Which Way Do Option Prices Move? Six Inputs, Four Options

Worked Solution

How to Think About It: Change one input at a time and ask two questions: does the change make the payoff at expiry larger, and does it change what the payoff is worth today (discounting, or the risk-neutral drift of the stock)? For American options add a third question: does the change give the holder more choices? A longer-dated American option always contains the shorter-dated one, so extra time can never hurt. A European holder has no such protection, and that is where the only ambiguous sign lives.

Quick Estimate: Under Black-Scholes-Merton with continuous dividend yield $y$, $$c = S e^{-yT} N(d_1) - K e^{-rT} N(d_2), \qquad p = K e^{-rT} N(-d_2) - S e^{-yT} N(-d_1),$$ and the signs of the partial derivatives are: $\partial c/\partial S = e^{-yT}N(d_1) > 0$, $\partial p/\partial S = -e^{-yT}N(-d_1) < 0$; $\partial c/\partial K < 0 < \partial p/\partial K$; $\partial c/\partial \sigma = \partial p/\partial \sigma = S e^{-yT}\sqrt{T}\,N'(d_1) > 0$; $\partial c/\partial r = TKe^{-rT}N(d_2) > 0$, $\partial p/\partial r = -TKe^{-rT}N(-d_2) < 0$; $\partial c/\partial y < 0 < \partial p/\partial y$. Only $\partial/\partial T$ has no fixed sign for European options.

Formal Solution:

*Step 1 -- Stock price $S$ increases.* The call payoff $(S_T - K)^+$ is nondecreasing in $S_T$ and the put payoff $(K - S_T)^+$ is nonincreasing, and a higher $S$ today shifts the whole distribution of $S_T$ upward. So calls (European and American) rise and puts fall.

*Step 2 -- Strike $K$ increases.* A higher strike lowers every call payoff and raises every put payoff, and it also raises the present value of what a put holder receives. Calls fall, puts rise, for both exercise styles.

*Step 3 -- Time to maturity $T$ increases.* American: a $T_2$-option ($T_2 > T_1$) can be exercised at any time up to $T_1$ as well, so it is worth at least as much. Both American calls and puts rise (weakly). European: extra time raises the dispersion of $S_T$ (good for both) but also changes the present value of the strike and the dividends lost. For a European call on a non-dividend stock, early exercise is never optimal, so the European and American calls coincide and the value rises with $T$. For a European put, longer maturity lowers $Ke^{-rT}$, which is what the holder ultimately receives; a deep in-the-money put is worth about $Ke^{-rT} - S$ and *falls* with $T$. Numerically, with $S = 50$, $K = 100$, $r = 10\%$, $\sigma = 20\%$, the European put is worth $47.5$ at $T = 0.25$, $40.5$ at $T = 1$, and $16.0$ at $T = 5$. For a European call on a stock paying large dividends before expiry, a longer maturity means more dividends are paid to shareholders rather than the call holder: with $S = 150$, $K = 100$, $r = 1\%$, $y = 8\%$, $\sigma = 20\%$ the call is worth $47.3$ at $T = 0.25$ but $39.9$ at $T = 1$. So the sign is ambiguous for European puts and for European calls on dividend-paying stocks.

*Step 4 -- Volatility $\sigma$ increases.* Both payoffs are convex in $S_T$ with a floor at zero: the holder keeps the upside of larger moves and is protected on the downside. More dispersion therefore raises both calls and puts, American or European. (The book's Table 6.1 lists this as an increase for every option type.)

*Step 5 -- Risk-free rate $r$ increases.* Two effects point the same way. The present value of the strike $Ke^{-rT}$ falls, which helps a call (pays $K$ later, cheaper) and hurts a put (receives $K$ later, cheaper). Also the risk-neutral drift of the stock rises, pushing $S_T$ up. Calls rise, puts fall.

*Step 6 -- Dividends increase.* On the ex-dividend date the stock drops by roughly the dividend, and the option holder does not receive it. Lower expected $S_T$ means calls fall and puts rise. (For an American call, large dividends can also make early exercise just before the ex-date optimal, but the price still falls as dividends increase.)

Answer:

(a) $S$ up: calls up, puts down. (b) $K$ up: calls down, puts up. (c) $T$ up: American calls and puts up; European call on a non-dividend stock up; European puts and European calls on dividend-paying stocks are ambiguous (deep in-the-money European puts fall with $T$). (d) $\sigma$ up: all four up. (e) $r$ up: calls up, puts down. (f) Dividends up: calls down, puts up.