Arbitrage Relationships for Options
 

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.

V₀(r,t) stands for the present value (at time 0) of a dollar to be received at time t when the risk-free interest rate for the time interval [0,t] is r.The form of the present value function V₀(r,t) depends upon which form the interest rate is expressed in. If r is the instanteous rate then V₀(r,t)=exp(−rt)=e^−rt; whereas if r is the effective annual rate then V₀(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.

Property 1: C₀ ≥ max[0, S₀ - V₀(r,T)X]

where, as was indicated above, C₀ is the value of the call option at time 0, S₀ is the price of the stock at time 0, X is the strike price, T is the time until expiration, and V₀(r,T)X is the present value of the exercise price X discounted back over the life of the option at the risk-free interest rate r.

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 non-negative. This means that the value of the portfolio at time 0 must be non-negative. Therefore

C₀ + V₀(r,T)X - S₀ ≥ 0,
(again where V₀(r,T)X is the present value at time 0 of an amount X to be received T years in the future and the interest rate is r) and hence
C₀ ≥ S₀ - V₀(r,T)X.

But C₀ ≥ 0 so

C₀ ≥ max[0,S₀ - V₀(r,T)X]

Property 2: If a stock pays a single dividend
of d₁ at time t₁, then

C₀ = max[0,B₁,B₂,B₃)
where

• B₁ = S₀-X
• B₂ = S₀-V₀(r,t₁)X
• B₃ = S₀-V₀(r,t₁)d₁-V₀(r,T)X

Proof:

C₀ ≥ B₁ because the call option could be exercised immediately yielding a payoff of S₀-X.

To prove that C₀ ≥ B₂ 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 ex-dividend 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 non-negative value at the ex-dividend date and a non-negative value at time 0. Therefore

C₀ ≥ S₀ - V₀(r,t₁)X

To prove that C₀ ≥ B₃ 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 d₁ discounted back from just before the ex-dividend 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 d₁ on ex-dividend day would have grown to d₁/V₀(r,(T-t₁) 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

W = C_T − (S_T+d₁/V₀(r,(T-t₁)) + X + d₁/V₀(r,(T-t₁)
which reduces to
W = C_T − S_T + X

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.,

C₀ + V₀(r,t₁)d₁ + V₀(r,T)X - S₀ ≥ 0.

Hence

C₀ ≥ S₀ - V₀(r,t₁)d₁ - V₀(r,T)X.

Property 3: Early exercise of a call option on a stock that pays no dividend prior to expiration is never optimal.

Proof: Property 1 means that at any time t

C_t ≥ S_t - V₀(r,(T-t))X.

Since V₀(r,(T-t))X < X if t<T, it follows that

C_t > S_t - X.

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.

Property 4: Exercising an American call option is optimal only on either the expiration date or just prior to the ex-dividend date.

Property 5: If the stock pays a dividend prior to expiration then an American call may be worth more than a European call.

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.

Property 6: If the maximum dividend is less than the interest on the strike price from the ex-dividend date to expiration, early exercise of an American call is not optimal.

Proof: Let D₁ be the maximum dividend that is possible. The actual dividend d₁ is less than or equal to D₁.

Exercise just before the ex-dividend would be optimal only if the intrinsic value at that time was greater than the expected payoff from holding to expiration; i.e.

S_t1^- - X > C_t1^-.

where t₁^- means just before the ex-dividend time.

If one exercises the call just before the ex-dividend 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/V₀(r,(T-t₁). Therefore, the amount of interest foregone on the strike price is

X/V₀(r,(T-t₁))-X,

which is equal to

[1/V₀(r,(T-t₁))-1]X.

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 ex-dividend date we must adjust them to the same date.

The present value of this foregone interest discounted back to the ex-dividend date is

V₀(r,(T-t₁)[1/V₀(r,(T-t₁))-1]X = [1 - V₀(r,(T-t₁))]X

If D₁ < X[1 - V₀(r,(T-t₁))] then d₁ < X[1 - V₀(r,(T-t₁))]. It follows from this that

d₁ + V₀(r,(T-t₁))X < X

and therefore

S_t1- - d₁ - V₀(r,(T-t₁))X > S_t1- -X.

If this holds, then by Property 2 applied at time t₁,

C_t1- ≥ S_t1- - V₀(r,(T-t₁))X -d₁,
so
C_t1- > S_t1- - X.

The LHS is the value of holding the call unexercised just prior to the ex-dividend date and the RHS is the value of exercising the call just prior to the ex-dividend date.

Property 7: The value of an American put on a stock that pays a certain (i.e., risk-free) dividend d₁ at time t₁ satisfies the condition:

P₀ ≥ max[ 0, X-S₀, V₀(r,t₁)(X+d₁)-S₀].

Proof: The value of the put obviously has to be non-negative and has to be at least equal to its intrinsic value X-S₀.

To prove that it must be worth at least V₀(r,t₁)(X+d₁)-S₀ consider a portfolio made up of one put, one share of the stock, and interest bearing debt which will have a value of X+d₁ on ex-dividend day. The payoff profile for this portfolio as a function of the stock price on ex-dividend day is a combination of these three components:

The combination is then

The portfolio has a non-negative value on ex-dividend day so the portfolio has a non-negative value at time 0. Its value at time 0 is

P₀ + S₀ - V₀(r,t₁)(X+d₁)
so
P₀ + S₀ - V₀(r,t₁)(X+d₁) ≥ 0.

Thus

P₀ ≥ V₀(r,t₁)(X+d₁) - S₀.

Property 8: If X₁, X₂, and X₃ are three strike prices such that X₂ = (X₁+X₃)/2 then the corresponding values of the calls and puts satisfy the following conditions:

C₂ ≤ (C₁+C₃)/2
and
P₂ ≤ (P₁+P₃)/2.

Proof: Create a portfolio by buying one call at X₁, selling two calls at X₂, and buying one call at X₃; i.e., sell a butterfly strike price spread. The payoff on this portfolio on expiration day is:

This portfolio has a non-negative value on expiration date so it has a non-negative value at time 0. It value at time 0 is

C₁-2C₂+C₃, so

C₁-2C₂+C₃ ≥ 0

C₁+C₃ ≥ 2C₂.
Thus
C₂ ≤ (C₁+C₃)/2.

To establish the condition for puts, create a butterfly spread by buying one put at X₁, selling two puts at X₂, and buying one put at X₃. This portfolio has the same payoff profile as the previous one; i.e.

Its value at time 0 is P₁-2P₂+P₃, hence

P₁-2P₂+P₃ ≥ 0,
and therefore
2P₂ ≤ P₁ + P₂
and hence
P₂ ≤ (P₁ + P₂)/2.

Thus

P₂ ≤ (P₁+P₃)/2.

Property 9 (Put-Call Parity): The value of a European put equals the value of a European call plus the present value of the strike price less the current price of the stock.

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.

Property 10: The value of a European put on a stock that pays a dividend equals the value of a European call plus the present value of the strike price plus present value of the dividend less the current price of the stock.

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

P₀ = C₀ + V₀(r,T)X + V₀(r,t₁)d₁ - S₀.

Property 11: For a stock that pays no dividend prior to expiration the value of an American put satisfies the following condition:

P₀ ≥ C₀ + V₀(r,T)X - S₀.

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.

Property 12: For a stock that pays no dividend prior to expiration the value of an American put satisfies the following condition:

P₀ ≤ C₀ + X - S₀.

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 non-negative value and hence

C₀ + X - P₀ - S₀ ≥ 0,
so
P₀ ≤ C₀ + X - S₀.

Reference: Peter Ritchken, Options: theory, strategy and applications, Scott-Foresman, 1987.