Consider two European call options on the stock of XYZ$.$ Both options emature one year from now. The first option $($Option #$1)$ has a strike price of $$30\mathrm{.}0$ and trades at $$22\mathrm{.}4$ today; the second option $($Option #$2)$ has a strike price of $$50\mathrm{.}0.$ Currently, the stock price is equal to $$50\mathrm{.}0,$ and the one$-$year continuously$-$compounded risk$-$free rate is $4\%.$ The stock does not pay dividends.

What is the lowest price of the second option $($Option #$2)$ consistent with absence of arbitrage?

$1\mathrm{.}96053$

$ parcialmente correcto

That's a good first step, but the answer is not correct yet because there is a higher bound. Hint: Try to construct a portfolio consisting of Option #$2$ and a discount bond that provides at least as high payoff at expiration as Option #$1.$ The price of this portfolio should be at least as high as the price of Option #$1,$ which would give you the lower bound on the price of Option #$2.$