> `!,VZ*t:*xh~xڝTMhA~3&VD,I!
=
nMXLuMHVxВ/"xC^V=YͳWEד`3IlH3̼}~O(]`MN $B>ΐc.@vAEws(τ`rF[_aXGga)&@GxIef_tϷ (%ˬrf#vb{FuVχfIFDpNbNuG;$C[M+:q{2DʰLeawYuE7>DD![)@7fY#S"9ر=:"7,LM}5ދꀰ\uR^^'% /ǋTq$mvz7Ƞދ" TgsJr2ʆ:Dq.#ܭ[/[Mf`5.]eU֜p.kODHHGuN;ݔR8fSWuV8?Ƞ 芮:CA8ņd'SbtTqkVsh;s:ow0I^2m"ma/{0{BLKV6sd}x `!ZI֤媝:3@hxڝTOQyۢ,%41)X54ՖdꪍmKHROƃ@ؓ5hƆǎvFp~ȉ1#nrŶm/ZٯghMz&MM%<:s+;"mM^gt$o[T^zHS2ʮco`!(ҹ$@BX63{V`U!liD&)/M1}ٱMɁ He]Heϓ~*Ww/i4<âƬ[)OfIIiCwMCHq2*%GUɸS6ҝ眭O$&ӧK9}vv,W_/^5Skf!_sBoX)͒:Raİs==acւq)=˻CDưqA'LOɖq9;p4ƳNƳQ
Tfo#6˃24ͨ
ى4(}[~ ល ZٚsqѩŲe)k,{U0Kq><`!l*Ft(Q$Xu#hxڝTMhA~3T
$B4hH*i^$`~lJaՍ&d7$Q)^ғxzr/5xrf^j빚gNE'ۉJ^OOmImc
PR9Iq$PAC7U\ܬ[BK!Zi8c@P؊"E\;5f o)9JWC0Ҁ1y*]Vfئ*f1ygkd1T5gMbDijEh؍Bٶ죂RonLvS4D}+>ҷy.("**RD""֙3gMv5͜39sf x??1Ln&!x #YZ%C1^nXjFV% 6 7Zň:f$lX {ӂe^Afϫߢl?~_TJdGg˕9^>>!Evu/.73&.ň<Umk2W翰/6aez1=c[Wz'^\Bx&1Ս\#V28Q>xkF;cѾY}(=R݉t>0OQ
y)$oE
~ "T{~#b?xw":'ꜯElm}Z=վ
;Xmc^f{ۮgPo"7ʌ>gI.OFg:ȓɲVFe%[Y7Vc>Kɜ&n짥_]zj7nY`!ΞQl ' ^;xY]h\E>gۻ7MZuV(Bu7IӆRRڴڇHkE"X
b((C_C$jL@ܙ3sw7uffswΜ{gh8Abۀ r/5]&j.aJԏa,2h売uK%[5/x?4
>mfnp6aӂl=̿?J0b
^d#cN^t{2%eԖ{ȎK!~qdgj\CqxMVf3xƥz
q1`'oljrLD+[ٷ%}ia=Q$$IӦ;v4",E&hl4Da6Oil%[ lyf!lr yja+aSHXj̫ ^#le¶Qvjc=Ow=PӁ^g/dE~e_DzoʾwxN5hA?q^81("hƕxShxYv:L0k=9Oa!",%vy5_,G'^\cjQZ\c}/woخRq\W^.}XXs'yH%mF4+I6%@# 0Hf$Md^6FB(gەN9rI!>6zkgKg]_wzG~[O>%PkVN;ž#uo?]Ot{3},\)4g,oAO"ˌ (CoLy<peVlɫv&;>PWx$3^XXRvQ{N0)ߍ]kރ9v_l>6
oN+kdxy:#A(
Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation
Equation Equation.DSMT40*MathType 5.0 Equation/00DArialngsRomanTT(uܖ0ܖ"DGaramondRomanTT(uܖ0ܖ DTimes New RomanTT(uܖ0ܖ0DWingdingsRomanTT(uܖ0ܖ
A.
@n?" dd@ @@``LD5l
21 "#%&'()+,./2442$,VZ*t:2$ZI֤媝:32$fFeOf2$l*Ft(Q
O$2$BIs$M,\2$ΞQl '0AA@8 w_I ʚ;lf8ʚ;g4dddd0Xppp@<4dddd k0Tu<4!d!d l0T80___PPT10
?
%76Early exercise and Monte Carlo obtaining tight bounds 7566RMark Joshi
Centre for Actuarial Sciences
University of Melbourne
www.markjoshi.comABermudan optionality /A Bermudan option is an option that be exercised on one of a fixed finite numbers of dates.
Typically, arises as the right to break a contract.
Right to terminate an interest rate swap
Right to redeem note early
We will focus on equity options here for simplicity but same arguments hold in IRD land.
BDZD[Why Monte Carlo?_Lattice methods are natural for early exercise problems, we work backwards so continuation value is always known.
Lattice methods work well for lowdimensional problems but badly for highdimensional ones.
Pathdependence is natural for Monte Carlo
LIBOR market model difficult on lattices
Many lower bound methods now exist, e.g. LongstaffSchwartzM
Buyer s priceHolder can choose when to exercise.
Can only use information that has already arrived.
Exercise therefore occurs at a stopping time.
If D is the derivative and N is numeraire, value is therefore
Expectation taken in martingale measure. A
0Justifying buyer s priceBuyer chooses stopping time.
Once stopping time has been chosen the derivative is effectively an ordinary pathdependent derivative for the buyer.
In a complete market, the buyer can dynamically replicate this value.
Buyer will maximize this value.
Optimal strategy: exercise whenSeller s priceSeller cannot choose the exercise strategy.
The seller has to have enough cash on hand to cover the exercise value whenever the buyer exercises.
Buyer s exercise could be random and would occur at the maximum with nonzero probability.
So seller must be able to hedge against a buyer exercising with maximal foresight.
AZA
0Seller s price continuedMaximal foresight price:
Clearly bigger than buyer s price.
However, seller can hedge.
!Hedging against maximal foresight""(Suppose we hedge as if buyer using optimal stopping time strategy.
At each date, either our strategies agree and we are fine
Or
1) buyer exercises and we don t
2) buyer doesn t exercise and we do
In both of these cases we make money!6D&D&The optimal hedge Buy one unit of the option to be hedged.
Use optimal exercise strategy.
If optimal strategy says exercise . Do so and buy one unit of option for remaining dates.
Pocket cash difference.
As our strategy is optimal at any point where strategy says do not exercise, our valuation of the option is above the exercise value.
6 4Rogers /HaughKogan methodEquality of buyer s and seller s prices says
for correct hedge Pt with P0 equals zero.
If we choose wrong , price is too low = lower bound
If we choose wrong Pt , price is too high= upper bound
Objective: get them close together. p0.C
JH,r}
Approximating the perfect hedge8If we know the optimal exercise strategy, we know the perfect hedge.
In practice, we know neither.
AndersonBroadie: pick an exercise strategy and use product with this strategy as hedge, rolling over as necessary.
Main downside: need to run subsimulations to estimate value of hedge
Main upside: tiny variance9Z9mImproving AndersonBroadienOur upper bound is
The maximum could occur at a point where D=0, which makes no financial sense.
Redefine D to equal minus infinity at any point out of the money. (except at final time horizon.)
Buyer s price not affected, but upper bound will be lower.
Added bonus: fewer points to run subsimulations at.
8Z8Provable suboptimality2Suppose we have a Bermudan put option in a BlackScholes model.
European put option for each exercise date is analytically evaluable.
Gives quick lower bound on Bermudan price.
Would never exercise if value < max European.
Redefine payoff again to be minus infinity.
Similarly, for Bermudan swaption. >1D
Breaking structures6Traditional to change the right to break into the right to enter into the opposite contract.
Asian tail note
Pays growth in FTSE plus principal after 3 years.
Growth is measured by taking monthly average in 3rd year.
Principal guaranteed.
Investor can redeem at 0.98 of principal at end of years one and two. @nZZncdNonanalytic break valuesTo apply Rogers/HaughKogan/AndersonBroadie/LongstaffSchwartz, we need a derivative that pays a cash sum at time of exercise or at least yields an analytically evaluable contract.
Asiantail note does not satisfy this.
Neither do many IRD contracts, e.g. callable CMS steepener.
> 6c e !Working with callability directly
We can work with the breakable contract directly.
Rather than thinking of a single cashflow arriving at time of exercise, we think of cashflows arriving until the contract is broken.
Equivalence of buyer s and seller s prices still holds, with same argument.
Algorithm model independent and does not require analytic break values.
OZOUpper bounds for callables Fix a break strategy.
Price product with this strategy.
Run a Monte Carlo simulation.
Along each path accumulate discounted cashflows of product and hedge.
At points where strategy says break. Break the hedge and Purchase hedge with one less break date, this will typically have a negative cost. And pocket cash.
Take the maximum of the difference of cashflows.*WZZWImproving lower boundsMost popular lower bounds method is currently LongstaffSchwartz.
The idea is to regress continuation values along paths to get an approximation of the value of the unexercised derivative.
Various tweaks can be made.
Want to adapt to callable derivatives.
2. The LongstaffSchwartz algorithm!!( Generate a set of model paths
Work backwards.
At final time, exercise strategy and value is clear.
At second final time, define continuation value to be the value on same path at final time.
Regress continuation value against a basis.
Use regressed value to decide exercise strategy.
Define value at second last time according to strategy
and value at following time.
Work backwards.0SPPPImproving LongstaffSchwartz
7We need an approximation to the exercise value at points where we might exercise.
By restricting domain, approximation becomes easier.
Exclude points where exercise value is zero.
Exclude points where exercise value less than maximal European value if evaluable.
Use alternative regression methodology, eg loess, *!LongstaffSchwartz for breakables""( Consider the Asian tail again.
No simple exercise value.
Solution (Amin)
Redefine continuation value to be cashflows that occur between now and the time of exercise in the future for each path.
Methodology is modelindependent.
Combine with upper bounder to get twosided bounds.
JKZzZVZZKzXEExample bounds for Asian tail Difference in bounds
ReferencesA. Amin, Multifactor cross currency LIBOR market model: implemntation, calibration and examples, preprint, available from http://www.geocities.com/anan2999/
L. Andersen, M. Broadie, A primaldual simulation algorithm for pricing multidimensional American options, Management Science, 2004, Vol. 50, No. 9, pp. 12221234.
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer Verlag, 2003.
M.Haugh, L. Kogan, Pricing American Options: A Duality Approach, MIT Sloan Working Paper No. 434001
M. Joshi, Monte Carlo bounds for callable products with nonanalytic break costs, preprint 2006
F. Longstaff, E. Schwartz, Valuing American options by simulation: a least squares approach. Review of Financial Studies, 14:113 147, 1998.
R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Economics 3, 125 144, 1976
L.C.G. Rogers: Monte Carlo valuation of American options, Mathematical Finance,
Vol. 12, pp. 271286, 2002
0PP2
h
9 _c 0` 3` ff>>\fg` J*T333` QYmx~3ft` \ғhEy`` cb^DDf`Y˵W` sg7xGr` K%ޯd{mG/` 33f>?" dd@,?nFd@ @ ` n?" dd@ @@``PR @ ` `<p>>`XP(
X
X
6ԁ #"`b `
>*
X
6쌘 #"`` `
@*xT ~
X"~\ {
X"{
X
cBBCDEFd@
bb
H
T W6Vw}\gFQ6<1++11 1L b6xQrq6\}N 
c
0
A
Q
g
S
6N KawF
bFy0 a*ly7lE;uz B

a
F
0!1<B<&lL
0
@`"T
X
cBC+DEFyd@
==gL6cI}\eA& w6m!W<!&<W!r6W}F6\}>68QNlX^XH8+Sgg@`"Jo
5
J
X
B!CDEF@
))\*l{FDNF
W% W
!!
!
6X
L
b x_ /aF6b}SA$l\\TX@`s"* T
u4
X
BC%DEF
ss&QW}k6D Q r 8
N
c
t
y
y
;t
Qi
gS
}8
T
:0\Q4*%%v*o(V}5q; S
<s}l\g}\,L
;n
&
Z}\[6 }WF<!e C
}hmW]
,t
B RxkqA,Rg@`"
{b
X
BCDEFy@
,,Lh6\}mWG
&a*zCxsh&jG]~5g6&LLZ\@`s"*
u(
X
BCDEFy@
__@%
Mf4@{ f0`lA}L\A+h
MB6ww FJl{
0V{J
*5KlGn a*:Pf\6/6KbfF
kP}5x}VvaP@@@`s"* =>
X
cBlCrDEF"d@
rlrl@`"~
X
6x " `}
T Click to edit Master title style!
!
X
6X #"``
@*$
X
0ԥ " `
RClick to edit Master text styles
Second level
Third level
Fourth level
Fifth level!
SB
Xs*h ? 3___PPT10i.
u%.+D=' =
@B + Streamb 0yqp\(
\xT ~
\"~\ {
\"{
\
cBBCDEFd@
bb
H
T W6Vw}\gFQ6<1++11 1L b6xQrq6\}N 
c
0
A
Q
g
S
6N KawF
bFy0 a*ly7lE;uz B

a
F
0!1<B<&lL
0
@`"T
\
cBC+DEFyd@
==gL6cI}\eA& w6m!W<!&<W!r6W}F6\}>68QNlX^XH8+Sgg@`"Jo
5
J
\
B!CDEF@
))\*l{FDNF
W% W
!!
!
6X
L
b x_ /aF6b}SA$l\\TX@`s"* T
u4
\
BC%DEF
ss&QW}k6D Q r 8
N
c
t
y
y
;t
Qi
gS
}8
T
:0\Q4*%%v*o(V}5q; S
<s}l\g}\,L
;n
&
Z}\[6 }WF<!e C
}hmW]
,t
B RxkqA,Rg@`"
{b
\
BCDEFy@
,,Lh6\}mWG
&a*zCxsh&jG]~5g6&LLZ\@`s"*
u(
\
BCDEFy@
__@%
Mf4@{ f0`lA}L\A+h
MB6ww FJl{
0V{J
*5KlGn a*:Pf\6/6KbfF
kP}5x}VvaP@@@`s"* =>
\
cBlCrDEF"d@
rlrl@`"~
\
< "F
T Click to edit Master title style!
!
\
0 " `
W#Click to edit Master subtitle style$
$
\
6 #"`` `
>*
\
6, #"`b
@*
\
6̘ #"`d `
@*B
\s*h ? 3___PPT10i.
u%.+D=' =
@B +0 0(
x
c$\p*
x
c$\` 0
H
0h ? 33___PPT10i.b%k~+D=' =
@B +
000(
x
c$0
X `}
x
c$DX `
H
0h ? 33___PPT10i.c%+D=' =
@B +
0@*(
x
c$`
X `}
r
SsX`0
H
0h ? 33___PPT10i.f%"++D=' =
@B ++
0B:P(
x
c$`
X `}
x
c$X@#
0
NA? ?XP
0 H
0h ? 33___PPT10i.f%@+D=' =
@B +\
0skh(
hr
h Sx[
X `}
r
h SPX
h
0
w%continuation value < exercise value
6%0nFZ
2&$H
h0h ? 3___PPT10i.%A+D=' =
@B +}
0l$(
lr
l S
X `}
r
l S@X `
H
l0h ? 3___PPT10i.%+D=' =
@B ++
0B:p(
pr
p SPߘ
X `}
x
p c$Xp
p0
TA? ?XpKH
p0h ? 3___PPT10i.%VV+D=' =
@B +}
0x$(
xr
x S
X `}
r
x S`X `
H
x0h ? 3___PPT10i.%p1+D=' =
@B +}
0$(
r
 S+
X `}
r
 SԖX `
H
0h ? 3___PPT10i.%:*+D=' =
@B ++
0B:(
r
S
X `}
x
c$X
0
TA? ?XuH
0h ? 3___PPT10i.%+D=' =
@B +}
0 $(
r
S<
X `}
r
SX `
H
0h ? 3___PPT10i.%S/+D=' =
@B ++
0B:0(
r
S$
X `}
x
c$8fX `
0
TA
? ?X I
H
0h ? 3___PPT10i.%1Q+D=' =
@B +}
0P$(
r
S
X `}
r
S/X `
H
0h ? 3___PPT10i.%^+D=' =
@B +}
0`$(
r
S/
X `}
r
S LX `
H
0h ? 3___PPT10i.%0l+D=' =
@B +}
0p$(
r
S
X `}
r
S*X `
H
0h ? 3___PPT10i.%+D=' =
@B +$
0$(
r
S%
X `}
r
SǖX `
H
0h ? 380___PPT10.% I$
0$(
r
SY
X `}
r
SpX `
H
0h ? 380___PPT10.%g$
0$(
r
S@
X `}
r
S@0X `
H
0h ? 380___PPT10.%pk$
0$(
r
SP.
X `}
r
S X `
H
0h ? 380___PPT10.&p8$
0$(
r
S
X `}
r
SX `
H
0h ? 380___PPT10.&$
0$(
r
SXL
X `}
L
r
S6LX `L
H
0h ? 380___PPT10.
&pU/$
0(
r
STL
X `}
L
r
S{X `
X
0A?P
H
0h ? 380___PPT10.
&F
0 F(
r
SL
X `}
TA ?XXH
0h ? 380___PPT10.&8~$
0@$(
r
S
X `}
r
SnX `
H
0h ? 380___PPT10.&'eExXkY?NR4k>u]XQ m*CLVҨq4ILA. iAVUaTXAsf6K%'=57=''}9XdHP(AMCpcbHCj*J>!`2QAxak*,,Pn=l[3ƟȨwc/^Ϩ5~Hl+<$C"f來_zm܇pGQ4QH`BTƙ'<Q< APؗȼr{vd.)_ݔ(,ՍRTA?Sj/וxUQ̹j^Yxg&;^
v"NC&H8eB%Gxja+̱z.ֹy[JY_A
t!4Zzwxpnqj5p
.HħEr&SI3xn*
k86I.0O@[[63KWu֙?A 22;
3`0<(d;ذ_a$ V<6G*%K쿚+l3xguQ zS3'{{
تjR;Atl@r,six%Vieg2jß)2i)&x@98ywtW<<ґ9K'Osvd M++qصkZpk(y=c蛦n@Su*ڦX6.l61EQp3'.ڜab_]8Ukpa9႑<}
,9MH04jZxr+#)CYn^n
mΩLR;k<QVvI0DB=3L}VVQema%y[KPV̤45tYL۶f>{Y+>q8*w5wҿtt>d*ߧpU&xY{lU?~غ! 2f$?F(['F7'Rhqݣ@cHWt@[<~fv=sEGxiQIs@<1 +lhOVD?tRa4Y%+Zwq쑁}GꜺZ3$d)I;b}ЎVY
и3c1{7ߣ߭k+S;7AR?lys@
.@l4S&ퟸ4?ZRO?s3lR?;d"ɇ^7\>DJ<̣xF@/$=Jz& $IJG~&ӥ,5n PF6?
0&@~0ͧ W#1^8D̓0b}Y_rq{&ע2w`e nIl9SWJPUqܖ/qr;82y@R#]bv,Bu5Ѩwlj]%T*!+tN17ik <VxmU;0+Ϭ3rVZkx}X m^Qkӛ];zRh֠I@g}3q2ٔ&&P4RX5צIWi`<^"nMEKdBItxy_"9rWwwG)aFs:,1ASaXYg C6a^X
/*Z^mO.OEAAyM]m+m7%]_AXy`0XvR*Aq<6DI2LI,)#+LqmWbt/\!̲y6z[ﲚ=;<(2H)2״u?uB\͝<];^r4*By.,Ykne.Xptպ!s\p3OuG9kx>U܀ MʱVEkPXI_푙Lv)A}*'uNIyDQceGnUmg5u;aе1ĲKp^~gc%P,ڹ{wC}vEzZUk
8Tk;+m@躗;b
&vymu=hiߊwYò(JE䳶(
)s2jx'(aQ^2rK?I~N?(xdޟdG)<tXfn7q/.n'ABSBeHEgmXbà6`0>k{X@>k.B{TM_27'ֈ{b6>02AgCƏ?fS1~x/bҺ{lx0wt5l?O=
#JCiJ?Sς&xXoh[U?.ZG״nĢC*KW' !sFM^m
H ~W}06h'yyMms}߹;ݷ+?Q`u5:/nPڨnummLkURtbi*lH`ʜ/jEy74g7X٥ShI<$gěkgeL#='`q
^,aE9,O,?UIOX5̷w7~~7f5.?
ҹ[aK*]X3s`ӱ@<IǞK"GѴ/O&"qlKNb568쓦Ti#ʸ]_b9
cl^ʴe敚zpX?r2"#NzK"gX9;A1l8Pն3_2ÜDb{z`uo.<_xV= ϱ\A^#gBtZE.tIdK>(&^L4q@ˑ3Ԡ.&>mR>q
a4t*?봷J$UƟ0^?m~={ _Pz+XQZS>TbSjg6 t\4˙0hǩg UCßI
9&t%L'oxFh~Du@WWx/vPgYov%_U.?cTKk;iWC3{^SV]nu
Y;TנN[ wj8AX2ĉK1RG52WjTVD#{?&^`͂MV4I&{Lck~#l!j u'ð3˕l,&4pZd{[:HqN$hFly?ѐ3ɸv_mlg\<ῐ',;ɷvoTߪoT)~YO>ɭ` m)6
MhHwH[
}\fX` RPוuisǯs;qC,w\k;Cm۹oޝ<&xY_L[Uι¥?26
n0e`$jf(_Lii>,= {1%NxƘ%}&≯,YX5FޮC`{9_wgȒ'A:(c`eT$`ڨʎÏNާ
5'U咢Y`,.L3=a7a
h:vgǿ}`=`/6ô`:f hQ<ޫXr@^ϯ#}!` ff}6?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrsuvwxyz{}~Root EntrydO)ڕ'PicturesCurrent User>SummaryInformation(t$UPowerPoint Document(;DocumentSummaryInformation8Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation
Equation Equation.DSMT40*MathType 5.0 Equation/00DArialngsRomanTTyܖ0ܖDGaramondRomanTTyܖ0ܖ DTimes New RomanTTyܖ0ܖ0DWingdingsRomanTTyܖ0ܖ
A.
@n?" dd@ @@``LD5l
21 "#%&'()+,./2442$,VZ*t:2$ZI֤媝:32$fFeOf2$l*Ft(Q
O$2$BIs$M,\2$ΞQl '0AA@8 w_I ʚ;lf8ʚ;g4dddd0(ppp@<4dddd k0Tx<4!d!d l0T80___PPT10
?
%76Early exercise and Monte Carlo obtaining tight bounds 7566RMark Joshi
Centre for Actuarial Sciences
University of Melbourne
www.markjoshi.comABermudan optionality /A Bermudan option is an option that be exercised on one of a fixed finite numbers of dates.
Typically, arises as the right to break a contract.
Right to terminate an interest rate swap
Right to redeem note early
We will focus on equity options here for simplicity but same arguments hold in IRD land.
BDZD[Why Monte Carlo?_Lattice methods are natural for early exercise problems, we work backwards so continuation value is always known.
Lattice methods work well for lowdimensional problems but badly for highdimensional ones.
Pathdependence is natural for Monte Carlo
LIBOR market model difficult on lattices
Many lower bound methods now exist, e.g. LongstaffSchwartzM
Buyer s priceHolder can choose when to exercise.
Can only use information that has already arrived.
Exercise therefore occurs at a stopping time.
If D is the derivative and N is numeraire, value is therefore
Expectation taken in martingale measure. A
0Justifying buyer s priceBuyer chooses stopping time.
Once stopping time has been chosen the derivative is effectively an ordinary pathdependent derivative for the buyer.
In a complete market, the buyer can dynamically replicate this value.
Buyer will maximize this value.
Optimal strategy: exercise whenSeller s priceSeller cannot choose the exercise strategy.
The seller has to have enough cash on hand to cover the exercise value whenever the buyer exercises.
Buyer s exercise could be random and would occur at the maximum with nonzero probability.
So seller must be able to hedge against a buyer exercising with maximal foresight.
AZA
0Seller s price continuedMaximal foresight price:
Clearly bigger than buyer s price.
However, seller can hedge.
!Hedging against maximal foresight""(Suppose we hedge as if buyer using optimal stopping time strategy.
At each date, either our strategies agree and we are fine
Or
1) buyer exercises and we don t
2) buyer doesn t exercise and we do
In both of these cases we make money!6D&D&The optimal hedge Buy one unit of the option to be hedged.
Use optimal exercise strategy.
If optimal strategy says exercise . Do so and buy one unit of option for remaining dates.
Pocket cash difference.
As our strategy is optimal at any point where strategy says do not exercise, our valuation of the option is above the exercise value.
6 4Rogers /HaughKogan methodEquality of buyer s and seller s prices says
for correct hedge Pt with P0 equals zero.
If we choose wrong , price is too low = lower bound
If we choose wrong Pt , price is too high= upper bound
Objective: get them close together. p0.C
JH,r}
Approximating the perfect hedge8If we know the optimal exercise strategy, we know the perfect hedge.
In practice, we know neither.
AndersonBroadie: pick an exercise strategy and use product with this strategy as hedge, rolling over as necessary.
Main downside: need to run subsimulations to estimate value of hedge
Main upside: tiny variance9Z9mImproving AndersonBroadienOur upper bound is
The maximum could occur at a point where D=0, which makes no financial sense.
Redefine D to equal minus infinity at any point out of the money. (except at final time horizon.)
Buyer s price not affected, but upper bound will be lower.
Added bonus: fewer points to run subsimulations at.
8Z8Provable suboptimality2Suppose we have a Bermudan put option in a BlackScholes model.
European put option for each exercise date is analytically evaluable.
Gives quick lower bound on Bermudan price.
Would never exercise if value < max European.
Redefine payoff again to be minus infinity.
Similarly, for Bermudan swaption. >1D
Breaking structures6Traditional to change the right to break into the right to enter into the opposite contract.
Asian tail note
Pays growth in FTSE plus principal after 3 years.
Growth is measured by taking monthly average in 3rd year.
Principal guaranteed.
Investor can redeem at 0.98 of principal at end of years one and two. @nZZncdNonanalytic break valuesTo apply Rogers/HaughKogan/AndersonBroadie/LongstaffSchwartz, we need a derivative that pays a cash sum at time of exercise or at least yields an analytically evaluable contract.
Asiantail note does not satisfy this.
Neither do many IRD contracts, e.g. callable CMS steepener.
> 6c e !Working with callability directly
We can work with the breakable contract directly.
Rather than thinking of a single cashflow arriving at time of exercise, we think of cashflows arriving until the contract is broken.
Equivalence of buyer s and seller s prices still holds, with same argument.
Algorithm model independent and does not require analytic break values.
OZOUpper bounds for callables Fix a break strategy.
Price product with this strategy.
Run a Monte Carlo simulation.
Along each path accumulate discounted cashflows of product and hedge.
At points where strategy says break. Break the hedge and Purchase hedge with one less break date, this will typically have a negative cost. And pocket cash.
Take the maximum of the difference of cashflows.*WZZWImproving lower boundsMost popular lower bounds method is currently LongstaffSchwartz.
The idea is to regress continuation values along paths to get an approximation of the value of the unexercised derivative.
Various tweaks can be made.
Want to adapt to callable derivatives.
2. The LongstaffSchwartz algorithm!!( Generate a set of model paths
Work backwards.
At final time, exercise strategy and value is clear.
At second final time, define continuation value to be the value on same path at final time.
Regress continuation value against a basis.
Use regressed value to decide exercise strategy.
Define value at second last time according to strategy
and value at following time.
Work backwards.0SPPPImproving LongstaffSchwartz
9We need an approximation to the unexercise value at points where we might exercise.
By restricting domain, approximation becomes easier.
Exclude points where exercise value is zero.
Exclude points where exercise value less than maximal European value if evaluable.
Use alternative regression methodology, eg loess>
*!LongstaffSchwartz for breakables""( Consider the Asian tail again.
No simple exercise value.
Solution (Amin)
Redefine continuation value to be cashflows that occur between now and the time of exercise in the future for each path.
Methodology is modelindependent.
Combine with upper bounder to get twosided bounds.
JKZzZVZZKzXEExample bounds for Asian tail Difference in bounds
ReferencesA. Amin, Multifactor cross currency LIBOR market model: implemntation, calibration and examples, preprint, available from http://www.geocities.com/anan2999/
L. Andersen, M. Broadie, A primaldual simulation algorithm for pricing multidimensional American options, Management Science, 2004, Vol. 50, No. 9, pp. 12221234.
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer Verlag, 2003.
M.Haugh, L. Kogan, Pricing American Options: A Duality Approach, MIT Sloan Working Paper No. 434001
M. Joshi, Monte Carlo bounds for callable products with nonanalytic break costs, preprint 2006
F. Longstaff, E. Schwartz, Valuing American options by simulation: a least squares approach. Review of Financial Studies, 14:113 147, 1998.
R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Economics 3, 125 144, 1976
L.C.G. Rogers: Monte Carlo valuation of American options, Mathematical Finance,
Vol. 12, pp. 271286, 2002
0PP2
h
9 _$
0$(
r
S#
X `}
#
r
S#X `#
H
0h ? 380___PPT10.&r 1
՜.+,0
$XOnscreen ShowThe University of Melbourne;Arial GaramondTimes New Roman
WingdingsStreamMathType 5.0 Equation7Early exercise and Monte Carlo obtaining tight bounds Bermudan optionalityWhy Monte Carlo?Buyers priceJustifying buyers priceSellers priceSellers price continued"Hedging against maximal foresightThe optimal hedgeRogers/HaughKogan method Approximating the perfect hedgeImproving AndersonBroadieProvable suboptimalityBreaking structuresNonanalytic break values"Working with callability directlyUpper bounds for callablesImproving lower bounds!The LongstaffSchwartz algorithmImproving LongstaffSchwartz"LongstaffSchwartz for breakablesExample bounds for Asian tailDifference in boundsReferencesFonts UsedDesign TemplateEmbedded OLE Servers
Slide Titles"_
0Mark JoshiMark Joshi