# Calculating the max. risk free interest rate with two given options

I have an excercise where we have two European Call Options, which have the same underlying, same maturity $$t = 3$$, same interest. The only difference is their price and their strike. The price of the first Call is $$C_A = 3$$ with a Strike of $$K_A = 50$$ and the second is $$C_B = 12$$ with Strike $$K_B = 40$$. Now I have to find out the maximal riskfree interest rate $$r_f$$. Does anyone how to find this $$r_f$$ thanks in advance

• it's not obvious what you mean by 'max potential risk free interest rate' given the context you describe. please clarify your question. – Chris May 15 at 21:18

Consider the 40/50 call spread. This has a maximum payoff of 10, and hence has a maximum value of $$10/(1+r)^3$$. Where r is the annual risk free rate. But we know it is priced at 12-3 =9, so the maximal risk free rate satisfies $$(1+r)^3=10/9$$ which gives r= 3.6%