Comparative statics of call and put prices

Explain in which direction the prices of European and American calls and puts move when each of the following increases: the stock price, the strike, the time to expiry, the volatility, the risk-free rate, and the dividends.

Calls rise with spot, volatility and rates and fall with strike and dividends; puts do the opposite, except both rise with volatility. Longer maturity always helps American options; it helps a European call on a non-dividend stock, but a European put (and a European call when dividends are paid) can lose value with more time. Rates help calls by lowering the PV of the strike and hurt puts for the same reason; dividends lower the expected stock price, hurting calls and helping puts.