93:, known as caplets, which exist for each period the cap agreement is in existence. To exercise a cap, its purchaser generally does not have to notify the seller, because the cap will be exercised automatically if the interest rate exceeds the strike (rate). Note that this automatic exercise feature is different from most other types of options. Each caplet is settled in cash at the end of the period to which it relates.
835:
Notice that there is a one-to-one mapping between the volatility and the present value of the option. Because all the other terms arising in the equation are indisputable, there is no ambiguity in quoting the price of a caplet simply by quoting its volatility. This is what happens in the market. The
78:
rate exceeds 2.5%. They are most frequently taken out for periods of between 2 and 5 years, although this can vary considerably. Since the strike price reflects the maximum interest rate payable by the purchaser of the cap, it is frequently a whole number integer, for example 5% or 7%. By comparison
53:
against interest rate fluctuations. For example, a borrower who is paying the LIBOR rate on a loan can protect himself against a rise in rates by buying a cap at 2.5%. If the interest rate exceeds 2.5% in a given period the payment received from the derivative can be used to help make the interest
897:
An important consideration is cap and floor (so called Black) volatilities. Caps consist of caplets with volatilities dependent on the corresponding forward LIBOR rate. But caps can also be represented by a "flat volatility", a single number which if plugged in the formula for valuing each caplet
917:
Imagine a cap with 20% vol and floor with 30% vol. Long cap, short floor gives a swap with no vol. Now, interchange the vols. Cap price goes up, floor price goes down. But the net price of the swap is unchanged. So, if a cap has x vol, floor is forced to have x vol else you have
847:, so the Black model became increasingly inappropriate (as it implies a zero probability of negative interest rates). Many substitute methodologies have been proposed, including shifted log-normal, normal and Markov-Functional, though a new standard is yet to emerge.
901:
Therefore, one cap can be priced at one vol. This is extremely useful for market practitioners as it reduces greatly the dimensionality of the problem: instead of tracking n caplet Black volatilities, you need to track just one: the flat
79:
the underlying index for a cap is frequently a LIBOR rate, or a national interest rate. The extent of the cap is known as its notional profile and can change over the lifetime of a cap, for example, to reflect amounts borrowed under an
887:
Caps based on an underlying rate (like a
Constant Maturity Swap Rate) cannot be valued using simple techniques described above. The methodology for valuation of CMS Caps and Floors can be referenced in more advanced papers.
771:
668:
269:
898:
recovers the price of the cap i.e. the net of the caplets still comes out to be the same. To illustrate: (Black
Volatilities) → (Flat Volatilities) : (15%,20%,....,12%) → (16.5%,16.5%,....,16.5%)
871:-maturity bond. Thus if we have an interest rate model in which we are able to value bond puts, we can value interest rate caps. Similarly a floor is equivalent to a certain bond call. Several popular
550:
201:
LIBOR rate with an expiry of 1 February 2007 struck at 2.5% with a notional of 1 million dollars. Next, if on 1 February the USD LIBOR rate sets at 3%, then you will receive the following payment:
158:
830:
86:
The purchaser of a cap will continue to benefit from any rise in interest rates above the strike price, which makes the cap a popular means of hedging a floating rate loan for an issuer.
923:
Assuming rates can't be negative, a Cap at strike 0% equals the price of a floating leg (just as a call at strike 0 is equivalent to holding a stock) regardless of volatility cap.
357:
The size of cap and floor premiums are impacted by a wide range of factors, as follows; the price calculation itself is performed by one of several approaches discussed below.
995:
442:
186:
317:
is the simultaneous purchase of an interest rate cap and sale of an interest rate floor on the same index for the same maturity and notional principal amount.
324:
The objective of the buyer of a collar is to protect against rising interest rates (while agreeing to give up some of the benefit from lower interest rates).
1126:
348:
Buyers can construct zero cost reverse collars when it is possible to find floor and cap rates with the same premiums that provide an acceptable band.
46:
is a derivative contract in which the buyer receives payments at the end of each period in which the interest rate is below the agreed strike price.
686:
586:
1144:
907:
Another important relationship is that if the fixed swap rate is equal to the strike of the caps and floors, then we have the following
1026:
206:
1002:
1150:
454:
1098:
1053:
301:. The buyer of the floor receives money if on the maturity of any of the floorlets, the reference rate is below the agreed
1120:
54:
payment for that period, thus the interest payments are effectively "capped" at 2.5% from the borrowers' point of view.
1072:
979:
110:
782:
1165:
345:
The buyer selects the index rate and matches the maturity and notional principal amounts for the floor and cap.
1139:
70:
in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed
31:
in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed
327:
The purchase of the cap protects against rising rates while the sale of the floor generates premium income.
1114:
843:
As negative interest rates became a possibility and then reality in many countries at around the time of
338:
is the simultaneous purchase of an interest rate floor and simultaneously selling an interest rate cap.
372:
The option seller must be compensated more for committing to a fixed-rate for a longer period of time.
364:
premiums are highest for in the money options and lower for at the money and out of the money options
28:
418:
197:
applies. For example, suppose that it is
January 2007 now and you own a caplet on the six month
273:
Customarily the payment is made at the end of the rate period, in this case on 1 August 2007.
427:
171:
1090:
876:
422:
90:
67:
1135:
8:
1130:
908:
844:
286:
933:
837:
190:
1094:
1068:
1049:
1020:
975:
879:
have this degree of tractability. Thus we can value caps and floors in those models.
74:. An example of a cap would be an agreement to receive a payment for each month the
35:. An example of a cap would be an agreement to receive a payment for each month the
872:
330:
A collar creates a band within which the buyer's effective interest rate fluctuates
969:
567:
396:
80:
50:
294:
83:. The purchase price of a cap is a one-off cost and is known as the premium.
1159:
1082:
1065:
Valuation of
Interest-Sensitive Financial Instruments: SOA Monograph M-FI96-1
580:
766:{\displaystyle d_{1}={\frac {\ln(F/K)+0.5\sigma ^{2}t}{\sigma {\sqrt {t}}}}}
1046:
Interest Rate Models - Theory and
Practice with Smile, Inflation and Credit
413:
The simplest and most common valuation of interest rate caplets is via the
302:
71:
32:
361:
The relationship between the strike rate and the prevailing 3-month LIBOR
414:
392:
385:
378:
663:{\displaystyle {1 \over \alpha }\left({\frac {P(0,t)}{P(0,T)}}-1\right)}
914:
Caps and floors have the same implied vol too for a given strike.
342:
The objective is to protect the bank from falling interest rates.
20:
264:{\displaystyle \$ 1M\cdot 0.5\cdot \max(0.03-0.025,0)=\$ 2500}
1043:
445:
298:
75:
36:
308:
545:{\displaystyle V=P(0,T)\left(FN(d_{1})-KN(d_{2})\right),}
417:. Under this model we assume that the underlying rate is
198:
785:
689:
589:
457:
430:
209:
174:
113:
89:
The interest rate cap can be analyzed as a series of
1087:
Valuation of fixed income securities and derivatives
1151:
Introduction to Caps, Floors, Collars and
Swaptions
1147:
Dr. Shing Hing Man, Thomson
Reuters Risk Management
824:
765:
662:
544:
436:
352:
263:
180:
152:
377:Prevailing economic conditions, the shape of the
96:In mathematical terms, a caplet payoff on a rate
1157:
974:. Global Professional Publishi. pp. 52–73.
448:expiring at t and paying at T has present value
402:floor premiums reveal the opposite relationship.
228:
126:
1062:
153:{\displaystyle N\cdot \alpha \cdot \max(L-K,0)}
825:{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {t}}}
583:of the rate. For LIBOR rates this is equal to
1081:
855:It can be shown that a cap on a LIBOR from
1048:(2nd ed. 2006 ed.). Springer Verlag.
836:volatility is known as the "Black vol" or
388:—caps will be more expensive than floors.
321:The cap rate is set above the floor rate.
309:Interest rate collars and reverse collars
882:
381:, and the volatility of interest rates.
1127:Martingales and Measures: Black's Model
1067:(1st ed.). John Wiley & Sons.
891:
1158:
1136:Bond Options, Caps and the Black Model
1044:Damiano Brigo, Fabio Mercurio (2001).
1025:: CS1 maint: archived copy as title (
276:
1145:Online Caplet And Floorlet Calculator
1115:Basic Fixed Income Derivative Hedging
967:
193:corresponding to the period to which
963:
961:
959:
957:
955:
953:
951:
949:
167:is the notional value exchanged and
57:
399:, the greater are the cap premiums.
13:
444:. Under this model, a caplet on a
255:
210:
14:
1177:
1108:
946:
863:is equivalent to a multiple of a
369:Premiums increase with maturity.
850:
391:the steeper is the slope of the
16:Type of interest rate derivative
1117:- Article on Financial-edu.com.
353:Valuation of interest rate caps
49:Caps and floors can be used to
1129:Dr. Jacqueline Henn-Overbeck,
988:
726:
712:
643:
631:
623:
611:
531:
518:
506:
493:
479:
467:
408:
249:
231:
147:
129:
1:
1140:University of Texas at Austin
1037:
336:reverse interest rate collar
7:
968:Coyle, Brian (2001-01-01).
681:is the standard normal CDF.
10:
1182:
927:
1063:David F. Babbel (1996).
939:
419:distributed log-normally
29:interest rate derivative
437:{\displaystyle \sigma }
181:{\displaystyle \alpha }
826:
767:
664:
546:
438:
265:
182:
154:
1166:Derivatives (finance)
971:Interest-rate Options
883:Valuation of CMS Caps
827:
768:
665:
547:
439:
266:
183:
155:
91:European call options
1121:Convexity Conundrums
892:Implied Volatilities
783:
687:
587:
455:
428:
315:interest rate collar
287:European put options
207:
172:
111:
1138:Dr. Milica Cudina,
1131:University of Basel
911:: Cap-Floor = Swap.
845:Quantitative Easing
283:interest rate floor
277:Interest rate floor
44:interest rate floor
39:rate exceeds 2.5%.
934:Interest rate swap
822:
763:
660:
542:
434:
261:
191:day count fraction
178:
150:
1100:978-1-883249-25-0
1055:978-3-540-22149-4
873:short-rate models
867:-expiry put on a
820:
761:
758:
647:
598:
64:interest rate cap
58:Interest rate cap
25:interest rate cap
1173:
1123:by Patrick Hagan
1104:
1089:(3rd ed.).
1078:
1059:
1031:
1030:
1024:
1016:
1014:
1013:
1007:
1001:. Archived from
1000:
992:
986:
985:
965:
877:Hull–White model
831:
829:
828:
823:
821:
816:
808:
807:
795:
794:
772:
770:
769:
764:
762:
760:
759:
754:
748:
744:
743:
722:
704:
699:
698:
669:
667:
666:
661:
659:
655:
648:
646:
626:
606:
599:
591:
551:
549:
548:
543:
538:
534:
530:
529:
505:
504:
443:
441:
440:
435:
270:
268:
267:
262:
187:
185:
184:
179:
159:
157:
156:
151:
1181:
1180:
1176:
1175:
1174:
1172:
1171:
1170:
1156:
1155:
1111:
1101:
1075:
1056:
1040:
1035:
1034:
1018:
1017:
1011:
1009:
1005:
998:
996:"Archived copy"
994:
993:
989:
982:
966:
947:
942:
930:
909:put–call parity
894:
885:
853:
815:
803:
799:
790:
786:
784:
781:
780:
753:
749:
739:
735:
718:
705:
703:
694:
690:
688:
685:
684:
627:
607:
605:
604:
600:
590:
588:
585:
584:
568:discount factor
525:
521:
500:
496:
486:
482:
456:
453:
452:
429:
426:
425:
411:
397:ceteris paribus
355:
311:
305:of the floor.
293:on a specified
285:is a series of
279:
208:
205:
204:
173:
170:
169:
112:
109:
108:
81:amortizing loan
60:
17:
12:
11:
5:
1179:
1169:
1168:
1154:
1153:
1148:
1142:
1133:
1124:
1118:
1110:
1109:External links
1107:
1106:
1105:
1099:
1079:
1074:978-1883249151
1073:
1060:
1054:
1039:
1036:
1033:
1032:
987:
980:
944:
943:
941:
938:
937:
936:
929:
926:
925:
924:
921:
920:
919:
912:
905:
904:
903:
893:
890:
884:
881:
875:, such as the
852:
849:
833:
832:
819:
814:
811:
806:
802:
798:
793:
789:
774:
773:
757:
752:
747:
742:
738:
734:
731:
728:
725:
721:
717:
714:
711:
708:
702:
697:
693:
682:
676:
670:
658:
654:
651:
645:
642:
639:
636:
633:
630:
625:
622:
619:
616:
613:
610:
603:
597:
594:
574:
553:
552:
541:
537:
533:
528:
524:
520:
517:
514:
511:
508:
503:
499:
495:
492:
489:
485:
481:
478:
475:
472:
469:
466:
463:
460:
433:
410:
407:
406:
405:
404:
403:
400:
389:
375:
374:
373:
367:
366:
365:
354:
351:
350:
349:
346:
343:
332:
331:
328:
325:
322:
310:
307:
295:reference rate
278:
275:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
177:
161:
160:
149:
146:
143:
140:
137:
134:
131:
128:
125:
122:
119:
116:
59:
56:
42:Similarly, an
15:
9:
6:
4:
3:
2:
1178:
1167:
1164:
1163:
1161:
1152:
1149:
1146:
1143:
1141:
1137:
1134:
1132:
1128:
1125:
1122:
1119:
1116:
1113:
1112:
1102:
1096:
1092:
1088:
1084:
1083:Frank Fabozzi
1080:
1076:
1070:
1066:
1061:
1057:
1051:
1047:
1042:
1041:
1028:
1022:
1008:on 2016-02-03
1004:
997:
991:
983:
981:9780852974421
977:
973:
972:
964:
962:
960:
958:
956:
954:
952:
950:
945:
935:
932:
931:
922:
916:
915:
913:
910:
906:
900:
899:
896:
895:
889:
880:
878:
874:
870:
866:
862:
858:
851:As a bond put
848:
846:
841:
839:
817:
812:
809:
804:
800:
796:
791:
787:
779:
778:
777:
755:
750:
745:
740:
736:
732:
729:
723:
719:
715:
709:
706:
700:
695:
691:
683:
680:
677:
675:is the strike
674:
671:
656:
652:
649:
640:
637:
634:
628:
620:
617:
614:
608:
601:
595:
592:
582:
581:forward price
578:
575:
573:
569:
566:) is today's
565:
561:
558:
557:
556:
539:
535:
526:
522:
515:
512:
509:
501:
497:
490:
487:
483:
476:
473:
470:
464:
461:
458:
451:
450:
449:
447:
431:
424:
420:
416:
401:
398:
394:
390:
387:
383:
382:
380:
376:
371:
370:
368:
363:
362:
360:
359:
358:
347:
344:
341:
340:
339:
337:
329:
326:
323:
320:
319:
318:
316:
306:
304:
300:
296:
292:
288:
284:
274:
271:
258:
252:
246:
243:
240:
237:
234:
225:
222:
219:
216:
213:
202:
200:
196:
192:
188:
175:
166:
144:
141:
138:
135:
132:
123:
120:
117:
114:
107:
106:
105:
103:
99:
94:
92:
87:
84:
82:
77:
73:
69:
65:
55:
52:
47:
45:
40:
38:
34:
30:
27:is a type of
26:
22:
1086:
1064:
1045:
1010:. Retrieved
1003:the original
990:
970:
886:
868:
864:
860:
856:
854:
842:
834:
775:
678:
672:
576:
571:
563:
559:
554:
412:
356:
335:
333:
314:
312:
303:strike price
290:
282:
280:
272:
203:
194:
168:
164:
162:
101:
97:
95:
88:
85:
72:strike price
63:
61:
48:
43:
41:
33:strike price
24:
18:
902:volatility.
838:implied vol
415:Black model
409:Black model
393:yield curve
386:yield curve
379:yield curve
1091:John Wiley
1038:References
1012:2016-01-30
918:arbitrage.
423:volatility
384:upsloping
297:, usually
100:struck at
68:derivative
813:σ
810:−
751:σ
737:σ
710:
650:−
596:α
510:−
432:σ
291:floorlets
256:$
238:−
226:⋅
220:⋅
211:$
176:α
136:−
124:⋅
121:α
118:⋅
1160:Category
1085:(1998).
1021:cite web
928:Compare
579:is the
189:is the
21:finance
1097:
1071:
1052:
978:
555:where
163:where
1006:(PDF)
999:(PDF)
940:Notes
446:LIBOR
421:with
299:LIBOR
241:0.025
76:LIBOR
66:is a
51:hedge
37:LIBOR
23:, an
1095:ISBN
1069:ISBN
1050:ISBN
1027:link
976:ISBN
776:and
570:for
259:2500
235:0.03
859:to
733:0.5
562:(0,
313:An
289:or
281:An
229:max
223:0.5
199:USD
127:max
104:is
62:An
19:In
1162::
1093:.
1023:}}
1019:{{
948:^
840:.
707:ln
395:,
334:A
1103:.
1077:.
1058:.
1029:)
1015:.
984:.
869:T
865:t
861:T
857:t
818:t
805:1
801:d
797:=
792:2
788:d
756:t
746:t
741:2
730:+
727:)
724:K
720:/
716:F
713:(
701:=
696:1
692:d
679:N
673:K
657:)
653:1
644:)
641:T
638:,
635:0
632:(
629:P
624:)
621:t
618:,
615:0
612:(
609:P
602:(
593:1
577:F
572:T
564:T
560:P
540:,
536:)
532:)
527:2
523:d
519:(
516:N
513:K
507:)
502:1
498:d
494:(
491:N
488:F
484:(
480:)
477:T
474:,
471:0
468:(
465:P
462:=
459:V
253:=
250:)
247:0
244:,
232:(
217:M
214:1
195:L
165:N
148:)
145:0
142:,
139:K
133:L
130:(
115:N
102:K
98:L
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.