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738:
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130:
having the feature that, if there is no exogenous infusion or withdrawal of money, the purchase of a new asset must be financed by the sale of an old one. This concept is used to define for example
600:
974:
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932:
1511:{\displaystyle V_{t}=\sum _{i=1}^{n}\left\{h_{0}^{i}S_{0}^{i}+\int _{0}^{t}h_{u}^{i}\mathrm {d} S_{u}^{i}\right\}=h_{0}\cdot S_{0}+\int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}
770:
743:
If there are transaction costs then only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that
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Hamel, Andreas; Heyde, Frank; Rudloff, Birgit (November 30, 2010). "Set-valued risk measures for conical market models".
1736:
467:(in physical units, i.e. the number of each stock) is self-financing (with trading on a finite set of times only) if
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54:
1642:. This means in particular that there have been no infusion of money in or withdrawal of money from the portfolio.
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32:
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642:
If we are only concerned with the set that the portfolio can be at some future time then we can say that
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409:{\displaystyle L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}({\mathcal {F}}_{T}):X\in K_{t}\;P-a.s.\}}
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234:{\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}
733:{\displaystyle H_{\tau }\in -K_{0}-\sum _{k=1}^{\tau }L_{d}^{p}(K_{k})}
1682:
1288:{\displaystyle h=\left((h_{t}^{1},\dots ,h_{t}^{d})\right)_{t}}
15:
891:
a d-dimensional predictable stochastic process such that the
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190:
176:
1562:
corresponds to the initial wealth of the portfolio, while
1615:{\displaystyle \int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}
1200:{\displaystyle V_{t}=\sum _{i=1}^{n}h_{t}^{i}S_{t}^{i}.}
1622:
is the cumulated gain or loss from trading up to time
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1100:. Denote the value process of the trading strategy
46:. Unsourced material may be challenged and removed.
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1705:(3rd ed.). Oxford University Press. p.
595:{\displaystyle H_{t}-H_{t-1}\in -K_{t}\;P-a.s.}
8:
1008:denote the number of shares of stock number
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106:Learn how and when to remove this message
1210:Then the portfolio/the trading strategy
1668:
969:{\displaystyle \forall \,i=1,\dots ,d}
514:{\displaystyle t\in \{0,1,\dots ,T\}}
7:
44:adding citations to reliable sources
1701:Arbitrage theory in continuous time
884:{\displaystyle h=(h_{t})_{t\geq 0}}
825:{\displaystyle S=(S_{t})_{t\geq 0}}
138:, the latter being fundamental for
1598:
1494:
1412:
941:
750:
168:
14:
460:{\displaystyle (H_{t})_{t=0}^{T}}
140:arbitrage-free derivative pricing
1555:{\displaystyle h_{0}\cdot S_{0}}
927:{\displaystyle h^{i}\cdot S^{i}}
20:
155:Assume we are given a discrete
31:needs additional citations for
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165:
1:
765:{\displaystyle \Delta t\to 0}
280:for the market. Denote by
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1080:the price of stock number
157:filtered probability space
55:"Self-financing portfolio"
1073:{\displaystyle S_{t}^{i}}
1028:in the portfolio at time
1001:{\displaystyle h_{t}^{i}}
602:with the convention that
631:{\displaystyle H_{-1}=0}
124:self-financing portfolio
146:Mathematical definition
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136:replicating portfolios
1697:Björk, Tomas (2009).
1652:Replicating portfolio
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263:
261:{\displaystyle K_{t}}
236:
132:admissible strategies
120:financial mathematics
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893:stochastic integrals
843:
784:
747:
646:
606:
525:
475:
420:
416:. Then a portfolio
284:
245:
162:
40:improve this article
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837:frictionless market
832:be a d-dimensional
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1737:Portfolio theories
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231:
201:
1716:978-0-19-877518-8
1635:{\displaystyle t}
1113:{\displaystyle h}
1093:{\displaystyle i}
1041:{\displaystyle t}
1021:{\displaystyle i}
274:transaction costs
272:(with or without
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978:
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935:
914:
901:
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895:
865:
855:
841:
840:
806:
796:
782:
781:
778:
776:Continuous time
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717:
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541:
528:
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522:
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37:
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12:
11:
5:
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1689:
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1659:
1657:Self-financing
1654:
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1393:
1388:
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1360:
1355:
1351:
1346:
1340:
1335:
1332:
1329:
1325:
1321:
1316:
1312:
1297:self-financing
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1268:
1263:
1259:
1255:
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1249:
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1231:
1227:
1222:
1219:
1208:
1207:
1196:
1191:
1186:
1182:
1176:
1171:
1167:
1161:
1156:
1153:
1150:
1146:
1142:
1137:
1133:
1109:
1089:
1067:
1062:
1058:
1037:
1017:
995:
990:
986:
976:. The process
965:
962:
959:
956:
953:
950:
947:
943:
921:
917:
913:
908:
904:
878:
875:
872:
868:
862:
858:
854:
851:
848:
834:semimartingale
819:
816:
813:
809:
803:
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795:
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758:
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729:
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531:
510:
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114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
1749:
1738:
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1734:
1732:
1718:
1712:
1708:
1703:
1702:
1693:
1690:
1684:
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1672:
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1662:
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1649:
1645:
1643:
1629:
1607:
1603:
1594:
1589:
1585:
1579:
1574:
1570:
1547:
1543:
1539:
1534:
1530:
1503:
1499:
1490:
1485:
1481:
1475:
1470:
1466:
1462:
1457:
1453:
1449:
1444:
1440:
1436:
1432:
1426:
1421:
1417:
1406:
1401:
1397:
1391:
1386:
1382:
1378:
1373:
1368:
1364:
1358:
1353:
1349:
1344:
1338:
1333:
1330:
1327:
1323:
1319:
1314:
1310:
1302:
1301:
1300:
1298:
1280:
1275:
1266:
1261:
1257:
1253:
1250:
1247:
1242:
1237:
1233:
1225:
1220:
1217:
1194:
1189:
1184:
1180:
1174:
1169:
1165:
1159:
1154:
1151:
1148:
1144:
1140:
1135:
1131:
1123:
1122:
1121:
1107:
1087:
1065:
1060:
1056:
1035:
1015:
993:
988:
984:
963:
960:
957:
954:
951:
948:
945:
919:
915:
911:
906:
902:
894:
876:
873:
870:
860:
856:
849:
846:
838:
835:
817:
814:
811:
801:
797:
790:
787:
775:
773:
759:
753:
741:
722:
718:
709:
704:
700:
694:
689:
686:
683:
679:
675:
670:
666:
662:
659:
654:
650:
625:
622:
617:
614:
610:
589:
586:
583:
580:
577:
574:
568:
564:
560:
557:
552:
549:
546:
542:
538:
533:
529:
521:we have that
505:
502:
499:
496:
493:
490:
487:
481:
478:
470:
469:
468:
452:
447:
444:
441:
431:
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400:
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388:
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371:
368:
365:
357:
340:
335:
331:
327:
324:
318:
310:
306:
297:
292:
288:
279:
275:
271:
270:solvency cone
253:
249:
225:
222:
217:
212:
209:
206:
196:
181:
171:
158:
151:Discrete time
150:
145:
143:
141:
137:
133:
129:
125:
121:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1700:
1692:
1671:
1521:
1296:
1209:
779:
742:
641:
277:
154:
123:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1683:1011.5986v1
1663:References
1295:is called
276:) at time
241:, and let
96:March 2024
66:newspapers
1595:⋅
1571:∫
1540:⋅
1522:The term
1491:⋅
1467:∫
1450:⋅
1383:∫
1324:∑
1251:…
1145:∑
958:…
942:∀
912:⋅
874:≥
815:≥
757:→
751:Δ
695:τ
680:∑
676:−
663:−
660:∈
655:τ
615:−
578:−
561:−
558:∈
550:−
539:−
500:…
482:∈
389:−
372:∈
328:∈
169:Ω
128:portfolio
1731:Category
1646:See also
471:for all
268:be the
80:scholar
1713:
1048:, and
934:exist
82:
75:
68:
61:
53:
1678:arXiv
126:is a
87:JSTOR
73:books
1711:ISBN
839:and
780:Let
134:and
122:, a
59:news
1299:if
1120:by
118:In
42:by
1733::
1709:.
1707:87
772:.
740:.
142:.
1719:.
1686:.
1680::
1630:t
1608:u
1604:S
1599:d
1590:u
1586:h
1580:t
1575:0
1548:0
1544:S
1535:0
1531:h
1518:.
1504:u
1500:S
1495:d
1486:u
1482:h
1476:t
1471:0
1463:+
1458:0
1454:S
1445:0
1441:h
1437:=
1433:}
1427:i
1422:u
1418:S
1413:d
1407:i
1402:u
1398:h
1392:t
1387:0
1379:+
1374:i
1369:0
1365:S
1359:i
1354:0
1350:h
1345:{
1339:n
1334:1
1331:=
1328:i
1320:=
1315:t
1311:V
1281:t
1276:)
1272:)
1267:d
1262:t
1258:h
1254:,
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850:=
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