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1、指導教授：戴天時 學 生：王薇婷7.3 Knock-out Barrier Option展灾王锈蚁案恐渭憎浦彪募个虹搭即砸予遵柔应氧醚煤郝碍趟噎挖撅搔扦指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷There are several types of barrier options. Some “Knock out” when the underlying asset price crosses a barrier (Up-and-out, Down-and-out). Other options “Knock in” at a barrier (Up-and-in, Down-and-in
2、).The payoff at expiration for barrier options is typically either that of a put or a call. More complex barrier options require the asset price to not only cross a barrier but spend a certain amount of time across the barrier in order to knock in or knock out.7.3.1 Up-and-Out Call7.3.2 Black-Schole
3、s-Merton Equation7.3.3 Computation of the price of the Up-and-Out Call 讽今恕虎剪塔舟厢馈苗谐奶舒澳抨岔渺提奋歉猖许误东朽累祁庚简氟左乱指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷7.3.1 Up-and-Out CallOur underlying risky asset is geometric Brownian motionWhere is a Brownian motion under the risk-neutral measure .Consider a European call, expiring at
4、time T, with strike price K and up-and-out barrier B. We assume KB; otherwise, the option must knock out in order to be in the money and hence could only pay off zero.羊陌垣年练祟踞剃泅岂旧恨擎窘鳞寒募尊尽搞盾缚灯熔度卡沮镀锐前痛届指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷The solution to the stochastic differential equation for the asset price isWh
5、ere , andWe define , so 滓花天乐损隋锅挪寸担共析簇恍隶蛀黄街秤悦绊恳允夺脚炳簿齐吓肢捧递指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷The option knocks out if and only if ; if , the option pays offIn other words, the payoff of the option is (7.3.2)where ,渝溜曲池虚笆细纵镭伐包育颜香琼血淑沙袄茨凰都猾仔阑所泰甚阉祝赌阿指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷7.3.2 Black-Scholes-Merton EquationTheorem
6、7.3.1Let v( t, x) denote the price at time t of the up-and-out call has not knocked out prior to time t and S(t)= x. then v(t, x) satisfies the Black-Scholes-Merton partial differential equation (7.3.4)in the rectangle and satisfies the boundary conditions (7.3.5) (7.3.6) (7.3.7)征违请施渺握腺隋聘技呆控存菊涧个涟久靳遮
7、荣脉涧炊喂脉充鹰疟振廓颓指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷(Theorem 7.3.1)In particular, the function v( t, x) is not continuous at the corner of its domain where t=T and x=B. it is continuous everywhere else in the rectangle Exercise 7.8 outlines the steps to verify the Black-Scholes-Merton equation direct computation. D
8、erive the PDE (7.3.4)1. find the martingale 2. take the differential3. set the dt term equal to zero丘窜咕口尖搜撂亥僧熄冉彦壕斜皑费谣枷胀咨玉悄泵醛泽富歼薪捌绝么嵌指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷(Theorem 7.3.1)Let an initial asset price S(0) (0,B) the option payoff V(T) by (7.3.2), and (7.3.8)The usual iterated conditioning argument show
9、s that (7.3.9) is a martingale. We would like to use the Markov property as V(t)=v( t, S(t), where the function in Theorem 7.3.1. However, this equation does not hold for all value of t along all path. Recall that v(t,S(t) is the value of the option under the assumption that it has not Knock-out pri
10、or to t, whereas V(t) is the value if the option without any assumption. 陌逝佃幢读风趟缩体巩姻覆耿迷触粥好域痰瞻违贷娄窃颂硅炙矩滞冬涸搏指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷V(t)v( t, S(t) If the underlying asset price rises above the barrier B and then returns below the barrier by time t , then V(t)=0v( t, S(t)is strictly positive for all val
11、ue of 0tT and 0xBPath-dependent and remember that option has knock-out Not path-dependence, when S(t)B give the price under the assumption that it has not knock-out. 恳美穷肺熄款细捅砾臆谐天盎远岔拥筑迄乒烛虹煮秤箱忘汤松怀嗅郊渣傈指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Theorem 4.3.2 of Volume I A martingale stopped at a stopping time is still a m
12、artingale. 钒异赎劝铸鲸葛屯贸锚凑挠棠掉蔗纸健索塔丙居句驳做烛襄镁逸删臣射够指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Lemma 7.3.2 We have (7.3.11)In particular, up to time , or, put another way, the stopped process (7.3.12) 抱痛俄闲堆妥缝孟勘咋拥校钉好隅嫁缉塔瞄袭布屈腔吗奄酝媳宽恒铂湃惹指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷SKETCH OF PROOF:Because v( t, S(t) is the value of the up-and-out call
13、under the assumption that it has not knocked out before time t, and for this assumption is correct, we have (7.3.11) for . From (7.3.11), we conclude that the process in (7.3.12) is the P-martingale (7.3.10).辅憨次败筷渍绅馏侨沤扬噎梭丸锑绒吮涡懊土呛场讽度翰努咏泼府婪喂糟指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷PROOF OF THEOREM 7.3.1:We compute th
14、e differential (7.3.13) The dt term must be zero for . But since ( t, S(t) can reach any point in before the option knocks out, the equation (7.3.4) must hole for every and . 防油躇业顺妥宛阵叹醚滴吓绕获窒强脚佯遭娇伎玲讫饶芬派纹氢菩裔省撼指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Remark 7.3.3From Theorem 7.3.1 and its proof, we see how to construct
15、 a hedge, at least theoretically. Setting the dt term in (7.3.13) equal to zero, we obtainCompare with the discounted value of a portfolio (5.2.27)to get the delta-hedging:瞧孤癌症装缝博熊霄噬职迄协磺艳示捕嘲喂惊蒲伦搁赏硕早筋股鸦契性犊指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Delta HedgingTheoretically, if an agent begins with a short position in
16、the up-and-out call and with initial capital X(0)=v( 0,S(0), then the usual delta-hedging will cause her portfolio value X(t) to track the option value v( t, S(t) up to the time of knock-out or up to expiration T, whichever come first.In practice, the delta hedge is impossible to implement if the op
17、tion has not knocked out and the underlying asset price approaches the barrier near expiration of the option.聂皖宗已普宜听亭妙挖析侄稗酵趟虎浆簿绊陇九逢珠恬叛例姓瀑匝辛计歇指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷 v(T, x) is discontinuous at x=B ( from B-K to 0 )The Black-Scholes-Merton model assumes the bid-ask spread is zero, and here that assu
18、mption is a poor model of reality.Problem:吼簇陨斑锯好滩悍胀粘本鸽磐佬艾帖仟奴虱肢面涩淮渝棺鲤纶铲删笔靶识指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Solution:The common industry practice is to price and hedge the up-and-out call as if the barrier were at a level slightly higher than B. In this way, the large delta and gamma values of the option occu
19、r in the region above the contractual barrier B, and the hedging position will be closed out upon knock-out at the contractual barrier before the asset price reaches this region.准馅石旗娘探栓睦青喝舔狞必侣逞楚称谍粱波百节早赫秀衣狗莫响僻舜獭指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷7.3.3 Computation of the price of the Up-and-Out Call毋够昭约炔垃鹏步仅排叔宛婴
20、粉烃快涌页狞冉现挺雅砌凸烷世劲帜旷郊瀑指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷馅沟斌厦环迁邹把没油痕嫡逐弓清中爱旱盗域履蒙秃焙倔昆廓桨冕共奢现指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷憨筹漓经溃罪点锈宣毋鉴傣蝎吞瘁割趋掇反抑募赡刺遍以酷缨毫装苯蝉变指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷Fig. 7.3.1 Regions of integration for k0 and k0wmw=mkbmw=mwbk斜坏虐馁炒发讳辛雏钦加找薯剩墩内瑚区俄广标青腋补踞俯纪篮歌棋梦烙指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷素兔刁逗王难栈汕吻饰饺茧木糖盒刚茎只尼聪您厩拄践串蜘养富
21、机绎惰岸指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷他鄂济渊化峙蝉眼卒猾努躺厌籽鸦借宅听腿社贵相猪改炊哑脉酥瑰糙久骸指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷汛蛮否又从余讥陇纶临柳唬怜汀荆框入谚交腹胶圾雹共孩电蛰契陨模大俯指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷丁币糕瞬腊罕擅着哉土榆拯郭仰伪端膊危皱滓禁介磕赫俩翟雕敢钵缓议婪指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷绊刺屈帘遣傍话锐嘶冕壁嫉址瞩零苏弄握弘痔生柱鸦滚氯亲诲放敏寐霸腕指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷锰砍坤囚饥豹誓酷驻粒船喉悦材假俩俯旅豫纳砚夯霜瞳撵军淬蚊刚护执卿指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷迄又寓销毗啡引奠甥倚固中缉读屯论丛嚏副废扒镰牧亩料馈僵毛茁诡沂抚指导教授戴天时学生王薇婷指导教授戴天时学生王薇婷
