Lecture 3 Finance Pro ject Somesh Jha 1 Maxim um Lik eliho - PDF document

Lecture 3 Finance Pro ject Somesh Jha 1

Maxim um Lik eliho o d estimation (MLE) � Assume that w e ha v e mortgages at time N t in the mortgage p o ol. t � The n um b er of pre-pa ymen ts at time (denoted b y ) follo ws a P oisson c t distribution. 2

MLE (Con td) � F ormally , probabilit y ( c = ) that is P k c t t equal to is: k � � ( X ;� ) N k ( � ( X ) N ) e t t ; � t t k � Chec k that the follo wi ng is true: 1 ( c = ) = � ( X ) N k P k ; � X t t t =0 k 3

F orm of � ( X ) ; � t � Recall that � ( X ) has the follo wi ng form: ; � t � ( t ) exp ( � � r f 7+ 0 1 � ln ( bur nout ) + � season ) � � 2 3 � Meaning of eac h co v ariate is giv en b elo w: 7 (Re�nancing opp ortunities) r f ( t ) (Age) � 0 season (seasonal) bur nout (burnout) 4

Problem � Supp ose w e a ha v e p o ol of mortgages that is to the p o ol underlying the MBS w e similar are trying to price. � W e ha v e historical data ab out this mortgage. 5

Problem (Con td) � W an t to �nd parameters that � ; � ; � b est 1 2 3 this historical data. �t � W e will use a tec hnique called Maxim um Lik eliho o d Estimation (or MLE) for this purp ose. 6

Basic idea of MLE � Assume a distribution (w e assume P oisson distribution for the n um b er of prepa ymen ts). � Estimate the probabilit y ( � ) of observing f the historical data. 7

Basic idea of MLE (Con td) � The (denoted b y lo g likeliho o d function L ( � )) is log ( � ). f � The parameters are giv en b y the solution � to the follo wi ng global-opti mization problem: max L ( � ) � 8

MLE (Con td) � Assume that w e ha v e historic pre-pa ymen t data for a p o ol of mortgages. � Also assume that the n um b er of prepa ymen ts at time only dep ends on the t n um b er of mortgages in the p o ol at time t and is indep enden t of the history . 9

MLE (Con td) � History is giv en for times 1 ; 2 ; � � � , and ; T the follo wi ng things are giv en: { (Num b er of pre-pa ymen ts at time t ). c t { (n um b er of mortgages remaining in N t the p o ol). � is the lifetime of the mortgage p o ol under T consideration. 10

MLE (Con td) � Probabilit y ( c ) that the n um b er of P t prepa ymen ts is at time is: c t t � � ( X ) N ;� c ( � ( X ) N ) e ; � t t t t t ! c t � The probabilit y of observing the en tire history (using indep endence here) is: T ( � ) = ( c ) f P Y t t =1 � Log lik eli ho o d function L ( � ) is: T ( c ln( �N ) � � ln( c !)) �N X t t t t t =1 � F or notational con v enience, In the expression for L ( � ) I ha v e suppressed X t and . � 11

MLE (con td) � The factor ln ( c !) is a constan t so w e ignore t it in the maximization problem. � W e ha v e to maximize the follo wing function with resp ect to (I ha v e suppressed the � X t and factors for notational con v enience): � T ( c ln ( �N ) � ) �N X t t t t =1 � Next w e discuss a metho d for maximization. 12

Steep est Ascen t � Let L ( � ) b e the log lik eli ho o d function. � Recall that is v ector of three parameters � ( � , , ). � � 1 2 3 � The of the log lik eli ho o d gr adient ve ctor L ( � ) @ function is (denoted b y ) is: @ � @ L 2 3 � 6 7 6 1 7 6 7 6 @ L 7 6 7 6 7 6 7 � 6 7 2 6 7 6 7 @ L 6 7 6 7 4 � 5 3 � In tuitiv ely , the gradien t v ector at � (denoted b y ( � )) is the direction in whic h g 13

the log lik eliho o d function increases most steeply (at the p oin t ). � 14

Steep est Ascen t (Con td) � Cho ose an initial v ector . � 0 � Let b e the old estimate. The new � i � 1 estimate is giv en b y the follo wing � i equations: � = ( � ) � � cg i � 1 i � 1 i k � � k = � k i � 1 i � is the step size. Notice that is k c determined b y the equations giv en ab o v e. Norm of the v ector � is denoted b y � � i � 1 i k � � k . � i � 1 i 15

Problems with Steep est Ascen t � Con v ergence v ery slo w near a lo cal maxim um. � V ariet y of metho ds for n umerical optimization. � Judge, George G., Willia m E. Gri�ths, R. Carter Hill, and Tsoung-Chao Lee. 1980. The The ory and Pr actic e of onometrics . New Y ork: Wiley . Ec � Quandt Ric hard E. 1983. \Computational Problems and Metho ds," in Zvi Grilic hes and Mic hael D. In triligator editors., onometrics , V ol 1. Handb o ok of Ec Amsterdam : North-Holland. 16

In teresting exercise � Assume that sto c k prices follo w the motion . ge ometric br ownian � Using MLE estimate the drift and v olatili t y of the sto c k. � Historical prices for man y sto c ks are a v ailable on n umerous w eb-sites. 17

In teresting exercise (Con td) � Using prices of v arious options on the sto c k �nd the curve . implie d volatility � Ho w far is the implie d volatility curve a w a y from the MLE estimate? � Let me kno w if y ou try this exercise. 18

Summary of MBS cash-�o ws � (mortgage pa ymen t at time t ) M P t � (in terest pa ymen t at time t ) I t � (principal pa ymen t at time t ) P t � (prepa ymen t at time t ) P P t � (service c harge at time t ) S t � (net in terest rate at time t ) N I t � (mortgage balance at time t ) M B t � (cash-�o w at time t ) C F t � (Single Mon thly Mortalit y Rate at S M M t time t ) 19

Summary (Con td) � is equal to M P t n � t +1 c (1 + c ) M B t � 1 n � t +1 (1 + c ) � 1 � , , , and follo w the equations I S P N I t t t t giv en b elo w: = I cM B t t � 1 = S sM B t t � 1 = � P M P I t t t = � N I I S t t t � is giv en b y the follo wi ng expression: M B t � � M B P P P t � 1 t t 20

Summary (Con td) � is giv en b y the follo wi ng form ula: C F t + + N I P P P t t t � is obtained from the pre-pa ymen t S M M t mo del. � Prepa ymen t at time is giv en b y the P P t t follo wi ng equation: ( M � ) S M M B S t t � 1 t 21

P ass-throughs � Supp ose a pass-through o wns p ercen t of x the mortgage p o ol. � Cash �o w of the pass-through at time is t giv en b y the follo wi ng equation: C F x t 100 22

CMOs � Supp ose there are tranc hes � � � m T ; ; T 1 m with par-v alues � � � . P ; ; P 1 m � A t time let the remaining par-v alue of t t tranc h b e . T P i i � Let j b e the least n um b er suc h that T is j not retired. � The cash-�o w of that tranc h is: t � 1 P j + + I P P P t t t M B t � 1 23

CMOs(Con td) t � The new par-v alue of tranc h is: P T j j t � 1 � � P P P P t t j t � 1 � If is equal to zero, the tranc h P r etir e j . T j � F or all tranc hes suc h that the T i > j i cash-�o w is t � 1 P i I t M B t � 1 t � 1 t � is equal to (Wh y?) P P i i 24

Stripp ed MBSs � The class gets + min us the P O P P P t t servicing fee. � The class gets min us the servicing fee. I O I t 25

High-Lev el Design Do cumen t � Query Phase Describ es the steps in whic h the user in teracts with the system. User c ho oses what instrumen t he/she w an ts to price and the v arious parameters. � Computation Phase High-lev el pro cedure to price these instrumen ts. Pro vide a description of the general tec hnique y ou are using (induction on lattices, sim ulation, �nite-di�erence sc hemes). � Pr esentation Phase What is the result presen ted to the user. 26

Ho w is the result presen ted to the user. 27

Query Phase � Ask the user what kind of MBS they need to price. � P ass-throughs, CMOs, or Stripp ed MBSs. � Ask the parameters of the mortgage p o ol asso ciated with the MBS (for description of parameters please see Lecture 1). � In case of CMOs ask the follo wing questions: { Num b er of tranc hes. { P ar-v alue of eac h tranc h. 28