I am going through the bond valuation and analysis section. However, I am getting a bit confused as I moving forward.

The backward induction and pathwise valuation derive the value of the bond today from the face value at maturity. However, what about the market value of the bond? What determines the market value of the bond today?

Also, correct me if I am wrong, different analysts may have different perspective about the forward rates and the volatility of the interest rate. Therefore, different analysts come up with different values through the model, and then they decide whether to buy/sell the bond based on the derived value?

Mary, who owns the bond, and Bob, who wants the bond, determine the market value of the bond today when they agree on a price and Mary sells it to Bob.