1、Chapter 7 Options: General Propertiesmain content7.1 Definitions7.2 Put-Call Parity7.3 Bounds on Option Prices7.3.1 European Options7.3.2 European and American Calls on Non-Dividend Paying Stock7.3.3 American Options7.4 Variables Determining Option Prices7.4.1 European Options7.4.2 American Options7

2、.5 Time Value of Options7.1 DefinitionsEuropean call option is a contract giving the holder the right to buy an asset,called the underlying, for a price X fixed in advance, known as the exercise price or strike price, at a specified future time T, called the exercise or expiry time.European put opti

3、on gives the right to sell the underlying asset for the strike price X at the exercise time T.American call or put option gives the right to buy or, respectively, to sell the underlying asset for the strike price X at any time between now and a specified future time T, called the expiry time. In oth

4、er words, an American option can be exercised at any time up to and including expiry.An option is determined by its payoff, which for a European call is S(T) X if S(T) X, 0 otherwise.This payoff is a random variable, contingent on the price S(T) of the underlying on the exercise date T. (This explai

5、ns why options are often referred to as contingent claims.) It is convenient to use the notation The gain of an option buyer (writer) is the payoff modified by the premium CE or PE paid (received) for the option. At time T ，European call the gain is European put the gain is ExerciseFind the stock pr

6、ice on the exercise date for a European put option with strike price $36 and exercise date in three months to produce a profit of $3 if the option is bought for $4.50, financed by a loan at 12% compounded continuously.7.2 Put-Call ParityTheorem (Put-Call Parity)For a stock that pays no dividends the

7、 following relation holds between the prices of European call and put options, both with exercise price X and exercise time T:ExerciseSuppose that a stock paying no dividends is trading at $15.60 a share. European calls on the stock with strike price $15 and exercise date in three months are trading

8、 at $2.83. The interest rate is r = 6.72%, compounded continuously. What is the price of a European put with the same strike price and exercise date?Option Parity with DividendTheorem (Put-Call Parity Estimates)The prices of American put and call options with the same strike price X and expiry time

9、T on a stock that pays no dividends satisfyModify Put-Call Parity Estimates Theoremfor a stock paying a dividend between time 0 and the expiry time T, where div0 is the value of the dividend discounted to time 0.for a stock paying dividends continuously at a rate rdiv.7.3 Bounds on Option PricesFor

10、European and American options with the same strike price X and expiry time T.The price of a call or put option has to be non-negative because an option of this kind offers the possibility of a future gain with no liability.7.3.1 European OptionsThe upper and lower bounds on the prices of European ca

11、ll and put options.European call option upper boundEuropean call option lower boundEuropean put option upper boundEuropean put option lower bound7.3.2 European and American Calls on Non-DividendPaying StockTheorem The prices of European and American call options on a stock that pays no dividends are

12、 equal, , whenever the strike price X and expiry time T are the same for both options.7.3.3 American OptionsThe prices of American call and put options on a stock paying no dividends satisfy the inequalitiesFor dividend-paying stock7.4 Variables Determining Option PricesThe option price depends on a

13、 number of variables. These can be variables characterising the option, such as the strike price X or expiry time T, variables describing the underlying asset, for example, the current price S(0) or dividend rate rdiv, variables connected with the market as a whole such as the risk-free rate r, and

14、of course the running time t.7.4.1 European OptionsDependence on the Strike PriceDependence on the Underlying Asset Price7.4.2 American OptionsDependence on the Strike PriceDependence on the Underlying Asset PriceDependence on the Expiry Time7.5 Time Value of OptionsDefinitionAt time t T the intrins

15、ic value of a call option with strike price X is equal to (S(t) X)+. The intrinsic value of a put option with the same strike price is (X S(t)+.The intrinsic value is zero for options out of the money or at the money. Options in the money have positive intrinsic value. The price of an option at expiry T coincides with the intrinsic value. The price of an American option prior to expiry may be greater than the intrinsic value because of the possibility of future gains. The price of a European option prior to the exercise time may be greater or smaller than the intrinsic