1Chapter 8 Properties of Stock Option Prices28.1 factors affecting options pricesn c : European call option pricen p : European put option pricen S0 :Stock price todayn K : Strike pricen T : Life of option n : Volatility of stock pricen C :American Call option pricen P : American Put option pricen ST :Stock price at option maturityn D :Present value of dividends during options lifen r : Risk-free rate for maturity T with cont compNotation3Effect of Variables on Option Pricing cpCPVariableS0 K T +?+48.2 basic option price relationship under no arbitrage conditionI.An American option is worth at least as much as the corresponding European optionC cP p II.upper and lower bounds for option pricesUpper boundsCall options:Put options:58.2 basic option price relationship under no arbitrage conditionII.upper and lower bounds for option pricesLower bond for European calls on no-dividend- paying stocksSuppose:S0=$20,K=$18,r=10%per annum, T=1year6Continued.nFormal prove:nPortfolio A: one European call option plus an amount of cash equal to nPortfolio B: one sharenAt time T:nA is worth:nB is worth: STnUnder no arbitrage condition, today:7II.upper and lower bounds for option pricesLower bond for European puts on no-dividend- paying stocks8.2 basic option price relationship under no arbitrage condition8Continued.nFormal prove:nPortfolio C: one European put option plus one sharenPortfolio D: an amount of cash equal to nAt time T:nC is worth:nD is worth: KnUnder no arbitrage condition, today:9Continued.nSuppose p=$1. Is there an arbitrage opportunity?Suppose:S0=$37, X=$40, r=5%per annum, T=0.5year10III.Put-Call Parity for European optionsuPortfolio A: one European call option plus an amount of cash equal touPortfolio C: one European put option plus one shareuBoth are worth at expiration date.uEuropean options; under no arbitrage condition : portfolios A and C must have identical values today.8.2 basic option price relationship under no arbitrage condition11Suppose:S0=$31, K=$30, r=10%per annum. The price of a three-month European call option is $3, and the price of a three-month European put option is $2.25.nIs there an arbitrage opportunity?Example:128.3 Early Exercise: Calls on a non-dividend-paying stocknUsually there is some chance that an American option will be exercised earlynAn exception is an American call on a non-dividend paying stocknThis should never be exercised early13nWhat should you do if 1.You want to hold the stock for more than two month? nKeep the option and exercise it at the expiration date.nNo income is sacrificedn We delay paying the strike pricen Holding the call provides insurance against stock price falling below strike price American call option14nWhat should you do if 2.You do not feel that the stock is worth holding for the next few months?nThe trader is better off selling the option than exercising it.American call option158.3 Early Exercise: Calls on a non-dividend-paying stocknIt is never optimal to exercise an American call option on a non-dividend-paying stock before the expiration date.nThe trader wants to keep the underlying asset for the remaining life of the option.nThe trader thinks the underlying asset is overpriced and want to sell the underlying asset.nOptions serve as insurances; Time value of money168.3 Early Exercise: Calls on a non-dividend-paying stockn在到期日之前，不付红利股票的看涨期 权不应该执行。因此，同一种不付股票 红利的美式看涨期权的价值与相同股票 的欧式看涨期权的价值相同。n但此结论对美式看跌期权不适用。178.4 relationship between American put and call pricesnThe relationship for American option prices when the stock pays no dividends.18Questions:1.A trader sells a European put on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances dose the trader make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the traders profit with the stock price at the maturity of the option.page221:9.16