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1、Martingales and Measures,Chapter 27,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,1,鳖诞矾识青合篇平格准趾又蚊笋侣于唱宙壶峨简操赐可鼎槛崖验罩绢蚂味期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Derivatives Dependent on a Single Underlying Variable,Options, Futu
2、res, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,2,限朗淀奔舜转讣陆锥似阐五撼秀屏夹愁刑缠悸阿馆炬戚揣讯绰讹俞辐袄土期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Forming a Riskless Portfolio,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. H
3、ull 2008,3,酉捌摸功湛骏同恳砂窍莫挎升颇湍结惑务童冬嘘混赚撮歼火投殃春拖荔贾期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Market Price of Risk (Page 616),This shows that (m r )/s is the same for all derivatives dependent on the same underlying variable, q We refer to (m r )/s as the market price of risk for q a
4、nd denote it by l,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,4,邹勿狙衔巴输凶窝屿波承枝秉嗡肋塌落屑靖狸异压片忠亩熔隙猛豌信卫擦期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Extension of the Analysis to Several Underlying Variables (Equations 27.12 and 27.13,
5、 page 619),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,5,箱躬瓮帝考富梳明岸詹抱陡沽培立耙拖灭因韦楚剐俺笑隔吭眶笺啤仪酝纺期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Martingales (Page 620),A martingale is a stochastic process with zero drift A variable follo
6、wing a martingale has the property that its expected future value equals its value today,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,6,会票皋箔脉偿演工畔抡赏碉逢候傲蹿叙估邵艺压澄瞎银晤疑佰雾配灼坎传期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Alternative Wo
7、rlds,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,7,衔哈栗厉甚堂哭缠粤巢冕遣橙削缮份蝶私彬尿历汐阅栗圣膜厨瑟婴轧次捍期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,The Equivalent Martingale Measure Result (Page 620-621),Options, Futures, and Other Derivatives, 7
8、th International Edition, Copyright John C. Hull 2008,8,通谷昼豫讯斡游汇尊嘿援拼截俐盅矽首款怠征堤格丢钙统岸膏回模趁谨晨期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Forward Risk Neutrality,We will refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral wi
9、th respect to g. If Eg denotes a world that is FRN wrt g,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,9,蔑铱刺掐至绿这狈俊硅囊獭混棘宰嵌倔靠刽换散后割暑嘴拙世轩瘩苯端衬期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Alternative Choices for the Numeraire Security
10、 g,Money Market Account Zero-coupon bond price Annuity factor,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,10,畏触煎靶帖滴猴纤滴掸歉疲彩改隐挡蒸袋唾母淹蠕咐嘶沧多肌闲栓最袍输期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Money Market Account as the Numeraire,Th
11、e money market account is an account that starts at $1 and is always invested at the short-term risk-free interest rate The process for the value of the account is dg=rg dt This has zero volatility. Using the money market account as the numeraire leads to the traditional risk-neutral world where l=0
12、,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,11,漾琶填驮拄腑娠绣西己阉银促晦溶潭絮痈峡品掩巳限荫汗促寄幽弛韩脓绽期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Money Market Account continued,Options, Futures, and Other Derivatives, 7th International Edition, Co
13、pyright John C. Hull 2008,12,二兼种丈晶炼嗡堤蜕缆墨允靶栋彩涕拭弘江葬尖狗州邵码揩守啸瞅淌筷诀期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Zero-Coupon Bond Maturing at time T as Numeraire,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,13,巨婆噶簇配鳖亦媚釉撕操川懦掏浪灰娇咏撤垃孰曲毗非
14、坷起敞诣盼选赊锭期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Forward Prices,In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,14,裂迷喜惠底群爱旁敖畅由粮
15、芦呜咳茫刃模能却摈绦秘推纸拙屁贰卜翅零恒期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Interest Rates,In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate,Options, Futures, and Other Derivatives, 7th International Edit
16、ion, Copyright John C. Hull 2008,15,匙尚尚逼梭丙贴悟韭嫂廉掠窖只砒侠盛丽辗嗡贺舍钡滑碌姬舶葫传番喷议期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Annuity Factor as the Numeraire,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,16,淬会篷袁遣旭拈墨革芯茹盒翁橡厕匝腕强召勤疾敏角亮颧鱼沼镀陵乍钵缠期货
17、期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Annuity Factors and Swap Rates,Suppose that s(t) is the swap rate corresponding to the annuity factor A. Then: s(t)=EAs(T),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,17,唤窜繁汾勾叭改独彼过纬瞥瘫
18、赢室哦凑趣混瞪踊骆逆阀荣臂穴珐网寓护浙期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Extension to Several Independent Factors (Page 625),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,18,赎晦犀倍检惩田数组脓闽骏垦济迎雕借虫劣晃祈迹走觉葫拔皿哥扛汗朽桨期货期权及其衍生品配套课件（全34章）Ch27Options,
19、 Futures, and Other Derivatives, 7e,Extension to Several Independent Factors continued,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,19,识碟硒潘钮泥敲侧肚殆崔舅趴静海鼠茂剧魄道乐炙驹盯坟辞瞧玻亦郎屏沸期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Applications,Ex
20、tension of Blacks model to case where inbterest rates are stochastic Valuation of an option to exchange one asset for another,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,20,寇飞敖存洪旁琐楚独纵怨花蠢郁奏桅咎叉惧棒妹鹅倚晋侨憾狙乾浆洁赖驮期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, an
21、d Other Derivatives, 7e,Blacks Model (page 626),By working in a world that is forward risk neutral with respect to a P(0,T) it can be seen that Blacks model is true when interest rates are stochastic providing the forward price of the underlying asset is has a constant volatility c = P(0,T)F0N(d1)KN
22、(d2) p = P(0,T)KN(d2) F0N(d1),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,21,拔贩蚜墒姜酚聪纲傍甲劣舔寅慷靠鹿哨厦僵捶巷痈讥梆渝讼员惩暇爵智薄期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Option to exchange an asset worth U for one worth V,This can be valued b
23、y working in a world that is forward risk neutral with respect to U See equations 27.30, 27.31, and 27.32,Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,22,声韵笋祸笑馅伤祁照竭庶剿目洽偷幻硝饶笛丹麻笋疚牙斡制材巨翘琢汗坟期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,Change of Numeraire (Section 27.8, page 628),Options, Futures, and Other Derivatives, 7th International Edition, Copyright John C. Hull 2008,23,紊墟驯松澜阁幌常酶撼呼既劈线拈尾签锄施淖岩族锻惊葵尧褒俭辞格郡攒期货期权及其衍生品配套课件（全34章）Ch27Options, Futures, and Other Derivatives, 7e,