San José State University
Department of Economics 

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The term arbitrage relationships is misleading in that they are relationships that hold if no arbitrage is possible. If they do not hold then arbitrage in some form is possible.
In what follows C_{t} and P_{t} stand for the value at time t of a call option and a put option, respectively. S_{t} stands for the price of the underlying stock at time t.
V_{0}(r,t) stands for the present value (at time 0) of a dollar to be received at time t when the riskfree interest rate for the time interval [0,t] is r.The form of the present value function V_{0}(r,t) depends upon which form the interest rate is expressed in. If r is the instanteous rate then V_{0}(r,t)=exp(−rt)=e^{−rt}; whereas if r is the effective annual rate then V_{0}(r,t)=1/(1+r)^{t}. For more on this matter see rates.
An expression of the form max[A, B, C] stands for the maximum of the arguments.
Proof: Consider a portfolio made up of one European call option at a strike price of X plus a short sale of a share of the stock (i.e., an owed share of stock) plus an savings account deposit, which bears interest r, equal to the present value of the strike price. On expiration day the savings account will have a value of X. The payoff profile for the portfolio is the same as that of a put option:
This portfolio is equivalent to a put and its expected payoff at time T must be nonnegative. This means that the value of the portfolio at time 0 must be nonnegative. Therefore
But C_{0} ≥ 0 so
C_{0} =
max[0,B_{1},B_{2},B_{3})
where
Proof:
C_{0} ≥ B_{1} because the call option could be exercised immediately yielding a payoff of S_{0}X.
To prove that C_{0} ≥ B_{2} consider, as in the proof of Property 1, a portfolio with one call, one owed short sale, and the present value of the strike price X discounted back from just before the exdividend date to time 0. Then, just as in the case of Property 1, the portfolio would be equivalent to a put option and thus has a nonnegative value at the exdividend date and a nonnegative value at time 0. Therefore
To prove that C_{0} ≥ B_{3} consider, as in the previous case, a portfolio with one call, one owed short sale, a bank account equal to the present value of the dividend d_{1} discounted back from just before the exdividend date to time 0, and a bank account equal to the present value of the strike price X discounted back from the expiration date to time 0. Consider the payoff on this portfolio at expiration day as a function of the stock price on expiration day. The bank account based upon the exercise price would be equal to X on expiration day. The other bank account which was equal to d_{1} on exdividend day would have grown to d_{1}/V_{0}(r,(Tt_{1}) by expiration day.
The dividend plus accumulated interest must be covered as well as the share of stock in covering a short sale. If the short sale is covered on expiration day the value of the portfolio would be
Thus the portfolio is equivalent to a put with exercise price of X.
The portfolio has a value of W at expiration so at time 0 it must a value at least equal to the present value of W, which is a positive number; i.e.,
Hence
Proof: Property 1 means that at any time t
Since V_{0}(r,(Tt))X < X if t<T, it follows that
Since (S_{t}X) is the value of the call if exercised at time t, this means it is not optimal to exercise the call at any t<T.
Proof: The American option allows early exercise, which in the case of a stock paying a dividend may be optimal, whereas the European option does not that option.
Proof: Let D_{1} be the maximum dividend that is possible. The actual dividend d_{1} is less than or equal to D_{1}.
Exercise just before the exdividend would be optimal only if the intrinsic value at that time was greater than the expected payoff from holding to expiration; i.e.
where t_{1}^{} means just before the exdividend time.
If one exercises the call just before the exdividend time instead of on the expiration date, one would have to take the strike price X out of an interest bearing account early. If X had been left in the account until expiration date it would have grown to be X/V_{0}(r,(Tt_{1}). Therefore, the amount of interest foregone on the strike price is
X/V_{0}(r,(Tt_{1}))X,
which is equal to
This is the value of the foregone interest on the expiration date. In order to compare it with the dividend which is given on the exdividend date we must adjust them to the same date.
The present value of this foregone interest discounted back to the exdividend date is
If D_{1} < X[1  V_{0}(r,(Tt_{1}))] then d_{1} < X[1  V_{0}(r,(Tt_{1}))]. It follows from this that
and therefore
If this holds, then by Property 2 applied at time t_{1},
The LHS is the value of holding the call unexercised just prior to the exdividend date and the RHS is the value of exercising the call just prior to the exdividend date.
Proof: The value of the put obviously has to be nonnegative and has to be at least equal to its intrinsic value XS_{0}.
To prove that it must be worth at least V_{0}(r,t_{1})(X+d_{1})S_{0} consider a portfolio made up of one put, one share of the stock, and interest bearing debt which will have a value of X+d_{1} on exdividend day. The payoff profile for this portfolio as a function of the stock price on exdividend day is a combination of these three components:
The combination is then
The portfolio has a nonnegative value on exdividend day so the portfolio has a nonnegative value at time 0. Its value at time 0 is
Thus
Proof: Create a portfolio by buying one call at X_{1}, selling two calls at X_{2}, and buying one call at X_{3}; i.e., sell a butterfly strike price spread. The payoff on this portfolio on expiration day is:
This portfolio has a nonnegative value on expiration date so it has a nonnegative value at time 0. It value at time 0 is
C_{1}2C_{2}+C_{3}, so
To establish the condition for puts, create a butterfly spread by buying one put at X_{1}, selling two puts at X_{2}, and buying one put at X_{3}. This portfolio has the same payoff profile as the previous one; i.e.
Its value at time 0 is P_{1}2P_{2}+P_{3}, hence
Thus
Proof: A portfolio made up of a call plus an interest bearing account deposit equal to the present value of the strike price plus a short sale of a share of the stock has the exact same payoff as a function of the stock price on expiration day as a put. Therefore they must be equal in value.
Proof: Construct a portfolio similar to the one used in proving Property 9, but including also an interest bearing account equal to the present value of the dividend. If the short sale is covered on expiration day it will cost the price of the stock plus the amount of the dividend plus the interest on the dividend from the time the dividend was paid on the stock. Thus the portfolio will have the same payoff as a put and hence
Proof: With no dividends the value of an American call is the same as the value of a European call. The PV expression on the right of the inequality is the value of a European put. An American put is always worth at least as much as a European put.
Proof: Consider a portfolio created by buying one call and selling one put at the same strike price together with an interest bearing security equal in value to the strike price and a short sale. If this portfolio were held until the expiration date the payoff would equal the interest on the strike price. Therefore the portfolio at time 0 must have a nonnegative value and hence
Reference: Peter Ritchken, Options: theory, strategy and applications, ScottForesman, 1987.
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