Can anyone point out what is the problem?
A single transistor used as a multiplier (modulator) is basically imperfect. Yes it will amplitude modulate the carrier but it will also inject the output with the modulation signal and this is what you have. As Brian Drummond says in his comment, there is amplitude modulation occurring AND regular additive mixing.
Look at the envelope valleys; the upper is about 2.2 big squares and the lower is about 3 squares: -
If these were equal there would be no modulation. Usually, to get rid of the base-band artifacts two emitter fed transistors are used with a differential output taken from both collectors. That differential output will contain identical base-band artifacts and therefore they will cancel out. However, to do this it's better to drive the bases differentially. This will work reasonably well: -
You can replace the transformer with collector resistors and look at the diff voltage across both collecotrs.
Or do what Brian says and high pass filter the output to remove the base-band signal.
You appear to have swapped the carrier and modulation inputs.
- The carrier is the primary signal and should be applied to C1.
- The modulation signal is applied to R4 and shifts the operating point of the transistor to modulate the amplification of the carrier signal.