## rate

In a very general sense the

*interest rate***is a number that tells us how much income can be earned by lending money, or equivalently how much cost is associated with borrowing money.**

*r*The higher the

**is, the more income one expects to earn by lending money.**

*r*It is not possible to define an

**mathematically so that it applies to all sorts of lending/borrowing operations.**

*r*One should rather speak of an

**in relation to a specific lending/borrowing contract.**

*interest rate*The simplest example of such a lending/borrowing contract is associates with a single time period and represented in Deriscope by the type Term Rate

In a more specialized way, the

**pertaining to certain short term deposits between banks is defined in a particular way and referred to as ibor rate**

*interest rate*The

**pertaining to interest rate swaps is defined in another way and referred to as swap rate**

*interest rate*Although not existing in the actual market, the

**pertaining to a theoretically conceivable instantaneous riskless deposit is defined appropriately and referred to as short rate**

*interest rate*Given a specific contract

**, the associated**

*C*

*interest rate***depends on the time**

*r***, when the value**

*t***is agreed between the parties entering into the contract**

*r***.**

*C*We may therefore speak of a function

**that maps each time**

*r(.)***to the respective**

*t***value**

*interest rate***.**

*r(t)*Assuming

**designates the time now, the value**

*t = 0***is not a simple number but rather a random variable, since it is not possible to know with certainty the**

*r(t), t > 0***that is going to prevail at the future tinme**

*interest rate***.**

*t*It follows, the function

**represents a mapping from**

*r(.)***to some random variable, and therefore is a stochastic process.**

*t*In the context of derivatives pricing, all what the so called "interest rate models" do, is to first choose a particular definition of

**and then assume a particular mathematical form of the process**

*r***.**

*r(t)*For example, in the so called "Market Model", either the

**or the**

*ibor rate***is chosen as the**

*swap rate***and a lognormal diffusion is assumed for the respective process.**

*r*On the other hand, in the so called "Short Rate Models", the

**is chosen as the**

*short rate***and any out of a great variety of possible diffusions can be adopted for the respective process.**

*r*