RestrictionsonOptionsPrices(衍生金融工具-人民.ppt

Lecture#6:,Basic No Arbitrage Restrictions on Options Prices,Some Option Contracts in Hong Kong,Introduction,Exact pricing formulas for options are more difficult to derive than formulas for forwards and futures.To arrive at a pricing formula for stock options,which we will do in a few lectures,we need to make assumptions on the dynamic behavior of the prices of the underlying stock.In what follows will derive some general restrictions on stock option price without assuming a dynamic model for stock price movement.The main purpose of doing that is to improve our understanding of option contracts.,Outline:A.NotationB.Basic intuitionC.Basic arbitrage relationsD.Arbitrage bonds on prices and Put-Call parityE.Effects on dividends on arbitrage restrictionsF.Conclusions,Notation,Basic Intuition,Effect on the price of a stock option of increasing one variable while keeping all others fixed:,Basic arbitrage relations:,Note:The following restrictions hold regardless of whether the underlying stock pays dividends or not.A.A call is never worth more than the stock and a put is never worth more than exercise price C(S,K,t,T)S(t)c(S,K,t,T)S(t)P(S,K,t,T)K p(S,K,t,T)K,B.European puts are never worth more than the present value of the exercise price.p(S,K,t,T)KB(t,T)K.Intuitively,this has to hold since th time-T payoff to European put holder is bounded(from above)by K.C.Options never has a negative value:C(S,K,t,T)0 c(S,K,t,T)0 P(S,K,t,T)0 p(S,K,t,T)0,D.American options are at least as valuable as European options:C(S,K,t,T)c(S,K,t,T)P(S,K,t,T)p(S,K,t,T)E.American options with more time to maturity are at least as valuable;i.e.,for T2 T1,C(S,K,t,T2)C(S,K,t,T1)P(S,K,t,T2)P(S,K,t,T1)Note:This does not always hold for European options.(Why?),F.An American option is worth at least its exercise value(what you would get if you exercise today).C(S,K,t,T)max0,S(t)-K P(S,K,t,T)max0,K-S(t)Example:Do we have an arbitrage opportunity if,for Intel stock with S(t)=$100,a call option with K=$90 and 6-month to maturity is trading at$9?Note:This rule does not always hold for European options.(Why?),More Arbitrage Bounds for Options on Non-Dividend-Paying Stocks:,Example:Same as on the previous page.Assume S(t)=$100,and the price of an Intel call with K=$90 and 6-month to maturity is$11.Assume that Intel will not pay any dividend within the next 6-month and assume that the risk free interest rate(a.c.c.)is 10%.Is there an arbitrage?,A.For a stock does not pay dividends:c(S,K,t,T)max0,S(t)-KB(t,T)C(S,K,t,T)max0,S(t)-KB(t,T)Proof:To prove this we only need to show(why?)c(S,K,t,T)S(t)-KB(t,T),We show this by contradiction.If c S-KB,we have an arbitrage.This implies that American calls on non-dividend-paying stocks will never be exercised earlier.(Intuition?),B.For European puts on non-dividend-paying stocks,a similar arbitrage argument shows that:Intuition?)p(S,K,t,T)max0,KB(t,T)-SC.Combining these rules implies that the value of a European call on a non-dividend-paying stock must lie in the region:max0,S(t)-KB(t,T)c(S,K,t,T)S(t).,D.Combining the rules for European puts,we see that the value of a European put on a non-dividend-paying stock must lie in the region:max0,KB(t,T)-S(t)p(S,K,t,T)KB(t,T),K-B(t,T),S(t),E.Is it possible to early exercise American Puts on non-dividend-paying stocks?Intuitions?Example:S(t)=$1,K=$25,T-t=6-month,r=9.5%(a.c.c),Put-Call Parity for Non-dividend-paying stocks,A.For European options:S(t)=c(S,K,t,T)-p(S,K,t,T)+KB(t,T)Intuition:a certain portfolio of bonds and options has the same payoff at maturity as a share of stock,so it must have the same price as a share of stock.,Example:K=50,S=50,r=0,T-t=1 month,c=4.5,p=4.0 Sc-p+KBWhat should you do if these were the true prices?,B.Static Replication with Put-Call ParityWe can make synthetic stock,call,put,and bond using the Put-Call Parity.For European options on a non-dividend-paying stock,we have:Synthetic stock:S=c-p+PV(K)Synthetic call:c=S+p-PV(K)Synthetic put:p=c-S+PV(K)Synthetic bond:PV(K)=S-c+p,Question:How is the Put-Call Parity related to the value of a forward contract on a stock(whose delivery price is equal to the strike price K)?C.Put-Call Parity for American OptionsP(S,K,t,T)+S(t)-KB C(S,K,t,T)C(S,K,t,T)P(S,K,t,T)+S(t)-K(Why?),Effects of Dividends on the Arbitrage Restrictions:,Note:We assume that the stock will pay a known dividend(or a known dividend yield in some cases)before the option maturity and extend our previous arbitrage restrictions.,A.Bounds for options on dividend paying stocks:C(S,K,t,T)c(S,K,t,T)S(t)-PV(D)-KBP(S,K,t,T)p(S,K,t,T)KB-S(t)+PV(D)Intuition:1.The value of a(European or American)call(or put)is higher than the value of a long(or short)position in a European forward with strike K and maturity T.That is,the value of a call(or put)is higher than the PV of the cash flow to the holder of the call(or put)who always exercise it at maturity.,2.The cash flow of a long position in a European forward with K and T is K-S(T)at time T and has a PV of KB-S(t)+PV(D).3.If either of these two rules is violated,one can construct an arbitrage by buying the option and shorting the right hand side of the inequality.,B.Early exercise decisions of American options American calls1.Given positive interest rates,it is never optimal to exercise an American call option between ex-dividend dates or prior to maturity.|-|-|Today(t)Ex-dividend Maturity Date()of Option(T),Consider two strategies:i.Exercise the option now,value is S(t)-Kii.Wait till just before the ex-dividend date and exercise for sure,even if the option is out-of-the-money.The value of the strategy at is Sc()-K,where Sc()is the cum-stock price just before the stock goes dividend.Hence the value of this strategy today is S(t)-KB(t,).What can you say?,2.The option will be exercised just prior to the ex-dividend date if,and only if,the exercise value exceeds the no exercise value,that is Se()+d-K C(Se(),K,T),where Se()is ex-dividend-day stock price.American Puts1.As it is pointed out earlier it may be optimal to early exercise American put option,even if the underlying stock pays no dividends.2.Dividends will tend to delay early exercise of an American put option.It never pays to early exercise an American put option just prior to an ex-dividend date.,Consider two strategies:i.Exercise the put option just before the ex-divided date.The value of he option is:K-(Se+d)ii.Exercise the put option just after the ex-dividend date.The value is K-Se.,C.Put-Call Parity for European calls on stock with known dividendS(t)=c(S,K,t,T)-p(S,K,t,T)+KB(t,T)+PV(D)D.Whats the Put-Call Parity for European calls on stocks with a continuous dividend yield q?E.Put-Call Parity for American calls on dividend paying stocks:S-PV(D)-KC-PS-KBTo prove this,we show that neither of the inequality can be violated,by considering two cases:,If the second inequality is violated;i.e.,C-PS-KB,then we can have the following arbitrage:Question:If the written call is exercised against you early what should be the value of your portfolio?If the first inequality is violated;i.e.,if C-P S-PV(D)-K,then we can have the following arbitrage:Question:We have only shown that the strategy is an arbitrage if your portfolio is held until maturity.What if the written put is exercised against you before maturity?,Summary,A.Arbitrage opportunities cannot exist in efficiently functioning financial markets.B.Based on the assumption of no-arbitrage,we can prove a number of rules about option prices without making any assumptions about the behavior of the underlying security over the time.,