RestrictionsonOptionsPrices(衍生金融工具-人民.ppt
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1、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 n
2、eed 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 contrac
3、ts.,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:,Bas
4、ic 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 wo
5、rth 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 valua
6、ble 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 exer
7、cise 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
8、?),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
9、 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
10、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 l
11、ie 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-pa
12、ying 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 sa
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