With the arbitrage free value p amer d there are

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with the arbitrage free value P Amer d . There are three alternatives: 1. The option is always out of the money ( d 2 S > K ) and therefore valueless (”No value”). 2. It is optimal to hold it for another period (better ”alive” than ”dead”). 3. It is optimal to exercise (better ”dead” than ”alive”). Always in-the-money : We start with case where it is always in the money ( udS < K ) and its value as ”alive” is P Eur d = Δ P d dS + Θ P d = π * u ( K - udS ) + π * d ( K - d 2 S ) R f = K R f - dS. Compare this value with immediate exercise and in most cases it is better to exercise K R f - dS < K - dS ⇐⇒ K R f < K = Exercise. (525) 146
However, for a particular value of the risk free rate the two alternatives are equal R f = 1 = K R f - dS = K - dS. Thus, it is always better to exercise if R f > 1. Notice, the reason is the possibility to get the exercise price today at t + 1 with a value of K , instead of tomorrow at t + 2 with a present value of K R f . Sometimes in-the-money : In the next case it is only in the money for the ”Down” state ( d 2 S K and udS > K ) and its value as ”alive” is P Eur d = Δ P d dS + Θ P d = π * d ( K - d 2 S ) R f . Is there any possible value for the underlying asset dS at the node d for which the put option has same value whether it is ”alive” or ”dead”? Such a value S is the solution to the following equation K - dS Dead = π * d ( K - d 2 S ) R f Alive dS = K R f - π * d R f - π * d d Dead = Alive (526) There are two different cases depending on whether d is above or below 1. In the first case (1 d ) the option has no value 1 d R f = R f - π * d R f - π * d d 1 = K - dS < 0 = P d = 0 . In the second case ( d < 1) the option has positive value d < 1 = R f - π * d R f - π * d d < 1 = K - dS > 0 = P d > 0 . Now it might be optimal to exercise dS < dS = π * d ( K - d 2 S ) R f < K - dS = Exercise . (527) If the stock is below dS at the node d , or S below S , then it is optimal to exercise at d . 147
We can conclude that an American put always has a value that is at least equal to the value of the identical European option P Amer P Eur . Thus, the possibility to exercise early has a non-negative price . 6.10 Put-Call-Parity Independently of the process followed by the stock we can construct an arbitrage free relation between the current prices of the four assets, stock, bond, call and put, where the two derivatives have the same exercise price and time to maturity ( n periods). Form the following portfolio Cash flow today S T K S T > K Short call: C 0 K - S T Long put: - P K - S T 0 Long stock: - S S T S T Borrow: KR - n f - K - K Total current value? 0 0 . The arbitrage free current value of this portfolio is zero C - P - S + KR - n f = 0 . The put-call-parity relationship is C = P + S - KR - n f . (528) Note that if S = K then C - P = K (1 - R - n f ) > 0 . The par value S = KR - n f gives C = P. 148
For a stock paying a dividend the relation is C = P + S - PV ( Div ) - KR - n f , (529) where PV ( Div ) is the present value of all dividends paid out before the expiration date. If the stock follows a binomial process we have the following relations C = Δ C S + Θ C P = C - 1) S + Θ C + KR - 1 f Δ P = Δ C - 1 0 (530) Θ P = Θ C + KR - 1 f 0 . (531) 6.11 The return and risk of a call option A call option, with a positive value, is a portfolio of two assets with the weights C = Δ · S + Θ C > 0 : 1 = Δ · S C + Θ C = w S + w B w S > 1 , w B < 0 .

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