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In a very general sense the *interest rate* *r* 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.

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

It is not possible to define an *r* mathematically so that it applies to all sorts of lending/borrowing operations.

One should rather speak of an *interest rate* in relation to a specific lending/borrowing contract.

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

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

Although not existing in the actual market, the *interest rate* pertaining to a theoretically conceivable instantaneous riskless deposit is defined appropriately and referred to as short rate

Given a specific contract *C*, the associated *interest rate* *r* depends on the time *t*, when the value *r* is agreed between the parties entering into the contract *C*.

We may therefore speak of a function *r(.)* that maps each time *t* to the respective *interest rate* value *r(t)*.

Assuming *t = 0* designates the time now, the value *r(t), t > 0* is not a simple number but rather a random variable, since it is not possible to know with certainty the *interest rate* that is going to prevail at the future tinme *t*.

It follows, the function *r(.)* represents a mapping from *t* to some random variable, and therefore is a stochastic process.

In the context of derivatives pricing, all what the so called "interest rate models" do, is to first choose a particular definition of *r* and then assume a particular mathematical form of the process *r(t)*.

For example, in the so called "Market Model", either the *ibor rate* or the *swap rate* is chosen as the *r* and a lognormal diffusion is assumed for the respective process.

On the other hand, in the so called "Short Rate Models", the *short rate* is chosen as the *r* and any out of a great variety of possible diffusions can be adopted for the respective process.