(Option bond) Exercise price = calculated price, exercise option?

If calculated price is equal to bond price exercise price, you are indifferent, you can or may not exercise since you can still buy the bond in the market if the exercise price = calculated price.

Hi there! Sorry for not replying earlier. I’m going to try to answer your question, and if my answer is not related to what you are asking, please let me know.  

Question: “Just a quick question as per the title, whether for call or put bonds, do we exercise the option if calculated price is equal to exercise price?” 

Answer: I assume you are talking about backward induction and the bond prices calculated at each node. For a call option, if your exercise price is $95 and you calculate at any given node a price = or greater to $95, then you exercise it! For a put option, if your exercise price is $95 and you calculate at any give node a price = or less than $95, then you exercise it as well. 

Question: “On the same topic of binomial tree of interest rates and backward induction, do we discount bond at t=maturity given the respective interest rates at each note or do the values are merely based on par rate + coupon?”

Answer: Well, it depends. For a straight bond, the price at each node is based on the respective interest rate. For a callable bond, if the bond price is equal or above the strike price, then it is based on the par rate + coupon. For a putable bond, if the bond price is equal or below the strike price, then it is based on the par rate + coupon.

Let me know if it is clear or if you have any more questions.