Assuming constant volatility across all strikes, how to use known premium of options to determine premium of options with another strike? e.g. suppose we know premium of \$40 call and put, \$50 call and put, how to determine premium of \$30 and \$60 options?
Under the Black-Scholes framework, you can calculate the implied volatility, given the option's price, underlying's price, time to maturity and the risk free rate. To calculate the implied volatility you have to use a root finding method, since there is not a closed form of the inverse of the B-S option pricing equation for volatility.
In the real world implied volatility varies across maturities (volatility smile). However, if you want to assume constant volatility as stated in the question you can use it to calculate option prices for other strikes.