MULTILATERAL VIRTUAL CURRENCY MODEL.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

ph. (513) 573-9348

Key words. Multilateral FX .

JEL code. C50, E40, F31.

Abstract. This is the project that represents a basic design of a multilateral FX market that synthetically connected to the real world bilateral FX market. The essence of the multilateral FX market is the new multilateral structures which can serve as ‘super’ currencies. We define exchange and interest rates of these super structures. These virtual currencies admit the real world single currency representation.

The goal of this paper is presentation of a new class of foreign exchange, FX instruments which we call multilateral structures. The idea of a multilateral structure is an alternative to the idea of Euro construction. We can see that the initial period stability of the Euro zone changed for the period of recession which then passed to the global financial crisis. The main reason of the deteriorating of the financial system is excessive cash surplus of the market. Two primary sources of the cash surplus are the global derivatives market and other is the expenses related to the wars. From classical economy we know that money is invested to produce something substantial that then will be purchased and generate some interest. Money invested in the wars does not fulfill its mission. Derivatives are often referred to as hedging instruments. Nevertheless to realize this function we need additional money that sooner or later will bring inflation and pricing instability. Beginning from 2007 different stimulus programs directed to rescue large companies and some countries have opened new source which overflow financial market.

To illustrate failing of the euro suppose that Greece is keeping the old currency drachma and drachma denominated bonds. Then by changing drachma interest rate one can make it more or less attractive for market investors to keep the balance which provides economic stability. The single currency market of the EU does not transfer such possibility for Greece. The one currency system looks less stable compare

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with old system. This observation suggests that one currency market globalization does not appropriate construction. It eliminates stability of the multicurrency market.

In addition note that surplus cash flows always will settle at the leaders of industries and therefore the policy of external trenches could not improve the general situation for minor economies of the Euro zone.

Our idea outlines other way of globalization. It synthetically presents a virtual currency which can be constructed with the help of single currencies. It remains a possibility for the governmental supervision of the local currencies as well as global supervision of the ‘super’ currencies which represented in the financial structures form.

Let n denote a number currencies in a group. For brevity 1 j ( t ), j = 1, 2, … n denote one unit currency j at the time moment t. For example we put 1 1 ( t ) = 1 $ ( t ), 1 2 ( t ) = 1 € ( t ), 1 3 ( t ) = = 1 £ ( t ). Notation EUR / USD is used to present the value of one euro called base currency in USA dollars which referred to as counter currency. This base/counter relationship at date t can be formally written in the form

1 k ( t ) = q k j ( t ) 1 j ( t ) (1)

where q k k ( t ) = 1 and k , j = 1, 2, … n. It follows from (1) that for any k, j, and t

q k j ( t ) = [ q j k ( t ) ] – 1 (2)

Equalities (1) and (2) imply that bid-ask spread is supposed to be equal to 0. Define multilateral structure

( FS ) G ( n ) ( t ) as a sum of

1 k ( t ) , k = 1, 2, … n. This structure admits single currency representation

as

G j( n )

( t ) = [ ∑

k = 1

n

q k j ( t ) ] 1 j ( t ) = Q j( n )

( t ) 1 j ( t ) (3)

Function Q j( n )

( t ) can be interpreted as exchange rate of the virtual currency G ( n ) in jth currency units ,

i.e. the price of the one unit of the virtual currency G ( n ) is equal to Q j( n )

( t ) units of the currency j. Let 0 < m < n and suppose that G ( m ) is a subset of the G ( n ). We can define exchange rate of the pair G ( n ) / G ( m ) . Indeed, let 1 j G ( m ) and therefore 1 j G ( n ). Note

G j( m )

( t ) = [ ∑

l = 1

m

q l j ( t ) ] 1 j ( t ) = Q j( m )

( t ) 1 j ( t )

Using j-th representation of the FS one define a rate

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Q j( m , n )

( t ) =

G j( m ) ( t )

G j( n ) ( t ) =

∑l = 1

m

q l j ( t )

∑k = 1

n

q k j ( t )

Let us show that exchange rate Q j( m , n )

does not depend on j. Bilateral exchange implies that

1 k ( t ) = q k j ( t ) 1 j ( t ) = q k j ( t ) q j i ( t ) 1 i ( t )

On the other hand, by definition

1 k ( t ) = q k i ( t ) 1 i ( t )

Hence, for any k , j , and i we have

q k i ( t ) = q k j ( t ) q j i ( t ) (4)

Hence,

Q j( m , n )

( t ) =

∑l = 1

m

q l j ( t )

∑k = 1

n

q k j ( t ) =

[ ∑l = 1

m

q l j ( t ) ] q j i ( t )

[ ∑k = 1

n

q k j ( t ) ] q j i ( t ) =

∑l = 1

m

q l i ( t )

∑k = 1

n

q k i ( t ) = Q i

( m , n )( t )

The index i on the right hand side is chosen arbitrary. It can be either belong to the structure or no. Therefore exchange rate of two FSs does not depend on its particular representations, i.e.

Q j( m , n )

( t ) = Q m n ( t )

Thus, for the group currencies G ( m ) G ( n ) one can define exchange rate

G ( n ) ( t ) = Q n m ( t ) G ( m ) ( t ) (5)

Now we show that equality (5) takes place for arbitrary groups of currencies regardless whether they have common elements or not. Let G ( u ) and G ( v ) be arbitrary groups of currency. Similarly to above for arbitrary j we have

G j( u )

( t ) = Q j( u )

( t ) 1 j ( t ) , G j( v )

( t ) = Q j( v )

( t ) 1 j ( t )

Therefore for any j and i it follows that

Q j( u , v )

( t ) =

Q j( u ) ( t )

Q j( v ) ( t ) =

Q j( u ) ( t ) q j i ( t )

Q j( v ) ( t ) q j i ( t ) =

Q i( u ) ( t )

Q i( v ) ( t ) = Q i

( u , v )( t ) = Q u v ( t )

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This equality proves that (5) is true for arbitrary groups of currencies.

The currency j is said strengthening with respect to currency k or k- currency is weakening with respect to currency j from t to T if q k j ( T ) < q k j ( t ). This inequality corresponds to the scenario when the value of the base currency k at the date t is higher in units of the currency j at the date T. We generalize this property for FS groups. We will say that super currency G ( u ) is strengthening with respect to G ( v ) from the moment t to T if Q u v ( t ) < Q u v ( T ). The latter inequality can be also rewritten as Q v u ( T ) < Q v u ( t ).

Let G ( n ) be an arbitrary super currency. Let us define the interest rate implied by this structure. The interest rate of the G ( n ) will depend on a particular choice of a single currency used for its representation. Let j be a currency from the group though it is possible to assume that j does not belong

to G ( n ). Define simple interest rate i j( n )

( t , T ) over [ t , T ] period by equality

G j( n )

( t ) [ 1 + i j( n )

( t , T ) ( T – t ) ] = G j( n )

( T )

where period T – t uses an appropriate year format. It is easy to verify that interest rate i j( n )

( t , T ) does not coincide with domestic interest rate i j ( t , T ) of the domestic bond of the j-th currency.

Domestic government bond by its definition does not depend on foreign exchange rates while i j( n )

shows this dependence explicitly. Indeed,

i j( n )

( t , T ) ( T – t ) =

G j( n ) ( T )

G j( n ) ( t ) – 1 =

∑k = 1

n

q k j ( T )

∑k = 1

n

q k j ( t ) – 1

Let 1 k G ( n ) , k = 1, 2, … n be the full set of components of the structure G ( n ). Then the value

1 k ( t ) [ 1 + i k ( t , T ) ( T – t ) ]

is the date-T return on one unit 1 k ( t ) investment at t. Multiplying this transaction on q k j ( t ) we

convert this transaction into j-currency. The value i j( n )

( t , T ) can be found from the equation

∑k = 1

n

q k j ( t ) [ 1 + i k ( t , T ) ( T – t ) ] = [ 1 + i j( n )

( t , T ) ( T – t ) ] ∑

k = 1

n

q k j ( t )

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Solving latter equation for rate i j( n )

we arrive at the formula

i j( n )

( t , T ) =

1T − t {

∑k = 1

n

e k j ( t ) [ 1 + i k ( t , T ) ( T – t ) ] - 1 }

where e k j ( t ) =

q k j ( t )

∑l = 1

n

q l j ( t ) , k = 1, 2, … n.

Let us introduce bond which covers FS in j-th currency

B j( n )

( t , T ) =

G j( n ) ( t )

G j( n ) ( T ) = [ 1 + i j

( n ) ( t , T ) ( T – t ) ] – 1

If 1 j G ( n ) then this bond denominated in j currency can be interpreted as virtual discount bond from T

to the date t . The correspondent implied forward rate F j( n )

( t ; T 1 , T 2 ) at t over the period [ T 1 , T 2 ] can be defined by the equality

B j( n )

( t , T 1 ) [ 1 + F j( n )

( t ; T 1 , T 2 ) ( T 2 – T 1 ) ] – 1 = B j( n )

( t , T 2 )

Solving this equation we arrive at the definition of the implied forward rate

F j( n )

( t ; T 1 , T 2 ) ( T 2 – T 1 ) =

B j( n ) ( t , T 1 )

B j( n ) ( t , T 2 ) – 1

Let us very briefly outline the implementation of the multilateral FX structure concept. We highlight it from the ECB perspective though one can present it from USD or other currency perspectives.

1. ECB can provide G 1( k )

( t ) where low index ‘1’ relates to euro currency and upper index ‘k’ be equal to say 3 or other number. The value ‘3’ for example can cover top 3 currencies such as

euro, dollar, and yen. Consider G 1( k )

( t ) as a synthetic virtual rate. We do not admit its physical realization in the form of paper money.

2. Along with data G 1( k )

( t ) it makes sense to provide values q l 1 ( t ) / G 1( k )

( t ) for each

component ‘l ‘of the G 1( k )

( t ). The value G 1( k )

( t ) is the common basis for any of its components.

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3. It makes sense to issue G 1( k )

( t ) treasures that can be converted to any single currency of the

G 1( k )

( t ) or roll over to the next maturity.4. A multilateral FX structure G ( j ) of the multilateral FX structure G ( k ) , j < k as well as a

single component of the G 1( k )

( t ) defines the swap rate which can be used as an important currency characteristic.

The multilateral FX structures would represent more stability to the FX market and also provide more degree of freedom to the present financial system. The load of dominance in the bilateral currency market should be transferred to the multilateral FX structures.

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