PUT$-$CALL PARITY

Consider a call c and a put option p written on the same underlying S with the same strike K $,$ both maturing in one year. The forward price of the underlying today also happens to be $\backslash (\backslash$mathrm$\{$F$\}=\backslash$mathrm$\{$K$\}\backslash ).$

$1.$ Draw the payoff diagrams of the call, the put, and the forward.

$2.$ Show graphically how to replicate the forward using the call and the put. Write the replicating portfolio.

$3.$ Using your portfolio above, derive put$-$call parity formula $($no further proof is necessary$).$ I.e$.,$ write an expression for the value of the forward, denoted by XO $,$ in terms of the prices of the call and put today, denoted byc$0$ and p$0.$ What result $($or "law"$)$ are you using?

$4.$ If the price of the put is $\backslash (\backslash$mathrm$\{$pO$\}=5\backslash ),$ what is the price of the call cO $?$ Hint $1$: Use put$-$call parity. Hint $2$: What is the price of the forward contract at inception?

$5.$ If the strike price is $\backslash (\backslash$mathrm$\{$K$\}=2\backslash )$ and price of the underlying is $10,$ what is the risk$-$free rate? Hint: What is the forward price?

$6.$ Suppose the price of the call increases by $2$ but the price of the underlying stays constant. What happens to the price of the put?