial center in Asia. In conclusion, despite the series of events impacting the Linked Exchange Rate System, Figure 5 demonstrates that Hong Kong’s currency board remains robust, with the interbank spot rate consistently maintained inside the official band up to March 2025, which provides strong evidence to support the prediction of the behavioral model and Tsang’s [5] [19] foresight.

Parenthetically, while this study employs a simple dummy variable approach to test the stability of the Hong Kong dollar, it may oversimplify its complex dynamics. For example, Krugman’s [9] target zone model does not assume government interventions within the band (Flood and Garber [48] ). However, under the current currency board arrangements in Hong Kong, the HKMA [47] may choose to intervene in foreign exchange markets when the Hong Kong dollar exchange rate is within the band, if it deemed necessary. Future research could benefit from more sophisticated empirical designs, such as structural equation modeling or time-series analysis, to capture the intricate interplay of factors influencing the credibility and stability of the hybrid exchange rate system in Hong Kong.

One final note is in order. It is important to emphasize that modern currency board arrangements, like the gold standard, are technical mechanisms designed to stabilize exchange rates. They cannot be substitutes for sound policies to address broader economic or financial challenges (Tsang [30] ). For instance, it has been widely recognized that the gold standard cannot serve as a remedy for inflation, if the political economy cannot commit to constrain inflation (Flood and Garber [49] ). The gold standard also cannot be served as instruments to cure excessive public debt, if the government cannot discipline its fiscal deficits (Cochrane [50] ).

Similarly, Argentina adopted a solid modern currency board in 1991 but was forced to abandon it in early 2002 due to poor economic fundamentals. By contrast, Hong Kong’s currency board has thrived, thanks to its vibrant and adaptable economy, as well as its determined political-economic commitment, which has successfully navigated the evolving global landscape over the past four decades. Likewise, Fernández-Villaverde and Sanches [13] present a successful gold standard with full commitment to price stability and fiscal discipline in their theoretical analysis. In conclusion, a stable exchange rate regime is underpinned by a resilient economy with sound macroeconomic policies, and vice versa. The two are inherently complementary.