Chapter 7
The Pricing of Second Generation Exotics
by JuÈrgen Hakala, Ghislain Perisse and Tino Senge
After vanilla options and the ®rst generation exotics some more exotic options are
of special interest for some clients. Here we present a formula catalogue for
computing the Theoretical Value (TV) of such options in the Black±Scholes model.
7.1 Introduction
The pricing and hedging of the second generation exotic options in the Black±
Scholes model works the same way as in the case of ®rst generation exotics.
One takes a geometric Brownian motion with a risk-neutral drift and computes
the discounted expected value of the respective option payo€s. Computing the
expectation results in the TV of the option. Trading at the TV is subject to
some model risk. However, for second generation exotics, the bid-ask spreads
are usually wider. Here we outline the valuation of some exotics such as
forward-start options in Section 7.2, ratchet options in Section 7.3, power
options in Section 7.4, instalment options in Section 7.5, stairs options in
Section 7.6, compound options on a forward-start strategy in Section 7.7,
options on the minimum/maximum in Section 7.8 and their generalisation in
Section 7.9.
All options are valued in the model for the exchange rate
dSt ˆ St ‰…r d
r f †dt ‡ sdWt Š
…7:1†
7.2 Forward-start options
A forward-start plain vanilla option is an option where the strike of the option
is derived at some date in the future relative to the spot at that date. It can be
seen as an option where it is paid for now but will start at some time in the
future. Let us consider two dates TF and Te , TF < Te , where
TF is the forward-start date,
Te is the expiry date.
The holder of the option receives at time TF an option with expiration date Te .
The strike of this option will be ®xed at TF to aSTF , where a is a constant which
FOREIGN EXCHANGE RISK
58
Chapter 7
is speci®ed at inception t ˆ 0 of the trade. The payo€ of a forward-start option
is de®ned as
‡
…7:2†
f STe aSTF
7.2.1 Pricing
In order to price forward-start options we have to distinguish between the time
intervals ‰0; TF Š and ‰TF ; Te Š.
First we consider the value of the option during interval ‰TF ; Te Š. The price of a
call option (f ˆ 1) is given by
v…t† ˆ vBS …St ; aSTF ; Te
t; s1;2 ; r d1;2 ; r f1;2 †; t 2 ‰TF ; Te Š
…7:3†
where vBS …† is the value of an European call option with spot St , strike aSTF ,
and time to maturity equal to Te t.
We now go on pricing the option within interval ‰0; TF Š. We start considering
the value of the option at time TF . At this time the value is equal to an
ordinary European option. In case of a call option (f ˆ 1) we get
TF ; s1;2 ; r d1;2 ; r f1;2 †
v…TF † ˆ vBS …STF ; aSTF ; Te
f
ˆ er 1;2 …Te
TF †
STF N …d‡ †
ˆ STF vBS …1; a; Te
d
er 1;2 …Te
aSTF N …d †
…7:5†
TF ; s1;2 ; r d1;2 ; r f1;2 †
where
"
STF
1
‡ …r d1;2
d ˆ
p ln
s1;2 Te TF
aSTF
TF †
…7:4†
#
r f1;2 †…Te
TF † …7:6†
p
1
s1;2 Te TF
2
…7:7†
Using risk-neutral valuation the value of a forward-start call option at time
t ˆ 0 is given by
h d
i
…7:8†
v…0† ˆ IEQ1 e r1 T1 STF vBS …1; a; Te TF ; s1;2 ; r d1;2 ; r f1;2 †
d f
is a martingale.
where Q1 is the probability measure such that e …r1 r1 †t St
t2‰0;TF Š
Since vBS …1; a; Te TF ; s1;2 ; r d1;2 ; r f1;2 † is deterministic, we use the martingale
property and obtain
v0 ˆ e
rf1 T1
S0 vBS …1; a; Te
TF ; s1;2 ; r d1;2 ; r f1;2 †
…7:9†
More generally, for t 2 ‰0; TF Š, we have
vt ˆ e
ˆe
rf1 …TF t†
rf1 …TF
t†
St vBS …1; a; Te
vBS …St ; aSt ; Te
TF ; s1;2 ; r d1;2 ; r f1;2 †
TF ; s1;2 ; r d1;2 ; r f1;2 †
The same calculation can be done for a forward-start put option (f ˆ
…7:10†
…7:11†
1).
REMARK 7.1 The rates r d1 and r f1 are the instantaneous interest rates. These
rates are used for discounting the option value within interval ‰0; TF †.
REMARK 7.2 The rates r d1;2 and r f1;2 are forward rates for interval ‰TF ; Te Š.
These rates will be used in the option pricing formula. In the same way s1;2 is the
forward volatility for time period ‰TF ; Te Š.
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The Pricing of Second Generation Exotics
59
7.2.2 Greeks
As in pricing we have to distinguish between two time intervals ‰0; TF Š and
‰TF ; Te Š.
During the interval ‰TF ; Te Š the sensitivity parameters are the same as in the case
of a plain vanilla option.
Let us focus on interval ‰0; TF Š. Within this period of time the sensitivity
parameters delta, gamma and theta for call options are
t ˆ e
t
r f1 …TF t†
TF ; s1;2 ; r d1;2 ; r f1;2 †
vBS …1; a; Te
…7:12†
ˆ0
t ˆ
…7:13†
e
r f1 …TF t† f
r 1 St
TF ; s1;2 ; r d1;2 ; r f1;2 †
vBS …1; a; Te
…7:14†
In order to calculate vega (V) and rho we have to consider dependencies in the
forward rates and forward volatility.
First we recall the de®nition of the forward volatility. Let s1 be the volatility
for the interval ‰0; TF Š and let s2 be the volatility within ‰0; Te Š. The forward
volatility s1;2 within ‰TF ; Te Š is given by
s
s22 T2 s21 T1
…7:15†
s1;2 ˆ
Te TF
if the term under the square root is non-negative. Using
@s1;2
1
s1 T1
ˆ p p
Te TF s22 T2 s21 T1
@s1
@s1;2
1
s2 T2
ˆ p p
2
Te TF s2 T2 s21 T1
@s2
…7:16†
…7:17†
we can calculate the vegas
V s1 t ˆ
e
rf1 …TF t†
St
V s1;2 BS…1;a;Te
V s2 t ˆ e
rf1 …TF t†
s1 …TF t†
1

p p
Te TF s22 …Te t† s21 …TF t†
TF ;s1;2 ;r d1;2 ;r f1;2 †
s2 …Te t†
1

St p p
Te TF s22 …Te t† s21 …TF t†
V s1;2 BS…1;a;Te
!
!
…7:18†
…7:19†
TF ;s1;2 ;r d1;2 ;r f1;2 †
d=f
Next we recall the de®nition of the forward rate. Let r1 be the rates for
d=f
d=f
interval ‰0; TF Š and let r2 be the rates for interval ‰0; Te Š. The forward rates r1;2
are given by
d=f
d=f
r T
r1 TF
d=f
r1;2 ˆ 2 e
…7:20†
Te TF
Recalling
d=f
@r1;2
d=f
@r1
d=f
@r1;2
d=f
@r2
FOREIGN EXCHANGE RISK
ˆ
ˆ
TF
Te
TF
Te
Te
TF
…7:21†
…7:22†
60
Chapter 7
d=f
we get as derivatives with respect to the rates r1
f
TF t
rhor d BS…1;a;Te
rhord t ˆ e r 1 …TF t† St
1
1;2
Te TF
f
TF t
rhor f BS…1;a;Te
rhorf t ˆ e r 1 …TF t† St
1
1;2
Te TF
f
er 1 …TF
t†
…TF
t†St PBS …1; a; Te
TF ;s1;2 ;r d1;2 ;r f1;2 †
…7:23†
TF ;s1;2 ;r d1;2 ;r f1;2 †
TF ; s1;2 ; r d1;2 ; r f1;2 †
…7:24†
d=f
As derivatives with respect to the rates r2 we can derive
f
Te t
rhord t ˆ e r 1 …TF t† St
rhor d BS…1;a;Te TF ;s1;2 ;r d ;r f †
2
1;2
1;2 1;2
Te TF
f
Te t
rhor f BS…1;a;Te TF ;s1;2 ;r d ;r f †
rhor f t ˆ e r 1 …TF t† St
1;2 1;2
2
1;2
Te TF
…7:25†
…7:26†
REMARK 7.3 From Equation (7.12) it can be seen that in the case of a forwardstart option the deltas of calls and puts are always positive on time interval ‰0; TF Š.
7.3 Ratchet options
A ratchet option consists of a series of forward-start options. The strike for the
next exercise date is set relative to the spot at the previous exercise date.
Ratchet options can be presented as a sum of forward-start options. The
exercise date of the ®rst component will be the forward-start date of the second,
the exercise date of the second component will be the forward-start date of the
third, . . . . Sometimes this type of option is called moving strike option or
cliquet option.
Ratchet options can be priced as the a sum of forward-start options. The price
of an n-period Ratchet option is given by
vRatchet …f; t† ˆ
n
X
iˆ1
vFPV …f; t; TFi ; Tei ; ai †
…7:27†
where
TF1 ˆ t; TFi‡1 ˆ Tei ; 1 4 i 4 n
1
…7:28†
and vFPV is the price of a forward-start plain vanilla as presented in Section 7.2.
7.4 Power options
A power option pays o€ the square of a plain vanilla with strike KS . Since this
would in general make the power option very expensive, it is capped at a level
KC . The power call pays
2
IIfKS <STe <KC g STe KS ‡IIfKC 4 STe g …KC KS †2
…7:29†
the power put has payo€
IIfKC <STe <KS g KS
2
STe ‡IIfSTe 4 KC g …KS
KC †2
…7:30†
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61
with expiration date Te and delivery date Td 5 Te . We denote by t the valuation
date (horizon) and assume that the premium is paid at the premium value date
Tp .
We model the exchange rate by
dSt ˆ mt;Te St dt ‡ st;Te St dW
…7:31†
or equivalently by
STe ˆ St e…mt;Te
4
1 s2
2 t;Te
†…Te
t†‡sWTe
…7:32†
t
r ft;Te .
with mt;Te ˆ rdt;Te
Risk-neutral valuation leads to the price of the power call
d
v…t† ˆ ert;Tp …Tp t†
h d
IEt e rTp ;Td …Td
d
Tp †
IIfKS <STe <KC g STe
2
KS ‡IIfKC 4 STe g …KC
KS †2
i
d
ˆ ert;Tp …Tp t† rTp ;Td …Td Tp †
i
h
2KS IEt IIfKS <STe <KC g STe
IEt IIfKS <STe <KC g S2Te
h
i
h
i
‡ K2S IEt IIfKS <ST <KC g ‡ …KC KS †2 IEt IIfKC 4 ST g
e
e
…7:33†
De®ning
1
S
p ln t ‡ mt;Te
st;Te Te t
KS
1
St
4
ˆ
‡ mt;Te
p ln
st;Te Te t
KC
p
4
ˆ d 0KS ‡ st;Te Te t
p
4
ˆ d 0KC ‡ st;Te Te t
p
4
ˆ d 0KS ‡ 2st;Te Te t
p
4
ˆ d 0KC ‡ 2st;Te Te t
4
d 0KS ˆ
d 0KC
d1KS
d1KC
d 2KS
d 2KC
1 2
st;Te …Te
2
1 2
…Te
s
2 t;Te
t†
…7:34†
t†
…7:35†
…7:36†
…7:37†
…7:38†
…7:39†
we can compute the expected values and obtain for the price of the power
option
d
d
v…t† ˆ ert;Tp …Tp t† rTp ;Td …Td Tp †
n
2
N fd2KC
S2t e…2mt;Te ‡st;Te †…Te t† N fd2KS
2KS St emt;Te …Te t† N fd1KS
N fd1KC
N fd 0KC
‡ K2S N fd 0KS
o
‡ …KC KS †2 N fd 0KC
where the binary variable f takes the values ‡1 in case of a call and
of a put.
FOREIGN EXCHANGE RISK
…7:40†
1 in case
62
Chapter 7
7.5 Instalment options
An instalment option is an iterative compound option with
.
N working dates T1 , T2 , . . ., TN (T0 ˆ t is the horizon date),
.
N strikes, K1 , K2 , . . . ;KN (Ki ˆ KTi ),
.
N put±call indicators, f1 , f2 , . . . ;fN taking the values ‡1 in case of a call
and 1 in case of a put.
On each period ‰Ti ; Ti‡1 Š, the spot Si ˆ STi is modelled by
dSi ˆ mi
1;i Si
dt ‡ si
1;i Si
dW
…7:41†
4
where the numbers mi 1;i ˆ ri 1;i d ri 1;i f are the forward drifts and si 1;i the
forward volatilities of the asset, which we take to be
s
s2i Ti s2i 1 Ti 1
…7:42†
si 1;i ˆ
Ti Ti 1
r T ri 1 Ti 1
ri 1;i ˆ i i
…7:43†
Ti Ti 1
The quantities r di ,r fi and si denote the usual forward rates and volatilities.
For N ˆ 2, this option is a regular compound option as discussed in Chapter 9.
The valuation of instalment options can be carried out using ®nite di€erences.
We describe this in detail in Section 7.3 of Chapter 22. Here we show how to
determine the value of an instalment option by iterative integration.
We introduce the notation
1
Si
1
2
p ln ‡ mi‡1 ‡ si‡1 …Ti‡1
si‡1 Ti‡1 Ti
2
x
p
4 i
i
d 2 …x† ˆ d 1 …x† si‡1 Ti‡1 Ti
4
d i1 …x† ˆ
and denote by
V N …SN 1 ; KN † ˆ IETN
1
…fN …SN
ˆ fN SN 1 emN
1;N …TN
KN ††‡
TN
1†
Ti †
…7:44†
…7:45†
N fN dN
1 …KN †
fN KN N fN dN
2 …KN †
the value of an undiscounted vanilla option at time TN
maturity TN when the spot is at SN 1 .
1
with strike KN and
Introducing the two functionals
4
IOfi ˆ‡1 …F †…Si 1 † ˆ IE …F …Si † Ki †‡ jSi 1
… ‡1
y2
i
1
ˆ
F …Si …yi †† p e 2 dyi
2p
xi‡1
4
IOfi ˆ 1 …F †…Si 1 † ˆ IE …Ki F …Si ††‡ jSi 1
… xi 1
y2
i
1
F …Si …yi †† p e 2 dyi
ˆ
2p
1
…7:46†
…7:47†
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with
Si …yi † ˆ Si 1 e…mi
xi‡1
xi 1
1;i
1 s2
2 i 1;i
†…Ti
Ti
1 †‡si 1;i
p
Ti Ti 1 yi
…7:48†
ˆ inffyi jF …Si …yi †† ˆ Ki g
…7:49†
ˆ supfyi jF …Si …yi †† ˆ Ki g
…7:50†
we can write down the price of an instalment option in the form
v…t† ˆ IOf1 . . . IOfN 1 …V N † . . .
…7:51†
7.6 Stairs options
A stairs option is an option which is working on N periods ‰t; T1 Š
‰T1 ; T2 Š; :::;‰TN 1 ; TN Š, with one, two or no knock-out barrier(s) on each period
and ®nal time payo€
‡
…7:52†
f STN K
with a strike K.
On each period ‰Ti 1 ; Ti Š we de®ne a lower barrier Li and a higher barrier Hi .
The asset spot is modelled by Equations (7.41), (7.42) and (7.43). Let Si denote
STi with T0 ˆ t. In the sequel we recall the formulae for values of vanilla and
barrier options as used in the setup of this section. For a derivation see
Chapters 1 and 6.
7.6.1 Last period functionals
Let us introduce
1
Si
1
2
p ln ‡ mi‡1 ‡ si‡1 …Ti‡1
si‡1 Ti‡1 Ti
2
x
p
4 i
i
d 2 …x† ˆ d 1 …x† si‡1 Ti‡1 Ti
1
x
1
4
2
i
d 3 …x† ˆ
p ln ‡ mi‡1 ‡ si‡1 …Ti‡1
si‡1 Ti‡1 Ti
Si
2
p
4 i
i
d 4 …x† ˆ d 3 …x† si‡1 Ti‡1 Ti
4
d i1 …x† ˆ
Ti †
…7:53†
Ti †
…7:54†
…7:55†
…7:56†
No barrier The value function of a non-discounted vanilla option is given by
V N …SN 1 ; K† ˆ IETN 1 …f…SN K††‡
ˆ fSN 1 emN 1;N …TN TN 1 † N fdN
fKN fdN
…7:57†
1 …K†
2 …K†
One barrier The value function of a non-discounted up-and-out call is given by
h
i
UOCN …SN 1 ; K† ˆ IETN 1 …SN K†‡ IIfmax‰TN 1 ;TN Š Ss <HN g
KN dN
ˆ SN 1 emN 1;N …TN TN 1 † N dN
1 …K†
2 …K†
N
SN 1 emN 1;N …TN TN 1 † N dN
1 …HN † ‡ KN d2 …HN †
2mN 1;N ‡1
HN s2N 1;N
mN 1;N …TN TN 1 †
SN 1 e
SN 1
2mN 1;N 1
2
HN s2N 1;N
N HN
N
†
N d3 …K† ‡ K
N d3 …
K
SN 1
2 HN
N dN
†
N dN
…7:58†
4…
4 …HN †
K
FOREIGN EXCHANGE RISK
64
Chapter 7
The value function of a non-discounted down-and-out put is given by
DOPN …SN 1 ; K† ˆ IETN
1
h
…K
SN †‡ IIfinf‰TN
SN 1 emN
1;N …TN
TN
1†
‡ SN 1 emN
1;N …TN
TN
1†
ˆ
SN 1 e
mN
1;N …TN
TN
N
1†
N
i
Ss >LN g
1 ;TN Š
dN
dN
1 …K† ‡ KN
2 …K†
dN
KN dN
1 …LN †
2 …LN †
LN
SN 1
2m2N
s
1;N
‡1
N 1;N
2 2mN 1;N
LN s2N 1;N
N LN
N
N d3
N d3 …K† ‡ K
K
SN 1
L2N
N
N dN
…
…L
†
†
N
d
N
4
4
K
Two barrier
given by
1
…7:59†
The value function of a non-discounted double-knock-out call is
DKOC…STN †
"
‡1
X
e …lN ‡sN
ˆ STN 1
1;N †
p
TN TN 1 …2nALHN †‡mN
1;N …TN
TN
1†
nˆ 1
n N A HN
N AK
‡1
X
p
TN TN 1 ‡ 2nALHN
…lN ‡ sN
…lN ‡ sN
e…lN ‡sN
1;N †
1;N †
o
p
TN TN 1 ‡ 2nALHN
1;N †
p
TN TN 1 …2AHN ‡2nALHN †‡mN
nˆ 1
n N
…AHN ‡ 2nALHN ‡ …lN ‡ sN
N AK
"
K
‡1
X
……lN ‡ sN
e
N AK
lN
TN
1†
p TN TN 1 †
1;N †
#
o
p
TN 1 ‡ 2AHN ‡ 2nALHN †
1;N † TN
p
lN TN TN 1 …2nALHN †
nˆ 1
1;N …TN
n N AHN
o
p
TN TN 1 ‡ 2nALHN
lN
‡1
X
p
TN TN 1 ‡ 2nALHN
elN
p
TN TN 1 …2AHN ‡2nALHN †
nˆ 1
n p N
…AHN ‡ 2nALHN ‡ lN TN TN 1 †
N AK
o
p
…lN TN TN 1 ‡ 2AHN ‡ 2nALHN †
#
…7:60†
and similarly for the non-discounted double-knock-out put
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65
DKOP…STN †
"
‡1
p
X
e lN TN TN 1 …2nALHN †
ˆK
nˆ 1
n p
N AK lN TN TN 1 ‡ 2nALHN
o
p
N ALN lN TN TN 1 ‡ 2nALHN
‡1
X
e lN
p
TN TN 1 …2AHN ‡2nALHN †
nˆ 1
n N AK
N ALN
"
‡1
X
St
p TN TN 1 †
#
o
p
…lN TN TN 1 ‡ 2AHN ‡ 2nALHN †
…2AHN ‡ 2nALHN ‡ lN
e
…lN ‡sN
p
TN TN 1 …2nALHN †‡mN
1;N †
1;N …TN
TN
1†
nˆ 1
n p
N AK …lN ‡ sN 1;N † TN TN 1 ‡ 2nALHN
o
p
N ALN …lN ‡ sN 1;N † TN TN 1 ‡ 2nALHN
‡1
X
e…lN ‡sN
1;N †
p
TN TN 1 …2AHN ‡2nALHN †‡mN
nˆ 1
n N AK
N AL
1;N …TN
1†
p TN TN 1 †
#
o
p
TN 1 ‡ 2AHN ‡ 2nALHN †
1;N † TN
…2AHN ‡ 2nALHN ‡ …lN ‡ sN
……lN ‡ sN
TN
1;N †
…7:61†
with
sN
ln SNK 1
p
TN 1
1;N TN
…7:62†
sN
ln SLNN 1
p
TN 1
1;N TN
…7:63†
sN
ln SHNN1
p
TN 1
1;N TN
…7:64†
sN
ln HLNN
p
TN 1
1;N TN
…7:65†
4
AK ˆ
4
ALN ˆ
4
AHN ˆ
4
ALHN ˆ
7.6.2 Within period functionals
No barrier We de®ne the functional
Qi …F †…Si 1 † ˆ IETi 1 ‰F …Si †Š
… ‡1
1
F …Si …yi †† p e
ˆ
2p
1
y2
i
2
dyi
…7:66†
Ti
…7:67†
One barrier We de®ne for s 2 ‰Ti ; Ti‡1 Š
1
4
Zli i ˆ
FOREIGN EXCHANGE RISK
si
1;i
ln
Si
ˆ li …s
Si 1
Ti 1 † ‡ Ws
1
66
Chapter 7
1
4
Ai … y† ˆ
4
li ˆ
si
mi
si
1;i
ln
1;i
1;i
y
Si 1
1
si
2
…7:68†
…7:69†
1;i
and use the joint density
1
IPQZ f max Zls 5 A and Zls 2 ‰y; y ‡ dyŠg ˆ p e
s2‰t;t‡TŠ
2pT
…2A y†2
2T
eyl
1 l2 T
2
dy
…7:70†
We now de®ne the up-and-out functional by
4
UOi …F †…Si 1 † ˆ IETi
1
ˆ IETi
1
h
h
F …Si †IIfsup‰Ti
i
1 ;Ti Š
S<Hi g
i
F …Si †IIfSi <Hi g IIfsup‰Ti 1 ;Ti Š S<Hi g
ˆ IETi 1 F …Si †IIfSi <Hi g
h
i
IETi 1 F …Si †IIfSi <Hi g IIfsup‰Ti 1 ;Ti Š S 5 Hi g
ˆ IETi 1 F …Si †IIfSi <Hi g
h
li
IETQiZ 1 F …Si 1 esi 1;i Zi †IIfZli <A …H †g IIfmax
i
i
‰Ti
i
… d i2 1 …Hi †
y2
i
1
ˆ
F …Si …yi ††c p e 2 dyi
2p
1
2m2i 1;i 1 … d i 1 …Hi †
4
Hi si 1;i
1
F …Si …yi †† p e
Si 1
2p
1
i
l
Zi
1 ;Ti Š i
y2
i
2
dyi
5 Ai …Hi †g
…7:71†
where in the ®rst integral
Si …yi † ˆ Si 1 e…mi
1;i
1 s2
2 i 1;i
†…Ti
Ti
†…Ti
Ti
1 †‡si 1;i
p
Ti Ti 1 yi
…7:72†
and in the second integral
Si …yi † ˆ
H2i …mi
e
Si 1
1;i
1 s2
2 i 1;i
1 †‡si 1;i
p
Ti Ti 1 yi
…7:73†
Similarly we de®ne the down-and-out functional by
h
i
DOi …F †…Si 1 † ˆ IETi 1 F …Si †IIfinf‰Ti 1 ;Ti Š S>Li g
h
i
ˆ IETi 1 F …Si †IIfSi >Li g IIfinf‰Ti 1 ;Ti Š S>Li g
h
i
ˆ IETi 1 F …Si †IIfSi >Li g
IETi 1 F …Si †IIfSi >Li g IIfinf‰Ti 1 ;Ti Š S 4 Li g
ˆ IETi 1 F …Si †IIfSi >Li g
h
i
li
IETQiZ 1 F …Si 1 e si 1;i Zi †IIfZ li < A …L †g IIfmax
li
5 Ai …Li †g
i i
‰Ti 1 ;Ti Š Zi
i
… ‡1
y2
i
1
F …Si …yi †† p e 2 dyi
ˆ
i
1
2p
d 2 …Li †
2m2i 1;i 1 … ‡1
y2
i
Li si 1;i
1
F …Si …yi †† p e 2 dyi
…7:74†
Si 1
2p
d i4 1 …Li †
RISK BOOKS
The Pricing of Second Generation Exotics
67
where in the ®rst integral
Si …yi † ˆ Si 1 e…mi
1;i
1 s2
2 i 1;i
†…Ti
Ti
†…Ti
Ti
1 †‡si 1;i
p
Ti Ti 1 yi
…7:75†
and in the second integral
Si …yi † ˆ
L2i …mi
e
Si 1
1 s2
2 i 1;i
1;i
1 †‡si 1;i
p
Ti Ti 1 yi
…7:76†
Two barriers The distribution of STi conditioned on the event that the path
reached neither an upper limit Hi nor a lower one Li on ‰Ti 1 ; Ti Š is known to
be
e
1 l2 …T
i
2 i
Ti
1 †‡ s
ST
i
Ti 1
ln S
0
!2 1
S
1
1
H
p 4exp@
ln Ti ‡ 2n ln i A
2
2s
…T
T
†
S
Li
2p
i
i 1
Ti 1
nˆ 1
i 1;i
0
1
!2 3
2
Hi
1
H
‡ 2n ln i A5IIfLi <STi <Hi g
exp@
ln
2s2i;i 1 …Ti Ti 1 †
STi STi 1
Li
‡1
X
2
li
i 1;i
…7:77†
We de®ne the double-knock-out functional by
h
i
4
…7:78†
DKOi …F †…Si 1 † ˆ IETi 1 F …Si †IIfLi <inf‰Ti 1 ;Ti Š S<sup‰Ti 1 ;Ti Š S<Hi g
… Hi
ST
li
1 l2 …T T
ln S i
i
i 1 †‡ s
Ti 1
i 1;i
ˆ
F …Si †e 2 i
Li
2 0
!2 1
‡1
X
STi
1 4 @
1
Hi A
p exp
‡ 2n ln
ln
2s2i 1;i …Ti Ti 1 †
STi 1
Li
2p
nˆ 1
0
1
3
!2
2
H
1
H
i
i
A5IIfL <S <H g
‡ 2n ln
exp@
ln
i
Ti
i
2s2i;i 1 …Ti Ti 1 †
STi STi 1
Li
… AH p 1 2
p
i
ˆ
F Si 1 esi 1;i Ti Ti 1 x e 2 li …Ti Ti 1 †‡li Ti Ti 1 x
ALi
1
1
2
p exp
…x 2nALHi †
2
2p
nˆ 1
1
2
dx
… x ‡ 2AHi ‡ 2nALHi †
exp
2
‡1 h p
X
ˆ
eli Ti Ti 1 …2nALHi †
‡1
X
nˆ 1
… AH
i
p
…li Ti Ti 1 ‡2nALHi †
F Si
p
ALi …li Ti Ti 1 ‡2nALHi †
p
li Ti Ti 1 …2AHi ‡2nALHi †
…
Hi
Li
2n !
e
p
…AHi ‡li Ti Ti 1 ‡2nALHi †
p
ALi …li Ti Ti 1 ‡2…AHi ‡2nALHi ††
F Si
Hi
Si 1
where in the integrals
Si …yi † ˆ Si 1 e…mi
FOREIGN EXCHANGE RISK
1;i
1 s2
2 i 1;i
†…Ti
Ti
1
p e
2p
1 †‡si 1;i
2 y2
i
2
Hi
Li
p
Ti Ti 1 yi
dyi
2n !
1
p e
2p
y2
i
2
dyi
…7:79†
68
Chapter 7
7.6.3 Price
We introduce ST ai i , where ai is taken from the set ai ˆ fLi ; Hi g, and consider
the following four cases
.
If Li ˆ 0, ST
.
If Hi ˆ ‡1, ST ai i ˆ DOi (Moreover, if i ˆ N, ST aNN ˆ DOPN ),
.
If Li ˆ 0 and Hi ˆ ‡1, ST ai i ˆ Qi (Moreover, if i ˆ N, ST aNN ˆ V N ),
.
If Li 6ˆ 0 and Hi 6ˆ ‡1, ST ai i ˆ DKOi (Moreover, if i ˆ N, ST aNN ˆ DKOPN
or DKOCN ).
ai
i
ˆ UOi (Moreover, if i ˆ N, ST aNN ˆ UOCN ),
Consequently each stair product is described by fai ji ˆ 1; :::; Ng and its price is
v…t† ˆ e ra …TN t† ST a11 . . . ST aNN
…7:80†
7.7 Compound on forward-start strategy
This product is a variant of the compound option on a strategy which will
include a strategy of forward-starting options all ®xing at the maturity of the
root option.
A compound option's intrinsic value is the di€erence of the strike and the
Market Value (MV) of the underlying option strategy
…f…MV
K††‡
…7:81†
This means that the holder of the options receives the underlying option
strategy for the price K at the maturity of the root option on exercise of the
root option. The relation is valid for both the option strategy being forwardstarting or a regular ®xed strategy.
In order to price a compound option on a forward-starting option strategy we
model the exchange rate by
dSt ˆ mt St dt ‡ st St dW
…7:82†
Next we consider the horizon date Th , the value date for the horizon Tv , the
maturity of the root option Tm , and the delivery date for the strike Td . The
®xing date for the forward-start strategy is the same as the maturity Tm . For
each option of the strategy there can be a di€erent maturity T and delivery Tid .
The product has a price
"
!!‡ #
N
X
Th
rTv ;Td …Td Tv †
…7:83†
f
P…STim † K
v…Th † ˆ IE e
iˆ1
We assume that the forward-start strategy is ®xed at maturity of the root option
and hence depends only on the spot at maturity. In particular, the price of a
forward-start option at the ®xing date is homogeneous of order one in the spot
(see Section 7.2) and can be written as
v…STm † ˆ STm v…1†
…7:84†
RISK BOOKS
The Pricing of Second Generation Exotics
69
In such a case the price of the compound is given by
v ˆ vf
vf ˆ
N
X
iˆ0
!
K
STh ; ; Tm ; Td ; rdTv Td ; r fTv Td ; sTh Tm vf
Pf
…7:85†
vf …1; ~
a; Tm ; Td ; T im ; T id ; r dTd T i ; r fTd T i ; sTm T im †
…7:86†
d
d
where vf denotes the value function of a plain vanilla, vf the unity-price of the
forward-start strategy at ®xing date and ~
a a vector of ®xing parameters relative
to the spot at maturity using the corresponding forward rates and volatilities.
This formula is valid for all strategies with a positive forward-start unity-price
bounded away from zero.
Examples of forward-start options which have a homogeneous price of order
one in the ®xing spot are
.
plain vanilla options with the strike ®xed with K ˆ aS,
.
single barrier options with the strike ®xed with K ˆ aS and the barrier ®xed
with B ˆ bS,
.
double barrier options with the strike ®xed with K ˆ aS and the barriers
®xed with L ˆ bS and H ˆ gS.
After maturity of the root option the product is either worthless (if expired) or
the sum of the ®xed forward-start options (if exercised).
7.8 Options on the minimum/maximum
This section concerns the products whose underlying is the maximum or
minimum of a basket of spots or exotic products. In the ®rst part, we introduce
the forward contracts on the maximum or the minimum of a basket of
currencies.
7.8.1 Best-of and worst-of currencies forwards
Best-of forward Let us consider n exchange rates with the same base currency
and n di€erent foreign currencies with their normalisers Ni . The payo€ of this
product is
S
S
1
…7:87†
max 1 ; ::; n
N1
Nn
We model the spots by
dSi ˆ mi Si dt ‡ si Si dW yi
…7:88†
where the Brownian increments dW yi are correlated (ie < dW yi ; dW yj >
ˆ …
T †ij dt ˆ rij si sj dt). After a Cholesky decomposition, we can rewrite this as
dSi ˆ mi Si dt ‡
FOREIGN EXCHANGE RISK
N
X
jˆ1
ij dWj
…7:89†
70
Chapter 7
where the Brownian increments dWi are independent. This implies
!
N
X
1 2
Si …T† ˆ Si …t† exp
mi
ij dWj
s …T t† ‡
2 i
jˆ1
…7:90†
In the following calculations we use the replacement Wi ˆ …T t†xi . The price is
given by
S1 …T†
Sn …T†
t
rpd …Td Tp †
1
max
; ::;
v…t† ˆ IE e
N1
Nn
ˆ e rpd …Td Tp † IEt S1 …T†IIfS1 …T†>S2 …T†;::;Sn …T†g ‡ . . .
‡ IEt Sn …T†IIfSn …T†>S1 …T†;::;Sn 1 …T†g
1
…7:91†
For the ®rst expectation we get
IEt S1 …T†IIfS1 …T†>S2 …T†;::;Sn …T†g t
… ‡1
1 2
s1 …T
exp m1
ˆ S1 …t†
2
x1 ;::;xn ˆ 1
1
IIfS1 …T†>S2 …T†;...;S1 …T†>Sn …T†g pn exp
2p
p
t† ‡ s1 T tx1
1 2
x1 ‡ :: ‡ x2n dx1 . . . dxn
2
…7:92†
We solve this integral by ®rst working on the region where S1 > S2 and hence
d 12 …x1 † > x2 with
p
S1 …t†
2
2
1
ln
‡
m
m
…s
s
†
…T
t†
‡
T t…
11 21 †x1
1
2
1
2
4
2
…t†
S
2
p
d12 …x1 † ˆ
…7:93†
22 T t
Moreover, S1 > S3 gives d 13 …x1 ; x2 † > x3 with
4
d 13 …x1 ; x2 † ˆ
ln SS13 …t†
‡ m1
…t†
m3
1
2
…s21
p
t† ‡ T t……
11
p
33 T t
s23 † …T
31 †x1 †
32 x2
(7.94)
Again, S1 > S4 gives d 14 …x1 ; x2 ; x3 † > x4 with
4
d4 …x1 ; x2 ; x3 † ˆ
ln SS14 …t†
‡
m1
…t†
m4
1 2
…s1
2
s24 † …T
p
t† ‡ T t……
11
p
44 T t
41 †x1 †
42 x2
(7.95)
Continuing this way, we get the price of the forward contract
… d i2 …xi †
… ‡1
… d i2 …x1 ;...;xn
n
p
X
1 2
Si …t†
e…mi 2 si †…T t†‡si T txi
...
v…t† ˆ e rpd …Td Tp †
xi ˆ 1
iˆ1
1
N …d i4 …x1 ; . . . ; xn 1 †† pn exp
2p
x1 ˆ 1
1 2
x ‡ . . . ‡ x2n
2 1
1
xn
dx1 . . . dxn
1ˆ
xi ˆ 1
1
N … d i4 …x1 ; . . . ; xn 1 †† pn exp
2p
x1 ˆ 1
1 2
x ‡ . . . ‡ x2n
2 1
1
1
1
Worst-of forward In the case of a worst-of forward, the value is
S1
Sn
rpd …Td Tp † t
v…t† ˆ e
IE 1 min
;...;
N1
Nn
… d i2 …xi †
…
…
n
‡1
p
X
1 2
ˆ e rpd …Td Tp † 1
Si …t†
e…mi 2 si †…T t†‡si T txi
...
iˆ1
43 x3
2†
1 …7:96†
d i2 …x1 ;...;xn
xn
1ˆ
2†
1
dx1 . . . dxn 1 …7:97†
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The Pricing of Second Generation Exotics
71
7.8.2 Best-of call and worst-of put
Best-of call
The value is given by
v…t† ˆ e
rpd …Td Tp †
S1
Sn
IE
max
; ::;
N1
Nn
‡ 1
t
…7:98†
De®ning for i ˆ 1; . . . ; n
4
d i1 ˆ
ln NSii ‡ mi 12 s2i …T
p
T tsi
t†
…7:99†
we obtain for the value
v…t† ˆ e
rpd …Td Tp †
n
X
Si …t†
… ‡1 e…mi
1 s2
2 i
†…T
p
t†‡si T txi
xi ˆd i1
iˆ1
1
N …d i4 …x1 ; . . . ; xn 1 †† pn exp
2p
1 2
x1 ‡ . . . ‡ x2n
2
1
… d i2 …xi †
1
x1 ˆ 1
...
… d i2 …x1 ;...;xn
xn
1ˆ
dx1 . . . dxn
2†
1
…7:100†
1
We list a numerical example in Table 7.1.
Table 7.1
Values of a one-year best-of call with spot US$/DM ˆ 1:6573, s ˆ 10:7%,
r d ˆ 3:1953%, r f ˆ 5:0223% and £/DM ˆ 2:754173, s ˆ 8:5%,
r d ˆ 3:1953%, r f ˆ 5:4923% for different numbers of steps using Gauss±
Legendre integration. The normalisers are the spot values, the strike is 1. The
Monte-Carlo simulation used 10 000 000 paths.
Number of Steps
Value
49
417.7713
99
416.7085
149
416.7879
199
416.8573
249
416.6880
299
416.8171
349
416.7681
399
416.7847
Monte Carlo
416.8990
Worst-of put Similarly, we obtain for the value of a worst-of put
v…t† ˆ e
ˆe
‡ S1
Sn
IE
1 min
; ::;
N1
Nn
i
…
…
n
d
p X
1
1 2
Tp †
Si …t†
1 e…mi 2 si †…T t†‡si T txi
rpd …Td Tp †
rpd …Td
t
iˆ1
xi ˆ 1
1
N … d i4 …x1 ; . . . ; xn 1 †† pn exp
2p
x1 ˆ 1
…
...
d i2 …x1 ;...;xn
xn
1 2
2
x1 ‡ . . . ‡ xn 1 dx1 . . . dxn
2
We list a numerical example in Table 7.2.
FOREIGN EXCHANGE RISK
d i2 …xi †
1ˆ
1
2†
1
…7:101†
72
Table 7.2
Chapter 7
Values of a one-year worst-of put with spot US$/DM ˆ 1:6573, s ˆ 10:7%,
r ˆ 3:1953%, r ˆ 5:0223%, spot £/DM ˆ 2:754173, s ˆ 8:5%,
r d ˆ 3:1953%, r f ˆ 5:4923% and spot Sfr/DM ˆ 1:211774, s ˆ 5%,
r d ˆ 3:1953%, r f ˆ 1:6588 for different numbers of steps using GaussLegendre integration. The normalisers are the spot values, the strike is 1. The
Monte-Carlo simulation used 25 000 000 paths.
Number of Steps
Value
9
605.7585
19
696.4858
29
699.7040
39
699.8150
49
699.5770
59
699.3771
69
699.0469
99
699.0175
Monte Carlo
699.0960
7.9 Generalised options on the minimum/maximum
We discuss the pricing of generalised options on the minimum/maximum.
Generalised means in this context an arbitrary set of spots with a possibly
di€erent overall payo€ currency.
The instruments Si with a base Bi and underlying currency Ui , which have
possibly no common base currency, enter a Foreign Exchange product with
payo€ in a pay-out currency B, ie, the payo€ is
‡
S S~
S S~
K
…7:102†
Q max 1 1 ; ; n n
N1
Nn
where the numbers Ni are normalisers and S~i stands for the conversion into a
common denominator currency and Q for conversion to the global base currency B.
Under the corresponding risk-neutral measure with respect to their domestic
measure Bi all the spots follow
dSUi Bi ˆ mUi Bi SUi Bi dt ‡ sUi Bi SUi Bi dW yi
…7:103†
There are three distinct cases
.
The base currency Bi is the global base currency B. In this case the pricing
formulae need not be altered.
.
The underlying currency Ui is the global base currency B. In this case the
drift is adjusted.
.
Neither base nor underlying currency is the same as the global base currency.
In order to derive expressions for the general case we will expand the number of
spots for these contracts with the corresponding set of Ui B and Bi B spots. For
these spots all of them share the same base currency and hence their dynamics
are described in the base currency measure. The di€erential equations are
RISK BOOKS
The Pricing of Second Generation Exotics
73
y
~
dSUi B ˆ mUi B SUi B dt ‡ sUi B SUi B dW
i
…7:104†
y
…7:105†
~
dSBi B ˆ mBi B SBi B dt ‡ sBi B SBi B dW
i
With the cross volatilities sUi Bj ; sUi Uj ; sBi Bj the correlation matrix of the
expanded system can be calculated. From there the Cholesky decomposition
leads to equations using orthogonal Brownian motions
X
~k
ik dW
…7:106†
dSUi B ˆ mUi B SUi B dt ‡ sUi B SUi B
k
dSBi B ˆ mBi B SBi B dt ‡ sBi B SBi B
Hence the dynamics of the spots SUi Bi ˆ
domestic measure
mUi Bi ˆ mUi B
sUi Bi sUj Bj rUi Bi =Uj Bj ˆ
X
SUi B
SBi B
sUi B sUj B ik jk ‡
k
~k
N‡ik dW
…7:107†
k
have correlation and drift in the
1 2
…sUi B
2
mBi B
X
X
s2Ui Bi
X
s2Bi B †
sBi B sBj B ik jk
…7:108†
…7:109†
k
X
sBi B sUj B ik jk
k
sUi B sBj B ik jk
k
Using this information one can derive a correlation matrix that gives rise again
to a Cholesky decomposition and the pricing formula with these changed
coecients and drifts can be used.
7.9.1 Example
We consider the case that all the involved currencies have a common underlying
currency which is the global base currency. The equation for the drift terms is
mUi Bi ˆ
mBi Ui ‡ …s2Ui Bi †
…7:110†
The correlations are determined as
rUi Bi =Uj Bj ˆ
rBi Ui =Uj Bj
rUi Bi =Uj Bj ˆ rBi Ui =Bj Uj
…7:111†
…7:112†
and are the same as in the non-quanto case.
For pricing the existing undiscounted formula is used with adapted risk-neutral
drift for each currency
Qe
rB T
ˆ Qe
PriceGeneralisedBW …Si ; rBi ; rUi ; mi ˆ rBi
rB T
PriceBW …Si ; rBi ; rUi ; mi ˆ rBi
FOREIGN EXCHANGE RISK
rUi ; sUi Bi ; rUi Bi =Uj Bj †
rUi ‡ s2Ui Bi ; sUi Bi ; rUi Bi =Uj Bj †
…7:113†