Chapter 11: Properties of Stock Options

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Factors Affecting Option Prices

Six factors influence option prices ( $↑\uparrow$ = increase, $↓\downarrow$ = decrease):

Factor	Call price	Put price	Reason
Stock price $S0↑S_0 \uparrow$	$↑\uparrow$	$↓\downarrow$	Higher $S_0$ makes call exercise more likely
Strike price $\uparrow$	$↓\downarrow$	$↑\uparrow$	Higher $K$ reduces call payoff
Time to expiration $\uparrow$	$↑\uparrow$ (usually)	$↑\uparrow$ (usually)	More time = more chance to move in-the-money
Volatility $σ↑\sigma \uparrow$	$↑\uparrow$	$↑\uparrow$	More uncertainty = more chance of large payoff
Risk-free rate $\uparrow$	$↑\uparrow$	$↓\downarrow$	Higher $r$ reduces PV of strike price
Dividends $↑\uparrow$	$↓\downarrow$	$↑\uparrow$	Dividends reduce stock price on ex-dividend date

Assumptions and Notation

No transaction costs, same borrowing/lending rates
No restrictions on short selling
Continuous trading, stock price follows a lognormal process
Notation: $S_0$ = current stock price, $K$ = strike, $T$ = time to expiration, $r$ = risk-free rate, $σ\sigma$ = volatility

Upper and Lower Bounds

Call Option Bounds

Upper bound: $\leq S_0$ and $\leq S_0$ (a call can't be worth more than the stock)

Lower bound for European call (no dividends):

$\geq S_0 - K e^{-rT}$

Since a call can never have negative value: $\geq \max(S_0 - K e^{-rT}, 0)$

Put Option Bounds

Upper bound: $\leq K$ and $\leq K$ (a put can't be worth more than the strike)

Lower bound for European put (no dividends):

$\geq K e^{-rT} - S_0$

Since a put can never have negative value: $\geq \max(K e^{-rT} - S_0, 0)$

Put–Call Parity

One of the most important relationships in options theory. For European options on a non-dividend-paying stock:

$c + K e^{-rT} = p + S_0$

Intuition: Portfolio A (one call + cash of $K e^{-rT}$ ) must equal Portfolio B (one put + one share) in value at expiration:

	Portfolio A at $T$	Portfolio B at $T$
If $S_T > K$	Call exercised: $S_T - K + K = S_T$	Put expires worthless: $S_T + 0 = S_T$
If $ST≤KS_T \leq K$	Call worthless: $0 + K = K$	Put exercised: $K - S_T + S_T = K$

For American options (no dividends):

$S0−K≤C−P≤S0−Ke−rTS_0 - K \leq C - P \leq S_0 - K e^{-rT}$

Put–Call Parity with Dividends

For European options on a dividend-paying stock:

$c + D + K e^{-rT} = p + S_0$

where $D$ is the present value of dividends during the option's life.

Calls on a Non-Dividend-Paying Stock

Early exercise is never optimal for an American call on a non-dividend-paying stock. A European call and an American call have the same value: $C = c$ .

Two reasons:

Insurance: Holding the call protects against stock price declining below the strike until maturity
Time value of money: Paying the strike later is better than paying it now

Puts on a Non-Dividend-Paying Stock

Early exercise can be optimal for American puts. When the stock price becomes very low, it may be optimal to exercise and earn interest on the strike price. Therefore:

$\geq p$

And: $\geq \max(K - S_0, 0)$ for an American put (can always exercise immediately).

Effect of Dividends

Dividends make early exercise of American calls potentially optimal right before the ex-dividend date. The call holder gives up the dividend by not exercising. If the dividend is large enough, it's worth exercising early to capture it.

Condition for early exercise of American call (single known dividend $D$ ):

$D>K(1−e−r(T−td))\text{Exercise just before ex-dividend date if } D > K \left(1 - e^{-r(T - t_d)}\right)$

where $t_d$ is the time of the dividend.