In order to do justice to this material and make it well understood, it is pertinent that we begin by introducing some vocabulary used in financial mathematics and offer clear definitions of some important notions which will largely form part of this work. This will be beneficial to any person who wishes to refresh his knowledge on preliminary concepts applicable in the field of finance. We will frequently refer to [6] and ([1], Chapters 2 and 10) in a bid to build up this section.

A financial market is a place which provides a platform offering a link between savings and investments. It involves trade in stock, equities, bonds, currencies, and derivatives. The basic instrument traded in any financial market is called an asset. An asset refers to any property which yields value in exchange and is said to have a positive economic value. There are three types of traders in a financial market. The first category includes those who risk more with the intention of gaining more, and are referred to as speculators. Since speculators bet on market movements, they are considered more vulnerable to gains or losses. The second category are the hedgers. These types of traders try their level best to make investments more certain by reducing risks associated with uncertainty. Last but not least, the arbitrageurs play with several markets in hope of finding an advantage to gain with no risk. In this market, a derivative security is any security whose price depends on an underlying asset. Since we are modeling interest rates, we shall later see the interconnection between interest rates and bonds. According to [7] and [8] a bond is a financial instrument which certainly guarantees the holder an interest payment in the future, i.e., a debt investment where the investor lends money to a loanee who promises to repay the principal and interest at a later date called the maturity date, e.g., government bonds. The issuer of the bond is also called the bond writer. It is worthwhile noting that bonds can be transferable in the secondary market, in which case, they become highly liquid. Bonds are generally issued in different pieces called coupons. Thus, a bondholder is entitled to receive these predetermined payments as long as the bond is still subsisting. The main difference between regular bonds and zero-coupon bonds is the fact that the former pays interest (coupons) to the bondholder up to maturity, while in the latter, the bondholder is only entitled to the face value of the bond at maturity.

2.1.1. Definition: Zero-Coupon

A Zero-Coupon Bond with maturity T > 0 is a contract which guarantees the holder a cash payment of one unit on the maturity date T [5]. The price at time t ∈ [ 0 , T ] of a zero-coupon bond with maturity T is denoted by B ( t , T ) > 0 , with B ( T , T ) = 1 .

Another type of financial derivatives is called options. They give the owner the right, but not the obligation to buy or sell an underlying stock at or before a predetermined period, for a fixed price called the strike price, determined a priori at time t . Thus, if S T is a risky asset, e.g., stock. A European call option [1] is a contract giving the holder the right to buy an underlying asset in the future, for a price K called the strike price, fixed in advance at time T . Its payoff (or value at time T ) is then

A European put option gives the right to sell the underlying asset for the strike price K at the exercise time T . Its payoff is

Remark.A European option differs from an American option in the sense that the former is exercised at the expiry time, while the latter may be exercised at any time before the expiry time.

2.1.2. Definition: Stock

A stock is any ownership in a company indicated by units called shares of stock, and which yields value when traded.

If at the initial time t 0 the stock price S ( t 0 ) is known to all investors and at time t , the price of a share of stock is S ( t ) , then:

and the rate of return on stock is given by:

2.1.3. Definition: Portfolio

In bond and stock markets, if an investor holds x amount of money in a given bond and y units of shares in a given stock over a given period of trading, we say that the vector h = ( x , y ) is the portfolio of that investor.

If x < 0 or y < 0 , at time t = 0 , we say that there is a short position in a bond or stock, while for x > 0 or y > 0 , one gets a long position in a bond or stock.

In our essay, we will make the following assumptions:

i) Short positions and fractional positions are allowed;

ii) the market is completely liquid; i.e., one can buy any amount of shares of stock;

iii) there are no transaction costs; i.e., no additional fee in buying/selling a stock;

iv) The selling price and buying price of all assets are equal.

It is factual that money put in a bank account earns interest over a given period of time. The main reasoning behind this is that the money is locked in a savings account and, as such, since it cannot be spent right away, the owner expects compensation for postponed consumption. This brings into perspective the concept of the time value of money. Consider an investment P made at time t = 0 , which is attracting an interest I ( t ) at a given rate. Then, at time t > 0 , at the end of the investment period, a total amount V ( t ) = P + I ( t ) is paid back to the lender or investor. Therefore, the interest earned from time t = 0 to time t is given by

We define interest rate r ( t ) as the proportion of the loan charged as interest to the borrower, i.e.,

and is normally expressed as a percentage of the outstanding loan.

2.2.1. Simple Interest and Bond Pricing

The simple interest is the money earned after investing an initial amount in a bank or financial institution; this initial amount is usually called the principal. Thus, simple interest is given by

The future value of the investment after time t constitutes the initial principal plus the total interest earned by this investment from the time the initial deposit was made, i.e.,

where,

V ( t ) : Value of investment at time t ;

r : Constant interest rate on the principal;

P : Principal amount.

Note: ( 1 + r t ) > 0 is typically called the growth factor,i.e., the quantity with which the loan or investment grows over time. The future value of the investment at time t ≥ s will be

Remark.Interest is said to be simple if it is attracted by the principal P only at a constant rate r > 0 .

From (1), given the future value of an investment, one can compute the present value of V ( t ) as follows:

Return on an investment starting at time t 0 and ending at time t , denoted by R ( t 0 , t ) , is given by the formula below:

In the case of simple interest,

2.2.2. Periodic Compounding Interest and Bond Pricing

Suppose a principal P , which earns interest at a constant rate r > 0 , is deposited in a bank account. We say that the interest is periodically compounded if the interest earned on the initial principal at the end of each compounding period is added to that principal, updating this principal and (cumulatively) attracting interest again at rate r in the next compounding period, and so on k times, until the last compounding period [1]. The future value after t years is given by

where the term ( 1 + r k ) t k is called the growth factor of periodic compound interest.

From (4), given the future value of an investment, one can compute the present value of V ( t ) using the following formula:

Using this, it is easy to see that the value of the investment at any earlier time s ≤ t is given by

2.2.3. Continuous Compounding Interest and Bond Pricing

The concept of continuous compounding borrows heavily from periodic compounding. Observe that (4) is equivalent to

Computing the limit of both sides of (6), when k approaches infinity, we obtain

which represents the future value of a principal attracting interest at a constant rate r > 0 , continuously compounded for t years. Noticing that e = lim k → ∞ ( 1 + 1 k ) k , then from (7), it follows that

that is,

which means that the rate of change of investment is directly proportional to the present value of the investment.

Deterministic interest rates depend only on the running time, and this is denoted by r = r ( t ) , while stochastic interest rates are deterministic interest rates that also depend on some randomness occasioned by the state of the world. Here, we shall consider a type of forward interest rate contract which gives its holder a loan decided at present time t over a future period of time [ T , T 1 ] . Companies often need to agree at present time t to a loan to be delivered over the given future period, t ≤ T ≤ T 1 , at a rate r ( t , T , T 1 ) . For instance, if a company purchases a bond priced B ( t , T 1 ) at time t , whose maturity is at T 1 , it ought to pay back e ( T 1 − t ) r ( t , T 1 ) B ( t , T 1 ) at maturity. However, if the company decides to extend the maturity of the tenor to T 2 , then it should repay e ( T 2 − t ) r ( t , T 2 ) B ( t , T 2 ) at time T 2 . Notice that there will be a floating rate r ( t , T 2 ) which is prone to increase at time T 2 . Thus, the company may arrange to pay back B ( T 2 , T 2 ) at an agreed rate f ( t , T 1 , T 2 ) . The following definition plays a crucial role in the analysis of deterministic and stochastic interest rates.

Definition: Forward Rate

According to [9], the forward rate in the above contract is the rate f ( t , T 1 , T 2 ) agreed upon at time t , , effected at T 1 , and ending at T 2 so that

To solve (9), we introduce the logarithm and proceed as follows:

i.e., − f ( t , T 1 , T 2 ) ( T 2 − T 1 ) = ln B ( t , T 2 ) − ln B ( t , T 1 )

thus we have:

which represent the forward rates.

If T 2 − T 1 is sufficiently small, such that T 2 ≈ T , then Equation (10) becomes,

which represents the instantaneous interest rates. Integrating this relation, one can recover the bond prices, i.e.,

In our study of forward rates, which are stochastic in nature, we introduce randomness of the floating rate f ( t , T ) by letting it be driven by Brownian Motion W on a filtered probability space. Applying the methodology of Health, Jarrow, and Morton [10], which is abbreviated as HJM, [1] assumes that the forward rates satisfy the equation

Obtained from the stochastic differential equation

where ω denotes the state of the world, α and σ are random adapted processes.

In a bid to meet the objective of our work, we shall not describe these kinds of models extensively, but rather, we will study similar models frequently used in the real-world market, for instance, the Euribor market model. The forward Euribor rates within this model imply what the market feels concerning the movements in the future interest rates. Since the Euribor interest rates, which we are modeling in this essay, depend both on time and on randomness, we will review the basics of random/stochastic processes and stochastic calculus in the next chapter.

In this section, we consider the concepts of conditional expectation, conditional probabilities, as well as conditioning of a σ -algebra G generated by a random variable. The majority of the definitions used here are adopted from [11].

3.1.1. Definition

Let Ω be a non-empty set. A family ℱ of subsets of Ω is called a σ -algebra of Ω if:

i) Ω ∈ ℱ ;

ii) A ∈ ℱ ⇒ A c ∈ ℱ ;

iii) If A 1 , A 2 , ⋯ , A n is a sequence of subsets in ℱ , , then ∪ i = 1 ∞ A i ∈ ℱ .

3.1.2. Definition

Let ( Ω , ℱ , ℙ ) and ( Ω * , ℱ * , ℙ * ) be two probability spaces. A function X : Ω → Ω * is ℱ measurable if and only if

If Ω * = ℝ , then X is a real random variable.

Given a probability space ( Ω , ℱ , ℙ ) , ℙ : ℱ → [ 0 , 1 ] is a probability measure on Ω.

A random variable X : Ω → ℙ is said to be integrable if ∫ Ω | X | d ℙ < ∞ . Thus, the expected value of X exists and is denoted by:

On the other hand, given two events A and B , the conditional probability of A | B is defined as ℙ ( A | B ) = ℙ ( A ∩ B ) ℙ ( B ) .

If A and B are independent, then

For an event B ∈ ℱ , the conditional expectation of X given B is defined by:

3.1.3. Definition

Let X be an integrable random variable on a probability space ( Ω , ℱ , ℙ ) and G be a sub σ -algebra contained in ℱ . Then, E ( X | G ) is measurable and

The conditional probability of an event A ∈ ℱ given a σ -algebra G is defined as

where 1 A is the indicator function of A.

Given two integrable random variables X and Y on a probability space ( Ω , ℱ , ℙ ) , G , ℋ ∈ ℱ , and a , b ∈ ℝ , then:

i) E ( a X + b Y | G ) = a E ( X | G ) + b E ( Y | G ) ;

ii) E ( X | G ) = E ( X ) if X is independent of G ;

iii) E ( X Y | G ) = X E ( Y | G ) if X is G measurable;

iv) If X ≥ 0 , then E ( X | G ) ≥ 0 ;

v) E ( E ( X | G ) | ℋ ) = E ( X | ℋ ) if ℋ ⊂ G .

Proof. We refer the reader to ([5], page 29). □

The concepts building up this section were adopted from [12].

3.2.1. Definition

A continuous time stochastic process is any collection { X t , t ∈ [ 0 , T ] } of random variables on a probability space, ( Ω , ℱ , ℙ ) , indexed by a continuous sub-interval [ 0 , T ] of ℝ + .

For a fixed ω ∈ Ω , X ( ⋅ , ω ) : [ 0 , T ] → ℝ is called a sample path or realization of X . If all the sample paths of X are continuous functions, then X is a continuous-time stochastic process. Further, if all the sample paths of X are right continuous and have left limits, then X is referred to as a process. Two stochastic processes are called versions of one another if ℙ ( X ( t ) = Y ( t ) ) = 1 . Since financial markets are often faced with uncertainties, trading in stock is subject to the random occurrence of events which may affect the prices in the future. In this paper, we shall see that for time increments, 0 ≤ t 1 < t 2 < ⋯ < t n , the value of stock ( S t i ) at different times t ∈ [ 0 , T ] , is a random vector and it is determined by its finite-dimensional distributions. We now define the notion of filtrations, a theory which is widely used in finance, especially in insider trading and credit risk.

3.2.2. Definition: Continuous-Time Filtration

A filtration in this context will represent an information structure available in the market. Let ( Ω , ℱ , ℙ ) be a probability space. A collection ( ℱ t ) t ∈ [ 0 , T ] of sub σ -algebras in ℱ is called a continuous-time filtration if:

i) ℱ 0 is trivial, i.e., it contains only ℙ -null sets and their complements, ℱ = { ∅ , Ω } ;

ii) ℱ s ⊆ ℱ t for all s ≤ t ≤ T (information accumulates);

iii) ℱ t = ∩ s > t ℱ s (right-continuous property);

iv) ℱ T = ℱ (total information).

A filtration therefore represents the evolution of information of the world with time. In mathematical finance, if the filtration is complete and right-continuous, then it is said to satisfy the usual hypothesis. Moreover, it is easy to make any filtration complete by adding all the ℙ null sets to it.

3.2.3. Definition: Adaptedness

A continuous-time stochastic process X is adapted to a given filtration ( ℱ t ) t ≥ 0 if X t is ℱ t -measurable for all t ∈ [ 0 , T ] , where [ 0 , T ] represents the time interval.

3.2.4. Definition:Continuous-Time Martingale

Let { X t } t ≥ 0 be a continuous-time stochastic process on a filtered probability space ( Ω , ℱ , ( ℱ t ) t ≥ 0 , ℙ ) . We say that { X t } t ≥ 0 is a martingale with respect to ℙ and ( ℱ t ) t ≥ 0 if

1) E [ | X t | ] < ∞ ∀ t ≥ 0 , and

2) the martingale property below holds

Remark.If we replace the equality in (3.2.1) with the inequality ≤ , then { X t } t ≥ 0 is called a ( ℙ , ( ℱ t ) t ≥ 0 ) -supermartingale. If we replace it with the inequality ≥ , then { X t } t ≥ 0 is called a ( ℙ , ( ℱ t ) t ≥ 0 ) -submartingale.

Assertion (iii) of definition (3.2.2) is applicable in martingale theory in the sense that we assume all martingales, being mathematical tools used for pricing market investments and derivative securities, have a regular right-continuous version. Now, we introduce the notion of Brownian motion and some of its basic properties.

3.3.1. Definition

A real-valued stochastic process { W t : t ≥ 0 } is called a standard Brownian motion or Wiener process if the following properties are satisfied:

i) W 0 = 0

ii) For all stationary time increments, 0 ≤ t 0 ≤ t 1 ≤ t 2 ≤ ⋯ ≤ t n , n ∈ ℕ , the random variables

follow a normal distribution with mean 0 and variance t n − t n − 1 ,i.e.,

which implies

iii) Almost surely, the function t ↦ W t is continuous.

The concept of almost surely (a.s), as used in probability theory, means an event occurs with certainty i.e., with probability 1. In other words, the probability of possible exceptions is zero.

Remark.With reference to the central limit theorem, under certain conditions, the distributions of independent random variables drawn from a large sample tend to converge to a normal distribution with mean μ and variance σ 2 . Thus, if

is a random variable, the probability density function of a normal distribution is given by the following formula

If μ = 0 and σ 2 = 1 , then (18) is called the standard normal distribution of a variate x and is denoted by

3.3.2. Lemma

For 0 ≤ s ≤ t , the covariance of W s and W t is s .

Proof.Guided by the definition of the covariance of two random variables, and with reference to [13], observe that

Thus, for some s , t ≥ 0 , C o v ( W s , W t ) = min ( s , t ) .

□

3.3.3. Theorem. Martingale Properties of Brownian Motion

Let { W t , t ≥ 0 } be a standard Brownian Motion on a probability space ( Ω , ℱ , ℙ ) . Then w, r, t its normal filtration F W = ( ℱ t ) t ≥ 0 , where ℱ t : = σ ( W s ; s ≤ t ) is the information available up to time t by observing all W s , s ≤ t . Then the following processes are ℱ -martingales:

i) the standard Brownian Motion W ;

ii) ( X t ) t , where X t = W t 2 − t , t ≥ 0 ;

iii) ( Y t ) t , where Y t = exp ( σ W t − 1 2 σ 2 t ) , for some constant σ > 0 .

Proof.We provide the proof of these properties, guided by the definition and properties of Brownian motion outlined in [11].

i) E ( W t | ℱ s ) = E [ W s + ( W t − W s ) | ℱ s ] = E [ W s | ℱ s ] + E [ ( W t − W s ) | ℱ s ] = W s + E ( W t − W s ) = W s ,     for   s < t .

This is because, for every time increment, the random variables are independent. Thus, W t is a martingale.

ii) X t = W t 2 − t , t ≥ 0 is a martingale for 0 < s < t , i.e., taking Δ W = W t − W s .

Because E ( Δ W ) = 0 and E ( Δ W ) 2 = E ( W t − W s ) 2 = t − s .

iii) The moment generating function of the random variable W t ~ N ( 0 , t ) is given by

This implies that e σ W t − 1 2 σ 2 t is integrable with E ( e σ W t − 1 2 σ 2 t ) = 1 . Furthermore, for s < t ,

It follows that, E [ e σ W t − 1 2 σ 2 t | ℱ s ] = e σ W s − 1 2 σ 2 s □

In the market, stock prices are influenced by various random dynamics such as transactions, decisions per unit time, etc., which we refer to as the state of the world. Thus, the concept of Brownian motion enables us to properly explain such dynamics.

The Brownian motion properties can be used to describe the market fluctuations affecting the price X ( t ) of an asset. As described in [13], we consider a one-dimensional Brownian motion on a probability space ( Ω , ℱ , ℙ ) equipped with the natural filtration F W = ( ℱ t ) t ≥ 0 of W t . We define the following important concepts:

A partition of the time interval [ 0 , T ] is a set π n = { t 0 , t 1 , ⋯ , t n } with 0 = t 0 < t 1 < ⋯ < t n = T nested, and δ t n = sup i | t i + 1 − t i | as the mesh size. For a non-random function f : [ 0 , T ] → ℝ , the variation of f over [ 0 , T ] is defined as:

If V [ 0 , T ] f < ∞ , then f is said to be of finite variation.

3.4.1. Definition

An F W adapted process ( X t ) t ∈ [ 0 , T ] is called a step process of time interval [ 0 , T ] if there exists a partition π n = { t 0 < t 1 < ⋯ < t n } such that

3.4.2. Definition

Let X : = ( X t ) t ∈ [ 0 , T ] ∈ C ( [ 0 , ∞ ) , ℝ ) be a stochastic process on a probability space ( Ω , ℱ , ℙ ) . Then the continuous quadratic variation of X over the interval [ 0 , T ] is defined as:

If Y : = ( Y t ) t ∈ [ 0 , T ] ∈ C ( [ 0 , ∞ ) , ℝ ) is another stochastic process in the same probability space, the quadratic covariation of X and Y is given by

If 〈 X , Y 〉 t exists and is continuous, then it is said to be of finite variation. This permits us to apply the polarisation formula, i.e.,

3.4.3. Definition

Let X t be an adapted step process as above. The Itô integral over [ 0 , T ] with respect to the Brownian motion is defined as

This can be extended to the space L 2 [ 0 , T ] of all continuous real-valued F W -adapted processes X t , such that ∫ 0 t X 2 d t < ∞ ∀ t ≥ 0 as follows:

If X t ∈ L 2 [ 0 , T ] , then, one can approximate the sequence of stochastic processes by a sequence of F W -adapted step processes ( X m ) m in L 2 [ 0 , T ] , i.e.,

3.4.4. Definition: Itô process

A real-valued stochastic process X t is called an Itô process if there exist two processes F t in L 1 [ 0 , T ] and G t ∈ L 2 [ 0 , T ] , such that for t ∈ [ 0 , T ] , the quantity

is obtained by integrating the stochastic differential equation

3.4.5. Proposition

Given an Itô process X t as defined above, X t is a ℙ martingale with respect to F W if and only if the drift part ∫ 0 t F s d s = 0 ∀ t almost surely (a.s.).

Proof. See [12] for details of the proof. □

3.4.6. Theorem. Itô Formula for Functions of Itô Processes

Let X t be an Itô process and f ( t , X t ) ∈ C 1 , 2 ( [ 0 , ∞ ) , ℝ ) , i.e., f is differentiable at least once in t and twice in x . Then, the process f ( t , X t ) is an Itô process with SDE

Proof. The details of the proof are given in [12]. □

As described in [12], stochastic differential equations (SDEs) arise when the coefficients of an ODE are perturbed by white noise, which is due to the randomness in the process. In finance, the application of the Black-Scholes model is important in the pricing of stock. For instance, for a market with a constant risk-free interest

rate r , the price of stock can be given by: S t = S 0 exp ( ( μ − 1 2 σ 2 ) t + σ W t )

which is a solution obtained by solving the following SDE

An equation of the form

where ( W t ) t ≥ 0 is a Brownian motion, μ ( t , X t ) and σ ( t , X t ) are given, and X t is the unknown process, is called an SDE.

3.5.1. Theorem

If we have two uniformly continuous Lipschitz functions f : ℝ → ℝ and g : ℝ → ℝ + , i.e., there exists a constant K > 0 such that for any x , y ∈ ℝ

Then the stochastic differential Equation (22) has strong solutions. Furthermore,

For any Brownian motion { W t } t ≥ 0 and its natural filtration F W = ( ℱ t ) t ≥ 0 , and

For a given starting point x 0 , there exists a unique adapted process X t with continuous paths such that

Proof.The proof, which parallels the main existence and uniqueness result for ordinary differential equations, can be found in [14]. □

3.5.2. Example

Show that X t = x 0 e a t + b W t , for t ≥ 0 , is a solution to the following SDE.

3.5.3. Definition

Given a Brownian motion, ( W t ) t ≥ 0 on a filtered probability space ( Ω , ℱ , ( ℱ t ) t ≥ 0 , ℙ ) . A strong solution of the SDE with initial condition x 0 ∈ ℝ is an adapted process X t = f ( t , W t ) with t → W t continuous such that ∀ t ≥ 0 .

3.5.4. Equation

SDEs occur in various forms and are important in asset pricing. We shall illustrate the existence of a strong solution by solving the Langevin equation and the Ornstein-Uhlenbeck process, which is given by:

If we take σ = 0 , then integrating the SDE (i.e., system (25)) yields X t = x 0 e − α t . Now consider Y ( t ) = X t e α t . This implies that

Substituting the equation of d X t given above, we obtain:

Thus,

Clearly

We consider a market economy where a finite sequence of assets, such as stochastic bonds with price processes B i ( t ) , i = 1 , ⋯ , n , are traded continuously on a finite time interval [ 0 , T ] and modeled under a filtered probability space ( Ω , ℱ , ( ℱ t ) t ≥ 0 , ℙ ) . We assume that each such bond is an Itô process governed by the following stochastic differential equation,

With μ ( t ) and σ ( t ) being two adapted processes verifying ∫ 0 T | μ ( t ) | d t < ∞ and ∫ 0 T σ ( t ) 2 d t < ∞ , almost surely.

A trading strategy is a predictable n-dimensional stochastic process.

Denote the holding in asset i with price B i ( t ) at time t .

The value of the trading strategy ξ ( t , ω ) is given by:

Thus, the value process of a self-financing trading strategy satisfies:

4.1.1. Definition

An arbitrage opportunity is a self-financing trading strategy ξ with ℙ [ V ( ξ , T ) ≥ 0 ] = 1 and V ( ξ , 0 ) < 0 , i.e., that has negative initial costs but, with probability 1, has a positive value at time T . The market is said to be arbitrage free if there is no arbitrage opportunity.

4.1.2. Definition

A numeraire is any asset B which has a strictly positive price for all t ∈ [ 0 , T ] . Examples of numeraires which we will study are bonds and bank accounts. Numeraires are used to discount other price processes; thus, relative prices are expressed as

In our case, we choose an appropriate numéraire that simplifies the pricing problem by transforming the discounted prices of other assets into martingales, which have a constant expected value as shown below.

Consider a filtered probability space ( Ω , ℱ , ( ℱ t ) t ≥ 0 , ℙ ) , with a set containing all probability measures. A probability measure ℚ is an equivalent martingale measure if:

i) ℚ is equivalent to ℙ , i.e., have the same null sets,

ii) the relative price processes B ′ i are martingales under ℚ , ∀ i ∈ ℕ , then

4.1.3. Theorem. Martingale

A continuous trading economy is said to be arbitrage-free if, for every choice of numeraire, there exists a (unique) equivalent martingale measure.

Proof. See [4]. □

This result means that if we have two numeraires and two different equivalent martingale measures, we can change from one measure to another as described below.

4.1.4. Definition. Choice, Change of Numeraire, and Girsanov’s Theorem

Generally, stochastic processes are not martingales under any probability measure. In this subsection, we will consider the Cameron-Martin-Girsanov theorem, which is used to change the given probability measure into another equivalent probability measure under which the processes become martingales. But first, we will outline aspects to consider when changing from one numeraire to another, i.e., we explore the Girsanov’s transformations.

4.1.5. Theorem: Change of Numeraire

If ℚ and ℚ * are two equivalent martingale measures associated with numeraires B and B * , respectively, then ∀ A ∈ ℱ we have:

where

is called the Radon-Nikodym derivative and is denoted by d ℚ d ℚ * .

Proof.See [4]. □

We now state an important result used to perform this transformation.

4.1.6. Theorem: Cameron-Martin-Girsanov Theorem

Let g = ( g ( t ) ) t ∈ [ 0 , T ] be a square-integrable stochastic process.

Consider the change of numeraire.

given by

where ℚ and ℚ * are two equivalent martingale measures associated with two numeraires B and B * , while W is a Brownian motion under the measure ℚ . Then, under the measure ℚ * , the stochastic process W * is defined by

is a Brownian motion and satisfies the SDE

Proof.See [15]. □

This section deals with the Euro Interbank Offered rate, which is considered “the benchmark giving an indication of the average rate at which banks in the Euro Market zone lend to each other unsecured funding” [3]. Approximately 43 banks currently on the panel range from co-operative banks, savings banks, international institutions, etc. Euribor, which is compiled by the European Banking Federation, is reported for 8 different maturities in Euro-currency.

To model the dynamics of the Euribor interest rates, we consider a market with a tenor structure T 1 < T 2 < ⋯ < T n < T n + 1 , and with n + 1 bonds with prices B ( t , T i ) 1 ≤ i ≤ n + 1 , at time t . . Each bond has maturity at time T i , i = 1 , 2 , ⋯ , n + 1 . We denote the constant time increment by α = T i + 1 − T i . Following a similar method for LIBOR interest rates as documented in [4] and [16], we are ready to define the notions of Euribor interest rates and processes.

4.2.1. Euribor Rate and Forward Euribor Rate Processes

Euribor interest rate, E ( T i ) , is a simple interest rate on the bond price B ( T i , T i + 1 ) , taken at time T i and lasting until time T i + 1 . This rate is reset at each time T i such that at maturity, the bond price B ( T i , T i + 1 ) yields 1 i.e.,

thus,

If the tenor is small, then using expression (12) we have

Suppose T 1 and T 2 are two maturity dates with T 1 ≤ T 2 . At time t , we denote by B ( t , T 1 ) and B ( t , T 2 ) the prices of bonds for the two maturity dates. If at time t ≤ T 1 , we invest an amount B ( t , T 2 ) discounted by B ( t , T 1 ) at time T 1 in a bank account whose payoff at maturity time T 2 is 1 unit, then the forward Euribor rate agreed at time t , , quoted on [ T 1 , T 2 ] and effected at time T 1 but lasting to time T 2 , satisfies

This, in turn, yields

which represents the expression for the forward Euribor interest rates.

4.2.2. Forward Euribor Processes under Equivalent Measures

The aim of this section is to express the forward Euribor rates under suitable martingale measures. We follow closely a similar methodology developed in [16].

We consider a market economy with n + 1 maturity dates T 1 < T 2 < ⋯ < T n < T n + 1 . At time t ≤ T i , let us assume we have n forward Euribor rates E i ( t ) = E ( t , T i , T i + 1 ) with tenors α i = T i + 1 − T i , i = 1 , 2 , ⋯ , n .

Further, we assume that under a given stochastic basis ( Ω , ℱ , ( ℱ t ) t ≥ 0 , ℙ ) , the associated bond price processes

can be expressed in the sense of Itô processes. Then, from (37), the corresponding forward Euribor rate processes are given by:

To this end, with reference to [17], we introduce the following result:

4.2.3. Theorem

If a market is arbitrage-free, then for i = 1 , ⋯ , n , there exists an equivalent martingale measure denoted by ℚ n + 1 , under which the Euribor rate process E i ( t ) is a martingale.

Proof.In reference to [17]). Take B i + 1 ( t ) as a numeraire discounting B i ( t ) , then, there exists an equivalent martingale measure. ℚ n + 1