# Fixed income basic notions and randomization

## The study of the notion of a fixed cash income. The definition of the advantages and disadvantages of the use of the interest rate. Description of notions about the bonds and pricing. The study of LIBOR forward rates on the modern money market.

Рубрика | Банковское, биржевое дело и страхование |

Вид | статья |

Язык | английский |

Дата добавления | 01.02.2013 |

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FIXED INCOME BASIC NOTIONS and RANDOMIZATION

Ilya Gikhman

6077 Ivy Woods Court

Mason OH 45040 USA

Ph. 513-573-9348

Abstract. In this paper, we outline a randomization of the primary fixed income notions. We present a construction of some stochastic interest rate models. We also consider forward rates which are implied by stochastic bond prices. We highlight to major drawbacks of the commonly used stochastic models. The first drawback is the theoretical possibility that bond admits higher than its face value prior to maturity. The second drawback is related to modeling itself. We model stochastic interest rate in such a way that it would be consistent with deterministic definition of the bond while some popular are not. In the next paper we will pay more attention to FX and a formal definition of the LIBOR rate. Here LIBOR rate is assumed to be known.

Key words. Stochastic interest rate, stochastic forward rate, LIBOR.

1. In this section, we present well known basic notions and definitions. Denote B ( t , T ) zero coupon default free bond price at t with expiration at date T and B ( T , T ) = 1. The simple example of the interest rate contract is a government debt security, which for brevity we will call bond. The simple interest rate model of the bond pricing is:

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

Here T - t is taking in 365 ( 360 ) day year format, i.e.

(T - t) year fraction = (T - t) days / 365

Thus, simple interest rate over [t , T] period is defined as:

i ( t , T ) ( T - t ) = - 1

In this formula the simple interest rate i (t , T) is completely defined by the value B (t , T) at t. 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)

t [0 , T] in which we put B (T , T) = 1. Function r (t , T) > 0 in (1) is called continuously compounded interest rate. Equation can be rewritten in integral form, which is useful when the annual interest rate is a random function. We can rewrite differential equation in integral form:

B ( t , T ) = 1 - r ( u , T ) B ( u , T ) d u (1)

Equation shows that the market value of the bond B (t , T) at the moment t depends on the values r (u , T); u (t , T); If r (u , T) is a measurable bounded function then the solution of the equation is:

B ( t , T , ) = exp - r ( u , T , ) d u (2)

If in (2) the function r ( u , T ) = r ( u ) does not depend on T then the rate r ( u ) is known as the short term rate of interest. It is by definition the interest rate over infinitesimal short time period [ t, t +, t ]. In a continuous time the notation r is referred to as to money-market account.

Let us briefly recall a discrete scheme bond equation solution construction. This construction can also be used in the stochastic setting. Let:

t = t _{0} < t _{1} < t _{2} < … < t _{n} = T

Be a partition of the interval [t , T] and assume for simplicity that step:

t = t _{j + 1} - t _{j}

Does not depend on j. For writing simplicity scenario symbol will be omitted. The continuous time interest rate model can be constructed recursively putting:

B ( t _{n - 1} , T ) [ 1 + r ( t _{n - 1} , T ) t ] = 1

B ( t _{j}_{ - 1} , T ) [ 1 + r ( t _{j} , T ) t ] = B ( t _{j }, T )

j = 1 , 2, … n or

B ( t , T ) [ 1 + r ( t _{j} , T ) t ] = 1

Taking into account that:

1 + r ( t _{j} , T ) t = exp r ( t _{j} , T ) t + o ( t )

Where:

= 0

We arrive at the formula:

B ( t , T ) [ 1 + r ( t _{j} , T ) t ] = B ( t , T ) exp [ 1 + r ( t _{j} , T ) t ] + ( t ) = 1

Where:

( t ) = 0

Passing to the limit when t 0 we get the formula

B ( t , T ) exp r ( u , T ) d u = 1

Solving this equation for B (t, T) we obtain the formula (2).

Note that the value B (t , T) is uniquely defined by the values of the interest rate r (u, T) , u (t , T). This observation is the basis of the formal randomization of the formula (1) and for a stochastic forward rate model which we present in the next section.

Similar to above we can outline bond pricing model based on the market money account. Indeed, assume that t is equal to day or other shorter or longer period. Let:

t [ t _{k - 1 }, t _{k} ], k = 1, 2

Then:

B ( t , t _{k} ) = B ( t , t _{k} ) - B ( t _{k} , t _{k} ) = [ 1 + r ( t _{k - 1 }, t _{k} ) ] -^{ 1}

Assume that the limit:

r ( t ) = r ( t , t + 0 ) = r ( t , t + t )

Is an integrable function. Hence:

B ( t , T ) = exp - r ( u ) d u (2?)

Note that bond pricing formula (2?) uses money market interest rate which does not depend on bond's maturity T. Next, let us assume that:

r ( t ) = в - л ( u , r ( u ) ) d u (3)

r ( t , T ) = в - л ( u , T , r ( u , T ) ) d u(3?)

Where:

в = в ( T ) > 0

Is a known constant that can be interpreted as instantaneous rate at date T which value actually does not known at t. It is common practice to use implied date-t forward rate as the estimate of the forward rate at T. Formally, in continuous time constant в is the limit of the interest rate over a small period [T , T + ] when the length of the period > 0 tends to 0. We return to this problem in grater details later when we will study this problem in stochastic setting.

Now suppose that the coefficient of the above equations (3), (3?) satisfy the conditions:

л ( t , 0 ) = 0 or л ( t , T , 0 ) = 0

In this case solutions of the equations (3) , (3?) are positive functions that guarantee that:

B ( t , T ) ? 1

Let us introduce a forward rate construction that is implied by the bond prices. In this construction we admit that the bond prices or spot rates are observable data. These two alternatives start points are equivalent to each other. We introduce first the forward rate concept. For given time moments 0 ? t ? T ? T + h we will distinct at t the rate:

r ( T T + h , )

Which will be known T and the date-t forward rate over the future interval [ T , T + h ]. Assuming that B ( t , T ) is a given function we define the implied forward rate f ( t , T , T + h ) at the moment t over a period [ T , T + h ] as a solution of the equation:

B ( t , T + h ) = B ( t , T ) [ 1 + f ( t , T , T + h ) h ] -^{ 1}

Then:

f ( t , T , T + h ) h = - 1 = - (4)

It follows from that date-t instantaneous h v 0 implied forward rate at T is:

f ( t , T ) = -

Denote:

t = t _{0} < t _{1} < ... < t _{n} = T

A partition of the interval [ t , T ] and let:

t _{j} = t _{j + 1} - t _{j} .

Bearing in mind (4) we note that:

B ( t , T ) = 1 + B ( t , t _{j + 1} ) - B ( t , t _{j}_{ }) = 1 + B ( t , t _{j + 1} ) f ( t , t _{j }, t _{j} ) t _{j}

Assume that there exists a continuous function f ( t , v ) , v [ t , T ] such that:

f ( t , t _{j }, t _{j} ) t _{j} = f ( t , v ) d v

Note that instantaneous forward rate f ( t , v ) is the an implied rate that relates to the particular bond.

Therefore:

B ( t , t _{j + 1} ) f ( t , t _{j }, t _{j} ) t _{j} = B ( t , t _{j }) f ( t , t _{j }) t _{j} + ( ) (5)

Where is a parameter that signifies the partition and:

( ) = [ B ( t , t _{j + 1} ) f ( t , t _{j }, t _{j} ) - B ( t , t _{j }) f ( t , t _{j }) ] t _{j}

If the function f ( t , v ) is absolutely integrable in v on [ t , T ] then:

( ) = 0

And it follows from (4) and (5) that:

B ( t , T ) = 1 + f ( t , v ) B ( t , v ) d v

The solution of the latter equation is:

B ( t , T ) = exp f ( t , v ) d v (6)

Assuming that interest rate and forward rate in formulas (2) and (6) are deterministic and sufficiently smooth functions one can easily verify the interest-forward rates parity:

r ( t , T ) = - f ( t , t ) + ( t , v ) d v

f ( t , T ) = - r ( T , T ) - ( u , T ) d u

In order to justify differentiation in the integrals on the right hand side of the above system one need also suppose that the integrands on the right hand side are absolutely integrable.

Now let us consider stochastic bonds pricing. If interest rates are random functions then bond prices as well as its implied forward rates are also random functions, i.e.

B ( t , T ) = B ( t , T , щ ) , f ( t , T , h ) = f ( t , T , h, щ )

We begin with remarks on existing models of the money market account that is also referred to as the bank account. These remarks also relate to interest rates with specified maturity. Let { , F , P } be a complete probability space and w ( t ) , t > 0 is a scalar Wiener process on this space. Elements щ are called in finance market scenarios. Define -algebra F _{t} , which by definition is the minimal -algebra generated by the values of Wiener process w ( s ) , 0 s t . Define two parametric set of the -algebras:

F_{ [ s , t ]}_{ }= { w ( u ) - w ( t ) , s u t }

This set can be used to define two types of filtrations. Filtration_{ }_{}_{( s , t ]}_{ }is defined for the fixed date t < + with respect to decreasing time s_{ } [ 0 , t ] and we put by definition:

_{}_{ s} = ( s + )

Other filtration:

_{} [ s , t ) _{}_{ }_{t}_{ }= _{}_{ }[ 0 , t )

Іs defined for the fixed moment s 0 with respect to increasing moment of time t,

t [ 0 + )

In applications filtrations are usually associated with the observable processes. It is known in the stochastic differential equations, SDEs that if the diffusion coefficient is non degenerative the filtrations generated by the Wiener process and observations over the values of the SDE solution are equivalent. Consider formula (2?) when money market rate is a random process r ( u , щ ). In this case we arrive at the market price of the bond, which depends on scenario:

B ( t , T , щ ) = exp - r ( u , щ ) d u

This formula holds on the original (real) probability space { , F , P }. We will distinct market price that depends on a market scenario and the spot price that is associated with a spot moment and is considered as a known number.

The spot price of the bond B_{spot} (t, T) at t can be defined in different ways. For example spot bond price can be associated with expected value of the market price, i.e:

B_{spot} ( t , T ) = E B ( t , T , щ )

Or a linear combination of the mean and standard deviation of the market price given that:

0 < B_{spot} ( t T ) ? 1, B _{spot} ( T , T ) = 1

Remarkably that there is no evidence why bond spot price should be assumed to be equal:

B _{spot} ( t , T ) = E ^{Q} { exp - r ( u , щ ) d u | _{}_{ }_{t}_{ }}

Where Q is so-called risk neutral measure. Some theoretical models admits that B (t, T, щ) is a diffusion process. We show that this assumption implies that B (t, T, щ) at any time t with a positive probability. This assumption should be rejected as far as it never be observed on the market and it is not clear whether we can use such model when we observe a conditionally sufficient good fit of a model to historical data and at the same time the probability of the event B ( t , T , щ ) > 1 say is about 5% . Other remark which is important to highlight here is related to measurability of the bond's SDE solution. We will clarify the essence of these problems later. Assume now for example that:

r ( t , щ ) = л ( t , щ ) + и ( t , щ ) ( t )

Where w ( t ) , t ? 0 is a Wiener process on { , F , P } and suppose that the functions л ( t , щ ) , и ( t , щ ) are _{}_{ }_{t}_{ }measurable , continuous in t , and uniformly bounded by a nonrandom constant. We put the stochastic interest rate directly in the formula (2?) and find the SDE that governs the bond evolution. Thus:

B ( t , T , щ ) = exp - [ л ( s , щ ) d s + и ( s , щ ) d w ( s ) ] (2??)

In this formula parameter t changes from 0 to T. Consider the difference:

B ( t - t , T , щ ) - B ( t , T , щ ) = B ( t , T , щ ) { exp - [ л ( s , щ ) d s + и ( s , щ ) d w ( s ) ] - 1 } = B ( t , T , щ ) { exp - л ( t , щ ) t - и ( t , щ ) w ( t ) + и ^{2} ( t , щ ) t }

Taking limit when t tends to 0 we arrive at the SDE:

d B ( t , T , щ ) = B ( t , T , щ ) { [ л ( t , щ ) - и ^{2} ( t , щ ) ] d t + и ( t , щ ) d w ( t ) }

This SDE can be rewritten in integral form:

B ( s , T , щ ) - B ( t , T , щ ) = B ( u , T , щ ) { [ л ( u , щ ) - и ^{2} ( u , щ ) ] d u + B ( u , T , щ ) и ( u , щ ) d w ( u )

Where 0 ? t ? s ? T. Putting s = T we obtain stochastic equation:

B ( t , T , щ ) = 1 - B ( u , T , щ ) { [ л ( u , щ ) - и ^{2} ( u , щ ) ] d u + B ( u , T , щ ) и ( u , щ ) d w ( u )

Note that the left hand side of this formula generally speaking is a F [t, T]

Measurable random process and the stochastic integral on the right hand side can not be interpreted as Ito stochastic integral as far as the integrand does not satisfy the condition of _{}_{ }_{t }-_{ }measurability even when the coefficients and are deterministic functions. We will introduce a randomization in another form. Consider stochastic equation:

B ( t , T , щ ) = 1 - B ( u , T , щ ) л ( u ) d u - B ( u , T , щ ) и ( u ) d( u ) (7)

We assumed that л ( u ), и (u) are _{}_{ }- measurable functions and stochastic integral on the right hand side is interpreted as backward Ito integral. It is easy to verify that the solution of the linear stochastic equation exists, unique, and for the all [t, T] with probability 1 the bond price is a positive random process. We show that the solution of the equation can be written in the form:

B ( t , T , ) = exp - ( t , T , )

Where:

( t , T , ) = [ ( u ) + ^{2} ( u ) ] d u + ( u ) d ( u )

Indeed, let:

t = t _{0}_{ }< t _{1} < … < t _{N} = T

Then from it follows that:

B ( t _{k - 1}_{ }, T ) - B ( t _{k }, T ) = B ( t _{k }, T , ) { exp - { л ( u ) + ^{2} ( u ) ] d u + и ( u ) d ( u ) } - 1 }

Suppose that the value:

= max t k

Is sufficiently small. Then from the latter equality it follows that:

B ( t _{k - 1}_{ }, T ) - B ( t _{k }, T ) = B ( t _{k }, T , ) { exp - { [ л ( t _{k} ) + ^{2} ( t _{k} ) ] t _{k} + и ( t _{k} , щ ) [ w ( t _{k - 1}_{ }) - w ( t _{k} ) ] + ^{2} ( t _{k} ) t _{k} + o ( ) }

Summing up the above equality for k = 1, 2, N and then putting 0 we arrive at the equation. Now for simplicity assume that functions л , и are deterministic and denote:

( t , T ) = [ л ( u , щ ) + и ^{2} ( u , щ ) ] d u

Then one can easy verify that with positive probability the random variable ( t , T , ) is negative:

P { ( t , T , ) < 0 } = P { ( u ) d w ( u ) < - } > 0

We can conclude that with positive probability the bond price admits values:

B ( t T ) > 1

For each moment t, t ? T. These scenarios do not make sense and the GBM model of the bond price in general can not be admitted. This remark suggests that we need to pay more attention using the lognormal model for the bond pricing. Nevertheless, for the small value and for a sufficiently short period of time the latter probability can be such small that one can ignore the chance that bond price exceeds its face value and therefore the validity of the model is not diminished. We have applied a `white' noise model for interest rate which does not fail to satisfy all properties of the bond definition.

Below we suggest other approach to bond price randomization. One can use a general nonlinear stochastic differential Ito equation for modeling the interest rate, which guarantees that the bond prices will satisfy inequality:

0 < B ( t T ) 1

For instance, we consider a nonlinear SDE for r ( t ) with deterministic or random coefficients. Note that in the real world we observe historical data that usually can be used to form filtrations. It is our choice to present a model in a simpler form. Let us apply backward time stochastic Ito equations BSDE for the bond pricing model. We interpret the bond price B (t, T, ) as a random _{}_{[ t}_{ , T}_{ ]}_{ }measurable function for each t, 0 ? t ? T with continuous with probability 1 in t sample paths. Note, that the primary formulas well known for the standard forward time Ito integrals hold without changes for its backward counterpart. Let us assume that r ( t ) is governed by the backward Ito equation

r ( t , ) = r _{T} - ( u , r ( u , )) d u - ( u , r ( u , )) d ( u )

Where the random variable:

r T = r ( T + )

Is instantaneous interest rate at T. Formally, r _{T} is a _{}_{ T} - measurable random variable which is defined as the limit of the interest rates over a small period [T, T + ) when value > 0 tends to 0 , i.e:

r T_{ }= ( ) -^{ 1} r ( T , T + )

The value of r _{T}_{ }is unknown at a spot date t, t < T. On the other hand in practice one can use date-t instantaneous forward rate:

в = в ( T )

As a convenient estimate of the rate r _{T}_{ }, where:

= [ B ( t , T + ) ] -^{ 1}^{ }[ B ( t , T ) - B ( t , T + ) ]

The use of this estimate implies market-model risk that stems from the fact that estimate в does not equal to the market future rate r _{T }. Stochastic integral on the right hand side (9) is stochastic backward Ito integral. Note that if we are interested in a particular interest rate with a fixed maturity T then the equation (9) can be rewritten as:

r ( t , T ) = r _{T} - ( u , T , r ( u , T )) d u - ( u , T , r ( u , T )) d ( u )

Let us assume that coefficients of the equation (9) are deterministic scalar continuous in (t, r)

[ 0 + ) Ч [ 0 + )

Functions and there exists a constant L > 0 such that:

| ( t , x ) - ( t , y ) | + | ( t , x ) - ( t , y ) | L | x - y | (10)

( t , 0 ) = ( t , 0 ) = 0

It follows from (9) when t ^ T that:

r ( T ) = r ( T , T + , щ )

Is a _{}_{T} -measurable random variable.

This is the real forward rate at T and it is unknown in prior moments and it does not equal to its date-t implied forward rate estimate. Conditions (10) provide existence and uniqueness _{}_{t} - measurable solution of the equation (9). Let us show that solution of the equation (9) is strictly positive and does not reach zero with probability 1 over a finite period of time [0, T]. Let > 0 and T < + be arbitrary constants and define the first stopping time _{} 0 for the inverse time process r (t, T, ) when it first hits the level , i.e.:

_{} = max { t 0 : r ( t , T , ) < }

Denote t s = max ( t , s ) and applying Ito formula to the function f ( x ) = x -^{ 1} , we see that for any s 0

r -^{ 1} ( _{} s ) -^{ 1} + r -^{ 2} ( _{} u ) | ( _{} u , r ( _{} u )) | d u + r -^{ 2} ( _{} u ) ( _{} u , r ( _{} u )) d ( u ) + r -^{ 3} ( _{} u ) ^{2} ( _{} u , r ( u , T )) d u

Taking expectation in this inequality and bearing in mind (9) we arrive at the inequality:

E r -^{ 1} ( _{} s ) -^{ 1} + 2 L E r -^{ 1} ( _{} u ) d u

From which it follows that:

E r -^{ 1} ( _{} s , T ) -^{ 1} exp 2 L T

Applying Chebyshev inequality we see that:

P { _{} s < s } = P { r -^{ 1} ( _{} s ) -^{ 1} } E r -^{ 1} ( _{} s , T )

Taking the limit when 0 we easy verify that:

P { _{} < s } 0

For any moment s 0.

Let us consider a linear case of the equation. In this case we suppose that the coefficients of the equation are linear in r, i.e.

( u , r ) = ( u ) r

( u , r ) = ( u ) r

Where and ( u ), ( u ) are deterministic, continuous in u functions. The `implied' version of the equation can be presented in the form

r ( t ; ) = - ( u ) r ( u ; ) d u - ( u ) r ( u ; ) d ( u ) (9)

The solution of the admits representation:

r ( t ; ) = exp - ( t , T , )

Where:

( t , T , ) = [ ( u ) + ^{2} ( u ) ] d u + ( u ) d ( u )

Indeed, let:

t = t _{0} < t _{1} < … < t _{N} = T

r ( t ; ) - r ( T ; ) = r ( t _{j} ; ) - r ( t _{j}_{ + 1} ; ) = exp - {[ ( u ) + ^{2} ( u ) ] d u + ( u ) d ( u ) } - 1 ] r ( t _{j}_{ + 1} ; ) = { -[ ( u ) - ^{2} ( u ) ] d u - ( u ) d ( u ) + ^{2} ( u ) d u } r ( t _{j}_{ + 1} ; ) + o ( )

Where:

= max ( t j + 1 - t j )

And the remainder on the right hand side satisfies the condition:

E | -^{ 1} o ( , ) | ^{2}^{ }= 0

Taking limit in the above equality when 0 , we arrive at the equation.

The second condition (10) guarantees that the interest rates is positive which implies that the bond values do not exceed its face value. Now we introduce other type of the condition that also guarantees the interest rate to be a positive function. Let ( t ; T , x ) be a solution of the Ito backward stochastic differential equation

( t T x ) = x - a ( u , ( u T x )) d u - b ( u ( u T x )) d ( u )

For simplicity let us suppose that coefficients of the equation are nonrandom and satisfy the standard conditions which provide existence and uniqueness of the continuous _{}_{t} - measurable solution:

( t ) , t [ 0 , T ]. Let ( x ) > 0

Be a smooth, uniformly bounded nonrandom function. Putting:

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

We arrive at the interest rate model governed by the backward Ito SDE

d( t ) = (( t )) d( t ) + ^{2} (( t )) b ^{2} ( t ,( t )) d t

t [ 0 T )

With the backward terminal boundary condition:

( T ) = = ( x )

Thus, by a choice of the functions:

( x ) , a ( t , x ) , b ( t , x )

We have a possibility to approximate the real world stochastic interest rate. Consider a randomization of the implied forward rate. Assume that interest rate of the bond is governed by the linear version of the equation:

r ( t , T ) = r _{T} - ( u , T ) r ( u , T ) d u - ( u , T ) r ( u , T ) d ( u )

With nonrandom, bounded, and continuous coefficients л , и . By definition the date-t implied forward rate F (t, v, v, ), v > 0 over a future period:

v v + v v t

Is defined by the equation

B ( t , v + v, ) = B ( t , v , ) [ 1 + F ( t , v , v , ) v ] -^{ 1}

Solving this equation we arrive at the formula:

F ( t , v , v , ) v = B ( t , v , ) B -^{ 1} ( t , v + v, ) - 1 = exp [ ( t , v + v , ) - ( t , v , ) ] - 1

Where expression in the brackets on the right hand side is:

( t , v + v , ) - ( t , v , ) = [ ( u , v + v ) + ^{2} ( u , v + v ) - ( u , v ) - ^{2} ( u , v ) ] d u + [ ( u , v + v ) - ( u , v ) ] d( u ) + [ ( u , v + v ) + ^{2} ( u , v + v ) ] d u + ( u , v + v ) d( u )

For a sufficiently small value v one can note that:

( t v + v ) - ( t v ) = { ( v v ) + ^{2} ( v v ) + [ ( u v ) + ( u , v ) ( u v ) ] d u + ( u v ) d ( u ) } v - ( v , v ) [ w ( v ) - w ( v - v) ] + o ( v , )

Where , are the partial derivatives with respect to v of the functions , and:

o ( v ) ( v ) -^{ 1}^{ } 0

In square mean sense when v 0. Hence:

exp [ ( t , v + v , ) - ( t , v , ) ] - 1 = { ( v , v ) + ^{2} ( v , v ) + [ ( u , v ) + ( u , v ) ( u , v ) ] d u + ( u , v ) d( u ) } v + ( u , v ) d ( u ) } v - ( v , v ) ( v )

It follows then that stochastic instantaneous forward rate at t can be defined by the formula:

F ( t , v , ) d v = - { ( v , v ) + ^{2} ( v , v ) + [ ( u , v ) + ( u , v ) ( u , v ) ] d u + ( u , v ) d( u ) } d v - ( v , v) d ( v )

Where t < T < H.

Stochastic integrals on the right hand side are interpreted as the backward Ito integral. The function F ( t , v , ) in (11) is the instantaneous forward rate at t over the infinitesimal future interval:

[ v v + d v ) v [ T H ]

Formula (11) defines forward rate as a _{}_{t} - measurable random process.

This is the market rate which depends on market scenario. In contrast to market rate the spot rate is a known at the date t number.

One can think that the spot forward rate is formed by the risk-reward expectation of the market and can be also interpreted as the `settlement' rate.

One can approximate implied (spot) date-t forward instantaneous rate as one of the relationship:

f ( t , T ) = E { F ( t , T , ) | _{t} }

f ( t , T , ) = a ( t , ) + b ( t , ) E { F ( t , T , ) | _{t} }

f ( t , T , ) = f ( 0 , T , ) + a ( u , E { F ( u , T , ) | _{u} } ) d u + b ( u , E { F ( u T ) | _{u} } ) d ( u )

Where a and b are deterministic or _{ t} - measurable functions that can be derived from the historical data. For illustrative simplicity assume that a = 0 and:

b ( u F ) = F

Then putting: Ft = t, and:

f ( t v ) = E { F ( t v ) | F _{t} }

We see that:

f ( t , v ) d v = E { F ( t , v , ) | F _{t} } d v = - E { ( v , v ) + ^{2} ( v , v ) + [ ( u , v ) + ( u , v ) ( u , v ) ] d u | F _{t} } d v

The equality (11) can be represented in differential form as:

f ( t , T ) = E { F ( t , T , ) | F _{t} } = - E { ( T , T ) + ^{2} ( T , T ) + [ ( u , v ) + ( u , v ) ( u , T ) ] d u | F _{t} }

Remark. The popular HJM model defines forward rate by the equation:

f ( t , T , ) = f ( 0 , T ) + б ( u , T ) d u + у ( u , T ) d w ( u ) (12)

Where б (u, T):

у ( u T )

Are known deterministic functions though one can assume that these functions are F _{u} - measurable in u and continuous in T. Putting in (12) T v t we arrive at the money market rate:

r ( t ) = f ( t t )

Formula:

r ( t ) = f ( 0 , t ) + б ( u , t ) d u + у ( u , t ) d w ( u )

This formula differs from the money market rate definition. Indeed, the only observable data that define r ( t ) is bond prices.

From it follows that:

B ( t , T ) = exp - r ( u ) d u = exp - { f ( 0 , u ) + б ( v , u ) d v + у ( v , u ) d w ( v ) } du

This model suggests that the value of the bond at some moment t depends on all observations in the past and in the future. This assumption is an inconvenient. From the market experience bond prices move at is depend on what would happen at t or what is expected in the next future and it does not actually depend on what was happen in the past.

We have defined instantaneous forward rate in terms of interest rate. Alternatively, we can begin with the forward rate and define the corresponding interest rate.

We need to note that it does not look reasonable as in practice we observe interest rate which then can be applied for the forward rate construction. Nevertheless, let us begin with a forward rate construction.

Let us assume that instantaneous stochastic forward rate is governed by the equation:

F ( t , T , ) = + ( t , v ) F ( t , v , ) d v + ( t , v ) F ( t , v , ) d ( v )

Where t is a current fixed moment and a future moment T, T ? t is variable. Constant is given, and coefficients:

( t ^{ } v )

( t ^{ } v )

Are _{v} - measurable, bounded with probability 1, measurable in t , v functions.

Given bond prices, interest rates, and forward rates data we chose in (13) the forward direction for the stochastic integral.

Under broad conditions there exists a unique _{v} - measurable solution of the equation. The stochastic integral on the right hand side is interpreted as a standard Ito integral. Solution of the equation can be written in the form:

F ( t , T , ) = exp ( t , T , щ )

Where:

( t , T , щ ) = [ ( t , v ) - ^{2} ( t , v ) ] d v + ( t , v ) d w ( v )

It follows from, that the process:

f ( t T ) = f ( t T )

Satisfies semi group property

f ( t ; T , , ) = f ( s ; T , f ( t ; s , , ) , )

With probability 1 for any:

0 ? t ? s ? T < +

At once. This property defines the stochastic flow. It is well known fact that bond price and its interest rate are inversely related. Taking into account that higher price of an interest rate contract implies lower interest rate we note that a buyer of the forward rate f ( t , T ) is at risk measured by the probability:

P { f ( t , T ) < F ( t , T , ) }

This value is a quantitative measure of scenarios for which the price of the forward rate f (t, T) is higher than can be observed in the real world at the settlement date T.

Let us consider for illustration Forward Rate Agreement ( FAR ) pricing. FAR is an interest rate contract between two parties at the time t on cash payment at maturity T, t < T. Party A pays at maturity T a fixed rate which is specified at t over a next future period [T, T + H] , H > 0 and receives at T a floating rate over the same future period [T, T + H] from counterparty B. Floating rate used for FRA contracts is usually LIBOR rate. Denote k ( t ) fixed FRA rate:

L ( T T + H щ )

LIBOR rate at T for the period [T, T + H]. Then cash payment at T from party B to A is equal to:

N [ L ( T T + H щ ) - k ( t ) ]

For scenarios щ when this amount is positive. Here N is a principal of the contract. If this amount is negative then B receives from A cash payment of:

N [ k ( t ) - L ( T T + H щ ) ]

The pricing of the FRA problem is to define value of k ( t ). The commonly used notation n Ч m FRA at date t means that:

T = t + n

And H = m where n an m are measured in month scale with 360 days year format. The value of the contract is known at T and is paid at T + H. Hence, the value of the contract at T should be equal to 0 that leads us to the equation:

N [ k - L ( T T + H щ ) ] = 0

The solution of this equation presents the market price of the FRA that depends on scenario. Date t estimate of the forward rate:

L ( T T + H щ ):l ( t T T + H )

Applying this rate we arrive at the implied interest of the contract:

k ( t ) = l ( t T T + H )

This solution of the pricing problem highlights the risk. Primary risk characteristics of the contract are the average buyer's loss and the correspondent volatility:

E _{b} Loss = N E [ k ( t ) - L ( T , T + H , ) ] { k ( t ) < L ( T , T + H , ) } V _{b} Loss = N E [ k ( t ) - L ( T , T + H , ) ] ^{2} [ k ( t ) < L ( T , T + H , ) ] - ( E _{b} Loss ) ^{2}

income pricing forward

The value of the contract at the settlement date T + H, is:

$ N [ l ( t ; T , T + H ) - L ( T , T + H , щ ) ]

This contract value represented in dollars at a British bank and to get equivalent value at t one needs to apply LIBOR rate as discount factor. Thus value of the contract is given by well known formula:

$ N [ l ( t ; T , T + H ) - L ( T , T + H , щ ) ] [ 1 + l ( t ; T , T + H ) ] -^{ 1}

References

1. I. Gikhman. Stochastic Differential equations and its Applications, LAP Publishing , 2011, 264 .

2. R. Jarrow, S Turnbull. Derivatives Securities, South-Western College Publishing, 2ed. 2000, 684.

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