We present a highly mechanical method for the construction of implied volatility surface and implied transition probability density function that satisfies the no arbitrage condition(NAC) under missing data environment. This paper assumes a Dupire's model which is a simple extension of the Black-Scholes model. The missing data problem for implied volatility is the first obstacle for constructing the volatility surface, the importance of the variance surface in relation to the thin plate spline was explained. With variance surface using TPS approximation, we present our main linear programming problem for the construction of implied transition density functions which are inherent in the market volatilities. Even though the constructed transition density function is a simple piecewise linear function, it reconstruct the market implied volatilities very accurate. We show detailed developing process by effectively overcoming the problems encountered, faithfully showing the process leading to the final result with helpful figures and separate algorithms.