Question

$2$

: Option Pricing Bounds

$($

$2$

$/$

$1$

$0$

$)$

$$Suppose a call option and a put option

have maturity

$$

$$and strike price

$$

$.$

$$The current market price of the underlying stock

$($

no

dividends

$)$

$$is

$$

$0$

$,$

$$and the continuously compounding interest rate is constant at

$$

$.$

$$Then we

have the payoffs of the two options satisfying

Call Option Payoff at

$$

:

$,$

$$

$$

$-$

$$

$<=$

max

$\{$

$0$

$,$

$$

$$

$-$

$$

$\}$

$<=$

$$

$$

Put Option Payoff at

$$

:

$,$

$$

$-$

$$

$$

$<=$

max

$\{$

$0$

$,$

$$

$-$

$$

$$

$\}$

$<=$

$$

$($

i

$)$

$$A fact states that if two quantities

$$

$$

$,$

$$

$$

$$at

$$

$$always satisfy

$$

$$

$<=$

$$

$$

$,$

$$then their

present value must satisfy

$$

$$

$0$

$($

$$

$$

$)$

$<=$

$$

$$

$0$

$($

$$

$$

$)$

$.$

$$Can you use this act to derive the

model

$-$

free bounds of the prices of call option

$$

$0$

$$and put option

$$

$0$

$?$

$($

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

$$

$0$

$$is larger than

$$

$0$

$,$

$$how can you arbitrage?

Suppose the price of the put option

$$

$0$

$$is larger than

$$

$$

$-$

$$

$$

$,$

$$how can you arbitrage?

$($

iii

$)$

$$Suppose the price of the call option

$$

$0$

$$is smaller than

$$

$0$

$-$

$$

$$

$-$

$$

$$

$,$

$$how can you

arbitrage? Suppose the price of the put option

$$

$0$

$$is smaller than

$$

$$

$-$

$$

$$

$-$

$$

$0$

$,$

$$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?