Options General Properties

Posted on March 22, 2015

Definitions

European call option is a contract giving the holder the right to buy an asset, called the underlying, for a price \(X\) fixed in advance, known as exercise price or strike price, at a specified time \(T\), called the exericse or expiry time. Similalry, a European put option gives the right to sell the underlying asset for the strike price \(X\) at the exercise time \(T\).
American call option or put option gives the right to buy or, respectively, to sell the underlying asset \(X\) for the strike price \(X\) at any time between now and a specified future time \(T\), called the expiry time.
Underlying asset: Apart from the typical assets such as stocks, commodities, or foreign currency, there are options on stock indices, interest rates, or even the snow level at a ski resort. Some underlying asset may be impossible to buy or sell.
The payoff of a European call is given by \((S(T)-X)^{+}\), where we use the notation \[x^{+} = \left\{ \begin{array}{cl} x& \text{if } x>0 \\ 0 & \text{otherwise.} \end{array} \right.\] For a put option, the payoff is given by \((X-S(T))^{+}\).
Since the investor will not lose money under any circumstance, a premium has to be paid to enter into an option. The prices of the calls and puts will be denoted by \(C_E\) and \(P_E\) (for European options) and \(C_A\), \(P_A\) (for American options).
The gain of an option buyer at time \(T\) is given by (for a European call)

\[ (S(T) - X)^{+} - C_{E}e^{rT}\]

Put-call parity

Theorem For a stock that pays no dividends the following relation holds between the prices of European calls and put options, both with exercise price \(X\) and exercise time \(T\):

\[ C_E - P_E = S(0) - Xe^{-rT} \]

The above result can easily be shown using the no-arbitrage principle. By making an equivalence with the forward contracts, we can prove the following results:

If the stock pays a dividend of \(d_0\) (\(d_0\) is the present value of dividend, i.e., if a dividend is paid at time \(t\), then \(d_0 = de^{-rt}\)) in between \(0\) and \(T\), then the put-call parity relation is given by

\[ C_E - P_E = S(0) - d_0 - Xe^{-rT} \]

If the dividends are paid continuously, at a rate \(r_d\), then

\[ C_E - P_E = S(0)e^{-r_dT} - Xe^{-rT} .\]

Theorem The prices of Americal put and call options with the same strike price \(X\) and expiry time \(T\) on a stock that pays no dividends satisfy

\[ S(0) - Xe^{-rT} \ge C_A - P_A \ge S(0) - X \]

For a stock that pays a dividend \(d_0\) (the present value of dividend), then then the prices of American call and put satisfies \(S(0) - Xe^{-rT} \ge C_A - P_A \ge S(0) - d_0 - X\).
For a stock that pays dividends continuously, \(S(0)- Xe^{-rT} \ge C_A - P_A \ge S(0)e^{-r_dT} - X\), where \(r_d\) is the rate at which dividends are paid.

Bounds on option prices

The following inequalities are obvious:

\[ C_E \le C_A, \qquad P_E \le P_A \]

One can also prove the following ones

\[ \begin{align*} C_E &< S(0) \\ S(0) - Xe^{-rT} &\le C_E \\ P_E &< Xe^{-rT} \\ -S(0)+Xe^{-rT} & \le P_E \end{align*}\]

For dividend paying stocks, the bounds are

\[ \begin{align*} \max\{0, S(0) - d_0 - Xe^{-rT}\} &\le C_E \le S(0) - d_0 \\ \max\{0, -S(0)+d_0+Xe^{-rT}\} &\le P_E < Xe^{-rT} \\ \end{align*}\]

European and American calls on Non-dividend paying stock

The prices of American and European call options on a stock that pays no dividends are equal, i.e., \(C_A = C_E\), whenever the strike price \(X\) and the expiry time \(T\) are same for both the options.

American options

One can easily show the following inequalities for American options that pays no dividends.

\[ \begin{align} (S(0)- Xe^{-rT})^{+} &\le C_A \le S(0) \\ (-S(0)+X)^{+} &\le P_A \le X \\ \end{align}\]

American options that pays dividends

The prices of American put and call options on a stock that pays no dividends satisfy the inequalities

\[ \begin{align} \max \{ 0, S(0) - d_0-Xe^{-rT}, S(0)-X \} &\le C_A < S(0)\\ \max \{0, -S(0) + d_0 + Xe^{-rT}, -S(0) + X \} &\le P_A < X \\ \end{align} \]

Variables that determine option prices

Here we shall analyze change of one variable keeping the other variables fixed.

European options

Strike price \(X\)

The function \(C_E(X)\) is a decreasing and \(P_E(X)\) is an increasing.
\(C_E(X)\) and \(P_E(X)\) are Lipschitz with Lipschitz constant \(e^{-rT}\). The following is true provided \(X' \le X''\):
\[\begin{align} C_E(X') - C_E(X'') \le e^{-rT}(X''-X')\\ P_E(X') - P_E(X'') \le e^{-rT}(X''-X') \end{align}\]
\(C_E(X)\) and \(P_E(X)\) are convex functions on \(X\).

Asset price \(S\)

Here we consider the asset as a portfolio and make conclusions:

Functions \(C_E(S)\) and \(P_E(S)\) are increasing and decreasing, respectively.
Suppose that \(S' \le S''\), then

\[ \begin{align} C_E(S'') - C_E(S') &\le S'' - S' \\ P_E(S') - P_E(S') & \le S'' - S' \end{align} \]

Thus \(C_E(S)\) and \(P_E(S)\) are Lipschitz functions with a Lipschitz constant \(1\).
The functions \(C_E(S)\) and \(P_E(S)\) are convex functions on \(S\).

American options

The relations of American options are more or less similar to that of European ones.

Strike price \(X\)

\(C_A(X)\) and \(P_A(X)\) are decreasing and increasing functions respectively.
Suppose \(X' < X''\). Then

\[\begin{align} C_A(X') - C_A(X'') \le X'' - X'\\ P_A(X'') - P_A(X') \le X'' - X' \end{align}\]

Thus the functions \(C_A(X)\) and \(P_A(X)\) are Lipschitz, with a Lipschitz constant \(1\).
The functions \(C_A(X)\) and \(P_A(X)\) are convex functions on \(X\).

Asset price \(S\)

\(C_A(S)\) and \(P_A(S)\) are increasing and decreasing functions respectively.
Suppose \(S' < S''\) then

\[ C_A(S'')- C_A(S') \le S'' - S' \\ P_A(S') - P_A(S) \le S'' - S'\]

The functions \(C_A(S)\) and \(P_A(S)\) are Lipschitz with a Lipschitz constant \(1\).
The functions \(C_A(S)\) and \(P_A(S)\) are convex.

Dependence on expiry time \(T\)

If \(T'< T''\), then

\[ \begin{align} C_A(T') &\le C_A(T''), \\ P_A(T') &\le P_A(T'') \end{align}\]

Time value of options

We use the following terminology. We say that at time \(t\) a call option with strike price \(X\) is

in the money if \(S(t)> X\)
at the money if \(S(t)=X\)
out of the money if \(S(t)<X\).

Similarly, we say that a put is:

in the money if \(S(t)<X\)
at the money if \(S(t)= X\),
out of the money if \(S(t)>X\).

The terms deep in the money and deep out of the money is used to say that the difference in the level is very high.

Intrinsic value: At time \(t\le T\), the intrinsic value of a call option with strike price \(X\) is equal to \((S(t)-X)^{+}\). The intrinsic value of a put option with the same strike price is \((X-S(t))^{+}\).

Time value: The time value of an option is the difference between the price of the option and its intrinsic value.

Proposition: For any European or American call or put option with strike price \(X\), the time value attains its maximum at \(S=X\).