I have the option price at a rate which is much smaller than the rate at which I have tick data for the underlying. If I have option price at times $t_1, t_3, t_5$ and I have tickdata at $t_1, t_2, t_3, t_4, t_5$ can I find the option price at $t_2, t_4$ ?
Why not? You can back out implied vol from the times you do have for the underlier price and then use that to price the options for the times you do not have. (This is assuming you are taking about pricing one particular options, not using options of one strike and expiry to price options at an other time, strike, and expiry.)
You could even do a linear interpolation and probably get very close.