Lecture 1 Finance Pro ject Somesh Jha 1 Goals of the - PDF document

Lecture 1 Finance Pro ject Somesh Jha 1

Goals of the course T eac h the studen ts to systematic al ly design � algorithms and systems for problems in the �nance domain. T aking �nance ideas from the ac ademic � domain and making them real. Maxim: Learn b y doing. � 2

Nature of the course Largely indep enden t in nature. � Je� and I are here to guide y ou, but y ou will � w ork as indep enden t teams. Think of the course as a structur e d � indep endent study c ourse . 3

What w e will do in class Je� and I will go through a systematic � design of a system for pricing mortgage b acke d se curities or MBSs . Use the lectures as a guide for y our pro ject. � There will b e �ve phases to the pro ject. � These phases will b e de�ned later. 4

Logistics Studen ts will form a team of 3-4. � Eac h team will select a pap er from a set � pro vided in class. Eac h team will design a system based on the � pap er they select. Eac h team will go through the �v e phases � (to b e describ ed later). 5

Some useful tips Pic k a b alanc e d te am . � Stic k to the sc hedule for eac h phase. � 6

Grading Grading will b e done dep ending on the � outcome of eac h phase. No tests and homew orks. � 7

Fiv e phases Description of these phases will b e pro vided later, but here they are. Requiremen ts phase ( Phase 1 ). � High-lev el design phase ( Phase 2 ). � Lo w-lev el design phase ( Phase 3 ). � 8

Phases (Con td) In class presen tation ( Phase 4 ). � Protot yp e ( Phase 5 ) � 9

Protot yp e This will b e a sc ale d b ack v ersion of the � design. Mak e as man y limiting assumptions as � p ossible, but state them carefully . Y ou can use to implemen t the � C,C++,JAVA protot yp e. Mak e sure y ou tell Je� the en vironmen t y ou are using. 10

Presen tation There will b e one presen tation p er team. � The presen tation will b e a synopsis of phase � 1,2, and 3. Eac h presen tation will b e 20 min utes long. � Last three lectures will b e all presen tations. � 11

P ap er 1 J. Hull and A. White, E�cien t Pro cedures � for V aluing Europ ean and American P ath-Dep enden t Options, Journal of Derivative, No 1, Pages 21-31 , 1993. 12

P ap er 2 J. Hull and A. White, V aluing Deriv ativ e � Securities Using the Explicit Finite Di�erence Metho d, Journal of Financial and Quantitative A nalysis, V ol 25. No 1, Pages 87-99 , 1990. 13

P ap er 3 J. Hull and A. White, Pricing � In terest-Rate-Deriv ativ e Securities, The R eview of Financial Studies, V ol 3, No 4, Pages 573-592 , 1990. 14

P ap er 4 A. Li, P . Ritc hk en, and L. � Sank arasubramanian, Lattice Mo dels for Pricing American In terest Rate Claims, Journal of Financ e, V ol L, No 2, Pages 719-736 , 1995. 15

P ap er 5 P . Ritc hk en and L. Sank arasubramanian, � V olatilit y Structures of F orw ard Rates and the Dynamics of the T erm Structure, Ma th ematic al Financ e, V ol 5, No 1, Pages 55-72 , 1995. 16

P ap er 6 P . Ritc hk en and L. Sank arasubramanian, � Pricing the Qualit y Option in T reasury Bond F utures, , Mathematic al Financ e, V ol 2, No 3, Pages 197-214 , 1995. 17

P ap er selection Don't c ho ose pap ers 5 or 6 unless y ou are � comfortable with sto c hastic calculus. F o cus on the tec hniques and algorithms in � the pap er. It is OK if y ou don't understand all the mathematical deriv ations. 18

Goals of reading the pap er Decide what �nancial instruments y ou � w an t to price after reading the pap er. Pic k 2-3 instrumen ts. Y ou will b e required to understand these instrumen ts completely . Y ou should ha v e a clear idea ab out the � algorithm prop osed in the pap er. Mak e a note of adv an tages/disadv an tages of � this tec hnique/algorithm. 19

Requiremen ts do cumen t Describ e the �nancial instrumen t in great � detail. Describ e the assets the instrumen ts dep end � up on. State the assumptions on the prices of these assets. Describ e the cash-�o w c haracteristics. Describ e the �nancial instrumen ts and there � cash-�o w c haracteristics. 20

Requiremen ts do cumen t In an abstract sense w e are describing what � is the seman tics of eac h op eration that the user can do. In this very sp e ci�c example this amoun ts � to de�ning the precise seman tics of mortgage b acke d se curities (MBSs) . 21

Describing Mortgages Fixe d R ate: The ann ual in terest rate of the � mortgage sta ys �xed through out the life of the mortgage. A djustable R ate Mortgages (ARMs): The � ann ual in terest rate can b e adjusted b y the loaning agency . 22

Fixed Rate Mortgages Let b e the original mortgage balance. M B � 0 Let b e the simple mon thly in terest rate. c � Let b e the mon thly mortgage pa ymen t. M P � Let b e the n um b er of mon ths. n � 23

Relationship b et w een and M P M B 0 The follo wing equation should hold b et w een � and : M P M B 0 n � i = (1 + c ) M B M P X 0 i =1 � n 1 (1 + c ) � = M P c Hence the mon thly mortgage pa ymen t M P � is giv en in terms of the mortgage amoun t using the follo wi ng form ula: M B 0 c (1+ c ) n = M P M B 0 (1+ c ) � 1 n 24

Principal at time t Let the remaining mortgage balance M B � t at time t . W e ha v e the follo wi ng relationship b et w een � and . M B M P t � ( n � t ) 1 (1 + c ) � = M B M P t c So w e ha v e the follo wi ng equation b et w een � and : M B M B 0 t n t (1 + c ) (1 + c ) � = M B M B t 0 n (1 + c ) 1 � 25

Breaking the mortgage pa ymen ts A t time the mortgage balance is t M B � t � 1 ( t 1). � The in terest on this is mortgage balance I � t is: cM B t � 1 The mortgage pa ymen t at time is M P t � brok en in to t w o parts: inter est p ayment I t and p ayment applie d towar ds princip al . P t W e ha v e the follo wi ng equation: = + M P I P t t 26

Scenarios A requiremen ts do cumen t for a large � soft w are system has a h uge n um b er of scenarios. Basically , sc enarios describ e what should � happ en in sp eci�c cases. F or example, in the requiremen ts do cumen t � for an online brok erage system a scenario migh t describ e what should happ en when a user logs on and buys a sto c k. 27

Examples In this case, scenarios are simply examples � of cash �o ws. Consider a mortgage of 100 ; 000, ann ual � mortgage rate (12 c ) of 9 : 5% and time p erio d of 30 y ears (360 mon ths). Chec k that the mon thly mortgage rate M P � is 840 : 85. 28

Example con tin ued Chec k that = 791 : 67 and = 49 : 19. I P � 0 0 Chec k that = 574 : 95 and I � 215 = 265 : 90. P 215 is a decreasing function of and is an I t P � t t increasing function of t . ( Why? ). 29

ARMs ARMs start out with an initial in terest rate. � ARMs in terest rate can b e adjusted b y a � margin at a frequency sp eci�ed in the m con tract. Lifetime c ap : This is an upp er b ound c � L that the in terest rate cannot exceed. Lifetime �o or : This is a lo w er b ound on c � F the in terest rate. 30

ARMs Let us the in terest rate is c ( t 1) at time � � 1 and w e are adjusting at time t , t � The new in terest rate t ) is giv en b y the c � ( follo wi ng cases: { if x ( t ) + c ( t 1) m > � min [ x ( t ) + c ( t 1) + ] m; c ; c � L P { if x ( t ) + c ( t 1) m � � min [ x ( t ) + c ( t 1) ] m; c ; c � � F P 31

Explanation of terms x ( t ) : Underlying index sp eci�ed in the � con tract. Tw o widely used indices are cost of funds index (COFI) and a constan t maturit y (one y ear or �v e y ear) T reasury index. and are the lifetime cap and �o or c c � L F resp ectiv ely . denotes the ARMs p erio dic cap, i.e., c � P cannot adjust b y more than this amoun t. 32