1876:
214:
653:
373:
708:
128:
459:
419:
496:
481:
1419:
994:
880:
will undo our change of numeraire, and give us the price in our original currency, which is the formula above. Alternatively, one can show it by the
1759:
1449:
85:, that we wish to price gives the buyer the right, but not the obligation, to exchange the second asset for the first at the time of maturity
236:
926:
1581:
1317:
721:
Margrabe's model of the market assumes only the existence of the two risky assets, whose prices, as usual, are assumed to follow a
1819:
987:
484:
1754:
1399:
42:
pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by
1901:
1794:
1332:
1186:
980:
825:
on the first asset (with its numeraire pricing) with a strike of 1 unit of the riskless asset. Note the dividend rate
1413:
725:. The volatilities of these Brownian motions do not need to be constant, but it is important that the volatility of
660:
1906:
1591:
1911:
1695:
1506:
722:
1814:
1809:
1464:
1409:
748:. The model does not require an equivalent risk-neutral probability measure, but an equivalent measure under S
1764:
1434:
1424:
1292:
1133:
1065:
959:
1713:
1561:
1546:
1511:
1454:
209:{\displaystyle \textstyle \sigma ={\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{1}\sigma _{2}\rho }}}
1774:
1625:
1541:
1118:
836:
760:
799:
Under this change of numeraire pricing, the second asset is now a riskless asset and its dividend rate
1728:
1685:
1675:
1665:
1386:
1327:
1262:
1216:
1211:
1085:
1045:
1012:
740:, is constant. In particular, the model does not assume the existence of a riskless asset (such as a
31:
1444:
941:
906:
1733:
1521:
1267:
806:
is the interest rate. The payoff of the option, repriced under this change of numeraire, is max(0,
488:
1784:
1769:
1738:
1723:
1690:
1556:
1347:
1312:
1075:
1040:
1003:
945:
910:
756:
424:
384:
1789:
1779:
1718:
1705:
1680:
1566:
1352:
1148:
648:{\displaystyle d_{1}=(\ln(S_{1}(0)/S_{2}(0))+(q_{2}-q_{1}+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}}
1670:
1660:
1650:
1609:
1604:
1586:
1516:
1282:
1277:
1249:
1201:
1080:
1020:
881:
741:
43:
39:
1880:
1850:
1845:
1799:
1635:
1630:
1576:
1486:
1394:
1367:
1307:
1302:
1272:
1221:
1206:
1123:
1103:
46:(PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles.
1855:
1840:
1640:
1551:
1501:
1478:
1459:
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1229:
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1191:
1171:
1095:
922:
466:
1895:
1835:
1804:
1645:
1571:
1531:
1526:
1362:
1234:
1181:
1176:
1158:
1055:
1035:
745:
1655:
1429:
1357:
1337:
1297:
1166:
1138:
1128:
1070:
1536:
1404:
1375:
1371:
1322:
1113:
1108:
822:
17:
965:
1860:
1496:
1491:
1257:
1143:
1050:
782:
230:
Margrabe's formula states that the fair price for the option at time 0 is:
1875:
1620:
1342:
1239:
1060:
220:
is the
Pearson's correlation coefficient of the Brownian motions of the
368:{\displaystyle e^{-q_{1}T}S_{1}(0)N(d_{1})-e^{-q_{2}T}S_{2}(0)N(d_{2})}
839:
with these values as the appropriate inputs, e.g. initial asset value
972:
796:
units of the second asset, and a unit of the second asset is worth 1.
862:, etc., gives us the price of the option under numeraire pricing.
832:
of the first asset remains the same even with change of pricing.
27:
Formula that calculates option prices for dividend-paying stocks
976:
785:'); this means that a unit of the first asset now is worth
942:"The Value of an Option to Exchange One Asset for Another"
907:"The Value of an Option to Exchange One Asset for Another"
74:, and that each has a constant continuous dividend yield
759:
by reducing the situation to one where we can apply the
132:
663:
499:
469:
427:
387:
239:
131:
1828:
1747:
1704:
1600:
1477:
1385:
1248:
1157:
1094:
1028:
1019:
767:First, consider both assets as priced in units of
702:
647:
475:
453:
413:
367:
208:
865:Since the resulting option price is in units of
703:{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T}}}
421:are the expected dividend rates of the prices
988:
8:
948:, Vol. 33, No. 1, (March 1978), pp. 177-186.
913:, Vol. 33, No. 1, (March 1978), pp. 177-186.
461:under the appropriate risk-neutral measure,
70:are the prices of two risky assets at time
1025:
995:
981:
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693:
681:
668:
662:
638:
630:
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607:
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581:
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498:
468:
445:
432:
426:
405:
392:
386:
356:
331:
316:
308:
292:
267:
252:
244:
238:
194:
184:
168:
163:
150:
145:
139:
130:
964:Rolf Poulsen, University of Gothenburg,
1820:Power reverse dual-currency note (PRDC)
1760:Constant proportion portfolio insurance
898:
958:Mark Davis, Imperial College London,
7:
1755:Collateralized debt obligation (CDO)
821:So the original option has become a
25:
1874:
485:cumulative distribution function
1582:Year-on-year inflation-indexed
627:
621:
574:
568:
565:
559:
541:
535:
522:
513:
362:
349:
343:
337:
298:
285:
279:
273:
89:. In other words, its payoff,
1:
1592:Zero-coupon inflation-indexed
927:"cites" page for this article
889:External links and references
1795:Foreign exchange derivative
1187:Callable bull/bear contract
454:{\displaystyle S_{1},S_{2}}
414:{\displaystyle q_{1},q_{2}}
1928:
1869:
1696:Stock market index future
1010:
872:, multiplying through by
723:geometric Brownian motion
1815:Mortgage-backed security
1810:Interest rate derivative
1785:Equity-linked note (ELN)
1770:Credit-linked note (CLN)
1765:Contract for difference
1066:Risk-free interest rate
774:(this is called 'using
755:The formula is quickly
108:If the volatilities of
1547:Forward Rate Agreement
704:
649:
477:
455:
415:
369:
210:
1775:Credit default option
1119:Employee stock option
837:Black-Scholes formula
761:Black-Scholes formula
705:
650:
478:
456:
416:
370:
211:
1902:Mathematical finance
1729:Inflation derivative
1714:Commodity derivative
1686:Single-stock futures
1676:Normal backwardation
1666:Interest rate future
1507:Conditional variance
1013:Derivative (finance)
966:The Margrabe Formula
661:
497:
467:
425:
385:
237:
129:
32:mathematical finance
1881:Business portal
1734:Property derivative
960:Multi-Asset Options
173:
155:
1739:Weather derivative
1724:Freight derivative
1706:Exotic derivatives
1626:Commodities future
1313:Intermarket spread
1076:Synthetic position
1004:Derivatives market
946:Journal of Finance
940:William Margrabe,
911:Journal of Finance
905:William Margrabe,
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36:Margrabe's formula
1907:Options (finance)
1889:
1888:
1790:Equity derivative
1780:Credit derivative
1748:Other derivatives
1719:Energy derivative
1681:Perpetual futures
1562:Overnight indexed
1512:Constant maturity
1473:
1472:
1420:Finite difference
1353:Protective option
935:Primary reference
744:) or any kind of
698:
643:
476:{\displaystyle N}
203:
16:(Redirected from
1919:
1912:Financial models
1879:
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1651:Forwards pricing
1425:Garman–Kohlhagen
1026:
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882:Girsanov theorem
851:, interest rate
742:zero-coupon bond
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44:William Margrabe
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1851:Great Recession
1846:Government debt
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1800:Fund derivative
1743:
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1661:Futures pricing
1636:Dividend future
1631:Currency future
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1596:
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1445:Put–call parity
1381:
1368:Vertical spread
1303:Diagonal spread
1273:Calendar spread
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489:standard normal
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1641:Forward market
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1522:Credit default
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1318:Iron butterfly
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1310:
1305:
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1290:
1288:Covered option
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1280:
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1265:
1260:
1254:
1252:
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1245:
1243:
1242:
1237:
1232:
1227:
1226:Mountain range
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1214:
1209:
1204:
1199:
1194:
1189:
1184:
1179:
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1169:
1163:
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1038:
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1023:
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1016:
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992:
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923:Google Scholar
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81:. The option,
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24:
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1836:Consumer debt
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1829:Market issues
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1805:Fund of funds
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1725:
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1703:
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1672:
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1649:
1647:
1646:Forward price
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1642:
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1627:
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1619:
1618:
1616:
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1608:
1606:
1603:
1602:
1599:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1557:Interest rate
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1553:
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1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1484:
1482:
1480:
1476:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1450:MC Simulation
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1415:
1411:
1410:Black–Scholes
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1392:
1390:
1388:
1384:
1377:
1373:
1369:
1366:
1364:
1363:Risk reversal
1361:
1359:
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1349:
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1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1293:Credit spread
1291:
1289:
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1279:
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1225:
1223:
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1217:Interest rate
1215:
1213:
1212:Forward start
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1208:
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1203:
1200:
1198:
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1190:
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1139:Option styles
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1056:Open interest
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1037:
1036:Delta neutral
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858:, volatility
857:
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835:Applying the
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747:
746:interest rate
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49:
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41:
37:
33:
19:
1656:Forward rate
1567:Total return
1455:Real options
1439:
1358:Ratio spread
1338:Naked option
1298:Debit spread
1129:Fixed income
1071:Strike price
970:
952:
951:
934:
933:
918:
901:
893:
892:
873:
866:
859:
852:
840:
826:
807:
800:
786:
775:
768:
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737:
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483:denotes the
229:
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93:, is max(0,
90:
86:
82:
75:
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63:
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53:
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1587:Zero Coupon
1517:Correlation
1465:Vanna–Volga
1323:Iron condor
1109:Bond option
823:call option
1896:Categories
1861:Tax policy
1577:Volatility
1487:Amortising
1328:Jelly roll
1263:Box spread
1258:Backspread
1250:Strategies
1086:Volatility
1081:the Greeks
1046:Expiration
953:Discussion
717:Derivation
1552:Inflation
1502:Commodity
1460:Trinomial
1395:Bachelier
1387:Valuation
1268:Butterfly
1202:Commodore
1051:Moneyness
783:numeraire
691:σ
688:−
636:σ
605:σ
588:−
520:
310:−
302:−
246:−
201:ρ
192:σ
182:σ
175:−
161:σ
143:σ
134:σ
1691:Slippage
1621:Contango
1605:Forwards
1572:Variance
1532:Dividend
1527:Currency
1440:Margrabe
1435:Lattices
1414:equation
1400:Binomial
1348:Strangle
1343:Straddle
1240:Swaption
1222:Lookback
1207:Compound
1149:Warrants
1124:European
1104:American
1096:Vanillas
1061:Pin risk
1041:Exercise
816:(T) - 1)
216:, where
54:Suppose
1610:Futures
1230:Rainbow
1197:Cliquet
1192:Chooser
1172:Barrier
1159:Exotics
1021:Options
379:where:
125:, then
99:(T) - S
50:Formula
1671:Margin
1537:Equity
1430:Heston
1333:Ladder
1283:Condor
1278:Collar
1235:Spread
1182:Binary
1177:Basket
860:σ
757:proven
738:σ
487:for a
218:ρ
120:σ
116:'s
40:option
38:is an
1542:Forex
1497:Basis
1492:Asset
1479:Swaps
1405:Black
1308:Fence
1167:Asian
1029:Terms
894:Notes
845:(0)/S
812:(T)/S
1376:Bull
1372:Bear
1114:Call
763:.
227:'s.
118:are
103:(T))
91:C(T)
62:and
1144:Put
925:'s
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849:(0)
781:as
105:.
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122:i
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97:1
95:S
87:T
83:C
78:i
76:q
72:t
66:2
64:S
58:1
56:S
20:)
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