12.1 Properties of Options

Properties of Options

FMP–13 is about model-free option properties: what must be true in a frictionless market regardless of Black–Scholes or any other pricing model. If a quoted price violates a bound or parity relation, an arbitrage portfolio restores the inequality. FRM candidates must know the six price factors, European and American bounds with and without dividends, put-call parity (and its forward form), and when early exercise can be rational.

Six Factors Affecting Option Prices

For a vanilla European or American option on a stock (or stock index), value depends on:

Intuition, not formulas: a call is a leveraged long-stock claim with a capped downside equal to the premium; anything that raises the forward of the stock relative to K (higher S0, lower K, higher r, lower dividends) helps calls and hurts puts. Volatility helps both because option payoffs are convex.

Worked directional table check

S0 = 100, K = 100, r = 5%, q = 0, σ = 20%, T = 1 year. A European call is worth about 10.45 under Black–Scholes (d1 = 0.35, d2 = 0.15); if σ jumps to 30%, the call and put both rise. If a discrete $5 dividend is announced just before expiration, the call falls and the put rises because the stock is expected to drop by about the dividend on the ex-date (all else equal).

Exam trap: treating “more time always raises European put value.” A deep ITM European put on a non-dividend stock can be worth less with longer T because the holder cannot exercise early to receive cash and earn interest, while the lower bound involves Ke^(-rT), which shrinks as T grows.

Intrinsic Value and Time Value

Intrinsic value of a call = max(S0 − K, 0); of a put = max(K − S0, 0). Time value = option price − intrinsic value (can be negative for some European puts that trade below intrinsic because early exercise is forbidden). American options satisfy price ≥ intrinsic value; otherwise exercise (or sell) dominates holding the mispriced contract.

Upper and Lower Bounds (No Dividends)

Assume no dividends, continuous compounding, European options unless noted.

Why Ke^(-rT) appears: a European call’s strike is paid only at T, so compare S0 with the present value of K. An American call can be exercised now, so its lower bound uses K, not Ke^(-rT). An American put’s upper bound is K (exercise for cash K), while a European put cannot be worth more than the PV of K.

Worked bound example (non-dividend)

S0 = 31, K = 30, r = 10%, T = 0.25 years. Ke^(-rT) = 30 × e^(-0.10 × 0.25) = 30 × e^(-0.025) ≈ 30 × 0.9753 = 29.26.

European call lower bound: max(31 − 29.26, 0) = 1.74. If the call quotes at 1.50, buy the call, short the stock, and lend 29.26 (or equivalently borrow less than S0): at expiration the package arbitrages the violation. American call lower bound is max(31 − 30, 0) = 1.00, which is weaker than 1.74 when rates are positive—so the European-style PV bound is the tighter no-arbitrage floor for the call when dividends are zero (and American call = European call on non-dividend stock; see early exercise below).

Bounds with Known Dividends or Yield

If the stock pays a known dollar dividend with present value D during the option’s life, replace S0 with S0 − D in European bounds:

• European call: max((S0 − D) − Ke^(-rT), 0) ≤ c ≤ S0 − D (more carefully, c ≤ S0, but the prepaid forward S0 − D is the natural underlying for Europeans).
• European put: max(Ke^(-rT) − (S0 − D), 0) ≤ p ≤ Ke^(-rT).

With continuous dividend yield q, the prepaid forward is S0 e^(-qT):

• c ≥ max(S0 e^(-qT) − Ke^(-rT), 0)
• p ≥ max(Ke^(-rT) − S0 e^(-qT), 0)

Dividends hurt calls and help puts; lower bounds shift accordingly.

Worked dividend bound

S0 = 40, K = 40, r = 9%, T = 0.5, expected dividend in two months with PV D = 0.50.

S0 − D = 39.50. Ke^(-rT) = 40 × e^(-0.09 × 0.5) = 40 × e^(-0.045) ≈ 38.24.

European call floor: max(39.50 − 38.24, 0) = 1.26. European put floor: max(38.24 − 39.50, 0) = 0.

Put-Call Parity: European Equality

For European options on a non-dividend-paying stock:

c + Ke^(-rT) = p + S0

Portfolio A: European call + cash Ke^(-rT). Portfolio B: European put + one share. Both worth max(ST, K) at expiration, so they must match today.

With continuous yield q:

c + Ke^(-rT) = p + S0 e^(-qT)

With discrete dividends PV = D:

c + Ke^(-rT) = p + (S0 − D)

Forward form

The forward price F0 = S0 e^((r−q)T) (or S0 e^(rT) if q = 0). Parity rearranges to:

c − p = e^(-rT) (F0 − K)

So the call–put difference is the discounted difference between forward and strike. If F0 = K (ATM-forward), then c = p for Europeans.

Worked parity example

S0 = 31, K = 30, r = 10%, T = 0.25, no dividends. Ke^(-rT) ≈ 29.26. Suppose c = 3.00. Then p = c + Ke^(-rT) − S0 = 3.00 + 29.26 − 31 = 1.26. If the put quotes at 1.00, buy put, buy stock, short call, short (lend opposite) the bond: lock the mispricing.

American Put-Call Relationships

American options can be exercised early, so equality fails. Standard inequalities (no dividends):

S0 − K ≤ C − P ≤ S0 − Ke^(-rT)

With dividends (PV = D):

S0 − D − K ≤ C − P ≤ S0 − Ke^(-rT)

(Exact textbook bounds vary slightly with notation; the exam point is that American parity is an inequality band, not a single price.)

Early Exercise: American Calls and Puts

American call on a non-dividend-paying stock should never be exercised early. Reason: C ≥ c ≥ S0 − Ke^(-rT) > S0 − K when r > 0 and T > 0. Live option value exceeds intrinsic value S0 − K, so selling dominates exercising. Interest on the deferred strike and remaining insurance value both argue against exercise.

American call on a dividend-paying stock may be exercised early immediately before an ex-dividend date if the dividend is large relative to remaining time value: you capture the dividend by holding the stock, and the option’s European-style value would drop on the ex-date.

American puts may rationally be exercised early even with no dividends. A deep ITM put is like a bond paying K: exercising early invests the strike proceeds at r. When interest on K exceeds the insurance value of waiting, early exercise can be optimal. That is why American puts are worth strictly more than European puts in general, and why European puts can trade below intrinsic value.

Synthesis for Exam Vignettes

When a vignette quotes c, p, S0, and K, first check parity and bounds before any model. When asked about early exercise, separate dividend policy from put versus call. When rates rise, remember calls benefit and puts suffer through the PV of strike—and American put early-exercise incentives strengthen as r rises.