LM curve: How interest rate is changing without OMOs by the central bank?
Hence interest rate is directly dependent on the OMOs.


While open market operations have a large influence on interest rates, there are other ways in which central banks influence interest rates. For example, they may increase or decrease the interest on excess reserves. Moreover, the IS-LM model does not necessarily assume the existence of a central bank. Interest rates have been set by the market far before central banks existed to influence them.  

So how do interest rates rise with a fixed real money supply? Increased transactions demand leads to a increase in demand for money. If real money supply stays constant, then interest rates should rise. The following is an example.

Imagine an economy with only one good, tomatoes, and two people, Buyer and Producer. Producer produces all tomatoes. Buyer owns all 10 credits and is disposed to consume everything possible. Imagine the Producer has produced 10 tomatoes, and is willing to sell at 1 credit per tomato. Thus real GDP is 10 credits. If the Buyer purchases all 10 tomatoes from Producer, Buyer will own 10 tomatoes and Producer will own 10 credits.

The next year Buyer has magically regained all 10 credits. Maybe Buyer stole them from Producer; it doesn't matter. Producer produces 20 tomatoes. Hence, real GDP is now 20 credits. Intuitively tomato prices should go down as there is more supply and a fixed demand (from Buyer) but that would imply that real money supply has gone down. So to better illustrate the LM curve, let us assume that Producer keeps prices at 1 credit per tomato. Buyer now purchases 10 tomatoes, and has no credits left. But Buyer is determined to consume everything possible, so Buyer strikes a deal with Producer to borrow the 10 credits that now Producer owns at 10% interest. Buyer then purchases 10 more tomatoes.

In short, real GDP doubled, real money supply stayed the same, and interest rates went up from 0% (no one borrowing) to 10%.


  Also, why would someone will buy the bond from central bank when the bond prices are going down?


Central banks will sometimes sell bonds that they have purchased previously to decrease the money supply. The announcement that a central bank will attempt to do so may attract front-running speculators. In other words, basic market dynamics will apply: if no one wants to purchase bonds, then prices will decline until someone believes prices have declined enough that they will eventually start rising again even under pressure from the central bank.