My editor (who knows nothing about options) gave me a poor-quality photocopy of a simple plain vanilla call payout using Black Scholes, and asked me to reproduce it in a spreadsheet so that she could insert it into an article. She insisted however that my call look exactly like hers, and that I provide the underlying data that generates the graph. In other words, all I know is the call price for various spot prices, but no other parameters. I need to solve for the implied vol, (implied) interest rate, and (implied) time to maturity, that uniquely describe my editor's diagram.Deriving just one unknown (e.g. implied vol) is straightforward via iterative methods. However I've never come across a problem where multiple parameters are missing simultaneously. Formally:1) Suppose you do not know sigma or r, but you know X, t, two call prices, and the two associated spot prices. Can you generate the entire payoff curve with just two data points? In other words, two equations and two unknowns, is it feasible to solve for the implied vol and interest rate simultaneously?2) Same idea, but suppose you do not know sigma and t, but you know X, r, two call prices, and the two associated spot prices. Any more difficult?3) Per the original problem, suppose you only know X and a handful of C and S values; you are missing sigma, t, and r. Are three call prices sufficient to solve for three unknowns, and generate the entire payoff curve, to a desired tolerance?Given that one equation and one unknown (volatility) itself has no closed-end solution, I would imagine it becomes extremely complicated to solve multiple equations for multiple unknowns, each iteratively, if this is even possible. Any thoughts?

Well this is impossible to do in any meaningful manner. What is this for?With a small number of strikes + prices pairs there might as well be unlimited solutions.Even with more and more pairs the solutions will still be way too many, but at least finite number.If all of the pairs are on the same underlying and you assume that the volatility smile doesn't exist and it's the same for all the options, then assume that t is a unknown too and r is a function of t (from a default yield curve) then you could solve this as a quadratic problem with the volatility linear for all options and the maturities changing. But then again not really meaningful for anything.

Probably this is routine. Just minimize the sum of the squared differences of the given price minus the Black-Scholesprice over the n unknown parameters. Nonlinear optimization is routine and best done in a computeralgebra system where it is a "one-liner".