Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain

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1 Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as DISCRETE ime o CONTINUOUS ime. A ype of pocess ha is of exensive inees in Finance is he Makov Pocess. An example of he Makov pocess is he well known Bownian Moion also called he Wiene Pocess. 1.1 Makov Pocess A Makov Pocess is a paicula ype of sochasic pocess whee only he pesen value of a vaiable is elevan fo pedicing he fuue. The pas hisoy of he vaiable and he way in which he pesen has emeged fom he pas ae ielevan. 1.2 Wiene Pocess A Wiene pocess is a paicula ype of Makov pocess. The behavio of a vaiable Z which follows a Wiene pocess, can be undesood by consideing he changes in is value in small inevals of ime. Conside a small ineval of ime and dene Z as he change in Z duing : Thee ae wo basic popeies Z mus have fo Z o be following a Wiene pocess. 1. Z is elaed o by he equaion Z = p whee N(; 1) 2. The values of Z fo any wo dieen sho inevals of ime ae independen. Popey 2 implies ha Z is a Makov Pocess. Now E(Z) =E V (Z) =V p p = E() = p =V () = Popey 2 implies ha Z is a Makov Pocess. Le us now conside he change in Z fo a elaively longe peiod of ime. This could be boken up ino smalle inevals of : Le hee be N inevals. Then N = T : Thus Z(T ), Z() = NX i=1 i p 1

2 and A funcion is o() if lim! O() if lim! whee c is a consan. E[Z(T ), Z()] = V [Z(T ), Z()] = f(),! f(),! consan NX i=1 NX i=1 p E(i )= V ( i )=N = T ) V [(Z) 2 ]=V[ 2 ] =[] 2 V [ 2 ] 1.3 Genealised Wiene Pocess {z } o(d) c z } { A genealized Wiene pocess foavaiable x can be dened in ems of dz as follows: whee a and b ae consans. 1.4 Io's Pocess dx = ad + bdz This is a genealized Wiene pocess whee he paamees a and b ae funcions of he value of he undelying vaiable x and ime : Algebaically i can be wien as dx = a(; x)d + b(; x)dz whee a(; x) is called he dif ae and b 2 (; x) is called he vaiance ae. 1.5 Io's Lemma Conside he Io's pocess The simples fom of Io's lemma is given by: dx = a(; x)d + b(; x)dz dx + 2 G 2 d Io's Lemma shows ha a funcion G of and x follows he pocess + + G 2 b2 bdz whee dz is he same Wiene pocess and G also follows an Io pocess wih he given dif and vaiance aes. 2

3 1.6 Some examples of Sochasic Inegaion 1. If dx() = (X();)d + (X();)dZ() hen 2. If 3. If ) Z dx(v) = ) X() =X() + ) X() =X() + Z Z (X(v);v)dv + Z Thee is no conained in and : Then X() =X() + (X(v);v)dv + (X(v);v)dv + Z Z (X(v);v)dZ(v) Z (X(v);v)dZ(v) (X(v);v)dZ(v) ) dx() = (X();)d + (X();)dZ() Z (X(v);v;)dv + Then Z ) dx()= (X();;)d + (X();;)dZ()+ 2 (X(v);v;)dZ(v) Z dz(v) d A geneal fom of em sucue models was poposed by Vasicek in He made hee assumpions which ae: 1. he spo ae follows a coninuous Makov pocess 2. he pice P (; s) of a discoun bond is deemined by he assessmen a ime ; of he segmen (); s of he spo ae pocess ove he em of he bond 3. he make is ecien; ha is, hee ae no ansacions coss, infomaion is available o all invesos simulaneously andevey inveso acs aionally. The Makov popey implies ha he spo ae pocess is chaaceized by a single sae vaiable namely is cuen value. Pocesses ha ae Makov and Coninuous ae called diusion pocesses.we now fomally pesen he model as d = (; )d + (; )dz (1) whee Z() is a Wiene pocess wih incemenal vaiance d: The funcions and ae he insananeous dif and vaiance especively of he pocess : All he models we sudy in his secion of he couse ae vaiaions of he above model. In he Meon as is he case hee is consan. In he CIR and Couadon howeve, may depend on (). 3

4 2.1 The Tem Sucue Equaion Assumpion (2) implies ha P = P (; ): Using Io's lemma o dieeniae his expession we ge dp + 2 P 2 2 d dz dp = P p (; ), P p (; )dz whee p and p ae he expessions given in he equaion above. Hence p and 2 p ae he mean and vaiance especively of he insananeous ae of eun on a bond wih mauiy dae s; given ha he cuen spo ae is : Now conside an inveso who a ime issues an amoun W 1 of a bond wih mauiy dae s 1 and simulaneously buys an amoun W 2 mauing a ime s 2 : The oal woh W = W 2, W 1 of he pofolio hus consuced changes ove ime accoding o he accumulaion equaion Le dw =(W 2 p (; s 2 ), W 1 p (; s 1 ))d, (W 2 p (; s 2 ), W 1 p (; s 1 ))dz W 1 = W 2 = W p (; s 2 ) p (; s 1 ), p (; s 2 ) W p (; s 1 ) p (; s 1 ), p (; s 2 ) subsiuing hese values in he equaion above we ge dw = W [ p(; s 2 ) p (; s 1 ), p (; s 1 ) p (; s 2 )] d p (; s 1 ), p (; s 2 ) Since he above equaion does no have a andom em, he inveso's pofolio is iskless. Hence he pofolio should povide a iskless eun. Hence dw = W()d p (; s 2 ) p (; s 1 ), p (; s 1 ) p (; s 2 ) p (; s 1 ), p (; s 2 ) = () p (; s 2 ) p (; s 1 ), p (; s 1 ) p (; s 2 )=()[ p (; s 1 ), p (; s 2 )] p (; s 2 ) p (; s 1 ), () p (; s 1 )= p (; s 1 ) p (; s 2 ), () p (; s 2 ) p (; s 2 ), () p (; s 2 ) = p(; s 1 ), () p (; s 1 ) Excess eun divided by sandad deviaion should be equal acoss all mauiies. Since he above equaion is valid fo abiay mauiy daes s 1 ;s 2 i follows ha he aio p(;s),() p(;s) is independen ofs: Le q(; ) denoe he common value of such aio fo a bond of a mauiy dae given ha he cuen spo ae is () =: Then q(; ) = p(; s; ), () p (; s; ) 4

5 whee s : The quaniy q(; ) is called he make pice of isk as i species he incease in expeced insananeous ae of eun on a bond pe an addiional uni of isk. We will now use he above equaion fo he pice of a discoun bond. q(; ) = p(; s; ), () p (; s; ) p (; s; ), () = p (; s; )q(; ) Subsiuing p and p fom he em sucue equaion we ge " P (; s; P, () =, 2 P + 2 P P 2 2, P =, q(; + q(; ), +[ + q(; + 2 P 2 2, P = fo s This is he basic equaion fo picing of discoun bonds in a make chaaceised by ou hee assumpions. This equaion is called he TERM STRUCTURE EQUATION(TSE). All single faco em sucue models ae special cases of equaion (2): The TSE is a paial dieenial equaion fo P (; s; ): Once he chaace of he spo ae pocess () is descibed and he make pice of isk q(; ) specied, he bond pices ae obained by solving he TSE subjec o he bounday condiion P (s; s; ) =1: The em sucue R(; T )ofinees aes is hen eadily evaluaed fom he equaion R(; T )=, 1 T log P (; + T;()) 2.2 Sochasic Repesenaion Of The Bond Pice Soluions of sochasic dieenial equaions of he ype such as he TSE can be epesened in an inegal fom in ems of he undelying sochasic pocess. Such epesenaion fo he bond pice as a soluion o he TSE and is bounday condiion is as follows: Z s P (; s) =E exp, ()d, 1 Z s Z s q 2 (;())d + q(;())dz() 2 fo s To pove he above dene Z u V (u) = exp, ()d, 1 2 Z u q 2 (;())d + Z u q(;())dz() Now le us Io dieeniae he pocess P (u; s)v (u): Le f = PV and hen df = d(pv) dp (PV) 2 dp dv dv 2 (PV) dp

6 Now and Hence 2 (PV) dv 2 = 2 2 (PV) dp 2 = 2 2 d(pv)=vdp + PdV + dp dv ) d(pv) = V 1 2 PVq2 du + V P du + 2dZ + PV +( + q) ,, 1 2 q2 du + PVqdZ +, P du + PVqdZ + dz d(pv)=pvqdz + dz by viue of he TSE. Inegaing fom o s and aking expecaion yields E (P (s; s)v (s), P (; s)v ())= because P (s; s) =1;V() =1and P (; s) =E[V (S)]: Hence poved. 6

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security 1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,