Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015

Master’s Thesis
Eﬀective Pricing of Cliquet Options
Peter Warken
December 2015
Supervisor: Prof. Dr. J¨orn Saß
University of Kaiserslautern
Department of Mathematics
Financial Mathematics Group

Abstract
This thesis provides cutting-edge conceptions for pricing equity-linked annuities.
A semi-closed-form expression of the price of cliquet options is developed in
a Black-Scholes market model and compared to a Monte Carlo approach for
pricing these path-dependent options. It is presented how the result can be
applied to comparable structured products like sum cap contracts. Finally, it is
demonstrated that the presence of stochastic volatility and stochastic interest
rates has a significant impact on the pricing behavior of cliquet options. Several
numerical experiments are performed to illustrate the influence of market models
and the associated financial parameters.

Contents
1 Cliquet option market 1
1.1 Contract specification . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Financial model and pricing concepts . . . . . . . . . . . . . . . . 9
2 A semi-closed-form solution 11
2.1 Pricing formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Vega of the cliquet option . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Numerical pricing 23
3.1 Implementing the semi-closed-form solution . . . . . . . . . . . . 23
3.2 Approximating the Vega of the option price . . . . . . . . . . . . 26
3.3 Monte Carlo technique I . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Comparison of the numerical techniques . . . . . . . . . . . . . . 30
3.5 On the choice of g and c . . . . . . . . . . . . . . . . . . . . . . . 37
4 The sum cap contract - a similar product 40
4.1 Pricing formula of sum cap contracts . . . . . . . . . . . . . . . . 40
4.2 Numerical pricing of sum cap contracts . . . . . . . . . . . . . . 42
5 The influence of stochastic volatility 45
5.1 The Heston stochastic volatility model . . . . . . . . . . . . . . . 45
5.2 Stock price simulation in the Heston model . . . . . . . . . . . . 46
5.3 Monte Carlo technique II . . . . . . . . . . . . . . . . . . . . . . 47
6 The influence of stochastic interest rates 52
6.1 The Black-Scholes-Hull-White model . . . . . . . . . . . . . . . . 52
6.2 Stock price simulation in the BSHW model . . . . . . . . . . . . 53
6.3 Monte Carlo technique III . . . . . . . . . . . . . . . . . . . . . . 53
6.4 Further short rate models . . . . . . . . . . . . . . . . . . . . . . 57
I

Introduction
Equity-indexed annuities - EIAs - are customized structured products, sold
for instance by insurance companies to provide savings and insurance benefits.
These EIAs differ from traditional variable annuities in several significant ways.
In particular, the return of an investment in an EIA is guaranteed not to fall
below a certain minimum level. This feature qualifies these investment products
as insurance products. An EIA provides a fixed return plus the possibility of
an additional return, based on the performance of the underlying. There is a
wide variety of possible designs. All of them have in common that the guarantee
is financed via a limitation of the returns associated with the underlying (cf.
[Pa06], [BB11]).
In the thesis the pricing of cliquet options is discussed. These contracts have
a European payoff at a fixed future maturity date. Cliquet options can be
interpreted as a series of forward-starting at-the-money options with a single
premium determined upfront which locks in any gains on specific reset dates.
At these dates the strike price is reset at the current level of the underlying
asset. Thus, any decline in the price of the underlying asset resets the strike
price to a lower level, while keeping earlier profits. Floors and caps are added
to fix the minimum and maximum returns. By construction, these structured
products provide a downside protection yet being affordable priced since the
payoff is capped locally.
In recent years, there has been an increasing interest in such path-dependent
options. Especially, turmoil in financial markets has led to a demand for invest-
ment solutions that reduce downside risk while still offering upside potential
(see [BBG11] for further details).

1 Cliquet option market
In this chapter cliquet options are introduced as in [BL13] and the financial
market model is described.
1.1 Contract specification
Let T, a future point in time, be the maturity date of the contract. The interval
[0, T] is divided into n different periods of length ∆ with T = n∆. For k = 0, ..., n
the dates tk = k∆ are called reset days. The initial investment is denoted by
K. By St the price of the underlying asset at time t ∈ [0, T] is denoted. Further
specifications include the guaranteed rate at maturity g as well as a local cap
c for each reset period tk − tk−1. The payoff of a minimum coupon cliquet is
given by
XT = K max 1 + g, 1 +
n
k=1
max 0, min c,
Stk
− Stk−1
Stk−1
.
The payoff of the cliquet option is paid at the fixed maturity date and thus the
option is of European type. The return of each period is furthermore locally
capped at c and the contract also consists of a local floor equal to 0. The
guaranteed rate of return is equal to g. As the contract is linked to periodical
returns, the notation is simplified by denoting by Rk the return of the underlying
in the kth period for k = 1, ..., n:
Rk :=
Stk
− Stk−1
Stk−1
.
A modified representation of the payoff of the cliquet option is achieved by
XT = K max 1 + g, 1 +
n
k=1
max (0, min (c, Rk))
= K (1 + g) + K max 0,
n
k=1
Zk ,
where Zk is defined as
Zk := max (0, min (c, Rk)) −
g
n
for k = 1, ..., n.
If the increments of the underlying asset price process are independent and
identically distributed (i.i.d.), then also the returns of the underlying Rk and
the modified Zk are i.i.d. random variables. Since their distribution does not
depend on k, denote by R and Z a corresponding independent and identically
distributed random variable.
1

Observe that in a single reset period k the returns are truncated at the floor
level 0 and capped at c. With the help of figure 1, the payout profile of the kth
period can thus be interpreted as 1
Stk−1
-call spreads, i.e. long calls with strike
Stk−1
and short calls with strike Stk−1
(1 + c). It is thus justified to interpret a
cliquet option as a series of forward-starting at-the-money options.
−0.1 −0.05 0 0.05 0.1
−0.1
−0.05
0
0.05
0.1
max(0,min(Rk
,c))
Rk
Figure 1: Payout profile in the kth period with c = 5%
2

For illustration purposes a 5-year minimum coupon cliquet with monthly reset
dates on the S&P 500 Index is considered. Let the local cap be equal to c = 0.6%
and the guaranteed rate is set to g = 16%. Based on a historical data set of
the closing prices since January 1999 a naive sampling algorithm with A = 105
trials is performed. The ﬁgure 2 displays the sampled distribution of the rate
of return of the cliquet option at maturity.
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32
0
100
200
300
400
500
600
Rate of return at maturity
Frequency
Figure 2: Sampled distribution of the rate of return at maturity
3

In the following consider a historical example of a yearly reseted cliquet option
on the S&P 500 Index, that started at the beginning of 2005 and matured at
January the ﬁrst, 2010. The chart below shows the performance of the under-
lying during this period.
1998 2000 2002 2004 2006 2008 2010 2012 2014
1000
1500
2000
2500
3000
3500
Years
Indexlevelatthebeginningoftheyear
S&P 500 Index Performance Chart
Figure 3: Performance chart S&P 500 Index
Let the local cap of the contract be equal to c = 8% and the guaranteed rate is
set to g = 16%. The yearly returns of the underlying as well as the ﬂoored and
capped yearly returns of the cliquet option are illustrated in the following bar
chart.
4

2005 2006 2007 2008 2009
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Years
Yearlyreturns
Yearly returns
yearly returns of the cliquet option
yearly returns of the underlying
Figure 4: Periodical returns
It can easily be recognized that the holder of the cliquet option is insured against
the large drawdown during the financial crisis in the year 2008. This insurance
is financed via the caps of the yearly returns. Thus, an investor is not able to
participate in the above 8% returns in the years 2006 and 2009. The final return
at maturity of the cliquet option is equal to 25.2%, whereas a simple buy-and-
hold investment in the underlying resulted in a return of 12.5% only. Thus,
especially during turmoils in financial markets cliquet options are investment
solutions that reduce downside risk while still offering upside potential.
To illustrate the in-depth product set-up of the cliquet contract a termsheet of
a cliquet note issued by Vontobel is provided below.
Source: https://derinet.vontobel.com/CH/DE/Produkt/CH0100562179
5

Bank Vontobel AG, Gotthardstrasse 43, 8022 Zürich: Financial Products, Tel. +41 (0)58 283 78 88, Fax +41 (0)58 283 57 67
Banque Vontobel Genève SA, Place de l’Université 6, 1205 Genève: Financial Products, Tél. +41 (0)22 809 91 91, Fax +41 (0)22 809 90 99
Internet: http://www.derinet.com
VONTOBEL CLIQUET NOTE ON THE DOW JONES EURO STOXX 50®
INDEX
CAPITAL PROTECTION 100% AT EXPIRY
2.00% ANNUAL MINIMUM COUPON
The Vontobel CLIQUET NOTE is characterized by capital protection at maturity and regular coupon payments. The coupon payments
are calculated on the annual coupon fixing dates, based on the sum of the monthly performances of the underlying (performance
component) and the minimum between 0 and the previous year performance component. In this respect, each monthly performance
has a cap. If the coupon calculated in this way is below the annual minimum coupon, then the annual minimum coupon is paid out.
This coupon calculation takes place annually.
PRODUCT INFORMATION
Issuer Vontobel Financial Products Ltd., DIFC Dubai
Lead Manager Bank Vontobel AG, Zurich
Calculation Agent Bank Vontobel AG, Zurich
Guarantor Vontobel Holding AG, Zurich (Standard & Poor's A; Moody's A2)
Underlying value Dow Jones EURO STOXX 50®
Index (no di valore svizzero: 846 480)
Issue Price per Note EUR 1000.00
Notional per Note EUR 1000.00
Reference price EUR 2317.36
Initial fixing April 27, 2009
Payment date May 4, 2009
Last trading day April 24, 2014 (12:00 CET)
Final fixing April 28, 2014
Repayment date May 5, 2014
Underlying per Note (Ratio) 0.4315
Swiss Sec. No./ISIN 1005 6217 / CH0100562179
Telekurs Symbol VQDJB
CAPITAL PROTECTION (NOTES)
Capital protection per Note EUR 1000.00 (100% of the issue price)
Net present value EUR 963.47
Taxes From the technical taxation aspect these CLIQUET NOTES are seen as a transparent capital
protected product with a non-predominantly one-off interest payment ("Non-IUP").
Accordingly, the difference between the capital protection and the cash value (EUR 1000.00 –
EUR 963.47 = EUR 36.53) will be subject to income tax for private investors in Switzerland only
at final redemption, the minimum annual coupon (2.00% p.a.) however on its respective due
date. (IRR: 2.79%)
The minimum coupon exceeding disbursements represent tax free capital gain.
TERMSHEET VONTOBEL CLIQUET NOTE
SSPA DESIGNATION: CAPITAL PROTECTION WITH
COUPON
+41 (0)58 283 78 88 or www.derinet.ch
Figure 5: Termsheet of a cliquet contract
6

TERMSHEET SEITE 2
No Swiss withholding tax, no stamp duties at issuance. For Swiss stamp duty purpose, the
product is treated as analogous to a bond. Therefore, secondary market transactions are in
principle subject to Swiss stamp duty (TK22).
For Swiss paying agents this product is subject to the EU taxation of savings income in the form
of interest payments. The guaranteed minimum coupon is liable to tax.
The taxation mentioned above applies on the issue date. The tax legislation and Internal
Revenue Service practice can change at any time.
COUPONS
Coupon frequency Annual
Coupon fixing dates 27.04.2010; 27.04.2011; 27.04.2012; 29.04.2013; 28.04.2014
Ex-dates First trading day following fixing date
Minimum annual coupon 2.00%
Yield cap per month 0.85% (10.20% p.a.)
Calculation of the coupon On each coupon fixing date (initially 27.04.2010) the coupon will be calculated based on the
sum of the monthly performances of the underlying (performance component) and the minimum
between 0 and the previous year performance component (supplementary component). In this
respect, each monthly performance has a cap. If the coupon calculated in this way is below the
annual minimum coupon, then the annual minimum coupon is paid out.
Coupon formula The annual coupon for year t is:
{ } 51t,B(t)A(t),2.00%MAXC t L=+=
where:
0)0( =A
∑
×
+−×= −
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
t
ti i
i
I
I
CapMintA
12
1)1(12 1
1,)(
"Performancekomponente"
( ))1(,0)( −= tAMintB "Supplementary component"
Cap = 0.85%
iI is the closing price of the underlying of the monthly observation date i
51t L= corresponding to 27.04.2010; 27.04.2011; 27.04.2012; 29.04.2013; 28.04.2014
The monthly observation date 0i = corresponds to the initial fixing date and the monthly
observation date 60i = corresponds to the closing fixing date. The monthly observation dates
are on the 27th day of each month, but if a particular day is not a business day, then the
monthly observation date is the first business day following the given date.
FURTHER INFORMATIONS
Reference Currency EUR
Issue Size 50'000 CLIQUET NOTES, the size may be increased any time
Repayment In addition to the last coupon, the investor will receive the capital protection per Note.
Secondary market The secondary market is guaranteed for the entire duration of the product.
The Cliquet Note is traded “flat”, that means accrued interest will be included in the price.
Clearing/Settlement SIX SIS, Euroclear, Clearstream
Sales restrictions USA, US persons, DIFC Dubai and United Kingdom
Listing Will be applied for in the main segment at the SIX Swiss Exchange.
Opportunities / Risks Vontobel CLIQUET NOTES give investors the opportunity to benefit from both capital protection
and participation in the performance of the equity index through regular coupons.
During the term the price can dip below the capital protection.
The value of structured products may depend not only on the development of the underlying
asset, but also on the credit rating of the issuer/guarantor. The investor is exposed to the
default risk of the issuer/guarantor.
Notice The product is not a collective investment within the meaning of the Federal Act on Collective
Investment Schemes (KAG); it is under no approval obligation and is not supervised by the Swiss
Financial Market Supervisory Authority FINMA (FINMA).
"Performance component"
7

TERMSHEET SEITE 3
Publication of notices All notices to investors regarding products and changes in product conditions (because of
corporate actions, for example) are published at www.derinet.ch; under the rules relating
to IBL (Internet Based Listing), notices concerning products quoted on the SIX Swiss
Exchange are also published at www.six-swiss-exchange.com. Term Sheets are generally not
amended.
The original version of this term sheet is in German; versions in other languages are non-binding translations.
This term sheet does neither constitute a Listing Prospectus in the sense of the Listing Regulation nor an Issuing Prospectus in the sense of Art. 652a and/or 1156 OR.
The alone relevant complete conditions as well as the detailed risk references to this product are contained in the appropriate Listing prospectus.
The Listing prospectus can be ordered free of charge at the Bank Vontobel AG, Financial Products Documentation, Dreikönigstrasse 37, 8022 Zurich (Tel.: 058 283 78 88) or
www.derinet.ch.
We would be glad to answer any questions you may have concerning our products on +41 (0)58 283 78 88 from 08.00-20.00 CET on bank workdays. Please note that all
conversations on this line are recorded. By calling, we assume that you agree to this business practice.
The list and details provided do not represent a recommendation on the specified underlying security; they are for informative purposes only and under no circumstances are
they to be used or considered as an offer to sell or a solicitation of any offer to buy any financial instrument. No responsibility is assumed for the completeness and accuracy
of the information provided herein. The information provided herein is not meant as a substitute for a consultation with your house bank which we consider indispensable
prior to entering any kind of derivatives transaction. Transactions of this nature should only be conducted once investors are fully aware of the risks involved and are in a
position to bear the possible related financial losses. Furthermore, we refer to our brochure «Special Risks in Securities Trading», which we will send you free of charge on
request.
Dow Jones EURO STOXX 50®
Index is owned by STOXX LIMITED and is a Service Mark of Dow Jones & Company Inc.
Zurich, April 27, 2009
8

1.2 Financial model and pricing concepts
The following subsection aims to give a short, yet comprehensive overview on
continuous-time models and option pricing. For proofs and additional details it
is recommended to follow [Sh04] or [Jo10].
Let (Ω, A, P) be a complete probability space. Suppose that W = (Wt)t∈[0,T ] is
a m-dimensional Wiener process w.r.t. some filtration F = (Ft)t∈[0,T ], satisfying
the usual conditions, with trivial F0 (augmented with null sets), and assume
that FT = A. The dynamics of the bond B = (Bt)t∈[0,T ] and stock prices
Si
= (Si
t)t∈[0,T ] (i = 1, ..., d and d ≤ m) are modeled according to
Bt = B0 exp
t
0
rsds
and
Si
t = Si
0 exp


t
0
µi
sds +
m
j=1
t
0
σij
s dWj
s −
1
2
t
0
σij
s
2
ds


with constant initial values B0, Si
0 > 0 for i = 1, ..., d.
(rt)t∈[0,T ], (µt)t∈[0,T ] and (σt)t∈[0,T ] are progressively measurable processes sat-
isfying
rt ≥ 0
and
T
0
rt + µt + σt
2
dt < ∞ a.s.,
where · denotes the Euclidean norm. For each t ∈ [0, T] it should hold that
σtσt is non-singular. Thus, the price processes are the unique Itô processes
satisfying
dBt = Btrtdt
and
dSi
t = Si
t

µi
tdt +
m
j=1
σij
t dWj
t

 i = 1, ..., d,
associated with the constant initial values. The discount factor at t ∈ [0, T] is
defined as
βt :=
B0
Bt
and the corresponding discounted price process ˜Si
= ( ˜Si
t)t∈[0,T ], i = 1, ..., d, are
given by
˜Si
t := βtSi
t.
Suppose that F is the filtration generated by W, augmented by the P-null
sets and that there exists a m-dimensional, progressively measurable process
θ = (θt)t∈[0,T ] such that
T
0
θt
2
dt < ∞ a.s.,
9

σtθt = µt − rt1d for all t ∈ [0, T] and
H = (Ht)t∈[0,T ] with
Ht = exp −
t
0
θs dWs −
1
2
t
0
θs
2
ds is a martingale under P.
A probability measure Q ∼ P is defined by the Radon Nikodym derivative
HT = dQ
dP . By the Girsanov Theorem
¯Wt = Wt +
t
0
θsds for t ∈ [0, T]
defines a m-dimensional Wiener Process ¯W = ( ¯Wt)t∈[0,T ] under Q. A probabil-
ity measure Q ∼ P under which the discounted price processes are local mar-
tingales, is called an equivalent martingale measure or a risk neutral probability
measure. Assuming d = m, the risk neutral measure is uniquely determined.
A portfolio process (πB
, π) = (πB
t , πt)t∈[0,T ], πt = (π1
t , ..., πd
t ) , is a (d + 1)-
dimensional, progressively measurable process with
T
0
(|πB
t + πt 1d||rt| + |πt (µt − rt1d)| + πt σt
2
)dt < ∞ a.s. .
πi
t is the amount of money invested in the ith stock and πB
t is the amount of
money invested in the bond. The corresponding wealth process XπB
,π
is defined
by
XπB
,π
t := πB
t + πt 1d
and the portfolio process is called self-financing, if for an initial capital x0 ∈ R
XπB
,π
t = x0 +
t
0
XπB
,π
s rsds +
t
0
πs dRs.
The portfolio process is then called admissible, if the discounted wealth process
˜XπB
,π
, defined by ˜XπB
,π
t := βtXπB
,π
t , satisfies ˜XπB
,π
t ≥ −K for some con-
stant K > 0 and all t ∈ [0, T]. The market model is arbitrage-free, i.e. there
exists no self-financing, admissible portfolio process π for which Xπ
0 ≤ 0 and
P(Xπ
T ≥ 0) = 1 and P(Xπ
T > 0) > 0.
A contingent claim CT is an FT -measurable random variable for which βT CT ≥
−K for some constant K > 0. An admissible, self-financing portfolio pro-
cess π with Xπ
T ≥ CT a.s. is called a superhedging strategy for the contin-
gent claim CT and the superreplication price of CT is defined as ¯p(CT ) =
inf{Xπ
0 : π superhedging strategy for CT }. CT is furthermore called attain-
able, if ¯p(CT ) < ∞ and there exists a superhedging strategy π with Xπ
T = CT
a.s.. π is then called a replication strategy for CT . The presented market model
is complete, i.e. any claim with ¯p(CT ) < ∞ is attainable. In the financial mar-
ket model with risk neutral measure Q it holds for any attainable claim that
¯p(CT ) = EQ[βT CT ] and there exists a martingale generating replication strat-
egy π with Xπ
t = β−1
t EQ[βT CT |Ft]. The arbitrage-free price of the contingent
claim CT at time t is then equal to
pt(CT ) := β−1
t EQ[βT CT |Ft] for t ∈ [0, T].
10

2 A semi-closed-form solution
In the following d = m = 1 and µ, r and σ > 0 are constant for t ∈ [0, T], i.e.
the complete, standard Black-Scholes market model is considered and the basic
concepts follow [BL13].
2.1 Pricing formula
0
0.1
0.2
0.3
0.4
0.5
0
0.01
0.02
0.03
1000
1050
1100
1150
1200
1250
1300
Volatility σLocal cap c
Optionprice
Figure 6: Cliquet option price surface w.r.t. σ and c
To derive a pricing formula for the price of the cliquet option, the expectation
of the discounted payoﬀ under the risk neutral measure Q has to be calculated.
Thus, it is crucial to calculate
E max 0,
n
k=1
Zk ,
where in the following the expectation is formed under Q.
In this context, characteristic functions provide an useful tool to derive a semi-
closed-form expression. The results below are based on [Jo13]. For a Rd
-valued
random variable X the characteristic function of X is deﬁned as
ϕX(t) := E[eit X
], t ∈ Rd
.
11

For independent, Rd
-valued random variables X1, ..., Xn it holds that
ϕ n
k=1 Xk
(t) =
n
k=1
ϕXk
(t).
Therefore, especially for i.i.d. Rd
-valued random variables (Xk)n
k=1 and X it
holds that ϕ n
k=1 Xk
(t) = ϕn
X(t). A C-valued function f is called µ-integrable,
if Re(f) and Im(f) are µ-integrable. Then
fdµ := Re(f)dµ + i Im(f)dµ
and it suﬃces to check, that |f| is integrable to verify the µ-integrability of f.
Especially, for f(t) = eit X
integrability w.r.t. a probability measure follows as
|eit X
| = 1.
Denote by ϕZ the characteristic function of the random variable Z and let
¯CT = max (0,
n
k=1 Zk). Given the FT -measurability and boundedness of βT
¯CT
from below, ¯CT forms a contingent claim.
Proposition 2.1. The time-0 price p0 of the contingent claim ¯CT is given by
p0
¯CT =
ne−rT
2
EZ +
e−rT
π
∞
0
t−2
(1 − Re (ϕn
Z (t))) dt.
Proof. Let x =
n
k=1 Zk. Then max (0,
n
k=1 Zk) = max (0, x). A simple case
analysis gives the representation max (0, x) = x+|x|
2 . Now a result from [Ha82]
is applied: Note that a substitution of u = |x|t yields
∞
0
t−2
(1 − cos(xt))dt = |x|
∞
0
u−2
(1 − cos(u))du.
Partial integration now implies that
∞
0
u−2
(1 − cos(u))du =
∞
0
sin(u)
u
du =
π
2
.
Therefore, |x| can be represented by
|x| =
2
π
∞
0
t−2
(1 − cos(xt))dt.
Let be a Rademacher random variable, i.e. ˜P( = ±1) = 1
2 under a probability
measure ˜P. It follows that for a Rademacher random variable it holds that
E ˜P [eixt
] = E ˜P [cos(xt )] + iE ˜P [sin(xt )] = cos(xt).
Then
|x| =
2
π
∞
0
t−2
(1 − E ˜P [eixt
])dt.
12

Thus
ΘZ : = E
n
k=1
Zk
=
2
π
∞
0
t−2
1 − E ˜P E ei( n
k=1 Zk)t
dt
=
2
π
∞
0
t−2
(1 − E ˜P [ϕn
Z(t )]) dt
=
2
π
∞
0
t−2
(1 − Re (ϕn
Z(t))) dt.
Here the positivity of the integrand as well as the boundedness of eitx
enable to
exchange the order of integration in the second step (e.g. [Jo13]). Combining
the earlier steps and using that the Zk are i.i.d. random variables, the result
follows immediately:
p0( ¯CT ) = e−rT
E max 0,
n
k=1
Zk
=
e−rT
2
E
n
k=1
Zk + E
n
k=1
Zk
=
ne−rT
2
EZ +
e−rT
π
∞
0
t−2
(1 − Re (ϕn
Z(t))) dt.
The pricing formula for the cliquet option follows directly:
Theorem 2.2. The time-0 price p0 of the cliquet option with payoﬀ
XT = K (1 + g) + K max 0,
n
k=1
Zk
is given by
p0 (XT ) = K (1 + g) e−rT
+ K
e−rT
2
(ΘZ + nEZ) ,
where ΘZ is deﬁned as
ΘZ =
2
π
∞
0
t−2
(1 − Re (ϕn
Z (t))) dt.
Proof. Let XT = K (1 + g) + K max (0,
n
k=1 Zk). Given the FT -measurability
and boundedness of βT XT from below, it forms a contingent claim. Then
p0 (XT ) = e−rT
K (1 + g) + KE ¯CT = e−rT
K (1 + g) + Kp0
¯CT .
The foregoing proposition
p0( ¯CT ) =
ne−rT
2
EZ +
e−rT
π
∞
0
t−2
(1 − Re (ϕn
Z (t))) dt,
yields the assertion.
13

An application of the earlier result is possible, if the expression ΘZ is well
understood. This one consists of the characteristic function ϕZ. Therefore, an
analysis of the characteristic function is needed.
Proposition 2.3. The characteristic function ϕZ of Z is given by
ϕZ(t) := E eitZ
= e−it g
n 1 + it
c
0
eitx
Q (R > x) dx .
Proof. Let Y be a non-negative random variable with finite expectation. The
use of Fubini’s theorem yields the following representation: (cf. [Jo13])
ϕY (t) : = E eitY
= E eit0
+ it
Y
0
eitx
dx
= 1 + it
∞
0
y
0
eitx
dxQ(dy)
= 1 + it
∞
0
eitx
∞
x
Q(dy)dx
= 1 + it
∞
0
eitx
Q (Y > x) dx.
The random variable Z = max (0, min (c, R))− g
n is - by construction - bounded
from below by − g
n . This is a consequence of the local floor of the cliquet option
at 0. Therefore, the random variable Z + g
n is non-negative. A case analysis
argument yields
Q Z +
g
n
> x =
0 if x > c
Q (R > x) if x ≤ c
.
Furthermore,
ϕZ(t) = E eitZ
= e−it g
n E eit(Z+ g
n ) = e−it g
n ϕZ+ g
n
(t).
By the use of the expression of the characteristic function of a non-negative
random variable, in this case Z + g
n , it follows that
ϕZ(t) = e−it g
n ϕZ+ g
n
(t) = e−it g
n 1 + it
c
0
eitx
Q (R > x) dx .
Denote the density of R under the risk neutral probability Q by fR. Then, a
modified expression for the expectation of Z under Q is obtained.
Proposition 2.4. The expectation of Z under the risk neutral measure Q is
given by
EZ = c −
g
n
Q (R ≥ c) +
c− g
n
− g
n
xfR x +
g
n
dx −
g
n
Q (R < 0) .
14

Proof. A simple case analysis yields the distribution of the random variable Z,
which is given by
Q (Z > x) =



0 if x > c − g
n
Q R − g
n > x if − g
n ≤ x ≤ c − g
n
1 if x < − g
n
.
Thus, Z has a mixed distribution with to mass points at − g
n and at c − g
n and
a density over − g
n , c − g
n . The expression of the expected value of Z then
follows immediately.
For a further analysis in the Black-Scholes setting, observe that the returns Rk
for k = 1, ..., n are independent and identically distributed random variables
(e.g. [Jo10]). Under the risk neutral measure Q the returns can be represented
as
Rk = exp r −
σ2
2
∆ + σ ¯Wtk
− ¯Wtk−1
− 1.
Let
ξ ∼ N mξ, σ2
ξ
with
mξ = r −
σ2
2
∆
and
σ2
ξ = σ2
∆.
Then
Rk ∼ eξ
− 1.
To price the cliquet options in the Black-Scholes model, the distribution of Zk
has to be calculated. The foregoing representation of the returns Rk = eξ
− 1
is used and denote by N(·) the cumulative normal distribution function. It is
now possible to calculate the three cases of the mixed distribution of Zk:
Q Zk = c −
g
n
= Q (Rk ≥ c) = Q (ξ ≥ ln(1 + c)) = N
mξ − ln(1 + c)
σ
√
∆
,
fZ(x) =
1
σ x + 1 + g
n
√
2π∆
e−
(ln(x+1+
g
n )−mξ)
2
2σ2∆ if x ∈ −
g
n
, c −
g
n
and
Q Zk = −
g
n
= Q (Rk ≤ 0) = Q (ξ ≤ 0) = N
−mξ
σ
√
∆
.
These results are now inserted in the expression of the expected value of the
random variable Z under the risk neutral measure Q:
15

EZ = c −
g
n
Q (R ≥ c) +
c− g
n
− g
n
xfR x +
g
n
dx −
g
n
Q (R < 0)
= c −
g
n
N
mξ − ln(1 + c)
σ
√
∆
+
1+c
1
y − 1 −
g
n
1
√
2π∆σy
e−
(ln(y)−mξ)
2
2σ2∆ dy −
g
n
N
−mξ
σ
√
∆
.
Analogously it holds for the characteristic function of the random variable Z
that
ϕZ(t) = e−it g
n E eit(Z+ g
n )
= eit(c− g
n )N
mξ − ln(1 + c)
σ
√
∆
+
1+c
1
eit(y−1− g
n ) 1
√
2π∆σy
e−
(ln(y)−mξ)
2
2σ2∆ dy + e−it g
n N
−mξ
σ
√
∆
.
All building blocks are now together to give a semi-closed formula for the time-0
price p0 of the cliquet option with payoﬀ
XT = K (1 + g) + K max 0,
n
k=1
Zk
in the Black-Scholes model.
Theorem 2.5. In the Black-Scholes model the price of a cliquet option with
payoﬀ
XT = K (1 + g) + K max 0,
n
k=1
Zk
is given by
p0 (XT ) = K (1 + g) e−rT
+ K
e−rT
2
2
π
∞
0
t−2
(1 − Re (ϕn
Z (t))) dt + nEZ ,
where
EZ = c −
g
n
Q (R ≥ c) +
c− g
n
− g
n
xfR x +
g
n
dx −
g
n
Q (R < 0)
= c −
g
n
N
mξ − ln(1 + c)
σ
√
∆
+
1+c
1
y − 1 −
g
n
1
√
2π∆σy
e−
(ln(y)−mξ)
2
2σ2∆ dy −
g
n
N
−mξ
σ
√
∆
and
ϕZ(t) = eit(c− g
n )N
mξ − ln(1 + c)
σ
√
∆
+
1+c
1
eit(y−1− g
n ) 1
√
2π∆σy
e−
(ln(y)−mξ)
2
n N
−mξ
σ
√
∆
.
16

2.2 Vega of the cliquet option
0.1
0.2
0.3
0.4
0.5
0.01
0.015
0.02
−1200
−1000
−800
−600
−400
−200
0
Vega
Figure 7: Vega surface of the cliquet option w.r.t. σ and c
To judge the sensitivity of the price of the cliquet option, partial derivatives of
the option price with respect to various parameters are calculated (cf. [KK01]).
Whereas responses to changes in the volatility parameter are well understood
in the case of plain vanilla options, it is first of all not clear how the price of
the cliquet option is going to behave under such variations. It turns out that
cliquet options are quite sensitive with respect to changes in volatility. It is
therefore natural to analyze the Vega of the cliquet option. The Vega of an
option is defined as the partial derivative of the option price with respect to
the volatility parameter σ. As Vega is no Greek letter, it is sometimes called
Lambda. Therefore, it is also convenient to use the notation Λ(t) for the Vega
at time t.
Proposition 2.6. The Vega of the cliquet option at time 0 is given by
Λ(0) : =
∂
∂σ
(p0(XT ))
= K
e−rT
2
∂
∂σ
ΘZ + n
∂
∂σ
EZ
= K
e−rT
2
−
1
π
∞
0
1
t2
∂
∂σ
(ϕn
Z(t) + ϕn
Z(−t)) dt + n
∂
∂σ
EZ .
Proof. The first expression follows immediately by differentiating
p0 (XT ) = K (1 + g) e−rT
+ K
e−rT
2
(ΘZ + nEZ) .
17

The partial derivative of ΘZ with respect to the volatility σ is calculated as
follows
∂
∂σ
ΘZ =
∂
∂σ
2
π
∞
0
t−2
(1 − Re (ϕn
Z (t))) dt
=
∂
∂σ
2
π
∞
0
t−2
1 −
ϕn
Z(t) + ϕn
Z(−t)
2
dt
= −
1
π
∞
0
1
t2
∂
∂σ
(ϕn
Z(t) + ϕn
Z(−t)) dt.
The expression is thus received by rewriting
Re (ϕn
Z (t)) =
ϕn
Z(t) + ϕn
Z(−t)
2
and interchanging integration and diﬀerentiation in the last step, which is al-
lowed due to the positivity of the integrand, cf. [Jo13].
Hence, it is required to calculate the partial derivative of the characteristic
function ϕZ with respect to the volatility parameter σ to obtain the Vega of the
cliquet option. In the Black-Scholes model the partial derivative is calculated
as follows.
Proposition 2.7. The partial derivative of the characteristic function ϕZ with
respect to the ﬁnancial parameter σ is given by
∂ (ϕZ(t))
∂σ
= ϕσ
Z1
(t) + ϕσ
Z2
(t) + ϕσ
Z3
(t),
where
ϕσ
Z1
(t) =
ln(1 + c) − mξ − σ2
∆
σ2
√
2πT
eit(y−1− g
n )e−
(mξ−ln(1+c))
2
2σ2∆ ,
ϕσ
Z2
(t) =
mξ + σ2
∆
σ2
√
2πT
e−it g
n e−
m2
ξ
2σ2∆
and
ϕ
σ
Z3
(t) =
1+c
1
1
σy
√
2π∆
e
it y−1−
g
n


−
1 + (ln(y) − mξ)
σ
+
ln(y) − mξ
2
σ3∆


 e
−
ln(y)−mξ
2
2σ2∆ dy.
Proof. The foregoing expression of the characteristic function is used to calculate
the partial derivative ∂
∂σ ϕZ:
ϕZ(t) = eit(c− g
n )N
mξ − ln(1 + c)
σ
√
∆
+
1+c
1
eit(y−1− g
n ) 1
√
2π∆σy
e−
(ln(y)−mξ)
2
n N
−mξ
σ
√
∆
.
18

2.3 Approximation
In the following, an approximation scheme is provided that enables to calculate
the integral in ΘZ. Such a method is needed, as an evaluation of the character-
istic function of the truncated lognormal distribution of Z is not possible in an
explicit way, see [Le91]. However, in the cliquet contract the periodical returns
are capped at c. Thus, the support of each characteristic function is bounded.
The two approximation steps are carried out below. The first step consists of
truncating the integral in ΘZ. In a second step the exponential function in the
characteristic function is approximated by a finite Taylor series.
Approximation. Define the truncated version of ΘZ as
ΘZ(U) :=
2
π
U
0
t−2
(1 − Re (ϕn
Z (t))) dt.
A finite Taylor series approximation of ϕZ yields
ϕZ,u(t) := e−it g
n 1 + it
c
0
u
k=0
(itx)k
k!
Q (R > x) dx .
Then, define the approximation as
ΘZ(U, u) :=
2
π
U
0
t−2
1 − Re ϕn
Z,u (t) dt.
It is now possible to show that the convergence of the foregoing approximation
to ΘZ is guaranteed, if u and U are chosen suitably large.
Proposition 2.8. For any positive integer U it holds, that
|ΘZ − ΘZ(U)| ≤
4
Uπ
.
For a fixed U, the convergence of the approximation follows:
lim
u→∞
|ΘZ(U) − ΘZ(U, u)| = 0.
Proof. The first result follows from the following inequality:
|ΘZ − ΘZ(U)| ≤
2
π
∞
U
t−2
(1 − Re (ϕn
Z (t))) dt
≤
2
π
∞
U
2
t2
dt
≤
4
Uπ
.
According to Proposition 2.3, ϕZ can be expressed as
ϕZ(t) = e−it g
n 1 + it
c
0
eitx
Q (R > x) dx .
By setting ϕZ,u equal to
ϕZ,u(t) = e−it g
n 1 + it
c
0
u
k=0
(itx)k
k!
Q (R > x) dx ,
19

the following estimation is obtained:
|ϕZ(t) − ϕZ,u(t)| ≤ t
c
0
eitx
−
u
k=0
(itx)k
k!
Q (R > x) dx
≤ 2
tu+2
(u + 1)!
c
0
xu+1
Q (R > x) dx
≤ 2
tu+2
(u + 2)!
cu+2
.
Hence,
lim
u→∞
|ΘZ(U) − ΘZ(U, u)| = 0
follows by using the inequality hereafter and the fact that U and n are ﬁxed.
|ΘZ(U) − ΘZ(U, u)| ≤
2
π
U
0
t−2
Re (ϕn
Z(t)) − Re ϕn
Z,u (t) dt
≤
2
π
U
0
t−2
ϕn
Z (t) − ϕn
Z,u (t) dt
≤
2
π
U
0
t−2
|ϕZ (t) − ϕZ,u (t)|
n−1
k=0
ϕn−k
Z (t) ϕk
Z,u (t) dt
≤
2
π
U
0
t−2
|ϕZ (t) − ϕZ,u (t)|
n−1
k=0
|ϕZ,u (t) |k
dt
≤
2
π
U
0
t−2
|ϕZ (t) − ϕZ,u (t)|
n−1
k=0
(1 + |ϕZ (t) − ϕZ,u (t) |)
k
dt
≤
2
π
U
0
2
tu
(u + 2)!
cu+2
n−1
k=0
1 + 2
tu+2
(u + 2)!
cu+2
k
dt.
A further investigation of the approximation is now carried out in the Black-
Scholes setting. As shown before, the periodical returns Rk are lognormally
distributed in this framework, i.e.
Rk = eξ
− 1
with
ξ ∼ N (mξ, σξ) .
For a local cap c and k > 0 deﬁne
µk := µ
(c)
k =
c
0
kxk−1
N
mξ − ln(1 + x)
σξ
and assume, that
µ
(c)
0 = 1.
20

The approximated characteristic function of Z can now be rewritten as
ϕZ,u(t) : = e−it g
n 1 + it
c
0
u
k=0
(itx)k
k!
Q (R > x) dx
= e−it g
n 1 + it
c
0
u
k=0
(itx)k
k!
N
mξ − ln(1 + x)
σξ
dx
= e−it g
n 1 +
c
0
u
k=0
(it)k+1
(k + 1)!
(k + 1)xk
N
mξ − ln(1 + x)
σξ
dx
= e−it g
n
u+1
k=0
(it)k
k!
µk.
Hence,
ϕn
Z,u(t) = e−itg
u+1
k=0
(it)k
k!
µk
n
= e−itg
n(u+1)
l=0
αl(it)l
,
where
αl :=
{(j1,...,jn)| n
s=1 js=l}
n
k=1
µjk
jk!
.
The real part of ϕn
Z,u is now expressed as
Re ϕn
Z,u(t) = cos(gt)
n(u+1)
2
l=0
α2l(−1)l
t2l
+ sin(gt)
n(u+1)−1
2
l=0
α2l+1(−1)l
t2l+1
.
Finally, it is possible to explicitly calculate the approximation
ΘZ(U, u) =
2
π
U
0
t−2
1 − Re ϕn
Z,u (t) dt.
Calculating the terms up to α2, the approximation is given as follows:
ΘZ(U, u) = ΘU,u
Z,0 + ΘU,u
Z,1 + ΘU,u
Z,2 + ΘU,u
Z,sin + ΘU,u
Z,cos,
where
ΘU,u
Z,0 =
2
π
U
0
t−2
(1 − cos(gt)) dt,
ΘU,u
Z,1 = −
2
π
U
0
sin(gt)α1
t
dt,
ΘU,u
Z,2 =
2
π
U
0
cos(gt)α2dt,
21

ΘU,u
Z,sin =
2
π
n(u+1)−1
2
l=1
U
0
sin(gt)α2l+1(−1)l+1
t2l−1
dt
and
ΘU,u
Z,cos =
2
π
n(u+1)
2
l=1
U
0
cos(gt)α2l(−1)l+1
t2l−2
dt.
Furthermore, the following simpliﬁcations and asymptotics for U → ∞ are valid:
ΘU,u
Z,0 =
2
π
U
0
t−2
(1 − cos(gt)) dt
= g −
2
π
∞
U
t−2
(1 − cos(gt)) dt
∼ g,
ΘU,u
Z,1 = −
2
π
U
0
sin(gt)α1
t
dt
= −α1 +
2
π
∞
U
sin(gt)α1
t
dt
∼ −α1
and
ΘU,u
Z,2 =
2
π
U
0
cos(gt)α2dt
=
2
π
sin(gU)α2
g
.
22

3 Numerical pricing
In the following chapter, the numerical pricing of a cliquet contract is analyzed.
The contract and the parameters are specified as follows:
T 5
n 60
K 1,000
g 0.30
c 0.01
r 0.02
These specification are set in such a way to represent a cliquet contract, heavily
traded in financial markets. To obtain the numerical result the pricing methods
are implemented in Matlab.
3.1 Implementing the semi-closed-form solution
The price of the cliquet option in the Black-Scholes model is given by Theorem
2.5. Therefore, a numerical result is obtained via the implementation of this
pricing formula. In the code, the procedure ”integral(·)” is used. This proce-
dure approximates the integrals in the pricing formula by using global adaptive
quadrature and default error tolerances.
The integral ΘZ is approximated by ΘZ(1, 000), i.e. by setting the upper
bound of integration equal to 1, 000, whereas the theoretical result demands
for an unbounded integration. A numerical evaluation of the integral ΘZ(U)
for varying U shows that the choice of U = 1, 000 is satisfactory. The following
chart illustrates the level of accuracy w.r.t. the upper bound of the integral
ΘZ(U). Choosing U larger than 1, 000 leads to an increased computational
effort, which is not justified by the additionally achieved level of correctness.
23

0 400 800 1200 1600 2000
0.028
0.029
0.03
0.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
Upper bound U
Θ
Z
(U)
Figure 8: Influence of the upper bound in the approximation ΘZ(U)
To verify the earlier statement on an additional computational effort for larger
upper bounds, the running time of the integration algorithm has to be checked.
By performing the evaluation for U ranging from 1, 000 to 2, 000, where each
calculation is done 1, 000 times, the average time needed serves as an indicator of
the computational effort. The experiment proves that the average time needed
to calculate ΘZ(U) increases sharply w.r.t. U and also the time fluctuations
escalate for larger upper bounds. This result is presented in the chart below.
24

1000 1200 1400 1600 1800 2000
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Upper bound U
AveragetimetocalculateΘ
Z
(U)
Figure 9: Average time needed to calculate ΘZ(U)
Another crucial component of the formula is the normal distribution. In the
code, the procedure ”randn(·)” is used to generate normally distributed pseu-
dorandom numbers.
25

In figure 10, the price of the cliquet contract is reported as function of the
volatility σ.
0 0.1 0.2 0.3 0.4 0.5
1015
1020
1025
1030
1035
Volatility σ
Time−0PriceoftheCliquet
Figure 10: Price of the cliquet contract w.r.t. σ
In this example, one recognizes, that the time-0 price is generally above the
initial investment of K = 1, 000. Furthermore, the graph displays a concave
structure. In particular, the price of the cliquet option is strongly increasing in
σ for small values of the volatility. But the slope of the function changes sign at
σ ≈ 0.30. Thus, a non-monotonicity of the price with respect to the volatility
parameter σ and a high sensitivity with respect to changes in this parameter
can be observed. Therefore, the corresponding Vega has to be studied in more
detail.
3.2 Approximating the Vega of the option price
An advantage of using the semi-closed-form solution of the price of the cliquet
option is the fact, that the Vega of the option price is approximated quite easily.
Recall, the Vega of the cliquet option is defined as the partial derivative of the
option price with respect to the volatility σ. Assuming that the option price
p0(XT ) is a function of σ (denoted by p0(XT )(σ)), the sensitivity with respect
to the volatility can be approximated via finite differences (cf. [Ja02],[Gl03]).
Therefore, set
Λ(0) ≈
p0(XT )(σ + κσ) − p0(XT )(σ)
κσ
,
26

where κ is chosen sufficiently small - in this case κ = 0.01. Having calculated
the price of the cliquet option with respect to the volatility σ in the first step,
the Vega is thus obtained by just computing this fraction.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−100
0
100
200
300
400
500
600
700
Volatility σ
Time−0VegaoftheCliquet
Figure 11: Vega of the cliquet option w.r.t σ
Observe that the figure 11 fits the interpretation from the foregoing subchapter:
the Vega of the cliquet option as function of σ is strictly decreasing, convex and
changes the sign for larger volatilities.
3.3 Monte Carlo technique I
In the following the price of the cliquet option is determined via Monte Carlo
simulation, see [Ja02] and [Gl03]. This numerical technique consists of two
steps. First of all, A independent realizations Xi
T
A
i=1
of the final payoff XT
are simulated. In the second step
e−rT 1
A
A
i=1
Xi
T
is chosen as an approximation for the time-0 price of the contract. It is clear,
that the foundation of Monte Carlo simulation is the strong law of large numbers,
which guarantees the convergence of the approximation to p0(XT ), the fair price
of the cliquet option. As the final payoff is a functional of the price process S, it
is necessary to simulate the path (St)t∈[0,T ] under the risk neutral measure Q to
simulate XT . In the case of cliquet options, only the returns of the underlying
price process contribute to the final payoff, thus the following approximation
procedure can be used for the purpose of simulating XT . First, n independent,
27

standard normally distributed random variables (Yk)n
k=1 are generated. The
returns (Rk)n
k=1 are then simulated by
Rk = exp r −
1
2
σ2 T
n
+ σ
T
n
Yk − 1.
This procedure is now independently repeated for i = 1, ..., A to generate the
returns Ri
k for k = 1, ..., n and i = 1, ..., A in order to simulate the payoffs
Xi
T
A
i=1
via
Xi
T = K max 1 + g, 1 +
n
k=1
max 0, min c, Ri
k .
The implementation of the Monte Carlo technique is thus straightforward. The
required independent, normally distributed random variables are given by the
Matlab procedure randn(·), which generates normally distributed pseudoran-
dom numbers.
In the examined pricing example A is chosen to be equal to 105
. During the
performance of the Monte Carlo technique for the given contract specifications
and parameters, the standard deviation s of the realized final payoffs is approx-
imately equal to
s :=
1
A − 1
A
i=1
Xi
T −
1
A
A
i=1
Xi
T
2
≈ 91.94.
28

In ﬁgure 12 the simulated price of the cliquet option is plotted as function of σ.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
CliquetOptionPrice
Volatility σ
The basic principle for approximating the Vega presented in the foregoing chap-
ter can also easily be applied in the concept of Monte Carlo simulation. Recall
that the Vega is approximated by
Λ(0) ≈
p0(XT )(σ + κσ) − p0(XT )(σ)
κσ
.
It is therefore crucial to calculate p0(XT )(σ + κσ) and p0(XT )(σ) in the Monte
Carlo setting. To obtain the approximated Vega, the standard technique, called
path recycling, is used. This method computes the prices of the cliquet option
at σ + κσ and σ by using the same random numbers.
29

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−100
0
100
200
300
400
500
600
700
800
Volatility σ
Figure 13: Vega of the cliquet option w.r.t. σ
3.4 Comparison of the numerical techniques
A comparison of the results presented earlier is provided in the two ﬁgures
hereafter. Observe that both pricing techniques lead to similiar prices and the
same particular behavior with respect to the volatility parameter can be seen.
Whereas the price of the cliquet option is rather smooth in the semi-closed-form
solution approach, the Monte Carlo technique also shows some smaller devia-
tions from a fully smooth function. Moreover, there is a tendency to slightly
lower prices in the semi-closed-form solution.
30

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163CliquetOptionPrice
Volatility σ
Semi−Closed−Form Solution
Monte Carlo Technique
The same behavior can also be observed by investigating the Vega. The Vega
as function of the volatility σ is smooth in both cases. The Monto Carlo tech-
niques leads to slightly lower Vegas, but the diﬀerences seem to be negligible.
31

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−100
0
100
200
300
400
500
600
700
Volatility σ
Semi−Closed−Form Solution
Monte Carlo Technique
Figure 15: Vega of the cliquet option w.r.t. σ
32

In terms of a comparison of different pricing techniques another important per-
formance measure is the computational effort needed to price the contracts.
Therefore, in the following experiment the running time of the implemented
semi-closed-form solution and the Monte Carlo technique (with σ = 0.20 and
A = 105
) is measured 100 times each for the exemplary cliquet option contract.
On average it takes 0.8156 min to price the contracts with a standard devia-
tion of 0.0097 using the semi-closed-form solution. The pricing procedure using
the Monte Carlo algorithm lasts slightly longer with 0.8725 min and a larger
standard deviation of 0.0209.
0 20 40 60 80 100
0.75
0.80
0.85
0.90
Iteration
Timetocalculateoptionprice
Figure 16: Time needed using the semi-closed-form solution
The two charts enable to verify the running time differences in more detail.
Besides the in general higher level of running time especially the higher fluctu-
ations, when using the Monte Carlo technique are easily observed.
33

0 20 40 60 80 100
0.75
0.8
0.85
0.9
0.95
1
Iteration
Timetocalculateoptionprice
Figure 17: Time needed using the Monte Carlo technique
It should be clear from the earlier considerations on cliquet options, that the
number of periods n, speciﬁed in the contract, has an enormous impact on the
running time of the algorithms. Interestingly, both methods behave diﬀerently,
when the number of periods is changed. In the experiment the pricing routine
has been run for n ranging from 1 to 60, where each contract is priced 100 times.
In the following charts the average time to perform one pricing routine for the
cliquet contract is compared.
34

0 20 40 60
0.5
1
1.5
2
2.5
3
3.5
Number of periods n
Averagetimetocalculatetheoptionprice
Figure 18: Average time needed using the semi-closed-form solution
Surprisingly, an increase in the number of periods n leads to a drop in the
time needed to price the cliquet options. To price the contract with a single
period takes 4 times longer than pricing the monthly reseted cliquet option. The
opposite is true when using the Monte Carlo pricer. The running time increases
almost linear from approximately 20 sec to more than 85 sec by increasing the
number of periods. Thus, it seems to be reasonable to use the Monte Carlo
technique in the case of only few periods, whereas the semi-closed-form solution
is superior for a larger amount of periods.
35

0 20 40 60
0
0.25
0.5
0.75
1
Number of periods n
Averagetimetocalculatetheoptionprice
Figure 19: Average time needed using the Monte Carlo technique
36

3.5 On the choice of g and c
In terms of product origination, it is crucial to understand the linkage between
the choice of g and c. By the construction of cliquet options an increase in the
guaranteed rate g, ceteris paribus, shifts the distribution of possible outcomes
upwards. On the other hand, if the local cap is decreased, then each periodi-
cal outcome is more limited and thus also the final payoff is influenced. This
parameter dependence is illustrated in the following two figures, where in each
case all other factors are held constant.
0
0.1
0.2
0.3
0.4
0.5
0
0.01
0.02
0.03
1000
1050
1100
1150
Optionprice
Figure 20: Option price surface w.r.t. c and σ
37

0
0.1
0.2
0.3
0.4
0.5
0
0.05
0.1
1000
1020
1040
1060
1080
Volatility σGuaranteed rate g
Optionprice
Figure 21: Option price surface w.r.t. g and σ
A product issuer might now be interested in the contract design of a cliquet
option with a fixed initial price p0(XT ), where the issuer is allowed to set the
parameters g and c. If the originator decides to offer a higher guaranteed rate,
this has to be financed via a limitation on the periodical returns, i.e. c has to
be chosen sufficiently small. Vice versa, if the issuer is interested in originating
a cliquet contract with a higher local cap c, the guaranteed rate g has to be
reduced in order to offer the same fixed initial price p0(XT ). As the parameter
dependence is slightly non-linear, the figure 22 shows the linkage between the
two components.
38

0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
Guaranteed rate g
Local cap c
Optionprice
Figure 22: Option price surface w.r.t. g and c
39

4 The sum cap contract - a similar product
The sum cap contract is a typical example of a globally floored and locally
capped contract. It thus appears to be a quite similar product compared to
the cliquet contract presented earlier. The only difference is the local floor in
the cliquet contract. This chapter provides a short overview on the semi-closed-
form solution for the price of the the sum cap contract. The proofs of the results
follow the same pattern as in the chapter on cliquet options and are thus sup-
pressed. The interested reader is advised to transfer the proofs presented in the
earlier chapter to this product or follow [BL13].
4.1 Pricing formula of sum cap contracts
The payoff of the sum cap contract is given by
SCT = K max 1 + g, 1 +
n
k=1
min (c, Rk)
= K (1 + g) + K max 0,
n
k=1
Lk ,
where Lk is defined as
Lk := min (c, Rk) −
g
n
for k = 1, ..., n.
As the increments of the underlying asset price process are i.i.d., the modified
Lk are i.i.d. random variables. Since their distribution does not depend on k,
denote by L a corresponding i.i.d. random variable.
For illustration purposes a 5-year monthly sum cap contract on the S&P 500
Index is considered - a similar experiment has been performed in the first chapter
on cliquet options. Let the local cap be equal to c = 8.5% and the guaranteed
rate is set to g = 10%. Based on a historical data set of the closing prices since
January 1999 a naive sampling algorithm with A = 105
trials is performed. The
figure 23 displays the sampled distribution of the rate of return of the cliquet
option at maturity.
40

0.2 0.4 0.6 0.8 1 1.2 1.4
0
500
1000
1500
2000
2500
3000
3500
4000
Rate of return at maturity
Frequency
Figure 23: Sampled distribution of the rate of return at maturity
It is possible to derive a semi-closed-form solution for the price of a sum cap
contract as in the case of cliquet options. Here, the time-0 price p0 of the sum
cap contract is given by
p0 (SCT ) = K (1 + g) e−rT
+ K
e−rT
2
(ΘL + nEL) ,
where ΘL is deﬁned as
ΘL =
2
π
∞
0
t−2
(1 − Re (ϕn
L (t))) dt.
The characteristic function ϕL of L is given by
ϕL(t) := E eitLk
= e−it(1+ g
n ) 1 + it
1+c
0
eitx
Q (R ≥ x − 1) dx .
The expectation of L under the risk neutral measure Q is equal to
EL = c −
g
n
Q (R ≥ c) +
c− g
n
−1− g
n
xfR x +
g
n
dx.
In the Black-Scholes model the price of a sum cap contract with payoﬀ
SCT = K (1 + g) + K max 0,
n
k=1
Lk
41

is given by
K (1 + g) e−rT
+ K
e−rT
2
2
π
∞
0
t−2
(1 − Re (ϕn
L (t))) dt + nEL ,
where
EL = c −
g
n
N
mξ − ln(1 + c)
σ
√
∆
+
1+c
0
y − 1 −
g
n
1
√
2π∆σy
e−
(ln(y)−mξ)
2
2σ2∆ dy
and
ϕL(t) = e−it(1+ g
n ) 1 + it
1+c
0
eitx
N
mξ − ln(x)
σ
√
∆
dx .
4.2 Numerical pricing of sum cap contracts
The aim is now to display differences between the particular behavior of the
prices of sum cap and cliquet contracts. Therefore a numerical example is
investigated. In the following a monthly sum cap contract specified by the
following parameters is considered.
T 5
n 60
K 1,000
g 0.10
c 0.085
r 0.02
Having introduced the numerical pricing of cliquet options, the implementation
of the semi-closed-form solution and also the Vega calculation of the sum cap
contract are now straightforward. In figure 24, the price of the sum cap contract
is reported as a function of the volatility parameter σ.
42

0 0.1 0.2 0.3 0.4 0.5
980
1000
1020
1040
1060
1080
1100
Volatility σ
Time−0Priceofthesumcapcontract
Figure 24: Price of the sum cap contract w.r.t. σ
Observe that the price has a very particular behavior with respect to the volatil-
ity parameter σ. Interesting facts include the non-monotonicity and the change
in the curvature of the price of the sum cap contract. Although the sum cap
and the cliquet contract seem to be quite similar at a first glance, it becomes
obvious that the non-existence of a local floor in the sum cap contract, changes
the outcome dramatically.
It is moreover interesting to examine the Vega of the contract. Of course, by
the foregoing figure the Vega changes the sign at σ ≈ 16.5%. But the shape of
the curve - see figure 25 - shows an intriguing behavior. The curvature of the
Vega of the sum cap contract with respect to σ changes several times.
43

0 0.1 0.2 0.3 0.4 0.5
−600
−400
−200
0
200
400
600
800
1000
Volatility σ
Time−0Vegaofthesumcapcontract
Figure 25: Vega of the sum cap contract w.r.t. σ
44

5 The influence of stochastic volatility
As presented in the analysis of the Vega of the cliquet option, the price of the
contract reacts quite sensitively to small changes in the volatility parameter.
Although it is appealing that a semi-closed form solution can be derived in the
Black-Scholes market model, many assumption like constant volatility do not
find justifications in financial markets. Relaxing the assumption of constant
volatility might therefore be important to receive market-consistent prices and
to account for the sensitivity of the option price to changes in volatility.
5.1 The Heston stochastic volatility model
The Heston model (introduced as in [Ha10], cf. [Ga06]) belongs to the class of
models, relaxing the constant volatility assumption by making volatility stochas-
tic and thereby incorporating phenomena like volatility clustering. Among the
stochastic volatility models this model stands out as a closed-form solution for
European call options is provided [He93].
The dynamics of the stock price in the Heston model (St)t∈[0,T ] under the risk
neutral measure Q are given by
dSt = St rtdt +
√
νtdWS
t
with constant initial level S0 ≥ 0. The stochastic variance process (νt)t∈[0,T ]
with constant initial value ν0 ≥ 0 is specified by
dνt = λ (ς − νt) dt + γ
√
νtdWν
t .
In this setting (WS
, Wν
) is a two-dimensional Wiener process under Q with
instantaneous correlation ρ, i.e.
dWS
t dWν
t = ρdt.
The model thus consists of the following parameters: the initial stock price
S0, the initial variance ν0, the long run variance ς ≥ 0, the mean reversion
rate λ ≥ 0, the volatility of the variance γ ≥ 0 and the correlation, called
leverage parameter, ρS,ν with |ρS,ν| ≤ 1. The denomination of ρS,ν is justified
as typically −1.0 < ρS,ν < −0.6, which implies that a decline in the stock price
correlates with a rise in the volatility, a phenomenon called leverage effect in
the traditional finance literature. For simplicity, it assumed that interest rates
are constant in the following, i.e. rt ≡ r.
The figure below illustrates a possible evolution of the stock price and variance
process in the Heston model for 200 trading days. The parameters are chosen
as follows:
S0 10
v0 0.20
r 0.02
λ 0.50
γ 0.15
ς 0.20
ρS,ν -0.70
45

0 20 40 60 80 100 120 140 160 180 200
10
12
14
Days
StockpriceS
Dynamics in the Heston stochastic volatility model
0 20 40 60 80 100 120 140 160 180 200
0,15
0,20
0,25
Days
Varianceν
Figure 26: Sample paths in the Heston model
The mean reverting property of the variance process as well as the leverage ef-
fect can easily be noticed in this particular sample path.
The variance process is well understood and several important results have been
shown (e.g. [BM01], [KT81]). For instance, the Feller condition guarantees that
the process is strictly positive, if 2λς > γ2
. On the other hand, if 2λς ≤ γ2
, the
origin is accessible and strongly reflecting, i.e. the process will not stay at zero.
Furthermore, the conditional distribution of the variance process is known to
be proportional to a non-central chi-squared distribution.
5.2 Stock price simulation in the Heston model
A naive implementation of the variance dynamics in order to simulate the stock
price in the Heston model might lead to difficulties. Using a Euler scheme for
the discretization of the dynamics might lead to negative values for the variance
process and one should be aware how to overcome these difficulties in a proper
simulation scheme.
For t > s ≥ 0 a naive Euler discretization of variance process leads to
νt = νs + λ(t − s) (ς − νs) + γ νs(t − s)Yν,
where Yν is a standard normally distributed random variable. Notice, that the
probability of νt becoming negative decreases in the time step t − s, but it is
strictly positive for every step size, unless γ = 0. In principal, there exist two
best practices: making zero an absorbing or reflecting boundary for the variance
process (cf. [Ha10]). The proposed, almost bias-free discretization scheme for
the variance, called full truncation, is specified by:
νt = νs + λ(t − s) ς − ν+
s + γ ν+
s (t − s)Yν.
Provided with the scheme for the variance process, the stock price process has
to be approximated. An application of the Itô formula yields the exact solution
46

of the stock price dynamics, which is given by
St = Ss exp
t
s
r −
1
2
νu du +
t
s
√
νudWS
u .
Therefore, the following log-Euler scheme for the asset price process is obtained:
log (St) = log (Ss) + r −
1
2
ν+
s (t − s) + ν+
s (t − s)YS,
where YS is a normally distributed random variable with correlation ρS,ν to Yν.
In an implementation the correlated random variables Yν and YS are generated
by setting Yν = Y1 and YS = ρS,νYν + 1 − ρ2
S,νY2, where Y1 and Y2 are two
independent random samples of a standard normal distribution. In Matlab the
multivariate normal random numbers generator mvnrnd(·, ·) is used for this
purpose. Thus, a simple and computing time eﬃcient discretization scheme is
implemented.
5.3 Monte Carlo technique II
Having introduced the Monte Carlo technique to price cliquet options in the
Black-Scholes model, an application to the Heston model is now straightfor-
ward. The simulation of the stock price is used to calculate the periodical
returns, the further steps are identical as only the dynamics of the underlying
asset price process change.
Consider a 5-year cliquet option with monthly resets. The local cap c is equal
to 1% and the guaranteed rate g is set to 30%. In the Heston model, the mean
reversion rate λ is set to 0.5 and the correlation ρ is given by −0.7. Moreover,
suppose that ν0 = ς. In the experiment, the inﬂuence of ς and γ on the price of
the cliquet option is analyzed. Therefore, the Monte Carlo technique is applied
for ς varying from 0.01 to 0.20 and the values for γ range from 0 to 0.5. The
associated option price surface is displayed below.
47

0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
1176
1178
1180
1182
1184
Volatility of the variance γ
Option price surface
Long run variance ς
Priceofthecliquetoption
Figure 27: Option price surface w.r.t. ς and γ
Under the assumption that ν0 = ς, the dynamics of the asset price processes in
the Black-Scholes and Heston model coincide for γ = 0 and σ =
√
ν0. Therefore,
the price of the cliquet option for this special case is the same in both market
models. But, it is not clear how the price of the option is inﬂuenced by an in-
crease in the volatility of the variance (leaving all other parameters unchanged).
The simulation supports the intuition, that an increase in γ should result in a
lower fair value of the cliquet contract. In this example ν0 is chosen to be equal
to 0.20.
48

0 0.1 0.2 0.3 0.4 0.5
1177
1178
1179
1180
1181
1182
1183
1184
Comparision of the price of the cliquet option
Price in the Heston model
Price in the Black Scholes model
Figure 28: Option price in the Black-Scholes and Heston model w.r.t. γ
In the Monte Carlo approach the final payoffs of the contract are simulated. For
each fixed value of γ the standard deviation of 105
simulated payoffs is calcu-
lated. Interestingly, the same qualitative behavior as for the price of the option
is recognized for the standard deviation.
49

0 0.1 0.2 0.3 0.4 0.5
4
6
8
10
12
14
16
Standarddeviation
Figure 29: Standard deviation of the simulated payoffs w.r.t. γ
Suppose now, in the Heston model it holds that γ = 0.25 for the variance pro-
cess. Based on the earlier numerical example one might expect the price in the
Heston model to be lower than the price in the Black-Scholes market. A varying
long-term variance ς has a significant effect on this relationship. Therefore the
difference in the prices of the contracts in the both models w.r.t. ς = ν0 = σ2
is
shown in the following figure. The graph has a humped shape with a maximum
at ς ≈ 0.05.
50

0 0.1 0.2 0.3 0.4 0.5
−1
0
1
2
3
4
5
6
Price difference in the market models w.r.t.ς
Long run variance ς
Black−Scholesprice−Hestonprice
Figure 30: Diﬀerence of the option price in the market models w.r.t. ς
51

6 The influence of stochastic interest rates
The theory for pricing equity derivatives in the Black-Scholes model is based on
the assumption of deterministic interest rates. From a practitioner’s point of
view this simplification might be harmless in most situations as the variability
of interest rates is negligible compared to the volatility in the equity markets.
Nevertheless, when pricing long-dated options the fluctuations in interest rates
have a stronger impact on the fair price of a contingent claim. It is therefore ad-
visable to relax the assumption of deterministic or even constant interest rates.
6.1 The Black-Scholes-Hull-White model
The Black-Scholes-Hull-White (BSHW) model (introduced as in [Ha10]) com-
bines the Black-Scholes model for the dynamics of the asset price process and
the Hull-White model for the dynamics of the short rate. Under the risk neutral
measure Q (using the bank account as numeraire), the dynamics of the stock
price in the BSHW model (St)t∈[0,T ] are given by
dSt = St rtdt + σdWS
t
with constant initial level S0 ≥ 0. The short rate (rt)t∈[0,T ] follows an Ornstein-
Uhlenbeck process with constant initial value r0, i.e.
drt = (τt − brt) dt + ψdWr
t .
In this setting (WS
, Wr
) is a two-dimensional Wiener process under Q with
instantaneous correlation ρ, i.e.
dWS
t dWr
t = ρdt.
τt is a time-dependent, deterministic function, that describes the long-term
mean level of the instantaneous interest rate and is in general chosen in such a
way, that it exactly fits the currently observed term structure of interest rates.
The parameter b ≥ 0 is the speed of reversion and characterizes the velocity of
regrouping at τt. ψ is interpreted as the instantaneous volatility. Thus, the short
rate follows a mean-reverting, stationary, Gaussian and Markovian process (see
[BM01]). For simplicity, it assumed that the long-term mean level is constant
in the following, i.e. τt ≡ τ (the Vaˇs´ıˇcek model for the dynamics of the short
rate).
The figure below illustrates a possible evolution of the stock price and interest
rate process in the BSHW model for 200 trading days. The parameters are
chosen as follows:
S0 10
r0 0.02
τ 0.04
b 2
ψ 0.10
σ 0.20
ρS,r 0.00
52

0 20 40 60 80 100 120 140 160 180 200
8
10
12
14
Days
StockpriceS
Dynamics in the BSHW model
0 20 40 60 80 100 120 140 160 180 200
−0.1
0
0.1
Days
Shortrater
Figure 31: Sample paths in the BSHW model
6.2 Stock price simulation in the BSHW model
The simulation of the processes in the BSHW model is straightforward. For
t > s ≥ 0 a naive Euler discretization of short rate process leads to
rt = rs + (τ − brs)(t − s) + ψ (t − s)Yr,
where Yr is a standard normally distributed random variable. The log-Euler
scheme then yields the discretization of the asset price process:
log (St) = log (Ss) + rs −
1
2
σ2
(t − s) + σ (t − s)YS,
where YS is a normally distributed random variable with correlation ρS,r to Yr.
6.3 Monte Carlo technique III
After revisiting the Monte Carlo technique in the Heston model, the basic idea is
obvious: the simulation of the stock price in the BSHW model is used to calcu-
late the periodical returns to simulate the ﬁnal payoﬀs of the cliquet contract.
But in comparison to the Black-Scholes and the Heston model, not only the
dynamics of the asset price change, also the discount factor has to be adapted.
Remember, that for constant rates r, discounting is achieved by multiplying by
exp(−rT). This corresponds to exp −
T
0
rtdt for time-dependent rates. Hav-
ing simulated a sample path for the short rate, the discount factor is obtained
via exp (−∆
n
k=1 rtk
). The resulting sample distribution of the discount factor
is displayed in the histogram below for A = 105
trials.
53

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0
1000
2000
3000
4000
5000
6000
7000
8000
Sample distribution of the discount factor
Discount factor
Frequency
Figure 32: Sample distribution of the discount factor
The sample distribution of the non-discounted final payoffs of the cliquet con-
tract can be found in the figure below.
1200 1250 1300 1350 1400 1450
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Sample distribution of the final payoff
Final payoff
Frequency
Figure 33: Sample distribution of the final payoff
The associated distribution of the time-0 price of the contract is displayed here-
after.
54

600 800 1000 1200 1400 1600 1800 2000
0
1000
2000
3000
4000
5000
6000
7000
8000
Sample distribution of the price of the cliquet option
Price of the cliquet option
Frequency
Figure 34: Sample distribution of the price of the cliquet option
In the following, the sensitivity of the price of the cliquet option w.r.t. the
instantaneous volatility ψ is investigated.
0 0.1 0.2 0.3 0.4 0.5
1140
1160
1180
1200
1220
1240
1260
1280
1300
1320
Price of the cliquet option
Instantaneous volatility ψ
Figure 35: Price of the cliquet option w.r.t. ψ
55

In this example, the price of the cliquet contract is strictly increasing and convex
in ψ. This qualitative behavior is unchanged for different correlations between
the stock price and short rate process. Nevertheless, there exist an influence of
the correlation of the Wiener processes, which is displayed in the chart hereafter.
0 0.1 0.2 0.3 0.4 0.5
−20
−15
−10
−5
0
5
10
15
Differencetonon−correlatedprocesses
perfectly positive correlated
perfectly negative correlated
Figure 36: Difference of the option price due to correlation
If the processes are perfectly positive correlated the price difference to the non-
correlated option price is positive and shows an upward-sloping trend, whereas
in the negatively correlated case the opposite is true.
In the following consider the model specified by the parameters hereafter.
S0 10
r0 0.03
τ 0.03
b 2
ψ 0.10
σ 0.20
ρS,r 0.00
The long-term mean level of the instantaneous interest rate is thus given by
r = τ
b = 0.015. Therefore, it is interesting to check how the feature of a
stochastic interest rate changes the option price compared to pricing with the
constant rate r.
56

Interest rate model Option price Standard deviation
Vaˇs´ıˇcek 1,211.4 15.09
Constant rates 1,212.2 14.61
In this case, the price of the cliquet option is slightly lower in the Vaˇs´ıˇcek model,
whereas the standard deviation increases by 3.3%.
6.4 Further short rate models
Interest rate modeling was traditionally based on assumptions on the dynamics
of the short rate r. Besides the earlier considered time-homogeneous Vaˇs´ıˇcek
model, there is a wide variety of possible designs (cf. [BM01] for the results
hereafter). In this subchapter the influence of the specific model on the resulting
option price is analyzed. Recall that the dynamics of the short rate in the
Vaˇs´ıˇcek model are given by
drt = (τ − brt) dt + ψdWr
t .
This model allows for non-positive interest rates and the short rate is normally
distributed. Therefore, it is interesting to check the pricing results in models
with different properties. In the Cox-Ingersoll-Ross model the short rate is
specified as follows
drt = (τ − brt) dt + ψ
√
rtdWr
t .
Here, the short rate is characterized by a non-central chi-squared distribution.
For parameters ranging in a reasonable region the model admits positive rates
only. In the Black-Karasinski model, the short rate is lognormally distributed
and does not allow for non-positive interest rates. In this case, the dynamics
are given by
drt = rt (o − h ln rt) dt + ψrtdWr
t .
The figure below illustrates a possible evolution of the interest rate processes
in the short rate model for 200 trading days. The parameters are chosen as
follows:
r0 0.02
τ 0.03
o -6
b 2
h 1.5
ψ 0.10
57

0 50 100 150 200
−0.1
−0.05
0
0.05
0.1Vasicek
Sample paths of the dynamics in the short rate models
0 50 100 150 200
0
0.01
0.02
0.03
Cox−Ingersoll−Ross
0 50 100 150 200
0.01
0.02
0.03
Days
Black−Karasinski
Figure 37: Sample paths in the short rate models
The assumptions on the short rate in the models lead to diﬀerent pricing results.
For the exemplary cliquet option contract the Monte Carlo approach yields the
following prices.
Interest rate model Option price Standard deviation
Vaˇs´ıˇcek 1,214.4 12.63
Cox-Ingersoll-Ross 1,209.5 14.06
Black-Karasinski 1,191.6 14.09
Interestingly, there is an approximately 2% price diﬀerence between the Vaˇs´ıˇcek
and the Black-Karasinski model. Furthermore, the standard deviation in the
pricing procedure increases slightly when the Vaˇs´ıˇcek model is not used.
Recall, an foregoing example showed that the price of the cliquet option is
increasing and convex in the instantaneous volatility ψ.
58

0 0.1 0.2 0.3 0.4 0.5
1200
1250
1300
1350
1400
Price in the Vasicek model
Figure 38: Price w.r.t. ψ in the Vaˇs´ıˇcek model
Observe that in this case the prices of the contract range from slightly above
1, 200 to almost 1, 400. In the other models, the same linkage w.r.t. the param-
eter ψ is illustrated in the two charts below.
59

0 0.1 0.2 0.3 0.4 0.5
1208
1210
1212
1214
1216
Price in the Cox−Ingersoll−Ross model
Figure 39: Price w.r.t. ψ in the Cox-Ingersoll-Ross model
But in both cases the price range is tighter. Pricing the option in the Cox-
Ingersoll-Ross model yields a 6 units wide price range (from 1, 209 to 1, 215). In
the Black-Karasinki model, the lowest price is 1, 191 and reaches a maximum of
1, 195.
60

0 0.1 0.2 0.3 0.4 0.5
1190
1192
1194
1196
Price in the Black−Karasinski model
Figure 40: Price w.r.t. ψ in the Black-Karasinski model
The qualitative behavior w.r.t. the parameter ψ is thus unchanged in the dif-
ferent short rate models, but the detailed analysis prevails the inﬂuences and
diﬀerences of the models.
61

Conclusion
In a publication from 2002, P. Wilmott has already described cliquet options
as ”the height of fashion in the world of equity derivatives” [Wi05]. As these
contracts provide a downside protection while simultaneously offering an enor-
mous upside potential, they might be a perfect fit during turmoils in financial
markets. In the aftermaths of the subprime crisis, these contracts were rarely
traded and cliquet options are drawing attention more heavily only since 2010.
Nevertheless, research literature on this topic is not widespread, neither in the
fields of finance nor in mathematical areas, as the pricing concepts are rather
proprietarily developed by investment banks.
This master’s thesis tries to capture recent research highlights on the pricing
concepts used for these type of contracts. The presented ansatz of developing
a semi-closed-form solution for the option price might serve as starting point
for further applications on various equity-linked derivatives. Besides the com-
putational tractability, the formula enables to calculate hedging parameters like
the Vega of the option price easily. From an investor’s perspective as well as
product issuer the choice of the local cap and the guaranteed rate is crucial
and changes the pricing results vehemently. Moreover, the numerical examples
showed the tremendous influence of the pricing parameters. These illustrations
prevail that exotic traders at investment banks face challenges when trying to
price these path-dependent structures. A reinforcement of these issues is showed
in financial market models allowing for stochastic volatility and stochastic in-
terest rates. The main difficulty in pricing and hedging the cliquet options in
a non-Monte Carlo fashion is the design of a expression for the expectation as
well as for the characteristic function of the truncated returns of the underlying
asset.
Cliquet options as equity-linked annuities provide interesting opportunities for
investors - also in terms of insurance products. But as many aspects regarding
the pricing of these structured products are not clarified yet, further research
should focus on these investment vehicles to offer an in-depth understanding of
the contracts.

References
[BBG11] C. Bernard, P. P. Boyle, W. Gornall (2011): Locally-capped investment
products and the retail investor, The Journal of Derivatives, Summer
2011, Vol. 18 (4): pp. 72-88
[BB11] C. Bernard, P. P. Boyle (2011): A natural hedge for equity indexed
annuities, Annals of Actuarial Science, September 2011, Vol. 5 (2): pp.
211-230
[BL13] C. Bernard, W. V. Li (2013): Pricing and hedging of cliquet contracts
and locally-capped contracts, SIAM Journal on Financial Mathematics,
April 2013, Vol. 4 (1): pp. 353-371
[BM01] D. Brigo, F. Mercurio (2001): Interest Rate Models - Theory and Prac-
tice, Springer Finance
[Ga06] J. Gatheral (2006): The Volatility Surface - A Practitioner’s Guide,
Wiley Finance
[Gl03] P. Glasserman (2003): Monte Carlo Methods in Financial Engineering,
Springer Finance
[Ha82] U. Haagerup (1982): The best constants in the Khintchine inequality,
Studia Mathematica, 1981, Vol. 70 (3): pp. 231-283
[Ha10] A. v. Haastrecht (2010): Pricing Long-Term Options with Stochastic
Volatility and Stochastic Interest Rate, PhD Thesis
[He93] S. L. Heston (1993): A closed-form solution for options with stochastic
volatility with applications to bond and currency options, The Review of
Financial Studies, 1993, Vol. 6 (2): pp. 327-343
[Ja02] P. Jaeckel (2002): Monto Carlo Methods in Finance, Wiley Finance
[KT81] S. Karlin, H. M. Taylor (1981): A Second Course in Stochastic Pro-
cesses, Gulf Professional Publishing
[KK01] E. Korn, R. Korn (2001): Option Pricing and Portfolio Optimization:
Modern Methods of Financial Mathematics, Oxford University Press
[Le91] R. B. Leipnik (1991): On lognormal random variables: I-the charac-
teristic function, The Journal of the Australian Mathematical Society,
1991, Vol. 32 (Series B): pp. 327-347
[Pa06] B. A. Palmer (2006): Equity-Indexed Annuities: Fundamental Concepts
and Issues, Insurance Information Institute
[Wi05] P. Wilmott (2005): The Best of Wilmott 1: Incorporating the Quanti-
tative Finance Review, Wiley Finance
[Jo10] J. Sass (2010): Financial Mathematics 1, Lecture Notes
[Jo13] J. Sass (2013): Probability Theory, Lecture Notes
[Sh04] S. E. Shreve (2004): Stochastic Calculus for Finance II, Springer Fi-
nance

Statutory Declaration
Peter Warken
I hereby declare, that I have authored the thesis
Eﬀective Pricing of Cliquet Options
independently and that I have not used other than the declared sources/resources.
Frankfurt am Main, December 6, 2015
Best regards,
Peter Warken

Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015

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