Question $2$: Option Pricing Bounds $(2/10)$ Suppose a call option and a put option

have maturity $T$ and strike price $K.$ The current market price of the underlying stock $($no

dividends$)$ is $S_0,$ and the continuously compounding interest rate is constant at $r.$ Then we

have the payoffs of the two options satisfying

Call Option Payoff at $T$:$,S_T-K$max$\{0,S_T-K\}{S}_T$

Put Option Payoff at $T$:$,K-S_T$max$\{0,K-S_T\}K$

$($i$)$ A fact states that if two quantities $A_T,B_T$ at $T$ always satisfy $A_T{B}_T,$ then their

present value must satisfy $P{V}_0(A_T)P{V}_0(B_T).$ Can you use this act to derive the

model$-$free bounds of the prices of call option $C_0$ and put option $P_0?($Hint: present

value of a stock without dividends is its current market price, and present value of a

risk$-$free amount is the amount discounted by risk$-$free rate.$)$

$($ii$)$ Suppose the price of the call option $C_0$ is larger than $S_0,$ how can you arbitrage?

Suppose the price of the put option $P_0$ is larger than $K{e}^{-rT},$ how can you arbitrage?

$($iii$)$ Suppose the price of the call option $C_0$ is smaller than $S_0-K{e}^{-rT},$ how can you

arbitrage? Suppose the price of the put option $P_0$ is smaller than $K{e}^{-rT}-S_0,$ how

can you arbitrage?

$($iv$)$ What is the equation for put$-$call parity? Suppose the price of the put option is larger

than what the put$-$call parity implies, how can you arbitrage?