Part II. Martingale measres and their constrctions 1. The \First" and the \Second" fndamental theorems show clearly how \mar tingale measres" are impo

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1 Albert N. Shiryaev (Stelov Mathematical Institte and Moscow State University) ESSENTIALS of the ARBITRAGE THEORY Part I. Basic notions and theorems of the \Arbitrage Theory" Part II. Martingale measres and their constrctions Part III. Qicest detection of the appearing of the arbitrage possibilities Appendix I. A. N. Shiryaev, A. S. Cherny. \Vector Stochastic Integrals and the Fndamental Theory of Asset Pricing". Appendix II. J. Kallsen, A. N. Shiryaev. \The Cmlant Pro cess and Esscher's Change of Measre". Lectres in IPAM (Institte for Pre and Applied Mathematics), UCLA 3{4{5 Janary 001

2 Part II. Martingale measres and their constrctions 1. The \First" and the \Second" fndamental theorems show clearly how \mar tingale measres" are important for solving the problem \Arbitrage orno Arbitrage". It is reasonable to start or discssion of the constrction of the martingale mea sres that are (locally) absoltely continos or eqivalent to the original basic mea sre P involved in the denition of the ltered probability space ( F (F n ) n>0 P)

3 ESSENTIALS OF THE ARBITRAGE THEORY 3 with a discrete (with respect to time) version a reslt established by I. Girsanov for processes of dision type, which became the prototype for a nmber of reslts for martingales, local martingales, semimartingales and so on. Set h n = ; n + n " n where n and n are F n;1 -measrable F 0 = f? g and " =(" 1 " :::) is a seqence of Fn-measrable iid random variables, " n N (0 1), n > 0. From these assmptions it follows that Law(h n j Fn;1 P) =N ( n n) which allowsonetocallh = (h n ) a conditionally Gassian seqence (with respect to P) with (conditional) expectation and variance Setting H n = n P h, A n = n P E(h n j F n;1 )= n D(h n j Fn;1) = n :, M n = n P " weget (1) H n = ; n + n W n where x n = x n ; x n;1,= 1, W n = " n, which one can regard as a discrete conterpart to the stochastic dierential () dh t = ; t dt + t dw t of some It^o process H =(H t ) t>0 generated by a Wiener process W =(W t ) t>0 with the local drift =( t ) t>0 and the local volatility =( t ) t>0. Or constrction of the measre P e is based on the seqence of (positive) random variables (3) Z n =exp nx " ; 1 nx Lemma 1. 1) W The seqence Z =(Z n ) n>1 is P-martingale with EZ n =1,n > 1. ) Let F = F n and assme that 1 1X (4) E exp < 1 (the \Noviov condition"). Then Z = (Z n ) n>1 is a niformly integrable martingale with limit (P-a.s.) Z 1 =limz n sch that X 1 (5) Z 1 = exp " ; 1 1X and Z n = E(Z 1 j F n ), EZ 1 =1. :

4 4 ALBERT N. SHIRYAEV Lemma. 1) Let h =(h n ) n6n be aconditionally Gassian seqence sch that Law(h n j F n;1 P) =N ( n n) n 6 N: Let FN = F and e P N be themeasre dened by formla e P N (d!) =Z N (!) P(d!). Then the seqence h =(h n ) n6n is conditionally Gassian with respect to e P N and If H n = n P Law(h n j Fn;1 e P N)=N (0 n) n 6 N: h then one may nd a (new) seqence of the iid random variables e" =(e" n ) n>1, e" n N (0 1), sch that (e P N -a.s.) H n = nx e" n 6 N so, with respect to the measre e P N the seqence H =(H n ) n6n is a local martingale (= a martingale transform). ) If n, n 6 N, are independent on!, then h =(h n ) n6n is a seqence of inde pendent Gassian random variables with respect to e P N : Law(h n j Fn;1 e P N)=N (0 n) n 6 N: 3) If F = W F n and EZ 1 = 1 (i.e. Z =(Z n ) n>1 is a niformly integrable mar tingale) then the seqence h =(h n ) n>1 is the conditionally Gassian with respect to e P1 where e P1 (d!) =Z 1 (!) P(d!) and the seqence H =(H n ) n>1 is a e P1 -local martingale.. Very similar formlations one may done for the case of continos time. Name ly, sppose that the process H =(H t ) t>0 is an It^o process with (6) dh t = ; t dt + t dw t t > 0 H 0 =0 where W =(W t ) t>0 is a P-Wiener process (a Brownian motion). We dene Z t Z (7) Z t = exp dw ; 1 t d and pt P et (d!) =Z T P(d!). If 1 (8) E exp 0 Z T 0 0 d < 1 then the process (H t ) t6t is a e P T -local martingale schthat(e P T -a.s.) (9) H t = Z t 0 df W t 6 T

5 ESSENTIALS OF THE ARBITRAGE THEORY 5 where (f W ) 6T is a P et -Wiener process. If 1 Z 1 (10) E exp d < 1 (\Noviov condition") 0 then Z = (Z t ) t>0 will be niformly integrable martingale i.e. eqivalent tothe property EZ 1 = 1 where Z 1 = lim t!1 Z t. Let's give a proof of the \Noviov condition" for case of the continos time. (The case of the discrete time can be considered by the similar way and, in fact, can be obtained from the reslt of the continos time.) At that, sing Dambis, Dbins{ Schwartz theorem ([KY]) this problem can be reformlated by the following way: Let (11) Z t () =e B t; t where R, B =(B t )isabrownian motion. We claim that the \Noviov condition" (1) Ee < 1 implies that (13) EZ () =1: For proof that (1))(13) let's assme rstly that instead (1) we have a little bit more stronger condition (14) Ee 1+" < 1 for some " > 0. Under sch condition one may prove (R. Liptser and A. Shiryaev) that EZ () =1.The proof is simple. Really, it is scient toshow that for some >0 (15) sp E; 1+ Zt^ () < 1: t>0 with Taing for simplicity = 1 we have ; Zt (1) 1+ = (1) t () t (1) t () t = e (1+)B t; p(1+) = e ( p(1+) ; 1+ )t t

6 6 ALBERT N. SHIRYAEV and p =1+", q = 1+".ByHolder ineqality " becase E( (1) t ) p =1. Tae schthat E; "(1)t 1+ = ; E( (1) t Then ( () ) q 6 e ( 1 +") and so ) p 1=p; (1 + ) 6 E( () t ) q ; 1=q = E( () t ) q 1=q " (1 + ")(1 + ") : sp E(Z (1) ^t )1+ 6 Ee ( 1 +")(^t) 6 Ee ( 1 +") < 1: t>0 From this it follows that the martingale (Z t^ (1)) t>0 is niformly integrable and as a corollary EZ (1) = 1: Thesameistreforany R: EZ () =1. Now sppose that Ee < 1. Let's pt " =(1; ") with 0 <"<1thenwe nd that Ee 1+" " = Ee (1+")(1;") 6 Ee < 1: From this and previos consideration we get (16) EZ ( " )=1: Then applying the Holder ineqality with 1=p = 1; ",1=q = " we nd that 1=EZ ( " )=EZ ; (1 ; ") = Ee (1;")B ; (1;") = Ee (1;")(B ; ) e (1;")" ; 6 EZ 1;"; (1;") () Ee " ; 6 EZ 1;"; () Ee " : If Ee < 1 then from (16) we get with limit passage " # 0that 1 6 EZ (): Bt EZ () 6 1 becase (Z t^ ()) t>0 is a nonnegative spermartingale with Z 0 () = 1. So, (17) Ee < 1 =) EZ () =1: It is interesting to note that from (16) it is easy to see that instead the condition Ee < 1 it is scient to assme only that (18) lim " log Ee (1;") =0: "#0 Indeed, from this condition it follows that for sciently small ">0 Ee 1+" " = Ee (1+")(1;") = Ee (1;")(1;" ) < 1: Ths, again 1 = EZ ( " ) and the ineqalities in (16) hold and by (18) we get that EZ () =1.

7 ESSENTIALS OF THE ARBITRAGE THEORY 7 3. Weintend nowtoprove the statements of the Lemma. By Bayes's formla (e P N -a.s.) EeP (e ih n j N Fn;1) =E ; (i = E e n + n ; (i e n + n = E = e ; n where we se the eqality e ih n Z n n )" n;i n ; 1 ( n n )" n; 1 (i n+ n Z n;1 F n;1 n ) j F n;1 n ) e 1 (i n+ n Ee (i n+ n n )" n; 1 (i n+ n n ) =1 n ) ;i n ; 1 ( n n ) j Fn;1 and the fact that the n are Fn;1-measrable. So, we obtain the eqality EeP (e ih n j Fn;1) =e ; n N which means that the seqence h =(h n ) remains conditionally Gassian with respect to the new measre e P N, bt has now a trivial \drift" component: (19) Law(h n j F n;1 e P N)=N (0 n) n 6 N: One can say that the transitions from P to the measre e P N eliminates (\ills") the drift =( n ) n6n of the seqence h =(h n ) n6n,btpreserves the conditional variance. We have already mentioned that from (19) it follows that if e" = (e" n ) n6n is a seqence of F n -measrable random variables e" n with (0) Law(e" n j F n;1 e P N )=N (0 1) (one can always constrct sch a seqence, althogh it may be necessary to enlarge or initial probability space), then (1) Law(h n n6 N j e P N )=Law( n e" n n6 N j e P N): Hence it is clear that the seqence (h n ) n6n \behaves" as a local martingale-dierence ( n e" n ) n6n with respect to e P N, while in terms of the original measre P a property similar to (1) can be expressed as follows: () Law(h n ; n n6 N j P) =Law( n " n n6 N j P):

8 8 ALBERT N. SHIRYAEV 4. Now we considere a more-to-earth sitation of the (positive!) prices (3) S n = S 0 e H n where H n = h h n, n > 1, and, in particlar, (4) h n = ; n + n " n as in the previos presentation. It wept (5) e Hn = H n + X 6n(e H ; H ; 1) X = 6n(e H ; 1) then we nd that (6) S n = S 0 E( b H)n where the stochastic exponential (7) E( b H)n = e e H n Y 6n (1 + H b X )e ; H b = H ) 6n(1+b : If b H > ;1 and the seqence b H =( b H ) >1 is a local martingale then (E( b H) ) >1 is also a local martingale bt becase E( b H)n > 0 for all n > 1 the process (E( b H) ) >1 is a martingale, indeed. Consider now the case (4). First, it is reasonable to consider the qestion of the conditions ensring that the seqence S =(S n ) n>1 is a martingale with respect to the original measre P. Wehave already seen that it is scient to this end that the seqence b H =( b Hn ) n>1 with Hn b = e H n ; 1 be a local martingale, i.e. E(j Hn b j j F n;1 ) < 1 and E( Hn b j F n;1 ) = 0, or, eqivalently, (8) E(e H n j F n;1 )=1 (P-a.s.): Since we assme that H n = ; n + n " n we can rewrite condition (8) as follows: (9) E(e ; n+ n " n j F n;1 )=1 which is eqivalent to the relation E(e n" n j F n;1 )=e n : The left-hand side here is eqal to e 1 n.ths, wearrive to the condition (30) n = n n > 1

9 ESSENTIALS OF THE ARBITRAGE THEORY 9 ensring that the logarithmically conditionally Gassian seqence nx (31) S n = S 0 exp (; + " ) n > 1 is a martingale with respect to P. Of corse, this is what one cold expect becase the seqence nx exp " ; is a martingale. Wenow proceed to the case when (30) fails. Assme that n 6 N. We shall constrct the reqired measre e P N on FN bymeans of the (conditional) Esscher transformation, in the following form: (3) e P N (d!) =Z N (!) P(d!) with (33) Z N (!) = and (34) z (!) = NY z (!) e a h E(e a h j F ;1 ) where we shall choose the F ;1 -measrable variables a = a (!)(heref 0 = f? g) sch that the seqence (S n ) n6n is a e P N-martingale. In or case when S n = S 0 e H n, this e P N -martingale property means that or EeP N (e H n j F n;1 )=1 n>1 (35) E(e h n(a n +1) j F n;1 )=E(e a nh n j F n;1 ): Bearing in mind that h n = n + n " n we see that the eqality (35) holds if i.e. ; n + n = ;a n n (36) a n = n n ; 1 : (If (30) holds for all n 6 N then a n = 0 and Z N =1,i.e.e P N = P.)

10 30 ALBERT N. SHIRYAEV Ths Choosing a n in accordance with (36) weobtain (37) z n = and (38) Z N =exp E(e a nh n j Fn;1) =exp ; n n + n : 8 e a nh n E(e a nh n j F n;1) = exp ; ; n + n " n ; 1 ; n + n n n ; NX n=1 ; n + n n Hence the seqence S =(S n ) n6n with " n + 1 ; n + n : n S n = S 0 e H n H n = h h n and h n = n + n " n is a e P N -martingale with EeP N S n = S Extensive exposition of the problem of the constrction of Martingale measres for the case of the discrete time can be fond in [ESF Chapter V, Section 3] and for the case of the continos time in [ESF Chapter VII, Section 3] and in the Appendix II. 6. In the two inclded below appendices (Appendix I: the paper by A. N. Shiryaev and A. S. Cherny and Appendix II: the paper by J. Kallsen and A. N. Shiryaev) read ers can nd detailed presentations on the problems of the \Arbitrage", \Fndamental Theorems" and \Esscher's Change of Measre". Reference [ESF] Shiryaev A. N. Essentials of Stochastic Finance. World Scientic, Singapore, [RY]Revz D., Yor M. Continos martingales and Brownian Motion (Third ed.). Springer, Berlin, 1999.