Question

The current price of a non-dividend paying asset is $65, the riskless interest rate is 5% p.a. continuously compounded, and the option maturity is five years. What is the lower boundary for the value of a European vanilla put option on this asset with strike price of $80?

Are there two lower bounds in this question? Is there any relevance to the option premium or the payoff?

Answer 1

Option Value whether of a call or put is composed of two parts, namely the intrinsic value and the time value. Intrinsic Value of a put option is given by IV (Intrinsic Value) = (Strike Price - Prevailing Asset Price) and Time Value = Put Premium - Intrinsic Value. Needless to mention the time value of an option is directly proportional to the options' remaining maturity. This is so because greater the time remaining to maturity greater is the chance of a price fluctuation in the underlying asset's price, thereby pushing the option into an in-the-money territory. In this context the current IV of the put option is (80 - 65) = $ 15. The IV of this option decreases towards zero as the asset price rises from $ 65 and becomes exactly zero at asset price of $ 80 (which is the option's strike price). Beyond this price, the option's IV will continue to be zero, thereby forming a lower bound of the option's IV. The overall option value, however, might not be zero owing to option's time value which would be substantial in this case as the put option has a remaining maturity period of 5 years. Therefore, the option's total value would be zero plus its time value which would itself decrease towards zero as remaining maturity decreases. Therefore, the option value would have two lower bounds, once when the option's IV becomes zero and once when it's IV plus time value becomes zero. The point at which the option's time value and intrinsic value are both zero, the option's premium should also be zero and so should be its payoff (as payoff and IV for an option are one and the same thing).