Question

what is the lower bound of European put equals to & please prove your argument through hypothesis portfolios.

Answer 1

Notation we use below will be as follows:

Upper and lower bounds for put options:

The value of a put option

In the worst situation of price of an asset, the price can go to zero. Hence, the value of the option cannot be higher than discounted value of the strike price. That is to say,

Let us now look at the lower bound for value of the European put option. As we said before, the future expected price of a share cannot be less than the compounded value of its current priced, compounded at risk free rate. To put it differently, the present prevailing price of a share cannot be less than the discounted value of future expected price, or discounted value of the strike price. If it is, the same should be reflected in the price of the put option. That is to say

Example

Imagine the following situation:

This is also inefficient, as there is an arbitrage opportunity. If I buy the share, and buy the call option, thereby investing 104, borrowing money at risk free rate1 , I need to return 109.33. Because of the call option, I have anyway locked my selling price to 110 – hence, I am assured of a risk free profit of 110-109.33. If the share price is more than 110, I make profit on the difference between the prevailing price and 109.33. Hence, the minimum value of the option should be (discounted value of 110 minus the present prevailing price).