SPONTANEOUS EUROIZATION IN THE CZECH REPUBLIC ( is it a problem and why not ? )

The paper offers a preliminary analysis of possible spontaneous euroization in the Czech economy. After a brief general introduction of the issue of currency substitution it specifically discusses two things. First, the transmission channels of potential spontaneous euroization, through which the process could possibly complicate the implementation of domestic monetary policy. Second, it analyses the degree of euroization. Among the transmission channels, attention is paid to interest rate and exchange rate channels. The circumstances under which the transmission would be sub-optimal are discussed. Besides the impact on the monetary policy transmission, another risk of progressive spontaneous euroization is seen in the shift of the exchange rate risk from bigger to smaller enterprises in the economy. The available data do not allow a precise measurement of the degree of euroization. Nevertheless, both the ratio of euro-denominated over koruna-denominated deposits and the CNB’s survey in 2003 suggest that euroization is not an obstacle for the Czech monetary policy at the moment.

Introduction
Euroization 1) is a process of substituting a foreign currency for a domestic one to fulfil the essential functions of money as a medium of exchange (currency substi-*) Czech National Bank, Na Příkopě 28, CZ -115 03 Prague 1 (e-mail: martina.hornikova@cnb.cz; jaromir.hurnik@cnb.cz). **) Czech Saving Bank, Na Perštýně 1, CZ -113 98 Prague 1 (e-mail: vkotlan@csas.cz). 1) Euroization is often called dollarization due to the origin of currency / asset substitution in South America. tution) and as a store of value (asset substitution). There are two forms of euroization -the official and the unofficial one. The first case, the official euroization, is a more proper definition according to which the euroisation is a form of official currency substitution where a reserve currency replaces the domestic legal tender. In such a case a country adopts de jure the currency of a foreign country to replace its domestic currency fully. Then the foreign currency has all the meanings of money and happens to be the authorized transaction medium, the store of value and the unit of account. It is generally regarded as the hardest form of pegged exchange rate regime, one that is more difficult to reverse than a currency board.
The official euroization 2) is not a pressing concern for the Czech Republic. However, euroization can take a spontaneous form as well. This is referred to as unofficial euroization. The process reflects a situation of firms and individuals spontaneously and voluntarily substituting a foreign currency for a domestic one as a means of payment (leading to a currency substitution) and/or choosing to hold foreign rather than domestic denominated monetary assets as a store of value (leading to an asset substitution).
The structure of this note is the following. First, the issue of euroization is explained, both the official and spontaneous forms. Second, the channels through which euroization may complicate the conduct of domestic monetary policy are discussed in a rather general manner. The focus lies in the challenges the euroization possess for the transmission of policy actions, although the impact on the transfer of currency risk in the economy is also briefly discussed. Third, the degree of euroization in the Czech economy is assessed using statistical data and survey results. The note ends with several concluding policy remarks.

Euroization
The discussion of the costs and benefits of abandoning the national currency in favour of the dollar or the euro has been taking place in many countries over the past two decades. This debate is especially effective in small open economies that are highly connected via the foreign trade with the European Union (in case of euroization) or with the United States (in case of dollarization). It is also a frequent point in discussions in the economies that have acceded the European Union recently under the obligation to join the monetary union in the future.
There are basically two ways how a foreign currency could be adopted instead of the domestic one. Either the country could adopt (let's say) the euro unilaterally or by prior arrangement with the given institution (in this case the European Central Bank -ECB or the European Commission). The latter is termed "consensual euroization", Meade et al. (2002). In this case, the euro would circulate in parallel to the domestic currency, or perhaps replace it altogether. In connection with the processes laid down for the EU Member States not participating in the euro area, this action would require authorities to fix the value of the domestic currency in terms of the euro, rather than participating in the ERM II target band arrangement for two years before entering the euro area.
2) For a detailed discussion on the topic of both the official and unofficial currency substitution see the papers presented at the Croatian National Bank's conference on Monetary Policy and Currency Substitution in the Emerging Markets (http://www.hnb.hr/dub-konf/8-konferencija-radovi/mazzaferro-thimann-winkler.pdf). For a discussion on official currency substitution (unilateral euroization) in the European countries see Bratkowski, Rostowski (2000).
Euroization is not particularly new. Such an official currency substitution has become known as "dollarization" even if the currency chosen to replace the domestic currency is the euro or some reserve currency other than the US dollar. Schemes of this kind, familiar from Hong-Kong or Argentina, have been introduced recently in several transition economies. Nowadays, currency boards can be found in Estonia, Lithuania (together with ERM II participation), Bulgaria, Bosnia and Herzegovina, and the euro as a legal tender in Kosovo and Montenegro. Euroization schemes have been discussed also in Poland and Bulgaria; prospective euroization has been advocated for some post-Yugoslav Republics. 3) Several econometric studies have shown that hard pegs such as currency boards lead to better inflation performance. In one of these, by Ghosh et al. (2002), the authors find that, keeping other factors constant, inflation is 10.5 % per year lower under a hard peg than under a floating exchange rate. More than one half of these, 5.5 % represents the credibility gain associated with the hard peg regime, and the remaining 4.5 % derives from lower money growth under the pegged regime.
In the best circumstances, euroization eliminates currency risk and possibly reduces default risk. The extent of these gains depends, of course, on how irreversible this substitution is perceived to be. Ideally, credibility gains from institutions' substitution will substantially narrow the spread on euro/dollar liabilities issued by the euroizing/dollarizing country relative to those issued by the respective foreign central bank, producing lower domestic interest rates.
Nevertheless, to follow the process of euroization successfully, other economic policies must be appropriate. In particular, euroization is a commitment by the monetary authority only and does not guarantee a responsible fiscal policy. Over the short-term, a lax fiscal policy can erode the credibility gain from such a currency substitution by putting upward pressure on domestic interest rates. Over the longer term, excessive deficits may undermine the monetary regime itself, by raising the incentive to monetize government debt.
In practice and as briefly mentioned above, there are more ways to adopt the foreign currency. On the one hand, unilateral dollarization or euroization, i. e. the country that is adopting the reserve currency does not require the agreement of the foreign issuer. On the other hand, broader forms of dollarization or euroization involve some set of on-going obligations for the issuer of the reserve currency and thus require mutual agreement by the two countries to a treaty or bilateral arrangement. This agreement could consider such issues as sharing of seigniorage revenue to fund a lender of the last resort facility. Finally, the most cooperative form of arrangement is a monetary union such as the euro area, in which participating countries share a currency and decision-making power. If such a currency substitution is to provide major credibility gains, it must be widely viewed as irreversible, as an irrevocable commitment to replace the domestic legal tender with the reserve currency.
As mentioned at the beginning of this chapter, some authors have advocated early euroization in the new EU Member States as an interim exchange rate arrangement during the transition to the EMU. A target band system, they argue, is crisis-prone and euroization would be the best regime to forestall a speculative attack. European officials do not see the ERM II as crisis-prone because of the wide bands around the central rate.
The unofficial (the so-called de facto) euroization is typically a response of economic agents to a loss of confidence in their domestic currency, often resulting from 3) A short summary on the issue of currency substitution in the European economies can be found in Nuti (2002). episodes of inflation or currency devaluation. It can be also related to a growth of unrecorded economic activities in which a foreign currency is often the preferred medium of exchange. Nevertheless, these are not the right reasons for the case of the Czech Republic. Domestic policies are considered as credible and the economic environment is stable. Yet, the high openness of the Czech economy to the European markets brings along the exchange rate risk for companies trading at least partially in the euro. Some of these companies may then prefer to conduct their transactions with domestic suppliers (residents) also in the euro for the sake of avoiding the foreign exchange risk. This de facto spontaneous euroization may then, depending on its degree, inhibit the effectiveness of monetary policy in influencing the domestic economy. It can also lead to distortion in various measures of macroeconomic activity, making the formulation of macroeconomic policy more difficult, and it can adversely affect the net foreign exchange risk exposure of a country.

The Impact of Euroization on Monetary Policy
Due to the changes in the transmission mechanism of a gradually euroizing economy the ability of monetary policy to influence the domestic economy is indisputably affected. Although the basic view of the transmission mechanism may remain the same, 4) the increasing use of the euro by domestic households and firms may decrease its importance in terms of transmitting policy changes to the economy. This may happen along the two most important transmission channels: the interest rate and the exchange rate channel. Spontaneous euroisation may also, somewhat counter-intuitively, increase the overall net foreign exchange risk exposure of a country.

1 Interest Rate Channel
Starting with the interest rate channel, we have to distinguish between the impact of monetary policy on households and on the corporate sector. In the case of households, we have to mention several critical points that may lead to a constrained ability of monetary policy to affect the behaviour of domestic households. These are receiving the incomes in the euro; saving in the euro; borrowing in the euro, and last but not least actually thinking in terms of the euro units.
The last point reflects the fact that the transmission mechanism of monetary policy is based on subjective perception of the households about a difference between the actual level of the interest rates and the equilibrium (normal) level of the interest rates, both adjusted for inflation expectations. For precise description of this mechanism see Woodford (2003) or Walsh (2003). Unless households undertake their internal decisions in the euro units, domestic monetary policy will continue to influence their behaviour. As the internal deliberations of households are unobservable, the only possible way we can follow are the three above-mentioned points. So we can only ask if the households receive their incomes in the euro, save and borrow in the euro. Bearing in mind that the households' consumption represents the biggest part of GDP, it means that if there is no observable increase in the euro saving and borrowing among households, the main part of the interest rate channel remains working.
In the case of firms, the situation is similar. Just the incentives to think and operate in the euro units are stronger because of the fact that many firms trade interna-4) For basic view on the transmission mechanism under the inflation targeting regime see Mahadeva and Sinclair (2001). tionally and receive a certain part of their revenues in the euro. The constraint on the domestic monetary policy starts with the effort of these firms to transfer the payment for their inputs into euro units. If they do so, additional firms will have to face the euro incomes and will be pushed to do the same and also transfer their inputs into the euro units. This would continue over and over again and as a result firms with revenues and costs in the euro will tend to observe variables, relevant from the monetary policy perspective, connected with the euro currency. On the other hand, bearing in mind that as long as majority of firms' employees and owners are residents, the wages and salaries will be paid in domestic currency and will continue being affected by domestic monetary policy.

2 Exchange Rate Channel
Simultaneously with the interest rate channel, the above-described processes have an impact also on the exchange rate channel. 5) Using the euro as an accounting unit weakens the role of the exchange rates in firms' decisions on production and investment. However, probably more than in the case of the interest rate channel, weakening of the exchange rate channel is hindered by the necessity to pay wages and salaries in domestic currency. In the case of households the switch from domestic currency to the euro in their decision making would also results in no role for the exchange rate channel of policy transmission. But just thinking in the euro units is not sufficient for households to avoid the exchange rate influence. The necessary condition for households is to face also the domestic prices in the euro. This is identical to the firms' cost problem. With domestic prices expressed in the domestic currency, the movements of the exchange rate can still result in a change of the relative prices and affect the households' decision making process.
Apart from the obvious effects spontaneous euroization may have on the transmission of monetary policy, there may have an important impact on the overall net foreign exchange risk in the economy. The gradual transfer of the foreign exchange risk 6) that comes with currency mismatch of revenues and costs, will likely result in smaller companies (suppliers) bearing more foreign exchange risk than if such a transfer did not take place. Since hedging entails some fixed costs, the relative cost of hedging rises for companies with smaller revenues. As a result the small companies can be forced either to decrease their profit margins or to hold the exchange rate risk without hedging. Both ways hamper the development of smaller companies and as such the country's overall production capacity. If the spontaneous euroization grows in scope large enough, the described process of risk transfer to smaller companies may be the reason for speeding up official euroization or euro-adoption.

Degree of Euroization in the Czech Republic
Induced changes in the transmission mechanism are connected with a switch of domestic households and enterprises from using the domestic currency to the euro in their transactions. However, the decision process behind this switch that is undertaken by households and enterprises is hardly observable and measurable in the reality. We can observe only the processes that we guess as peripheral expressi-5) The impact of exchange rate on the houselds decision making is precisely described in Obstfeld and Rogoff (1999). 6) For basic explanation of the problematics of the foreign exchange rate risk see Mishkin (2001).
ons of the internal decisions to use the domestic currency or the euro in transactions.
Since the above described problem is a matter of currency substitution, not asset substitution, measuring the degree of euroization would ideally require measuring the volume of transactions conducted in the euro among residents. As the balance of payments statistics only includes transactions with non-residents, and there is no other direct measure of transaction in the economy available, we use the deposits in the euro as an approximation. After this discussion, we will refer to a recent survey conducted by Czech National Bank's (CNB) branch offices.

1 Data on Deposits
Specifically, we calculate the ratios of the demand euro deposits over the demand deposits in the domestic currency and the euro. This, using the CNB's data set, results in the ratio of only 4 % and 11 % for domestic households and domestic enterprises, respectively. Calculating similar results for Hungary we receive the ratio of 16 % for households and 19 % for enterprises. 7) This appears to be no problem, however, especially in comparison to the country as Croatia where the ratio of foreign currency deposits to the domestic deposits and currency in the circulation is 150 %. The numbers for the Czech economy do not point to any major euroization going on. The development in time can be seen in Graph 1 and 2. 7) As we do not have detailed information, the ratios are calculated using foreign currency deposits up to two years and similarly deposits up to two years denominated in domestic currency. From the graph above we can clearly see that in the case of the Czech households there was minimum percentage of the euro demand deposits of the total demand deposits till September 2001 (stabilized at the level of 1 %). During the last quarter of 2001, there was a sudden sharp increase of the euro deposits held by the Czech households as a fraction of their total demand deposits. Nevertheless, this sharp increase is easy to explain. It was clearly connected with the announced in-troduction of the euro in circulation. The easiest way for the Czech households how to cope with this announced change was to save foreign European currencies they were holding and have it changed into the euro automatically. Still, the percentage at this time was only 7 % and since then it has followed a decreasing trend again. Recently the level is stabilized, slowly decreasing to the level of 4 %. All these factors are clearly supporting our argument that the euroization is not nowadays a problematic issue in the Czech Republic.

Graph 1 Percentage of Demand Deposits in EUR of the Total Demand Deposits for the Czech Households (in %)
In Graph 2 we have analysed the situation with the Czech enterprises. Even though the level of euroization is permanently higher in this group, compared to the previous one, the trend is recently fluctuating around the level of 12 % and is still decreasing. Again, for the same reasons as with households, there was a sharp increase from 8 % to 14 % level during the last quarter of the year 2001 but then it decreased a bit to 12 % and has stayed more or less at this level. There is no visible effect of the 2002 -2003 period of sharp koruna appreciation that could, at least in theory, motivate enterprises to switch their payments into the euro.
One of the reasons for the low degree of euroization even at times of sharp exchange rate appreciation may rest in the costs connected with euro transactions among residents. First, since such a trade relationship is considered as a foreign one, the payments must be settled on the nostro accounts at correspondent foreign banks. Second, such a foreign operation is much more time demanding, with the koruna trade the settlement is up to 24 hours but with the euro it can take even a week.

2 2003 Czech National Bank's Survey
Our findings can be also supported by several surveys carried out by the branches of the CNB during November 2003. 8) These surveys were conducted by six regional CNB's branches (with the exception of the branch in Prague) to find out the scope of the bilateral euro invoicing among domestic companies (those belonging to the region of the CNB's branch). The research was run in quite a liberal way, both with respect to the branches (that could decide on the way of conducting the research in their region) 9) and to the interviewed companies (in the sense that there were no direct questions asked). During 2004, a more in-depth systematic research was run periodically based on direct questionnaires to a wider sample of the Czech enterprises and so the information base improved over time.
The 2003 survey reveals that the companies may be divided into several groups: those with domestic ownership structure, foreign ownership structure, with position on domestic or on foreign markets. According to a group the company belongs to, we can draw some major conclusions about the volume of euro transactions. For those companies that are trading almost entirely with domestic partners there is still majority of operations run in domestic currency and there is no foresight of any change even in the near future. The fact that clearing of bank accounts among domestic companies is done in CZK only makes it too expensive and meaningless to use another currency.
On the other hand, the foreign oriented group (both export and import oriented) carries out the majority of transactions in a foreign currency. According to the surveys around 80 % of transactions among both export and import oriented companies is run in the euro. Enterprises with either strong export or import only use foreign currency invoicing in 52 % of their financial transactions. Domestic enterprises trading almost entirely with other domestic partners are even more domestic currency specified and use the foreign currency just in 5 % to 10 % of their transactions. The overall finding based on the conducted survey was that 77 % of the Czech companies are sometimes facing payments in a foreign currency and for 25 % of these such a payment is quite a usual means of transactions. On the other hand, the rest 23 % of the Czech companies did not meet such a phenomenon at all, yet. One sixth of the interviewed companies has the impression of further increase of the euroization. The mentioned ratios are drawn in Graph 3. 9) Some of the branches were contacting companies only, some were also interviewing the Economic Chambers in their region and commercial banks' branches. Also the way of running the research was not directly specified (over telephone, in written forms, e-mail, etc.).

3 Several Survey Findings
Southern Bohemia -the euro is used mainly by companies with foreign shares that import from their foreign suppliers and pay back in the euro. At the same time usually these big companies export majority of their products and again request the payment back in the euro. Nevertheless, still the services in these companies are paid in the domestic currency entirely. Taking into account other big companies in this region their invoicing is carried out in the euro not more than in 10 %. On the other hand, majority of leasing contracts on machinery is dealt in the foreign currency as a protection against the exchange rate risks. At the same time, foreign companies that have property shares in domestic companies have recently been interested in credits from domestic banks.
Eastern Bohemia -out of 24 interviewed companies, 3 did not use the euro at all, 15 only rarely and wouldn't consider euroisation to be an important issue in these days and just 6 analysed companies considered the euro transactions to be meaningful.
The analysed export oriented company is operating in the Southern Bohemian region. Because of its close connection to foreign markets, hedging against the exchange rate risk is important for them to survive in the European market conditions. The structure of their production is the following: 20 % for the export in the euro, 50 % for domestic trade inside the Czech Republic but in the euro and just 30 % in CZK. The share of USD or other foreign currencies is not more than 5 % of the total income. Since 2001, regarding the percentage of the euro encashment and the exchange rate risk when converting the euro back to CZK, the company has begun to calculate the balance in the euro. Impelling on their suppliers they have more or less forced them to trade in the euro as well and in this way they hedge against possible risks due to currency fluctuations. The euro exchange rate on incoming or outgoing payments is settled on daily basis according to the actual exchange list by the CNB. Still the salaries, both for the Czech and foreign employees, are realised entirely in CZK and the same is valid for payments of different services (mainly because domestic companies provide those). This article has identified various types of euroization and their causes with a special focus on unofficial, the so-called spontaneous euroization. It was shown that gradual unofficial euroization may pose a constraint on the ability of monetary policy to influence the domestic economy. Possible constraints are stemming from weakening of both the interest rate and the exchange rate channels of the monetary policy transmission. Apart from the impact on transmission, spontaneous euroization may also lead to transferring the exchange rate risk from larger to smaller enterprises. We argue that this process could increase the overall risk in the economy and, should it become widespread, it could be an argument in favour of early official euro introduction.
After this general part, we discussed the degree of euroization in the Czech economy. The ratio of demand deposits held in the euro to the total sum of the households' demand deposits supports the idea that the unofficial euroization is not an important phenomenon among the households. The same measure for the corporate sector shows that the unofficial euroization could be more important among firms but still it is remaining at relatively low levels. The ratios are either stable or even declining and without any relation to exchange rate movements.


extfieldsofstudy: Economics Physics

text:
A characteristic time scale of tick quotes on foreign currency markets

This study investigates that a characteristic time scale on an exchange rate market (USD/JPY) is examined for the period of 1998 to 2000. Calculating power spectrum densities for the number of tick quotes per minute and averaging them over the year yield that the mean power spectrum density has a peak at high frequencies. Consequently it means that there exist the characteristic scales which dealers act in the market. A simple agent model to explain this phenomenon is proposed. This phenomena may be a result of stochastic resonance with exogenous periodic information and physiological fluctuations of the agents. This may be attributed to the traders' behavior on the market. The potential application is both quantitative characterization and classification of foreign currency markets.

Introduction
Empirical analysis of high-frequency financial data have been attracting significant interest among physicists as well as economists during a decade (Mantegna andStanley 2000, andDacorogna 2001). Many features of financial markets have been clarified by many successive studies.
Actually it is well-known that the markets have a characteristic time scale in long period (daily, weekly, and monthly). However recent studies (Takayasu 2003, Ohnishi 2004) on time-series analyses in financial markets show that the market has a characteristic time scale in short period and propose the reason why traders are mainly using strategies with weighted feedbacks of past prices. Furthermore using the self-modulation process Takayasu et al. have found that the characteristic time scale is about 2 minutes in the JPY/USD market (Takayasu 2003) (abbreviated as MT).
On the other hand Baninec and Krawiech and Ho lst proposed a possibility that stochastic resonance occurs in markets (Babinec 2002 andKrawiech 2003) through an Ising-like agent model. They suggest that a periodicity in the market results from exogenous periodical information (abbreviated as BKH).
In order to clarify the mechanism of this characteristic time I think that we should examine it on a different standpoint from MT and BKH. Both studies focus on prices or price returns. However, in this article, we focus on the number of tick quotes in foreign currency rates (USD/JPY) and investigate the statistical properties of them by utilizing the power spectrum technique. As the results of examining the number of tick quotes in USD/JPY market it is found that the power spectrum density (PSD) has some peaks at about 2 minutes (the peak frequency depends on the currency markets).
In order to explain this phenomena a simple agent model based on doublethreshold noisy device (Sato 2004) is proposed. From a result of numerical simulations of the model it is found that the high periodicity of the number of tick quotes may happen. This result leads to a hypothesis that this periodicity is caused by common exogenous periodical information.
The purposes of this study are as follows: (1) to examine the number of high-frequency quotes lead us to deeply understand microscopic market activities. (2) this may provide useful information for market players to consider their trading strategy.

Data Analysis
The number of ask quotes per minute in USD/JPY is counted for a period of 1998 to 2000. Utilizing the data we calculated three PSDs for 2,048 points in weekday and average them over the year. The averaged power spectrums on the semi-log scale are shown in Fig. 1. They all have a peak at 0.4 (1/min), namely 2.5 minutes. We consider that these peaks exhibit characteristic time scales of dealers' activities, i.e., the dealers act having the periodicity of 2.5 minutes.

Dealer model
We introduce a simple agent model based on double-threshold noisy devices in order to understand the characteristic time scales found in the averaged power spectrum density. This model contains N dealers and the ith dealer has double-threshold (θ (1) i > θ (2) i ) to decide buy(1), sell(-1) and wait(0), and noise source ξ i (t) to model an uncertainty in their mind. We assume that the ith dealer must choose a decision (output) into the three ones y i (t) = {1, 0, −1} based on information (input) x i (t) with an uncertainty ξ(t) in his/her mind: . (1) Here we assume that ξ i (t) is identically independent Gaussian distribution, where σ i are standard deviations of the ith dealer. It is assumed that the input of each dealer is exogenous periodic information s(t) = A sin(2πf t), where A represents an amplitude, and f a frequency. For s(t) > 0 the dealers feel it good news and tend to decide a buy, while for s(t) < 0 they do it bad news and to decide a sell.
Furthermore the number of tick quotes per unit time X(t) is defined as For simplicity assume θ Obviously the activity X(t) is always zero if σ = 0 and A < θ, so that, there is no uncertainty of the dealers in their mind and the exogenous information is weaker than the threshold for the dealers to decide their action. However if there is uncertainty σ > 0 then the activity X(t) can exhibit periodicity despite of A < θ due to stochastic resonance (see Gammaitoni Hänggi Jung and Marchesoni (1998)).
As shown in Fig. 2 it is found that the PSD has some peaks from numerical simulations of the dealer model for σ > 0 and A < θ. This peak is caused by stochastic resonance.

Discussion and Conclusion
We empirically investigate the number of the tick quotes per unit time in foreign currency market (USD/JPY). It is found that the power spectrum densities of them for a period 1998 to 2000 all have a peak at 0.4 [1/min]. From the results it is conclude that a periodical action of dealers exists.
In order to explain this phenomena a simple dealer model based on the double-threshold noisy devices is proposed. Under a hypothesis that the mechanism of this periodicity is stochastic resonance the market activity in the model shows periodicity due to uncertainty of dealers' decision even though the exogenous periodical information is weaker than the threshold for dealers to decide their action. In fact this model is a feedforward one, however, real markets contains complicated (positive and negative) feedbacks. The future work is to consider the feedbacks to improve the dealer model.
Moreover the source of this periodicity is open problem. One possibility is an endogenous feedback mechanism of dealers as shown in MT. The other is an exogenous periodical information as shown in this paper. More detailed data analyses let us clarify the mechanism of this phenomena. To consider this problem is expected to contribute to a deep understanding of fluctuations and structure in the market.


extfieldsofstudy: Mathematics Economics

s2fieldsofstudy: Economics Mathematics

text:
On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs

In this note, we consider a general discrete time financial market with proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual formulation for the set of initial endowments which allow to super-hedge some American claim. We show that this extends the result of Chalasani and Jha (2001) which was obtained in a model with constant transaction costs and risky assets which evolve on a finite dimensional tree. We also provide fairly general conditions under which the expected formulation in terms of stopping times does not work.

Introduction
We consider a discrete time financial market with proportional transaction costs. These markets have already been widely studied. In particular, a proof of the fundamental theorem of asset pricing was given in Kabanov and Stricker [4] in the case of finite Ω and further developed in Kabanov et al. [5], [6], Rásonyi [8], Schachermayer [10] among others. In these papers, a super-replication theorem is also provided for European contingent claims. The aim of this paper is to extend this theorem to American options. It is well known that, for frictionless markets, the super-replication price of an American claim admits a dual representation in terms of stopping times. However, it is proved in Chalasani and Jha [1] that, in markets with fixed proportional transaction costs and with assets evolving on a finite dimensional tree, this formulation does not hold anymore. In their setting, they show that a correct dual formulation can be obtained if we replace stopping times by randomized stopping times, which amounts to work with what they call "approximate martingale node-measures". In this paper, we provide a new dual formulation for the price of American option which works in the general framework of C-valued processes as introduced in Kabanov et al. [5], and extends the dual formulation of Chalasani and Jha [1].
The rest of the paper is organized as follows. In Section 2, we describe the model and give the dual formulation. The link between our formulation and the one obtained by Chalasani and Jha [1] is explained in Section 3. Section 4 is devoted to counter-examples. The proof of the dual formulation is provided in Section 5.

Model and main result 2.1 Problem formulation
Set T = {0, . . . , T } for some T ∈ N \ {0} and let (Ω, F, P) be a complete probability space endowed with a filtration F = (F t ) t∈T . In all this paper, inequalities involving random variables have to be understood in the P − a.s. sens. We assume that F T = F and that F 0 is trivial. Given an integer d ≥ 1, we denote by K the set of C-valued processes K such that R d + \ {0} ⊂ int(K t ) for all t ∈ T. 1 Following the modelization of Kabanov et al. [6], for a given K ∈ K and x ∈ R d , we define the process V x,ξ by where ξ belongs to and, for a random set E ⊂ R d P − a.s. and G ⊂ F, L 0 (E; F) (resp. L 0 (E; G)) is the collection of F-adapted processes (resp. G-measurable variables) with values in E.
The financial interpretation is the following: x is the initial endowment in number of physical units of some financial assets, ξ t is the amount of physical units of assets which is exchanged at t and −K t is the set of affordable exchanges given the relative prices of the assets and the level of transaction costs.
Before to go on, we illustrate this modelization through an example (see also Section 3, Kabanov and Stricker [4] and Kabanov et al. [6]). 1 Here, we follow Kabanov et al. [6] and say that a sequence of set-valued mappings (Kt) t∈T is a C-valued process if there is a countable sequence of R d -valued F-adapted processes X n = (X n t ) t∈T such that, for every t ∈ T, P − a.s. only a finite but non-zero number of X n t is different from zero and Kt = cone{X n t , n ∈ N}. This means that Kt is the polyhedral cone generated by the P − a.s. finite set {X n t , n ∈ N and X n t = 0}.
Example 2.1 Let us a consider a currency market with d assets whose price process is modelled by the (0, ∞) d -valued F-adapted process S. Let M d + denote the set of d-dimensional square matrices with non-negative entries. Let λ be a M d + -valued F-adapted process and consider the C-valued process (K t ) t∈T defined by where π ij t := (S j t /S i t )(1 + λ ij t ) for all i, j ≤ d and t ∈ T. In the above formulation, a ij stands for the number of units of asset j obtained by selling a ij π ij t units of assets i. λ ij t is the coefficient of proportional transaction costs paid in units of asset i for a transfer from asset i to asset j. If ξ t ∈ −K t , then we can find some financial i.e. the global change in the portfolio position is the result of single exchanges, η ij t , between the different financial assets, after possibly throwing away some units of these assets. The random set K t denotes the so-called solvency region, i.e. V t ∈ K t means that, up to an immediate transfer ξ t , V t can be transformed into a portfolio with no short-positionṼ t = V t + ξ t ∈ R d + . Observe that we can assume, without loss of generality, that Indeed, if this condition is not satisfied then any "optimal" strategy would induce an effective transaction cost equal toλ ij t := (1 + λ ik t )(1 + λ kj t ) − 1.
The set of all portfolio processes with initial endowment x is given by corresponds to the collection of their values at time t ∈ T.
It is known from the work of Kabanov and Stricker [4], Kabanov et al. [6], Kabanov et al. [5] and Schachermayer [10], see also the references therein, that, under mild no-arbitrage assumptions (see N A s (K) and N A r (K) below), the set A T (x; K) can be written as where Z(K) is the set of (F, P)-martingales Z such that and K * t (ω) denotes the positive polar of K t (ω), i.e.
The operator "·" denotes the natural scalar product on R d and L 1 (E; P) (resp. L 1 (E; F, P)) is the subset of P-integrable elements of L 0 (E; F) (resp. L 0 (E; F)).
In this paper, we are interested in the set of processes which are dominated by a portfolio in the sense of K: V t − ϑ t ∈ K t , for all t ∈ T. The relation V t −ϑ t ∈ K t means that there is an immediate financial transaction ξ t ∈ −K t such that V t + ξ t = ϑ t . Hence, A s (x; K) can be interpreted as the set of American claims ϑ, labelled in physical units of the financial assets, which are super-hedgeable when starting with an initial wealth equal to x.
More precisely, our aim is to provide a dual formulation for the set of initial holdings x that allow to super-hedge ϑ.

Dual formulation
In analogy with the standard result for markets without transaction cost, one could expect that Γ(ϑ; K) can admit the dual formulation where T (T) is the set of all F-stopping times with values in T. If we assume that ϑ is bounded from below, component by component, this formulation is obtained by taking the supremum over all stopping times in (2.1) as in the frictionless case. However, this characterization does not hold true in general, as shown in Section 4. This phenomenon was already pointed out in Chalasani and Jha [1] in a model consisting of one bank account and one risky asset evolving on a finite dimensional tree. In Chalasani and Jha [1], the authors show that a correct dual formulation can be obtained if we replace stopping times by randomized stopping times.
In our general framework, this amounts to introduce a new set of dual variables, see Section 3 for an interpretation in terms of randomized stopping times. ForP ∼ P, the associated set of dual variables, D(K,P), is defined as the collection of processes Example 2.2 In the model of Example 2.1, we have It follows that D(K,P) is the collection of processes Z ∈ L 1 (R d + ; F,P) such that In the following, we shall say that a subset We then have the following characterization of A s (K) := A s (0; K).
is closed in measure and that the no-arbitrage condition holds. Then, for allP ∼ P, there is Z ∈ D(K,P) with values in (0, ∞) d . Moreover, the following assertions are equivalent : for all Z ∈ D(K,P) such that (ϑ · Z) − ∈ L 1 (R; F,P).
this immediately provides a dual formulation for Γ(ϑ; K).
Following the approach of Kabanov et al. [5] and Kabanov et al. [6] the closure property of A s (0; K) can be obtained under the general assumption ξ ∈ A(K) and The financial interpretation of the assumption K 0 t = {0} is that there is no couple of assets which can be exchanged freely, without paying transaction costs. In the model of Example 2.1, it is easily checked that it is equivalent to λ ij t + λ ji t > 0 for all t ∈ T.
2. In the case where K 0 t may not be trivial, (2.4) holds under the robust no-arbitrage condition introduced by Schachermayer [10] and further studied by Kabanov et al. [6], N A r (K) : N A(K) holds for someK ∈ K which dominates K, In finance, this means that there is a model with strictly bigger transaction costs (in the directions where they are not equal to zero) in which there is still no-arbitrage in the sense of N A.
3 Interpretation in terms of "approximate martingale node-measures" and "randomized stopping times" In this section, we show that the dual formulation of Corollary 2.1 is indeed an extension of the result of Chalasani and Jha [1]. To this purpose, we consider the example treated in the above paper. It corresponds to a financial market with one non-risky asset S 1 and one risky asset S 2 . For sake of simplicity, we restrict ourselves to this 2-dimensional case, although it should be clear that the above arguments can be easily extended to the multivariate setting. We also assume that the interest rate associated to the non-risky asset is equal to zero and normalize S 1 ≡ 1, otherwise all quantities have to be divided by S 1 which amounts to considering S 1 as a numéraire and working with discounted values. Here, S 2 is a (0, ∞)-valued F-adapted process. Given µ and λ in (0, 1), the model considered in Chalasani and Jha [1] corresponds to the C-valued process (K t ) t∈T defined by It follows that D(K, P) is the collection of processes Z ∈ L 1 (R 2 + ; F, P) such that S 2

"Approximate martingale node-measures"
We first provide an alternative dual formulation in terms of what Chalasani and Jha [1] call "approximate martingale node-measures". Although we are not considering a finite probability space, we keep the term "node measure" used in the above paper for ease of comparison.
Given Z in D(K, P), let us defineẐ 1 = Z 1 /S 1 = Z 1 andẐ 2 = Z 2 /S 2 so that (3.1) can be written equivalently in 2) where 1 denotes the constant process equal to 1, and, for a bounded from below process α , for all t ∈ T. Here, we use the convention 0/0 = 0. Hence, any element Z in D(K, P) with Z 1 = 0 is such that (χ Z , q Z ) belongs to the set Q(K, P) of elements (χ, q) of L 1 (R 2 + ; F, P) such that, for all t ∈ T, is defined as above with q in place of q Z . Recall from Theorem 2.1 that there is some Z ∈ D(K, P) with values in (0, ∞) 2 , so that Q(K, P) = ∅.
The set Q(K, P) coincides with the set of approximate martingale node-measure defined in Chalasani and Jha [1] (see Definition 6.3). More precisely, for (χ, q) ∈ Q(K, P), the node-measure is defined by the map where P(T) is the collection of subsets of T. The term χ does not appear in the formulation of the above paper because the authors do no take into account the transaction costs that are possibly paid when the option is exercised and the hedging portfolio is liquidated.
Conversely, given (χ, q) ∈ Q(K, P) it is clear that we can find Z ∈ D(K, P) such that (χ Z , q Z ) = (χ, q). It follows from Corollary 2.1 that for ϑ ∈ L 0 (R 2 ; F, P) such that ϑ 1 and ϑ 2 are uniformly bounded from below, we have h(ϑ; K) := inf{x 1 ∈ R : (x 1 , 0) ∈ Γ(ϑ; K)} = sup This extends the dual formulation in terms of node-measures obtained by Chalasani and Jha [1], see Theorem 9.1, to the general discrete time case where we take into account the transaction costs that are possibly paid when the option is exercised. Since ϑ corresponds in our framework to a claim labelled in units of the assets, the corresponding amounts are given by the process (ϑ 1 , ϑ 2 S 2 ).

"Randomized stopping times"
In Theorem 9.1 of Chalasani and Jha [1] one can also find an equivalent formulation in terms of randomized stopping times. A randomized stopping times X is a non negative F-adapted process such that t∈T X t = 1. We denote by X the set of all randomized stopping times. Observe that for a stopping time τ , the process defined by X := (1I τ =t ) t∈T belongs to X . Chalasani and Jha [1] show that there is a one to one correspondence between node measures and pairs (X, Q) where X is a randomized stopping times and Q is a P-absolutely continuous probability measure (see Theorem 5.4 in [1]). This result can be easily extended to our framework as shown below.
Given an adapted process χ such that (1 − µ) ≤ χ t ≤ (1 + λ) for all t ∈ T, the P-equivalent measure Q is called a (χ, X)-approximate martingale measure if for all t ∈ T Observe that X + t = t≤s≤T X s since X is F-adapted and t∈T X t = 1. Denoting by P(X; K, P) the associated set of pairs (χ, Q) such that the above inequalities hold, we then obtain as in Chalasani and Jha [1] that Here again, the χ is added to the formulation of Chalasani and Jha [1] to take into account the transaction costs that are possibly paid when the option is exercised.
To obtain the last equality first, we argue in two steps : 1. First, take (χ, q) ∈ Q(K, P) and define for all 0 ≤ t ≤ T with the convention F −1 = F 0 . Then, let the F-adapted processes H and X be defined inductively by (H 0 , X 0 ) = (1, q 0 ), H t+1 = H t N t+1 /D t+1 1I D t+1 =0 + H t 1I D t+1 =0 and X t = q t /H t for all 1 ≤ t ≤ T . Then, H is a martingale starting from 1 and we can define the associated equivalent probability measure Q by dQ/dP = H T . Moreover, one easily checks by using an inductive argument that k j=0 X T −j = N T −k /H T −k for all 1 ≤ k ≤ T . Indeed, since q T = N T and D T = N T −1 − q T −1 one has since on {D T = 0} one has q T = 0 and therefore q T −1 = N T −1 . Using the identities D t = N t−1 − q t−1 , t ≥ 1, the same argument provides the above result. For k = T , this shows that T t=0 X t = N 0 /H 0 = 1 since N 0 = E t∈T q t = 1. Hence, X is a randomized stopping time. Rewriting q in terms of (H, X) in (3.3)-(3.4), one obtains (3.5) and the expectation entering in the definition of h(X ). This shows that h(X ) ≥ h(ϑ; K). 2. Conversely, observe that for X ∈ X (K, P) and (χ, Q) ∈ P(X; K, P), then (χ, q) ∈ Q(K, P) with q defined by q t := X t H t /E s∈T X s H s and H t := E [dQ/dP | F t ] for t ∈ T. In view of (3.4), this shows that h(X ) ≤ h(ϑ; K).

Counter examples
In this section, we first show that the duality relation D(K) : Γ(ϑ; K) = Θ(ϑ; K) for all ϑ ∈ L 0 (R d ; F) does not hold for a large class of C-valued process K ∈ K (recall the definitions of Γ and Θ in equations (2.2) and (2.3)). For x ∈ R d , let us define (4.1) In financial terms, c t (x) is the minimal amount, in terms of the first asset, necessary to dominate x in the sense of K t at time t. If the first asset is interpreted as a numeraire, it corresponds to the constitution value of x in terms of this numeraire.
The proof is postponed to the end of the section. In that case, we assume that there is randomness enough so that the probability that y still dominates x at time 1 is less than 1.
On the contrary, we can also show that D(K) does not only hold in the case where K t = K 0 t + R d + , i.e. there is no transaction costs.
Proof. We take Ω trivial, i.e. |Ω| = 1 with F 0 = F T = {Ω, ∅}, and put K = K 0 constant. Then, x ∈ Θ(ϑ; K) reads sup This example shows that, for D(K) to be wrong, we need not only to have non zero transaction costs but also enough randomness in the direction where transaction costs are positive.

Proofs
In this section, we first provide the proof of Theorem 2.1. It follows from standard arguments based on the Hahn-Banach separation theorem. For ease of notations, we simply write A(K) and A s (K) in place of A(0; K) and A s (0; K). We denote by L 0 the set of F-adapted processes with values in R d and by L 1 (P) (resp. L ∞ ) the subset of these elements which areP-integrable,P ∼ P, (resp. bounded). Observe that L 0 (resp. L ∞ ) can be identified as a subset of L 0 (R d×(T +1) ; F) (resp. L ∞ (R d×(T +1) ; F), the set of bounded elements of L 0 (R d×(T +1) ; F)).
Proof. Since A s (K) ∩ L 1 (P) is closed in L 1 (R d×(T +1) ; F,P) (when identified with a subset of L 1 (R d×(T +1) ; F,P)) and convex, it follows from the Hahn-Banach separation theorem, N A(K) and the fact that A s (K)∩L 1 (P) is a cone which contains −L ∞ , that, for all φ ∈ L 1 (R d By possibly replacing η φ t by E η φ t | F t , we can assume that η φ is F-adapted. Then, using a standard exhaustion argument, one can find some η ∈ L ∞ with values in (0, ∞) d such that we deduce from the above inequality that where we definedη This shows thatη t ∈ K * t for all t ∈ T. For an arbitrary bounded element ξ t in L 0 (K t ; F t ), the process V 0,ξ s = −1I {s=t} ξ t , s ∈ T, belongs to A s (K). In view of (5.1), this implies that η t ∈ K * t . ✷ Proposition 5.2 Let the conditions of Theorem 2.1 hold. FixP ∼ P and ϑ ∈ L 1 (P). If for all Z ∈ D(K,P) such that ϑ · Z ∈ L 1 (P), then ϑ ∈ A s (K).
By the same arguments as in the proof of Proposition 5.1, we can assume that η is and Σ t+1 is closed, we can find someξ ∈ A(K) such that τ s=t+1ξ s − (ϑ τ − ξ t ) ∈ K τ for all t + 1 ≤ τ ≤ T .
This shows that ϑ ∈ Σ t . 2. If P Ω < 1, then we can assume without loss of generality that P Ω = 0. Following line by line the proof of Lemma 2 in Kabanov et al. [6] and using the K s 's closure property, we can find someξ ∈ A(K) with ξ t = 1 such that κ τ := τ s=tξ s ∈ K τ for all t ≤ τ ≤ T .

(5.2)
Since ξ t = 1, there is a partition ofΩ into disjoint subsets Γ i ∈ F t such that Γ i ⊂ {(ξ t ) i = 0} for i = 1, . . . , d. We then defině ξ n s − β n,i tξ s 1I Γ i s ∈ T with β n,i t = (ξ n t ) i /(ξ t ) i on Γ i , i = 1, . . . , d. In view of (5.2) and definition of ξ n , we have τ s=tξ n s − ϑ n τ ∈ K τ for all t ≤ τ ≤ T , since K τ − K 0 τ ⊂ K τ , τ ∈ T. We can then proceed as in Kabanov et al. [6] and obtain the required result by repeating the above argument with (ξ n ) n≥1 instead of (ξ n ) n≥1 and by iterating this procedure a finite number of times. ✷