Properties of Stock Options

Transcription

1 Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 Factors a ecting option prices 2 Upper and lower bounds for option prices 3 Put-call parity 4 E ect of dividends

2 Assumptions There are no transaction costs. Borrowing and lending are possible at the risk-free interest rate. Market participants are prepared to take advantage of arbitrage opportunities as they rises. Notation S 0 :Thecurrentstockprice S T :Thestockpriceatmaturity K: Thestrikeprice T : The time to expiration : The volatility of the stock price r: Therisk-freeinterestrate D: Thedividendsthatareexpectedtobepaid c: European call option price p: European put option price C: American call option price P : American put option price

3 E ects of factors on option prices E ects on the option of a stock price of increasing one variable while keeping all others fixed (Hull, 2014; Table 11.1) Upper and lower bounds for option prices The option can never be worth more than the stock, that is, c apple S 0 and C apple S 0. Otherwise, an arbitrageur can make a riskless profit by buying the stock and selling the call option. An American put option can never be whorth more than its strike price K, thatis,p apple K. For European options, the option cannot be worth more than K at maturity, hence pe rt apple K.

4 Lower bound for calls on non-dividend-paying stocks c max(s 0 Ke rt, 0). (1) Proof. Consider the following two portfolios: Portfolio A: one European call option plus a zero-coupon bond that provides a payo of K at time T Portfolio B: one share of the stock. At time T,ifS T >K, the call option is exercised and portfolio A is worth S T ;ifs T <K, the call option expires worthless and the portfolio is worth K. Hence,attimeT, portfolio A is worth max(s T,K), and portfolio B is worth S T. Hence, portfolio A is always worth as much as or more than portfolio B at time T. It follows that in the absence of arbitrage opportunties this must also be true today, that is, c + Ke rt S 0, or c S 0 Ke rt. Note that the option value cannot be negative, i.e., c proved. 0. Hence (1) is Lower bound for European puts on non-dividendpaying stocks Proof. Consider the following two portfolios: p max(ke rt S 0, 0). (2) Portfolio C: one European put option plus one share of the stock Portfolio D: a zero-coupon bond that provies a payo of K at time T At time T,ifS T <K, the option in portfolio C is exercised and the portfolio becomes worth K; ifs T >K, the put option expires worthless and the portfolio is worth S T. Hence, portfolio C is worth max(s T,K) in time T, which is always worth as much as or more than portfolio D in time T. Then it follows that in the absence of arbitrage opportunties portfolio C must be worth as least as much as portfolio D today, that is, p + S 0 Ke rt or p Ke rt S 0.

5 Put-call parity c + Ke rt = p + S 0. (3) Proof. We consider the two portfolios used before Portfolio A: one European call option plus a zero-coupon bond that provides a payo of K at time T Portfolio C: one European put option plus one share of the stock. As we argued before, portfolio A is worth max(s T,K) at time T,and portfolio C is also worth max(s T,K) at time T. These two portfolios have identical values at time T,theymusthaveidenticalvaluestoday. Hence (3) is proved. E ects of dividends We now examine the impact of dividends on option prices. Assume that the dividends that will be paid during the life of the option are known. In the calculation of D, a dividend is assumed to occur at the time of its ex-dividend date. We can redefine portfolios A-D as follows. Portfolio A: one European call option plus an amount of cash equal to D + Ke rt Portfolio B: one share of the stock. Portfolio C: one European put option plus one share Portfolio D: an amount of cash equal to D + Ke rt. Using similar arguments, we can show that c max(s 0 D Ke rt, 0) (4) p max(d + Ke rt S 0, 0) (5) (put-call parity) c + D + Ke rt = p + S 0 (6)