Lower bounds of Option Prices and the possibility of early exercise.

Lower bounds for American and European Option Prices

E_C0 ≥ Max{0, [S₀ – PV (CF,o,T)] – X/(1+r)^T}
E_P0 ≥ Max {0, X/(1+r)^T – {S₀-PV(CF,0,T)}

Where: C = Call, S = Strike price, PV = Present value, CF = Cash Flow, r = interest rate, T= Time to expiration of the option

Cash flows for underlying assets are as follows:
· Stocks pay dividends - in formula terms FV (D,O,T) or PV (D,O,T)
· Bonds pay interest - in formula terms FV (CI,O,T) or PV (CI,O,T)
· Currency pays interest
· The underlying price is reduced by the PV of the cash flows of the underlying; therefore, the put-call parity relationship is calculated as:

E_C0 +X/(1+r)^T = E_P0 +{S₀ – PV(CF,0,T)}

The Relationship between American and European Options
· At any given point in time, the value of a call option or a put option cannot exceed a particular price.
· A call option can never be worth more than the stock price; therefore, the value of a call option should be lower or equal to the stock price.
· In the case of a put option, the upper bound is the strike price at which the contract has been entered. The value of a put will be lower than or (at most) equal to the strike price of the option. If this condition is violated, an investor can make use of the arbitrage opportunity by writing the option and investing the proceeds at the risk-free rate of interest.
· The lower bounds for an American call option are the same as the lower bounds for a European call option. Prices of American and European options differ mainly in whether they can be exercised before expiration, as is the case with American calls. As such, in most cases American calls and puts will be worth more than European calls and puts and their lower bounds will also differ.
· When the underlying asset does not make cash payments such as dividends or interest payments, the value of an American call option is equal to a European call option.
· When cash payments are involved, the value of an American call option tends to be higher than a European call option.
· American put options are almost always worth more than European put options.
· Cash flows on the underlying security and their effect on the put-call parity and the lowers bounds of options prices


Notes:
· Currency pays interest
· Commodities have carrying costs