Objection 1. It would seem that despair is not contrary to hope. Because "to one thing there is one contrary" (Metaph. x, 5). But fear is contrary to hope. Therefore despair is not contrary to hope.
Objection 2. Further, contraries seem to bear on the same thing. But hope and despair do not bear on the same thing: since hope regards the good, whereas despair arises from some evil that is in the way of obtaining good. Therefore hope is not contrary to despair.
Objection 3. Further, movement is contrary to movement: while repose is in opposition to movement as a privation thereof. But despair seems to imply immobility rather than movement. Therefore it is not contrary to hope, which implies movement of stretching out towards the hoped-for good.
On the contrary, The very name of despair [desperatio] implies that it is contrary to hope [spes].
I answer that, As stated above (Question 23, Article 2), there is a twofold contrariety of movements. One is in respect of approach to contrary terms: and this contrariety alone is to be found in the concupiscible passions, for instance between love and hatred. The other is according to approach and withdrawal with regard to the same term; and is to be found in the irascible passions, as stated above (Question 23, Article 2). Now the object of hope, which is the arduous good, has the character of a principle of attraction, if it be considered in the light of something attainable; and thus hope tends thereto, for it denotes a kind of approach. But in so far as it is considered as unobtainable, it has the character of a principle of repulsion, because, as stated in Ethic. iii, 3, "when men come to an impossibility they disperse." And this is how despair stands in regard to this object, wherefore it implies a movement of withdrawal: and consequently it is contrary to hope, as withdrawal is to approach.
Reply to Objection 1. Fear is contrary to hope, because their objects, i.e. good and evil, are contrary: for this contrariety is found in the irascible passions, according as they ensue from the passions of the concupiscible. But despair is contrary to hope, only by contrariety of approach and withdrawal.
Reply to Objection 2. Despair does not regard evil as such; sometimes however it regards evil accidentally, as making the difficult good impossible to obtain. But it can arise from the mere excess of good.
Reply to Objection 3. Despair implies not only privation of hope, but also a recoil from the thing desired, by reason of its being esteemed impossible to get. Hence despair, like hope, presupposes desire; because we neither hope for nor despair of that which we do not desire to have. For this reason, too, each of them regards the good, which is the object of desire.
Source: Summa Theologica (New Advent)