MARKET RISK OF THE FIXED RATES CONTRACTS.

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Email: ilyagikhman@yahoo.com

Abstract. In this paper we focus on the concept of a discount rate. In [1] one expressed some concerns regarding the models that present randomization of the discount rate. This paper proposed a new approach to the construction of variable deterministic and stochastic interest rates. This approach is based on the concept of forward rate. In [1,2] we introduced a new analytical model of popular rate LIBOR. This rate is commonly used in the construction of price instruments that include the euro-dollar exchange rate components. LIBOR is by definition is the average deposit rates of the world's major banks. Valuations of the LIBOR is based on survey Alternative construction of the euro-dollar rate contract may use a synthetic approach. This approach follows the model of LIBOR proposed in [1]. The synthetic approach involves the price value, which consists of the price of its constituent components. This interpretation implies that the LIBOR rate to be charged for deposit of one dollar in the risk free Bank at the initial time which then immediately converts it to the British pounds. The resulting amount is invested then in risk-free British bonds for the period of the contract. At the end of this period, cumulated amount is converted back into dollars. Calculated interest rate represents an analytical representation of a dollar deposit to the risk free Bank while LIBOR rate is the average of the top panel of banks. In [2] we synthetically constructed price associated with LIBOR. Hence, the standard modeling of the LIBOR deals with empirical data rather than with its formal definition.

JEL classification code: G13.Keywords: market risk, default, default bond, reduced form, credit risk, liquidity, randomization.

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I. Basic notations and definitions. Denote B ( t , T ) , 0 ≤ t ≤ T the price of the zero

coupon default free bond price at date t with expiration at T and B ( T , T ) = 1. The simple interest i s rate and discount rate i d are defined as

B ( t , T ) = [ 1 + i s ( t , T ) ( T – t ) ] – 1 = 1 - i d ( t , T ) ( T – t ) (1)

Here T – t is expressed in appropriate 365 or 360 day year format. Continuous time model of the bond price is governed by the equation

d B ( t , T ) = r ( t , T ) B ( t , T ) d t (1′)

Here the function r ( t , T ) > 0 is called annual interest rate. The value of a coupon bond at t which pays coupon c at the moments t 1 < t 2 < … < t n = T is equal to

B c ( t , T ) = ∑j = 1

n

c B ( t , t j ) + F B ( t , T )

Here F is a face value of the bond. It is useful to define a financial contract with the help of the cash flow notion. For example the value of the coupon bond B c ( t , T ) can be interpreted as the present value at t of the cash flow

CF = ∑j = 1

n

c χ ( t = t j ) + F χ ( t = T )

Here χ ( A ) is the indicator of the event A. It is equal to 1 when A is true and 0 otherwise. Thus,

B c ( t , T ) = PV t { CF }.

II. Forward rate agreement, FRA. FRA pricing is well known and we pay attention to some

details that usually are missed. FRA is a two party OTC contract for a future transaction. The value of the transaction is the difference between the realized reference rate and its estimate called implied forward rate multiplied by a notional principal of the contract.

Let t denote initiation date and the fixed and realized forward rates are assigned over a future period [ T , T + H ], where T > t and H > 0. A FRA contract can be specified as follows. Introduce a set of time moments t 0 < t spot < t fixing < t settle < t matur . The date t 0 is called trade or deal date. On this date the FRA contract is specified as following:

Notional principal N,

FRA ( T , T + H ; t 0 ) denotes the FRA fixed rate,

period specification m × H,

and the FRA time moments.

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The spot date t spot is usually equal t 0 + 2 is the beginning and t settle = T is the end of the contract, i.e.

t settle - t spot = m (months)

date t fixing = T - 2 is usually two business days prior t settle = T is the date at which value of the floating (reference) rate over the period [ T , T + H ] , t matur = T + H. At the settlement date T the settlement (netted) sum

N [ L ( T , T + H ) – FRA ( T , T + H ; t 0 ) ] H

is paid. If the latter value is positive then the FRA seller pays this amount to the FRA buyer (FRA holder). If the value (2) is negative then the FRA buyer pays this value to the FRA seller.

The FRA pricing problem is the determination of the fixed rate FRA ( T , T + H ; t 0 ). It is common practice to study a simplified scheme of this pricing problem. In such simplification one assumes that

t = t 0 = t spot and T = t fixing = t settle . Consider the valuation problem in such simplified setting. A FRA contract sometime is interpreted as a forward loan over [ T , T + H ]. Indeed, at the date T + H a borrower of the fund should pay the interest specified by the reference rate that usually LIBOR or other similar rate. The rate L ( T , T + H ) is known as T and therefore we can admit that FRA payoff is scheduled on the date T. One variation of the FRA admits payoff at T + H. Then date-T or date-( T + H ) value of the contract is

V ( t , T , T + H ) = N [ L ( T , T + H ) – FRA ( T , T + H ; t ) ] H (2)

The rate FRA ( T , T + H ; t ) should be determined at t while the rate L ( T , T + H ) is unknown at this date. It is the market practice to approximate unknown value L ( T , T + H ) by the date-t implied forward rate l ( t , T + H ; t ) over the period [ T , T + H ]. The implied forward rate l ( t , T + H ; t ) is defined as

l ( T , T + H ; t ) =

1H

[1 − L ( t , T + H ) ( T + H − t )

1 − L ( t , T ) ( T − t )− 1 ]

Here L ( t , T ) denotes date-t spot reference interest rate with expiration at T. Applying implied forward rate l ( T , T + H ; t ) as an approximation of the unknown L ( T , T + H ) we replace the real transaction (2) by its implied approximation

v ( t , T , T + H ) = N [ l ( T , T + H ; t ) – fra ( T , T + H ; t ) ] H (2′)

Here, fra ( T , T + H ; t ) denotes ‘no-arbitrage’ solution of the reduced FRA pricing problem corresponding to еру payoff (2′). As far as the value of the implied form contract at the settlement for either buyer or seller should be equal to zero then

fra ( T , T + H ; t ) = l ( T , T + H ; t )

Then the buyer price at t is the discounted payment of the contract at T. Hence,

N L ( t , T ) l ( T , T + H ; t ) H

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If the settlement date is the date T + H then in the latter expression discount factor L ( t , T ) should be replaced by the L ( t , T + H ). Note that no-arbitrage solution of the reduced FRA with the payoff (2′) does not coincide with the real world transaction at T defined by the payoff (2). This inequality implies the market risk of the FRA contract. The value of the risk stipulated by the value

δ l =def

L ( T , T + H ) - l ( T , T + H ; t ) ≠ 0

effective at the settlement date T. The market risk from the buyer perspective is associated with the market scenarios for which δ l < 0 while the scenarios for which δ l > 0 specify the seller risk.

Let us now consider randomization of the FRA pricing problem. There are two functions are available for randomization. These are L ( T , T + H ) and l ( T , T + H ; t ). Suppose that

L ( T , T + H , ) = L ( t , t + H ) + ∫t

T

( s ) L ( s , s + H ) d s +

(3)

+ ∫t

T

( s ) L ( s , s + H ) d w ( s )

where coefficients , are deterministic or random functions, which satisfy the standard conditions that ensure the existence and the uniqueness of the solution of the Ito equation (3). The solution of the equation (3) depends on the parameter H. Assume that there exists the limit in (3) when H tends to 0. Then the instantaneous forward rate

L ( T , ) = L ( T , T + 0 , ) = P.lim

H ↓ 0 L ( T , T + H , )

is the solution of the equation

L ( T , ) = L ( t ) + ∫t

T

( s ) L ( s , ) d s + ∫t

T

( s ) L ( s , ) d w ( s )

There exists another way to present a model for the future rate L ( T , T + H , ). It is based on the implied forward rate. At the fixed date t the function l ( T , T + H ; t ) is known. For the fixed t , T , and H let us consider a random function l ( T , T + H ; u ) of the variable u , which represents the value of the implied forward rate over the period [ T , T + H ] at a future date u [ t , T ]. Suppose that

l ( T , T + H ; u ) = l ( T , T + H ; t ) + ∫t

u

( v ) l ( T , T + H ; v ) d v +

(3′)

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+ ∫t

u

λ ( v ) l ( T , T + H ; v ) d w ( v )

Putting in (3′) u = T and bearing in mind that l ( T , T + H ; T ) = L ( T , T + H , ) we arrive at the equation that defines unknown rate L ( T , T + H , ). Given randomization of the future rate L ( T , T + H ) in the form (3) or (3′) we enable to present market risk in the form P { δ l > 0 } , P { δ l < 0 } for the FRA contract. One can define the cumulative distribution to calculate primary market risk characteristics such as average profit and losses as well as its standard deviations.

Remark 1. Let us highlight the difference between the benchmark and our approaches to FRA valuation. Let us recall the benchmark FRA pricing following [6, p.87]. It is stated

“ FRA can be valued if we:

1. Calculate the payoff on the assumption that forward rates are realized ( that is, on the assumption that R M = R F ).

2. Discount this payoff at the risk-free rate. “

Applying our notations we get the following R F = l ( T 1 , T 2 ; t ) , R M = L ( T 1 , T 2 ). Benchmark approach to FRA valuation makes sense in deterministic problem setting and it is ignoring the fact that R M ≠ R F in stochastic case. In stochastic setting we interpret the date-t known rate R F = l ( T 1 , T 2 ; t ) as an approximation of the unknown at t rate R M = L ( T 1 , T 2 ) . Then the random variable

[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) > L ( T 1 , T 2 ) }

represents the market risk of the FRA from the buyer perspective. Similarly the value

[ l ( T 1 , T 2 ; t ) - L ( T 1 , T 2 ) ] { l ( T 1 , T 2 ; t ) < L ( T 1 , T 2 ) }

specifies market risk of the FRA’s seller. Note that lim

t → T 1 l ( T 1 , T 2 ; t ) = L ( T 1 , T 2 ). Nevertheless in stochastic setting it is easy to verify using real data that with probability 1 implied forward rate does not equal to the value of the correspondent future rate, i.e. l ( T 1 , T 2 ; t ) ≠ L ( T 1 , T 2 ). The rate L ( T

1 , T 2 ) is the real rate known at T 1 while the rate l ( T 1 , T 2 ; t ) is an approximation or a statistical estimate of this rate. In stochastic setting the first step suggested for the valuation is incorrect as far as R F = l ( T 1 , T 2 ; t ) is a known constant at t while the rate R M = L ( T 1 , T 2 ) is assumed to be a random variable. Therefore, in general P { R M = R F } = 0. It is also important to note that in theory the probability distribution of the random variable L ( T 1 , T 2 ; ) is assumed to be known at any moment t before T 1 . This assumption is crucial for understanding pricing in stochastic setting. It leads us to a new comprehension of the price notion and it is the basis of derivatives pricing. The revision of the price notion comes from the fact that a continuous distribution prescribes probability 0 to any particular number representing price regardless whether we interpret it as ‘arbitrage free’ or other meaningful value making sense in modeling. Complete price of the contract is a specific spot market price l ( T 1 , T 2 ; t ) along with the probability P { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) }. The last probability represents a buyer's

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market risk. Primary risk characteristic from the buyer perspective is the mean of the losses that is equal to

M п = E L ( T 1 , T 2 ; ) { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } =

= ∫0

l ( T 1 , T 2 ; t )

y P{ L ( T 1 , T 2 ; ) d y }

It is easy to present formula for the corresponding standard deviation of the losses

E [ L ( T 1 , T 2 ; ) { L ( T 1 , T 2 ; ) < l ( T 1 , T 2 ; t ) } - M п ] 2

The latter remark on the stochastic pricing is usually ignored as far as market risk is always omitted from derivatives pricing concept.

The refinement regarding the price notion of the financial instruments in the stochastic market has not fully comprehended in the modern theories while the market risk is almost completely excluded from the pricing concept of the derivatives. This phenomenon takes its origination from the first steps of the derivatives pricing.

Remark 2. Randomization problem was studied also in the popular HJM and LMM models. In [1] we discussed the randomization problem of the instantaneous forward interest rates, which is by definition

the rate f ( t , T ) = lim

H ↓ 0 l ( t ; T , T + H ; 0 ). The HJM model is the basis of the LIBOR market model presents a randomization of the instantaneous implied forward rate in the form

f ( t , T , ) = f ( 0 , T ) + ∫0

t

α ( u , T ) d u + ∫0

t

σ ( u , T ) d w ( u )

where α ( u , T ) , σ ( u , T ) are supposed to be known functions. Putting in the latter equation T ↓ t we arrive at the money market rate r ( t ) = f ( t , t + 0 ), which satisfies the equation

r ( t ) = f ( 0 , t ) + ∫0

t

α ( u , t ) d u + ∫0

t

σ ( u , t ) d w ( u )

Note that this formula does not consistent with the money market rate definition. Indeed, the bond prices is only observable data that define rate r ( t ). Applying latter expression for r ( t ) to the bond definition it follows that

B ( t , T ) = exp – ∫t

T

r ( u ) d u =

= exp – ∫t

T

{ f ( 0 , u ) + ∫0

u

α ( v , u ) d v + ∫0

u

σ ( v , u ) d w ( v ) } du

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This formula suggests that the value of the bond B ( t , T ) at any moment t depends on all observations of the bond prices B ( u , T ), u [ 0 , T ]. This model assumption does not look accurate. Indeed, from the bond definition we know that

B ( t , T ) = 1 – ∫t

T

r ( u ) B ( u , T ) d u

from which, in particular follow that

r ( t ) =

dB ( t , T )B ( t )

Therefore, the value of the bond and the instantaneous interest rate at some date t depend on the values of the bond on ( t , T ]. In order to avoid pointed above contradictions in [1] was suggested to use the backward Ito equations for randomization problem does not lead to the pointed above contradictions.

III. Interest rate swap (IRS) valuation. In this section we outline market risk coming up by

using swap rate which represents implied value of the IRS. Let us recall the well-known valuation of IRS. Market values the IRS is defined as the PV-reduction of the netted transactions at the reset dates of the swap. The market risk of this reduction stems from the replacement of the real world values transactions by its implied counterpart. A similar reduction was used for FRAs valuations. Generic IRS is a financial contract to exchange a fixed rate c for a floating rate L based on a notional principal. Let t = t 0 < t 1 < t 2 < … < t n = T be a known sequence of the reset dates and N notional principal. The buyer of a swap pays fixed rate payments N c to a swap seller and receives the amount N L ( t j – 1 , t j ) at the date t j , j = 1, 2, … , n. As far as fixed and floating transactions are scheduled on the same reset dates the only netted values of transactions are taking place. The floating rate L is one of the basic market rates like a Treasury rate, or LIBOR, or other similar rates. Future rate L ( t j – 1 , t j ) are known at the date t j – 1 and therefore the only rate that is known at t is the rate L ( t 0 , t 1 ). Real world cash flow from the swap buyer perspective A to the swap seller B can be represented as

CF A → B ( t , T ) = ∑j = 1

n

[ L ( t j – 1 , t j ) - c ] χ ( t = t j ) (4)

Positive terms on the right hand side in (4) correspond to the payments B to A while negative terms in (4) signify payments by A to B. The swap valuation problem is determination a fixed rate c that promises to counterparties A and B equality of their positions in the deal. The equality of the A and B positions in the deal should be specified. With the deterministic setting we are dealing with known or implied data. The market risk is ignored and the problem has a unique solution. The fixed rate c is a solution implied by the equality of the present values of the two legs of the swap. In stochastic setting we need to take into account the market risk characteristics. There are a few types of the dollar denominated basic interest rates. Two primary rates are Treasury and LIBOR rates. They are intensively used by the market participants. Treasury rates are specified by the T-bonds. It is a common practice to use one of these rates

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as discount factor. In this case we can interpret bond price B ( t , T ) at t as a discount factor from date T to the date t. The LIBOR rate L ( t , T ) can be also used to form other discount factor D ( t , T )

D ( t , T ) = [ 1 + L ( t , T ) ( T – t ) ] – 1

which can also be interpreted as a value of a virtual bond at t that promises $1 at T. The forward rate implied by arbitrary discount factor D can be defined as

l ( t j – 1 , t j ; t ) =

1Δ t j

[D ( t , t j − 1 )

D ( t , t j )− 1 ]

From which it follows that

D ( t , t j – 1 ) - D ( t , t j ) = D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j

Hence, the present value of the floating leg is equal to

N ∑j = 1

n

[ D ( t , t j – 1 ) - D ( t , t j ) ] = N [ 1 - D ( t , T ) ] = N ∑j = 1

n

D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j

It is common practice to interpret equality of fixed and floating rate cash flows as equality its present

values, PVs. The PV of the fixed rate cash flow is ∑j = 1

n

c D ( t , t j ) Δ t j . In order to present PV of the floating leg we need first replace unknown at t rates L ( t j – 1 , t j ) by its date-t estimates l ( t j – 1 , t j ; t ). Then the PV of the ‘implied’ floating rate cash flow can be presented in the form

N

∑j = 1

n

D ( t , t j ) l ( t j – 1 , t j ; t ) Δ t j = N [ 1 - D ( t , T ) ]

The date-t value V ( t , T ) of the swap that by definition is the difference between the PVs of floating and fixed legs is

V ( t , T ) = N [ 1 - D ( t , T ) - ∑j = 1

n

c D ( t , t j ) Δ t j ] (5)

The swap rate c is the solution of the equation V ( t , T ) = 0. Hence,

c =

1 − D ( t , T )

∑j = 1

n

D ( t , t j ) Δ t j (6)

The value c = c ( t , T , n ) is called the swap spread. Equality (6) we can resolve with respect to the discount rate D ( t , T ) = D ( t , t n ). Indeed, from (5) it follows that

8

1 - D ( t , t n ) - ∑j = 1

n

c D ( t , t j ) Δ t j = 0

Solving this equation for D ( t , t n ) we arrive at recursive formula

D ( t , t n ) =

1 − ∑j = 1

n − 1

c D ( t , t j ) Δ t j

1 + c Δ t n

Remark 3. Note that the formula (6) presents implied swap rate, which is known at t number. Similarly to FRA valuation the market risk of the IRS stipulated by the stochastic rates L ( t j – 1 , t j ) can be represented by the cash flow

∑j = 1

n

N [ L ( t j – 1 , t j , ) - l ( t j – 1 , t j ; t ) ] Δ t j { t = t j }

Expressions in brackets can be either signs positive or negative. Assume for example, that rate L ( T , T + H , ) is governed by the equation (3). Then L ( T , T + H , ) can be written in the form

L ( T , T + H , ) = L ( t , t + H ) ρ ( t , T )

where

ρ ( t , T , ) = exp {∫t

T

[ ( s ) -

σ 2 ( s )2 d s ] +

∫t

T

( s ) d w ( s ) }

Assume that Δ t j = Δ t . Then PV of the real floating leg cash flow is

N ∑j = 1

n

D ( t , t j ) L ( t j – 1 , t j ) Δ t j = N L ( t , t + Δ t ) ∑j = 1

n

D ( t , t j ) ρ ( t , t j – 1 , ) Δ t

The real world PV of the swap can be written as

V ( t , T , ) = N [ L ( t , t + Δ t )∑j = 1

n

D ( t , t j ) ρ ( t , t j – 1 , ) Δ t - ∑j = 1

n

Q D ( t , t j ) Δ t ] (5′)

Therefore, the market realized swap spread depends on a market scenario and equal to

Q ( ) =

1 − ∑j = 1

n

D ( t , t j ) ρ ( t , t j − 1 ) Δ t

∑j = 1

n

D ( t , t j ) Δ t (6′)

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The market risk of the swap from the buyer perspective is associated with the negative terms in (4). Hence, swap buyer's market risk is

P { Q ( ) < c } = P { ∑j = 1

n D ( t , t j )D ( t , T ) ρ ( t , t j – 1 , ) Δ t > 1 }

It represents the value of the chance that the buyer of the swap pays the higher rate than it is implied by the market. Note that right hand side of the latter equality can be approximated by a probability, i.e.

P { ∑j = 1

n D ( t , t j )D ( t , T ) ρ ( t , t j – 1 , ) Δ t > 1 } P {

∫0

TD ( t , u )D ( t , T ) ρ ( t , u , ) d u > 1 }

Remark 4. We should remark that represented formulas are defined on the initial probability space while it is a common tradition to use risk-neutral probability space in which real rate of return replaced by its risk free counterpart. This initiated by the Black Scholes option price formula that uses virtual underlying which has the same volatility as original stock while real rate of return is replaced by the risk free rate. One can be embarrassed by the fact that an option promised at maturity underlying stock regardless whether expected rate of return is say +12% or – 10%. One can ask a legal question what the notion of the price can lead to such unexpected fenomen. Recall briefly main concept of the Black and Scholes pricing theory. More details are discussed in [3,4].

Call option gives the right to buy underlying stock at T for a predetermine price K. Thus if S ( T ) > K then a call option holder can make a profit at T which value is S ( T ) - K. If S ( T ) < K, the call option holder does not exercise the right. To buy the right issued at initiation moment t the call buyer should makes a payment C ( t , S ( t )) and the option pricing problem is discovery of the initial payment at t. The Black and Scholes did not provided definition of the price they intended to discover. They outlined a strategy that suggested a numeric construction of the price. They assumed that underlying stock follows SDE

d S ( t ) = S ( t ) d t + σ S ( t ) d w ( t )

with constant coefficients and . They considered a portfolio [6]

Π ( t , S ( t )) = − C ( t , S ( t )) +

∂ C ( t , S ( t ) )∂ S S ( t ) (П1)

and supposed that

d Π ( t , S ( t )) = − d C ( t , S ( t ) ) +

∂ C ( t , S ( t ) )∂ S d S ( t ) (П2)

Assume that C ( t , x ) is a sufficiently smooth function. Applying Ito formula note that

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d C ( t , S ( t ) ) = [

∂ C ( t , S ( t ) )∂ t +

σ 2 S 2( t )2

∂ 2 C ( t , S ( t ) )∂ S 2

] d t +

∂ C ( t , S ( t ) )∂ S d S ( t )

Then one can easy remark that

d Π ( t , S ( t ) ) = – [

∂ C ( t , S ( t ) )∂ t +

12

∂ 2 C ( t , S ( t ) )∂ S 2

σ 2 S 2 ( t ) ] d t (П3)

From the equality (П3) it follows that change in value of the portfolio over infinitesimal period of time d t does not contain risk factor associated with the term d w ( t ). If an instrument does not hold risk then a unique chance to eliminate arbitrage opportunity to assume that rate of return of this instrument is risk free rate. Therefore

d Π ( t , S ( t ) ) = r Π ( t , S ( t ) ) d t (П4)

Taking into account formulas (П1) and (П2) we arrive at the Black Scholes equation

∂ C ( t , S ( t ) )∂ t +

12

∂ 2 C ( t , S ( t ) )∂ S 2

σ 2 S 2 ( t ) = r [ C –

∂ C ( t , S ( t ) )∂ S S ( t ) ] (П5)

Let function C ( t , S ) is a smooth solution of the Black Scholes equation

∂ C ( t , S )∂ t + r S

∂ C ( t , S )∂ S +

12

∂ 2 C ( t , S )∂ S 2

σ 2 S 2 = r C (BSE)

in the area ( t , S ) [ 0 , T ) ( 0 , + ∞ ) with the boundary condition C ( Т , S ) = max { S - K , 0 } then equality (П5) holds. The construction of the portfolio in the form (П1) which satisfies (П3) and provides (П4) is the basis of the modern theory of the derivatives pricing.

The easiest way to make sure that Black Scholes pricing concept is wrong is to take the close form solution of the (BSE) and construct portfolio Π ( t , S ( t )) as it is defined in (П1). Then it is easy to verify

that equality (П2) is incorrect. The non-zero term S ( t ) d

∂ C ( t , S ( t ) )∂ S is lost on the right hand side of

the formula (П2). As a conclusion of the error is the fact that the call option price defined by the (BSE) does not admit perfect hedging by a certain portion of the underlying stocks. Also note that calculation of derivative prices on risk neutral spaces looks faulty. Additional mistake that relates to the outlined error is the concept of the derivative price. The loss of the market risk from pricing concept leads to oversimplified interpretation of the derivative price.

IV. Liquidity. Let us outline liquidity concept. Broadly speaking liquidity is a premium problem.

In other words what premium should be added (subtract) to a bid (ask) security price to arrive at the perfect liquidity price. Latter is associated with a single price models which implies equality bid and ask prices. In theory the liquidity premium exists when we study pricing in the bid-ask format. For the last

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years the liquidity problem attracted theoretical as well as practical attention. A measure of the liquidity premium is its bid-ask spread or half of the spread if we deal with mid-price models. The perfect liquidity is interpreted as a trade with no delay, which implies by the equality of the bid and ask prices at any moment time t. Indeed, in such a model a buyer who wishes to buy an asset at t pays at this moment the price while a seller receives exactly the same amount at this moment. Thus, equality bid and ask prices implies perfect equality of the buyer and seller interests. The liquidity problem is the value of adjustment of a single price given by the perfect model of liquidity in order to take into account illiquidity of the asset. It is clear that illiquidity of an asset stems from the fact that ask price always higher than its bid counterpart.

There are two primary types of liquidity. These are trading and funding liquidities. Trading liquidity shows the ease of the asset trading while funding liquidity is a characteristic of access to funding.

Consider here the trading liquidity problem. Let us define the value of the bond illiquidity at a moment t by the spread

λ 1, B B ( t , T ) = B ask ( t , T ) - B bid ( t , T )

It represents instantaneous losses buying bonds and selling it at t. Note that this indicator also depends on T, which specifies liquidity for this maturity. Instruments with different parameters t or T have different liquidity. We can also introduce other definition of liquidity which in general similar to the previous one. Define revenue of the bond d B ( s , t ; T ) from time s to time t

d B ( s , t ; T ) =

B bid ( t , T )B ask ( t , T ) (7)

It shows the portion of losses buying bond for ask price at s and selling it for bid price later at the date t.

The bond expiration date is the date T. The liquidity is the we defined as λ B ( t ; T ) = lims → t d B ( s , t ; T ),

λ B ( t ; T ) =

B bid ( t , T )B ask ( t , T ) (7)

The value of illiquidity we define as a portion of losses one arrives at from immediate buy and sell transaction of the bond. The perfect liquidity corresponds to the value λ B ( t , T ) = 1 while the absolute illiquidity corresponds to λ B ( t , T ) = 0.

We will say that asset A is more liquid than asset G at the moment t if λ A ( t ; T ) < λ G ( t ; T ). This definition relates to a particular moment of time. If the statement of definition holds for any moment of time during some period [ 0 , T ] then asset A is more liquid than G on this period. Assume that G is more liquid than A on a subinterval of the [ 0 , T ] and A more liquid than G on its complement . In order to state that A is more liquid than G on the interval [ 0 , T ] our definition of liquidity should be refined. One way to adjust the definition is to consider normalized bid-ask spread. For example asset A is more liquid than asset G on [ 0 , T ] in average if

12

1T

∫0

T

λ A ( t , T ) d t ≤

1T

∫0

T

λ G ( t , T ) d t (7′)

where λ A , λ G denote the liquidity of asset A and G correspondingly. This definition of liquidity does not comprise variety of the real market components associated with liquidity. Broadly speaking, one can consider a liquidity adjustment by adding volumes of trades during a particular time period. Indeed, let us imagine that assets A and G look similar in terms of (7), (7′) while the number of trades G is visibly larger than A over [ 0 , T ] then it might be makes sense to think that G is more liquid than A on this interval. The risk free bond price at t with expiration at T is defined by its bid-ask prices

B bid ( t , T ) < B ask ( t , T ) , t [ 0 , T )

B bid ( T , T ) = B ask ( T , T ) = 1

Obviously, that in the format of the single price which is the middle of the bid and ask prices

B ( t , T ) =

12 [ B bid ( t , T ) + B ask ( t , T ) ] (8)

does not represent real world perfect liquidity for either buyer or seller of the bond. In the construction the middle prices have not been used to present the perfect liquidity for either seller nor buyer of the bond. The buyer’s perfect liquidity is associated with the bid price while the seller’s perfect liquidity is associated with the ask price of the bond. Bearing in mind this remark the real world liquidity is defined for the buyer and seller separately. Therefore, the only real world prices B bid ( t , T ) , B ask ( t , T ) represent real world liquidity or illiquidity of the bond.

Assuming zero chance of default randomization of the liquidity problem can be studied by considering the pair of the random processes for bid and ask prices for which with probability 1

0 < B bid ( t , T ; ) < B ask ( t , T ; ) ≤ 1

B bid ( T , T ; ) = B ask ( T , T ; ) = 1

We put by definition

B k ( t , T ; ) = exp – ∫t

T

r k ( u , ) d u (9)

k = bid, ask. Here, the instantaneous rate r k ( u ) is defined by equality

r k ( t , ) = lim

Δ t → 0

1 − B k ( t , t + Δ t ; ω )Δ t =

= lim

Δ t → 0

B k ( t + Δ t , t + Δ t ) − B k ( t , t + Δ t ; ω )Δ t

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From (9) it follows that

B k ( t , T ; ) = 1 + ∫t

T

r k ( u , ) B k ( u , T ; ) d u (9′)

where the rate r k ( u , )

r k ( t ) =

1B k( t , T )

d B k ( t , T )d t

might depend on T too. From formulas (9′) we note that the price B k ( t , T ; ) as well as the r k ( t , ) at each moment t [ 0 , T ] are depended on the values of the bond over the interval ( t , T ]. For the stochastic bond price one usually assumes that price is governed by the linear Ito equation. In this case one can only guarantee the fulfillment of the condition B ( t , T ; ) > 0 while the condition B ( t , T ; ) 1 is not guaranteed. For simplicity we do not specify bid and ask prices. In this case we can only talk about approximation of the bond prices and whether the approximation is good one should

to verify that the value of the probability P {sup

t ∈ [ 0 , T ]B ( t , T ) > 1 } is sufficiently small. Following [1] let us assume that the function r ( t , ) is the solution of the stochastic Ito equation with inverse time

r ( t , ) = r T – ∫t

T

( u , r ( u , )) d u – ∫t

T

( u , r ( u , )) d w←

( u ) (10)

Here the stochastic integral on the right hand side is the backward Ito integral [5]. The random variable r T = r ( T + ) on the right side (10) represents the instantaneous rate at T. This variable will be known at the moment T. The choice of the inverse time in the model follows from the relationship between price and rate specified by formulas (9), (9′). If we suppose that the filtration of the -algebras are generated by the observations over the bond prices then B ( t , T ; ) and r ( t , ) are completely defined by the observations over [ t , T ]. The existed models of the interest rates assume that the value r ( t , ) is defined by the values r ( u , ), u t which explicitly implies on the fact that r ( t , ) depends on B ( u , T ; ), u t. This construction contradicts formulas (9), (9′). One should bring an example which explicitly represent such possibility. In stochastic models of the interest rates one should make consistent stochastic rate and stochastic price of the bond. Consider equation (10). Let

F←

[ t , T ] = { B ( u , T ; ) , t u T } , F←

t = F←

[ t , + , + ∞ )

Here T is fixed while t is variable on [ 0 , T ]. Recall that direction of the filtration can be chosen such

that it is better match to the model. Then ‘initial’ data r T is F←

t measurable random variable and

r T = lim

Δ ↓ 0 ( ) – 1 r ( T , T + ) (11)

and where r ( T , T + ) is the interest rate defined in (1′) for random bond prices. Suppose that current rate r ( t , T ) is known for all moments T > t. Following market rules to avoid arbitrage opportunity we

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replace random future rates by its implied values. The date-t implied forward rate r ( T , T + H ; t ) of the unknown, random rate r ( T , T + H ) is defined by the equality

[ 1 + r ( t , T + H ) ( T + H - t ) ] = [ 1 + r ( t , T ) ( T - t ) ] [ 1 + r ( T , T + H ; t ) H ]

Solving it for r ( T , T + H ; t ) we get the formula

r ( T , T + H ; t ) =

1H

r ( t , T + H ) ( T + H − t ) − r ( t , T ) ( T − t )1 + r ( t , T ) ( T − t )

Taking into account this definition we arrive at the market estimate of the random variable r T

r ( T , T + 0 ; t ) = lim

H ↓ 0

1H

r ( t , T + H ) ( T + H − t ) − r ( t , T ) ( T − t )1 + r ( t , T ) ( T − t )

Nonrandom variable r T ( t ) = r ( T , T + 0 ; t ) is known at the moment and exchange random variable r T for non-random r T ( t ) implies the modeling risk stipulated by the fact that r T ≠ r T ( t ). Let r ( t ; T , r T ) denote solution of the equation (10) and r ( t ; T , r T ( t )) denote the solution of the same equation (10) with the boundary condition at T r ( T ; T , r T ( t )) = r T ( t ). Buyer risk is that he pays more at t than it implies by the market. This model risk can be expressed by the scenarios

{ : r ( t , T ; r T ( )) < r ( t , T ; r T ( t )) }

This scenarios implies that bond price with unknown at t rate r ( t , T ; r T ( ) is less than the price of the bond with the implied forward rate r ( t , T ; r T ( t )). Assume that random function r k ( t , ) is a solution of the equation

r k ( t , ) = r T , k – ∫t

T

k ( u , r ( u , )) d u – ∫t

T

( u , r ( u , )) d w←

( u ) (10′)

Here, k = ask, bid , r T , ask < r T , bid , and аsk ( u , r ) < bid ( u , r ). Here we used without proof a comparison theorem for SDEs with inverse time in order to state that r ask ( t , ) r bid ( t , ) which

implies that B bid ( t , T ) B ask ( t , T ) 1 with probability 1 for t [ 0 , T ]. The F←

T measurable random variables r T , k , k = ask, bid are defined by (11). It is a market rule to replace random forward rates by its implied forward rates r T , k ( t ) = r k ( T , T + 0 ; t ). The solution

r k ( t , ) = r k ( t ; T , r T , k ( t ))

of the equation (10′) is uniquely defined by its timing data at T,

r T , k ( t ) = r k ( Т ; t , ) = r k ( T , T + 0 ; t )

k = ask, bid. If market does not have unpredictable perturbations one can assume that spot prices are formed as following

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r spot , k ( t ) = E { r k ( t , ) | F→

t } (12′)

where F→

t = F [ 0 , t ) = σ { B ( u , T ) , u < t }. To complete close form of the implied stochastic interest rate we replace the forward rate r ( u , ), u [ t , T ] in right hand side (10) by its implied forward rate r u ( t ) = r ( u , t ). In this case we arrive at implied current stochastic interest rate

r k( im )

( t , ) = r T , k – ∫t

T

k ( u , r u ( t )) d u – ∫t

T

( u , r u ( t )) d w←

( u ) (10′′)

k = ask, bid. The upper index ‘im’ at r k( im )

highlights the implied characteristic. The r k( im )

( t , ) is a Gaussian random process with mean and volatility equal to

r T , k – ∫t

T

k ( u , r u ( t )) d u , ∫t

T

Е [ ( u , r u ( t )) ] 2 d u

In according to (12) we put

r spot , k( im )

( t ) = E { r k( im )

( t , ) | F→

t } (12′′)

Equalities (12′), (12′′) represent basic formulas for pricing bonds with no risk of default. There two types if the risk were admitted in construction. First is the modeling risk. It represented by assumptions (10), (12′), (12′′). The second type of the risk is connected to applications of the implied forward rates, which were used to replace forward unknown rates (10′′).

Remark. It is common to use risk neutral distribution for spot values of the real world prices or interest rates. Recall that neutral distribution came with the necessity to connect underlying of the Black Scholes option price with the real stock which is underlying of the option. One can see that the statement that derivatives values are defined by the underlying actually is incorrect for Black Scholes pricing model. Black Scholes option price takes its value from the heuristic underlying having drift and volatility ( r , 2 ) while real underlying of the option is defined by ( , 2 ) coefficients. The sense of the risk neutral valuation can be explained as following. The stock price by definition is the solution of the linear SDE with coefficients ( , 2 ). It is defined on original probability space { , F , P }. In order to find connection between real stock and underlying of the Black Scholes formula one should consider original ( , 2 ) pricing process with respect to new risk neutral measure d Q = d P with appropriate Girsanov exponent. Such technique looks like an adjustment of the real world to Black Scholes pricing formula. Latter remark is also relates to the binomial scheme which presents realization of the Black Scholes pricing in discrete space-time setting.

Recall definition of the implied forward rate using a discount factor. Let D k and f k denote discount and implied forward rates k = ask, bid correspondingly. Then by definition

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f k ( T , T + H ; t ) =

1H

[1 − D k ( t , T + H ) ( T + H − t )

1 − D k ( t , T ) ( T − t )− 1 ]

where k = ask , bid. Note that we can use discount rates also for the interest rates that do not associate with a bond. Dealing for example with LIBOR rate we put in the latter formula D k = L k and f k = l k. Hence,

fra k ( T , T + H ; t ) = l k ( T , T + H ; t )

and FRA bid-ask spread of the FRA contract by definition is a difference

λ l ( t , fra ( T , T + H ; t ) ) = l ask ( T , T + H ; t ) - l bid ( T , T + H ; t )

This formula enables to compare liquidity of FRA contracts at t for different T , H. Recall the inverse relationship between prices and corresponding interest rates, l bid > l ask . As it was already pointed out, the implied rates can be interpreted as statistical estimates in stochastic market setting. A forward rate over a future period real rate depends on a market scenario. Therefore, the use of the corresponding implied rate is subject to market risk in a particular market model. This risk represents the fact that the date-t implied forward rates l bid or l ask do not equal to its real values at the future moment T. In other words, the liquidity market risk of the FRA contract implied by the bond is stipulated by the inequality

f ask ( T , T + H ; t ) - f bid ( T , T + H ; t ) ≠ B ask ( T , T + H ) - B bid ( T , T + H )

and the probability of the inequality

P { f ask ( T , T + H ; t ) - f bid ( T , T + H ; t > B ask ( T , T + H ) - B bid ( T , T + H ) }

is a measure of the illiquidity risk of the FRA contract. Here the FRA contract has been used to illustrate stochastic effect of the implied forward liquidity. The date-t implied liquidity spread λ f = f ask - f bid represents a statistical estimate which can be larger or smaller than its date-T real market rate λ l = λ ask ( T , T + H ) - λ bid ( T , T + H ). Higher future liquidity spread suggests that market predicts a higher risk of additional buyer’s expenses while lower future illiquidity suggests that market predict more liquidity in the future. It will reduce risk of additional expenses.

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References.

1. I. Gikhman. FX Basic Notions and Randomization. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964307. 2. I. Gikhman. A Comment On No-Arbitrage Pricing.

3. J. Hull, Options, Futures and other Derivatives. Pearson Education International, 7ed. 814 p. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2195310. 4. I. Gikhman. STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPPLICATIONS: Stochastic analysis of the dynamic systems. ISBN-10: 3845407913, LAP LAMBERT Academic Publishing, 2011, p. 252.

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