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Pricing FX Quanto Options underStochastic Volatility
A dissertation submitted to theWARWICK BUSINESS SCHOOLUNIVERSITY
OF WARWICK
in partial fulfillment of the requirements for the degree ofMSc
in Financial Mathematics
submitted byTANMOY NEOG (0850324)
supervised byProf. Dr. Nick Webber
7 September, 2009
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All the work contained is my own unaided effort and conforms to
the University guidelines on
plagiarism
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Acknowledgment
I would like to begin by thanking my supervisor Prof. Nick
Webber. I thank him for his patient
guidance and his enthusiasm to answer my questions. This
dissertation introduced me to the
intricacies of derivative pricing. It was the enthusiasm of my
supervisor that gave me the impetus
to try to improve my results. Hopefully I did not do very badly.
I also thank the WBS authorities
who ensured that we had the adequate facilities to work
efficiently.
I thank all the faculty members involved with the Financial
Mathematics course. I could learn a
lot from the rigour involved in this course. I thank my
batchmates Vineet Thakkar, Zenon, Ravi
Ganesan and Piyush Singh for several discussions related to
Financial Mathematics. Coming
from a background where I had no knowledge of Computational
Finance; these individuals
helped me to learn a lot of things in lesser time than I would
have taken. Last but not the least
special thanks to my dear friend Vallu with whom there was never
a dull moment.
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Contents
List of Figures vi
List of Tables viii
Abstract x
1 Literature Review 4
1.1 Bennett and Kennedy’s methodology for pricing the FX Quanto
Option . . . . . 4
1.2 Pricing the FX quanto option under different frameworks . .
. . . . . . . . . . . 7
2 Pricing FX Quanto Options in the Black Scholes Framework
14
2.1 The standard market practice in pricing FX Quanto Options .
. . . . . . . . . . 14
2.1.1 Pricing using the Black Scholes Model . . . . . . . . . .
. . . . . . . . . . 15
3 Pricing of FX Quanto under the Heston Model 17
3.1 Revisiting the Heston model . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 17
3.2 Option Pricing under the Heston model . . . . . . . . . . .
. . . . . . . . . . . . 18
3.2.1 The Heston Pricer using Fast Fourier Transform . . . . . .
. . . . . . . . 19
3.2.2 The Heston Trap . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 21
3.3 The Heston model with jumps . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 23
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3.4 Pricing the FX Quanto option . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 24
4 Pricing of FX Quanto under the GARCH Option Pricing Model
26
4.1 Duan’s GARCH Option Pricing Model . . . . . . . . . . . . .
. . . . . . . . . . . 26
4.2 Analytical Approximation of the GARCH Option Pricing Model .
. . . . . . . . 28
5 Modelling the Dependence Structure Using a Copula 31
5.1 Archimedean Copulas . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 33
5.1.1 Archimedean Copulas and their Measures of Dependence . . .
. . . . . . 33
5.1.2 Identifying the Right Copula from the Archimedean Family
to Model the
Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 35
5.2 The T Copula . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 37
5.2.1 Calibration of the T Copula . . . . . . . . . . . . . . .
. . . . . . . . . . . 38
5.3 Pricing using the copula . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 39
6 Data 41
6.1 Discount Factors, Implied Volatility quotes . . . . . . . .
. . . . . . . . . . . . . 41
6.2 Recovering Strikes and Prices of FX vanilla options . . . .
. . . . . . . . . . . . . 42
7 Implementation of Pricing Methods 45
7.1 Implementation of the Heston Stochastic Volatility Model
with Jumps . . . . . . 45
7.1.1 Calibration of the Model . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
7.1.2 Finding Implied Volatility . . . . . . . . . . . . . . . .
. . . . . . . . . . . 49
7.2 Implementation of the GARCH Option Pricing Model . . . . . .
. . . . . . . . . 50
7.2.1 Calibration Results . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 53
7.3 Choosing a model to price the Quanto option . . . . . . . .
. . . . . . . . . . . . 56
7.4 Results of pricing the quanto option . . . . . . . . . . . .
. . . . . . . . . . . . . 56
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7.5 Implementing a copula to price the FX quanto option . . . .
. . . . . . . . . . . 58
7.5.1 Parameter estimation for the T Copula . . . . . . . . . .
. . . . . . . . . 59
7.5.2 Parameter estimation for the Frank Copula . . . . . . . .
. . . . . . . . . 60
7.5.3 Calibration of marginals . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 60
7.5.4 Numerical Results for pricing using copula . . . . . . . .
. . . . . . . . . . 62
7.6 Monte Carlo Errors . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 64
8 Conclusion 69
Bibliography 71
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List of Figures
5.1 Frank Copula with ® = 1.5 . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 37
5.2 Clayton Copula with ® = 1.5 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 37
5.3 Gumbel Copula with ® = 1.5 . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 37
7.1 Time series observations of USD/JPY spot exchange rate . . .
. . . . . . . . . . 46
7.2 Market and model volatility(in %) for USD/YEN call options
with maturity 1
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 50
7.3 Market and model volatility(in %) for USD/YEN call options
with maturity 3
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 50
7.4 Market and model volatility(in %) for USD/YEN call options
with maturity 6
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 51
7.5 Market and model volatility(in %) for USD/YEN call options
with maturity 12
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 51
7.6 Market and model volatility(in %) for USD/YEN call options
with maturity 24
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 52
7.7 Market Implied Volatility Surface . . . . . . . . . . . . .
. . . . . . . . . . . . . . 53
7.8 Heston with Jumps Implied Volatility Surface . . . . . . . .
. . . . . . . . . . . . 53
7.9 Canonical Log likelihood function (Mashal and Zeevi) . . . .
. . . . . . . . . . . 60
7.10 Market and model volatility (in %) for EUR/USD call options
with maturity 1
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 62
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7.11 Market and model volatility(in %) for EUR/USD call options
with maturity 6
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 62
7.12 Market and model volatility(in %) for EUR/USD call options
with maturity 12
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 63
7.13 Market and model volatility(in %) for EUR/USD call options
with maturity 2
years . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 63
7.14 Market and model volatility(in %) for EUR/JPY call options
with maturity 1
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 64
7.15 Market and model volatility(in %) for EUR/JPY call options
with maturity 3
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 64
7.16 Market and model volatility(in %) for EUR/JPY call options
with maturity 6
months . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 65
7.17 Market and model volatility(in %) for EUR/JPY call options
with maturity 2 years 65
7.18 Frank Copula with ® = 0.0075 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 66
7.19 T Copula with º = 14 and ½ = 0.6 . . . . . . . . . . . . .
. . . . . . . . . . . . . 66
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List of Tables
5.1 Archimedean Copulas and their Generators . . . . . . . . . .
. . . . . . . . . . . 33
5.2 Archimedean Copulas and Measures of Dependence . . . . . . .
. . . . . . . . . . 34
6.1 Discount factors Di0,T corresponding to currency i and
maturity T . . . . . . . . 42
6.2 EUR/USD implied volatility quotes of call options
corresponding to standard
values of Black Delta and maturity T . . . . . . . . . . . . . .
. . . . . . . . . . 42
6.3 EUR/JPY implied volatility quotes of call options
corresponding to standard
values of Black Delta and maturity T . . . . . . . . . . . . . .
. . . . . . . . . . 43
6.4 USD/JPY implied volatility quotes of call options
corresponding to standard
values of Black Delta and maturity T . . . . . . . . . . . . . .
. . . . . . . . . . 43
7.1 Parameters of Heston with jumps calibration on USD/JPY rates
. . . . . . . . . 48
7.2 Parameters of GARCH calibration on USD/JPY rates . . . . . .
. . . . . . . . . 53
7.3 Implied volatilities for maturity of 1 month . . . . . . . .
. . . . . . . . . . . . . 54
7.4 Implied volatilities for maturity of 3 months . . . . . . .
. . . . . . . . . . . . . . 54
7.5 Implied volatilities for maturity of 6 months . . . . . . .
. . . . . . . . . . . . . . 55
7.6 Implied volatilities for maturity of 1 year . . . . . . . .
. . . . . . . . . . . . . . 55
7.7 Implied volatilities for maturity of 2 years . . . . . . . .
. . . . . . . . . . . . . . 55
7.8 Root mean squared errors for the implied volatility fits on
USD/JPY FX rate . . 56
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7.9 Quanto prices for maturity of 1 month. Spot price is 0.0083
dollars. All the prices
are in the Euro currency.Strike prices in dollars. . . . . . . .
. . . . . . . . . . . . 57
7.10 Quanto prices for maturity of 3 months.Spot price is 0.0083
dollars. All the prices
are in the Euro currency.Strike prices in dollars. . . . . . . .
. . . . . . . . . . . . 57
7.11 Quanto prices for maturity of 6 months. Spot price is
0.0083 dollars. All the
prices are in the Euro currency.Strike prices in dollars. . . .
. . . . . . . . . . . . 58
7.12 Quanto prices for maturity of 1 year. Spot price is 0.0083
dollars. All the call
prices are in the Euro currency.Strike prices in dollars. . . .
. . . . . . . . . . . . 58
7.13 Quanto prices for maturity of 2 year. Spot price is 0.0083
dollars. All the prices
are in the Euro currency.Strike prices in dollars. . . . . . . .
. . . . . . . . . . . . 59
7.14 Parameters for the T copula . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 59
7.15 Parameters for the Frank copula . . . . . . . . . . . . . .
. . . . . . . . . . . . . 60
7.16 Parameters for EUR/USD and EUR/JPY after calibration . . .
. . . . . . . . . 61
7.17 Root mean squared errors for the implied volatility fits on
EUR/USD and EUR/JPY
FX rates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 61
7.18 Relative Difference (RD)with Black Scholes for the models
for maturity of 1
month. Values have been rounded off. . . . . . . . . . . . . . .
. . . . . . . . . . 66
7.19 Relative Difference(RD)with Black Scholes for the models
for maturity of 3 months.
Values have been rounded off. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 67
7.20 Relative Difference(RD)with Black Scholes for the models
for maturity of 6 months.
Values have been rounded off. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 67
7.21 Relative Difference(RD)with Black Scholes for the models
for maturity of 1 year.
Values have been rounded off. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 68
7.22 Average Standard Monte Carlo errors.Each error is an
average of 25 simulations
for 25 different quanto options. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 68
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Abstract
This dissertation looks at the pricing of FX quanto options
using stochastic volatility models.
There are a lot of stochastic volatility models available.
However we look only at the applicability
of the Heston model with jumps and the GARCH Option Pricing
Model. By a no arbitrage
condition on FX rates we can view the FX quanto as a multi
currency option. In a Black type
of model the pricing of the FX quanto depends on implied
volatilities of three exchange rates.
This encourages the use of a copula function to model the
dependency. We do so by pricing
the quanto with a T Copula; member of the elliptic family and
the Frank Copula; a member of
the Archimedean Copula family with the marginals as Heston with
jumps process. We observe
that under stochastic volatility there is a considerable
difference in option prices from the Black
Scholes model. This is more so for options which have a low
delta. Further given our data set
the Heston model with jumps fits the market behaviour for plain
vanilla FX options better than
the GARCH Option Pricing model.
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Introduction
In this dissertation we take up the problem of pricing a
European style FX quanto option
under stochastic volatility. An FX quanto option has as its
underlying an exchange rate with a
domestic and foreign currency. The payoff at maturity is
converted into a third currency. This
third currency is called the quanto currency. An individual
would buy a quanto call option if he
believes that the exchange rate would appreciate. Further he
would expect the quanto currency
to appreciate more than the foreign currency for that particular
exchange rate.
In the past various authors have priced these options in a non
stochastic volatility framework.
The FX quanto is viewed as a multi asset option as volatility of
the underlying exchange rate
is dependent on the volatilities of the quanto domestic and
quanto foreign rates. We have a
closed form solution under the Black Scholes model for pricing
FX quanto options. There have
been authors who have taken a different approach such as
employing a copula to model the
dependence structure between exchange rates.(please refer to the
literature review) The joint
distribution of asset prices is extracted from market implied
volatilities. The price of the quanto
option is then evaluated as an integral involving the joint
density of asset prices.
There have been expressions for quanto options under discrete
time GARCH models which we
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consider as well. While pricing a multi asset product like a
quanto option one needs to take
into account multivariate models which can handle the
co-movement of the underlying price
processes. The multivariate normal distribution is a very easy
way of analyzing returns on
multiple assets. In a multivariate normal distribution the
dependency between the margins is
measured by linear correlation. The actual association between
the different assets may not
be so. The use of a copula is alternative measure of association
between assets. The basic
idea of copulas is to separate the dependence structure between
variables from their marginal
distributions.
Through the choice of copula, we can influence control on the
association of certain parts of the
distribution.For instance at the tails. To give an example it
may so happen that the returns of
two stocks might be correlated in the extreme tails, but not
elsewhere in the distributions, and
there are copulas which can model this behaviour.
The aim of this dissertation would be to investigate as to how
different quanto prices are under
a stochastic volatility framework. At first we do not
incorporate a dependency between the
quanto, foreign and domestic rates. After this we do incorporate
dependency through a copula.
It is unlikely that single copula family model can take care of
the asymmetry of the underlying
asset process. If not then the answer could lie in perturbation
of the copula family as used by
Bennett and Kennedy (2003) or mixture copulas.
I extend the standard FX quanto option pricing in two ways.
First the marginals are assumed to
follow a Heston wih jumps process and the dependence is modelled
using a copula. We choose a
t-copula from the elliptic family and the Frank copula from the
Archimedean family. The choice
of a Frank copula is based on best fit to historical data.
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When we model the individual FX rates as Geometric Brownian
Motion as in the Black’s
formulae the log returns follow a Normal distribution. From an
empirical point of view we
observe that the log returns of exchange rates we consider in
this paper have a high excess
kurtosis of around 9. We need to replace GBM dynamics. Hence I
take the marginals as non
linear GARCH processes. The GARCH process allows fatter tails.
We use the Duan’s GARCH
Option Pricing Model for our purpose.
When using a pricing model it is important for the model to fit
the market implied volatility
quotes for plain vanilla options. We observe that given our
dataset the Heston with jumps model
gives us a much better fit to market quotes as compared to the
GARCH model.
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Chapter 1
Literature Review
Over the years valuation of contingent claims has been an area
of extensive research. The
seminal work done by Black and Scholes (1973) and Cox et al.
(1979) introduced us to risk
neutral pricing models. However there is a shortcoming of these
models. The assumption of
a fixed volatility is violated in real time markets. The problem
of pricing FX quanto options
in a non stochastic volatility model has been elaborately
explained by Bennett and Kennedy
(2003). This paper is like a starting point for our
dissertation. We give a brief review of the
methodology adopted by the authors in pricing the FX quanto
options.
1.1 Bennett and Kennedy’s methodology for pricing the FX
Quanto Option
Benett and Kennedy express the payoff of a plain vanilla FX
option in terms of two other
exchange rates. This is obtained from the triangular no
arbitrage condition for currencies. The
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price of this European multi asset option is written as an
integral involving the joint density
of the asset prices at expiry. Hence the calculation of the
price of the option involves the
integration of a joint density function of the two exchange
rates. The authors then express the
joint density as a product of the marginal density functions of
the individual exchange rates
and a copula function. This is done by using Sklar’s Theorem.
The copula function determines
a joint density function of the two exchange rates. The use of
the copula allows the authors
to separate the modelling of the marginal distributions from the
modelling of the dependence
structure. The marginal densities are taken as mixture of
lognormal distributions. The initial
calibration of marginal distributions to implied volatility
quotes is done by a weighted non
linear least squares optimisation. This process of using option
implied densities is similar to
Dupire (1994). A Gaussian copula is then used which is perturbed
by a cubic spline to get a
dependence structure between the three currency pairs.The
modification of the upper and lower
tail dependence characteristics through this perturbation is to
allow calibration to a smile in
implied volatilities of the FX rates.
At the outset Bennett and Kennedy have considered a Gaussian
copula. However to calibrate
the joint distribution to the implied volatility smile on the FX
rates the dependence structure
associated with the Normal copula is perturbed. With this
perturbation the tail dependence
characteristics are modified. The authors have used the
following result which is due to Gen-
est(2000):
Theorem 1.1. Let ' : [0, 1] −→ [0, 1] be a continuous, twice
differentiable concave function
such that '(0) = 0 and '(1) = 1. Then
C'(u, v) = '−1(C('(u), '(v))) (1.1)
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is a copula if C(.,.) is a copula.
For Benett and Kennedy the starting point is a bivariate Normal
distribution with correlation
parameter ½. A transformation function is then used to modify
tail dependence. I will give a
review of how the authors obtained ' later. I proceed by giving
a mathematical form of the
transformation function.
The Normal Copula has the density function as
CNormal(u, v∣½) = N½(N−1(u), N−1(v)) (1.2)
In equation (1.2) N½ denotes the standard bivariate normal
distribution with correlation ½. N
denotes the standard univariate Normal distribution function. To
obtain the density of the
transformed copula C' the partial derivatives of equation (1.1)
with respect to both arguements
are taken. Bennett and Kennedy obtain c'(u, v), the density of
the transformed copula as
c'(u, v) ='′(u)'′(v)'′(C'(u, v)
(c('(u), '(v))− '
′′(C'(u, v))['′(C'(u, v)]2
dC
du
dC
dv
)(1.3)
The final step which we can observe from equation (1.3) is to
find the transformation function
'. From equation (1.1) we can say that the behaviour of '(x)
near x = 1 controls the upper tail
dependence of the transformed copula. The lower tail dependence
of the transformed copula can
be observed from the behaviour of '(x) near x = 0. The
correlation ½ is fixed. The authors then
change the endpoints of the transformation which enables them to
change the tail dependence
characteristics. Bennett and Kennedy observe that increasing the
size of the second derivative
of ' at the end points can change the tail dependence
characteristics. The authors specify ' as
a cubic spline with k + 1 suitable predefined knot points pi ∈
(−1, 1). The size of the second
derivative is an indicator of the increase in tail dependence.
Hence the second derivative of the
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spline at each knot point is first specified. The co-efficients
of the spline are obtained by making
use of the requirements '(0) = 0 and '(1) = 1.Hence ' is
found.
Hence if the knot points vector be denoted by p as p = (p0, p1,
......, pk)T with p0 = 0 and pk = 1.
The polynomial between the jtℎ and the (j + 1)tℎ nodes is given
by
Sj(x) =3∑
i=0
ai,j(x− pj)i (1.4)
for x ∈ [pj , pj+1] and coefficients ai,j which are determined
from an optimisation proceedure.
The cubic spline takes the value of the polynomial function
Sj(x) between each of the knot
points. The final outcome is that ' is determined and the
vanilla call options are priced. Once
the parameters from the plain vanilla calls are obtained; these
parameters are then used to price
the quanto FX option.
The resulting quanto prices under a number of real scenarios is
generally close to prices under
the Black formulae. The Black model often gives lower prices for
the quanto call options and
higher for put options. The relative difference is occasionally
large, that is in the region of 10
to 15 percent for standard strikes furthest away from
at-the-money.
1.2 Pricing the FX quanto option under different frameworks
We have a standard approach to pricing quanto FX options which
is based on the Black type
model. In this dissertation we will price the FX quanto option
under two diffferent frameworks.
First we consider pricing the quanto in the continuous time
Heston Stochastic Volatility Model
with jumps and then under the GARCH Option Pricing Model.
Comparisons with the the Black
Type model is then done.
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The shortcomings of the Black Scholes model in pricing foreign
currency options has been
shown by Melino and Turnbull (1990) . It makes a strong
assumption that returns are normally
distributed with known mean and variance. Stochastic volatility
models have been used by Hull
and White (1989), Scott (1987). However the methodology adopted
by them to price options
was computationally intensive in the absence of a closed form
solution. Heston (1993) then
came up with a semi closed form solution to price European call
options when the spot asset is
correlated with the volatility. We have a more elaborate
discussion with the relevant equations
on the Heston model and the Heston model with jumps in Chapter
3.
Some other continuous time stochastic volatility models are the
Stochastic Alpha Beta Rho
(SABR) model and the Variance Gamma model. The SABR model was
developed by Hagan
et al. (2002). The inspiration behind the SABR model was that
the behaviour of market smile
produced by local volatility models was opposite to the market.
Hagan et al. (2002) say that due
to this discrepancy the delta and the vega hedges derived from
local volatility models could be
unstable. In this two factor model the forward asset price F̂
and its volatility â are correlated.
dF̂ = âF̂ ¯dW1
dâ = ºâdW2
dW1dW2 = ½dt
Hagan et al. (2002) obtain the option prices from a perturbation
technique and further provide
a closed form formula for the implied volatility as a function
of the forward price F̂ and strike.
Madan et al. (1998) have used three parameter option pricing
method termed as the Variance
Gamma (VG) Process. At any random time a Brownian motion with
drift is evaluated by using
a gamma process. The drift of the Brownian motion and the
volatility of time change form the
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other two parameters.
In continuous time stochastic models it is difficult to obtain a
volatility variable from a set of
discrete observations of spot prices. There has been a lot of
research in estimating volatility in
continuous time models from historical observations. We can
refer to Cvitanic et al. (2006).
This problem is not inherent in discrete time autoregressive
conditional heteroscedastic models
(ARCH) introduced by Engle (1982) and Bollerslev (1986). Engle
(1982) was successfully able
to model properties of assets reurns having properties as fat
tails and time varying variances.
There have been many extensions to Engle (1982) like the
exponential GARCH model, the
non linear asymmetric GARCH models among others. The first
attempt to price European
style options under the GARCH framework was by Duan (1995). He
introduced the local risk
neutral valuation relationship for univariate GARCH processes.
Duan (1996) successfully used
his GARCH option pricing model to fit the volatility smile and
the term structure of implied
volatilities. Chaudhury and Wei (1996) did a simulation study of
comparing Duan’s GARCH
model with the Black Scholes model. They found out that the
GARCH model has least error
with the Black Scholes model when pricing out of the money
options with a maturity of 30 days
or less.
For our situation we are interested in the applicability of the
GARCH framework to the pricing
of options on foreign assets. The first work in this direction
was done by Duan and Wei (1999).
In this paper the authors have modelled the foreign exchange
rate and the underlying asset
process as a bivariate nonlinear asymmetric GARCH process. For
our purpose of pricing the FX
quanto option the underlying is also a foreign exchange rate.
The pricing framework incorporates
stochastic volatility, unconditional leptokurtosis and a
correlation between the lagged return and
9
-
the conditional variance for both the exchange rates in
question.
The inherent problem with Duan’s GARCH Option Pricing Formulae
was that one had to use
a Monte Carlo scheme to price the option.There were people who
worked around this problem.
We know that the discounted asset process is a martingale under
the risk neutral measure. Duan
and Simonato (1999) examine that in a Monte Carlo simulation
this property is violated. They
propose a technique called as the Empirical Martingale
Simulation (EMS). This imposes the
martingale property on the collection of simulated sample paths.
The EMS method generates
asset prices at future times t1, t2, ....., tn using the
following system
S̃i(tj , n) = S0Zi(tj , n)
Z0(tj , n)(1.5)
where
Zi(tj , n) = S̃i(tj−1, n)ˆSi(tj)
ˆSi(tj−1)(1.6)
and
Z0(tj , n) =1
ne−rtj
n∑
i=1
Zi(tj , n) (1.7)
Here ˆSi(t) denotes the ith simulated asset price at time t
before the EMS adjustment. S̃i(t0, n),
ˆSi(t) are set equal to S0. The steps followed in the simulation
are as follows
1. The return from time period tj−1 to tj i.e.
fromˆSi(tj)ˆSi(tj−1)
is first simulated.
2. Equation 1.7 is used to calculate the temporary asset price
Zi(tj , n)
3. Equation 1.8 is used to calculate the discounted sample
average.
4. Finally the EMS asset price at time tj is calculated by using
equation 1.6.
Duan et al. (1999) then followed up the work with an analytical
approximation for the GARCH
Option Pricing Model. This approximation uses the higher order
moments of the distribution
10
-
of log returns of the asset.We will discuss this method and the
GARCH Option Pricing Model
in detail in Chapter 4.
Around the same time Heston and Nandi (2000) introduced a closed
form solution to the GARCH
model. They derived the risk neutral transformations of the
parameters. The characteristic
function of the asset is taken to be in a log linear form. They
derived recursions for the involved
terms and finally provided an analytic expression in the Fourier
domain. A good study on the
capabilities and limitations of various GARCH option pricing
models is availabe in Christoffersen
and Jacobs (2004).
In the later part of the dissertation we try to extend the
Heston with Jumps option pricing
framework with a copula. A similar approach has been taken by
Chiou and Tsay (2008) in the
pricing of index correlation options with Duan’s GARCH Option
Pricing Framework. Chiou
and Tsay (2008) extend the univariate risk neutral pricing of
Duan(1995) to a multivariate case
under the copula framework. As mentioned in the introduction we
use a T Copula and a Frank
Copula. In the second part of the paper the copula based model
is used to assess the risk of a
portfolio comprising of assets which have the underlying of the
options as investment.The assets
in this case are investments on the NYSE and TAIEX index. The
copula is used to measure
the tail dependence of the asset returns. The authors come to an
interesting conclusion that
no matter what kind of dependence measure is used , holding a
portfolio of both indices always
has a higher chance to gain over 5 per cent than to lose more
than 5 per cent. The copula
based model is then used to calculate the Value-at-Risk of the
portfolio. This paper illustrates
an interesting use of the copula based approach to pricing
derivatives.
There are three methods by which we could have fit an copula to
our bivariate exchange rate
11
-
data. They are the Exact Maximum Likelihood Method(EML), the
Inference Function for
Margins Method (IFM) and the Canonical Maximum Likelihood Method
(CML). In our case of
fitting the T Copula we use the CML method and its application
by Mashal and Zeevi (2002).
We use this method because here because we do not have to make
any assumption about the
distributional form of the marginals. We review this method in
detail in Chapter 5. In the
EML method we have a n-dimensional vector µ of parameters to be
estimated. We denote the
parameter space with Θ. We denote the log likelihood function of
the observation at time t by
lt(µ). The log likelihood function l(µ) is hence
l(µ) = ΣTt=1lt(µ)
By a direct application of Sklar’s Theorem which enables the
seperation of univariate margins
and the dependence structure we have
l(µ) = ΣTt=1 = ln c(F1(x
t1), ...., FN (x
tN )
)+
T∑
t=1
N∑
n=1
ln fn(xtn) (1.8)
Here c denotes the density function of the copula and Fi(; )
denotes the distribution function
of the marginals and f(; ) the density function of the
marginals. xti denotes the time series
observation for the itℎ asset. The maximum likelihood estimator
is defined as the vector µ̂ such
that
µ̂ =(
ˆµ1, ....., µ̂k
)= argmax {l(µ) : µ ∈ Θ}
The Inference Function for Margins method is based on equation
(1.8). It is a two step fitting
proceedure in which one first finds the parameters of the
univariate marginals ¯ and then the
vector of copula parameters ®̃.
12
-
1. At the first stage the EML method is used to find ¯ = (¯1,
......, ¯n) as
ˆ̄i = argmax
¯t
T∑
t=1
ln fi(xti;¯i)
2. The copula parameter vector ® is estimated using ˆ̄ =(ˆ̄1,
...., ˆ̄n
)
ˆ̃®IFM = argmax®̃
T∑
t=1
ln c(F1(x
t1;
ˆ̄1), ...., FN (x
tN ;
ˆ̄N ); ®̃
)
The IFM estimator is hence(ˆ̄, ˆ̃®IFM
).
13
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Chapter 2
Pricing FX Quanto Options in the
Black Scholes Framework
2.1 The standard market practice in pricing FX Quanto
Options
In this section we will look at the Black Scholes model used in
pricing FX quanto options. Our
currency set is given by A = {f, d, q}. Here f denotes the
foreign currency, d the domestic
currency and q the quanto currency. We take i , j , k ∈ A Let us
denote by Xi,jt the exchange
rate between currency i and currency j . We begin with some
notations:
Dit,T : The time t value in currency i of a zero coupon bond
with maturity T
Qi : Equivalent martingale measure associated with the numeraire
Dit,T
Xi,jt : The value in currency i of one unit of currency j at
time t.
K : The strike price of the Quanto FX Option in
consideration
14
-
· denotes the quanto conversion factor which is predetermined.
In all our calculations we will
take this value to be 1.
M i,jt,T =Djt,TX
i,jt
Dit,Tis the forward FX rate.
In an arbitrage free economy the following relations hold:
Xi,jt = (Xj,it )
−1
Xi,jt =Xk,jtXk,it
Bennett and Kennedy (2003) consider a triangular no arbitrage
condition to price FX quanto
options. The numeraire considered will be in the quanto
currency. In the pricing formula we
take the risk neutral expectation with respect to the numeraire
in the quanto currency. The
payoff of the FX quanto call option is ·[Xi,jt −K]+. The price
of the FX quanto call option at
initial time:
Cquanto0 = ·Dq0,TE
qQ[X
i,jt −K]+ (2.1)
Here we use q because we want to denote that the discount factor
Dq0,T and the risk neutral
measure Qq is with respect to the quanto currency. Hence i
refers to the foreign currency and
j to the domestic currency.
2.1.1 Pricing using the Black Scholes Model
We assume that the forward FX rates follow correlated Brownian
motions
dM i,jt,T = ¾i,jMi,jt,TdW
i,jt (2.2)
15
-
Here ¾ij is the implied volatility of a vanilla FX option on the
particular exchange rate with the
particular option parameters as the quanto FX call.
dW k,it dWk,jt = ½dt (2.3)
½ =¾2k,i + ¾
2k,j − ¾2ij
2¾k,i¾k,j(2.4)
Here ½ is the implied correlation which we recover from the
implied volatilities of vanilla FX
options on the particular exchange rate. After evaluating the
expectation in equation (2.1)
Bennett and Kennedy (2003) find the price of the quanto FX
option under the Black Scholes
Model as
Cquanto0 = ·Dk0,T [MÁ(d̃1)−KÁ(d̃2)] (2.5)
M = M i,j0,T e(¾2k,i−½¾k,i¾k,j)T
d̃1 =ln(M i,j0,T )/K
¾ij√T
+1
2¾i,j
√T
d̃2 = d1 − ¾i,j√T
We use equation (2.5) to price FX quanto options under the Black
Scholes model in the disser-
tation.
16
-
Chapter 3
Pricing of FX Quanto under the
Heston Model
The Heston (1993) Stochastic Volatility model relaxes the
assumption of constant volatility in the
classical Black Scholes model. An instantaneous short term
variance process is incorporated. In
this section we will look at the key aspects adopted while
pricing the options under the Heston
stochastic volatility model. However we price our FX quanto
options by incorporating jump
processes in the Heston model.
3.1 Revisiting the Heston model
We shortly formalise the model to take care of the notations.
The dynamics of the stock process
{St, t ≥ 0} are given as :dStSt
= (r − q)dt+√vtdWtS0 ≥ 0 (3.1)
17
-
The instantaneous variance process vt is taken as a mean
reverting square root stochastic process
which is also known as Cox Ingersoll Ross (CIR) process. The SDE
is given by
dvt = ·(´ − vt)dt+ ¸√vtdW̃tv0 = ¾20 ≥ 0 (3.2)
Here W = Wt where t ≥ 0 and W̃ = W̃t where t ≥ 0 are two
correlated standard Brownian
motions with correlation ½. The parameters in the equation are:
initial volatility, ¾0 > 0, mean
reversion rate · > 0, the long run variance ´ > 0, the
volatility of variance ¸ > 0. The variance
is always positive and by the Feller condition it has been shown
that it cannot reach zero if
2·´ > ¸2. The variance process vt is non centrally Chi-Square
distributed.
3.2 Option Pricing under the Heston model
As shown in the Black and Scholes model (Black and Scholes
(1973))the value of a derivative
is dependent on the underlying tradeable assets. As the assets
are tradeable the option can be
hedged by trading in the underlying. When this happens we say
that the market is complete.
This means that every derivative can be replicated.
In the Heston model the price of a derivative would depend on
both the randomness of the
asset process (St, t ≥ 0) and its volatility (Vt, t ≥ 0).The
volatility is not a tradeable asset and
hence under the Heston model we do not work in a complete market
setting. An implication of
this fact to option pricing under the Heston model is that we do
not find a unique equivalent
martingale measure (EMM). Under the Heston model, the value of
any option U(St, Vt, t, T )
18
-
must satisfy the partial differential equation
1
2V S2
∂2U
∂S2+½¾V S
∂2U
∂S∂V+
1
2¾2V
∂2U
∂V 2+ rS
∂U
∂S+·[µ − V ]− Λ(S, V, T )¾
√V∂U
∂V− rU + ∂U
∂t= 0
(3.3)
Λ(S, V, T ) is called the market price of volatility risk. In
his paper Heston makes the assumption
that the market price of volatility risk is proportional to
volatility.
Heston has derived a closed form solution which makes it easier
for practitioners to price options
using stochastic volatility. However for our purpose of pricing
we will refrain from using the
closed form solution given by Heston. Instead we will use the
approach used by Carr and Madan
(1998). Carr and Madan have considered a transformation of the
option pricing formulae and
then applied Fourier inversion techniques to compute option
prices. We discuss this method as
follows.
3.2.1 The Heston Pricer using Fast Fourier Transform
We consider an asset with value Xt at time t and an option
written on the asset with strike K.
Let xt = ln(Xt) ; the logarithm of the underlying asset value at
time t. For our purpose the
asset is a foreign exchange rate. Let K̃ =ln (K); the logarithm
of the strike price. Then the
value of a European Call Option with maturity T as a function of
K̃ is given by
C(T, K̃) = e−rT∫ ∞k
(exT − eK̃)fT (xT )dxT (3.4)
Here fT (x) is the risk neutral density of x. r denotes the
riskfree interest rate. For our FX option
it is the riskfree interest rate in the foreign currency. We
observe from the above equation that
C(T, K̃) tends to the initial spot of the underlying as K̃ tends
to −∞. Hence the call pricing
19
-
function is not square integrable. As such Carr and Madan modify
the call price as
C̃(T, K̃) = e®kC(T, K̃)k ≥ 0 (3.5)
Carr and Madan then obtain an analytical expression for the
Fourier transform of C̃(T, K̃) as
³(u) =
∫ ∞−∞
eiuk̃C̃(T,K)dK̃. (3.6)
In terms of the characteristic function Á of the logarithm of
the stock price as
³(u) =e−(r−q)TÁ(u− (®+ 1)i)®2 + ®− u2 + i(2®+ 1)u. (3.7)
Here r denotes the risk free rate in foreign currency and q the
riskfree rate in the foreign currency.
This is followed by the numerical computation of the call prices
using the inverse transform
C(T, K̃) =e−®K̃
2¼
∫ ∞−∞
e−iuK̃Ã(u)du. (3.8)
The Fast Fourier Transform(FFT) is an efficient algorithm to
evaluate summations of the form
P (e) = Σj=Nj=1 e−i 2¼
N(j−1)(e−1)x(j) (3.9)
Carr and Madan note that (3.8) can be integrated using the FFT
algorithm. The FFT algorithm
is as follows
(a) We discretize (3.8) using the Trapezoid Rule and set uj =
´(j − 1). This gives us
C(T, K̃) ≈ e−®K̃
¼ΣNj=1e
−iujK̃³(uj)´ (3.10)
(b) We take a regular grid for K̃ of size ¸ to obtain K̃u =
−b+¸(s−1) for s = 1, 2, ...., N giving
log strike levels from -b to b where b = N¸2 .
20
-
(c) We substitute step (b) in step (a) to obtain
C(T, K̃u) =e−®K̃
¼
j=N∑
j=1
e−i2¼N
(j−1)(s−1)eibuj³(uj)´
3(3 + (−1)j − ±j−1). (3.11)
Here ±n is the Kronecker delta function.
(d) Finally we use the FFT to compute the call prices.
Carr and Madan provide the optimum parameter values as N = 4096,
´ = 0.25 and ® = 1.5.
As we see to evaluate ³ we need to use the characteristic
function Á of the logarithm of the
underlying asset.
In the Heston framework we need to take special care while
evaluating this characteristic func-
tion.
3.2.2 The Heston Trap
In this section we talk about the use of an alternative
characteristic function as given by Al-
brecher et al. (2007). For the log asset price distribution the
characteristic function is given
by
Á(u, t) = E[exp(iu ln(St))∣S0, ¾20
](3.12)
However there are two formulas for the characteristic function.
We can find the first formula in
Heston (1993). It is
Á1(u, t) = exp(A)× exp(B)× exp(C). (3.13)
21
-
where
A = iu(lnS0 + (r − q)t).
and
B =
(´·¸−2(·− ½¸iu+ d)t− 2 ln
(1− g1edt1− g1
)).
and
C =
(¾20¸
−2(·− ½¸iu+ d)(1− edt)1− g1edt
).
To explain notations further
d =√(½¸ui− ·)2 + ¸2(iu+ u2).
g1 = (·− ½¸iu+ d)/(·− ½¸iu− d).
The second one is as discussed in Albrecher et al. (2007).
Á2(u, t) = exp(A)× exp(D)× exp(E). (3.14)
where
D =
(´·¸−2(·− ½¸iu− d)t− 2 ln
(1− g1e−dt1− g1
)).
E =
(¾20¸
−2(·− ½¸iu+ d)(1− e−dt)1− g2e−dt
).
where
g2 =1
g1.
22
-
We note that the difference between B and D is that we have a
negative sign before dt in D while
a positive sign in B. Albrecher et al. (2007) have shown that
the options are mispriced when
using the value of Á1 in the Carr Madan Formulae for option
pricing. Our Heston parameter
space has to be restricted to use Á1 as a characteristic
function. However Á2 can be applied to
the whole unrestricted parameter space and is a better
choice.
3.3 The Heston model with jumps
One of the primary requisites of option pricing is that we can
fit our model prices to the market
smile. While the Heston stochastic variance model fits the long
term behaviour of the asset price
it does not adequately describe the short term behaviour as
shown by Weron et al. (2004). As
such we extend the Heston stochastic volatility model with jumps
in the asset price process as
by Bates (1993) and Bakshi et al. (1993).This model is a jump
diffusion model. Carr and Wu
(2004) say that in this model the desired smile is created for
short maturities by jumps while at
longer maturities the effect is created by stochastic
volatility.
dStSt
= (r − q − ¸¹J)dt+ ¾tdWt + JtdNt. (3.15)
Here N = {Nt, t ≥ 0} is an independent Poisson process with
parameter ¸ ≥ 0. Jt is the
percentage jump size which is assumed to be lognormally and
identically distributed with time
and with unconditional mean ¹J . ln(1 + Jt) is normally
distributed with mean ln(1 + ¹J)− ¾2J2
and variance ¾2J . As far as our pricing methodology goes the
only change with the Heston model
would be in the characteristic function. The characteristic
function of the Heston with jumps is
actually a product of the characteristic function of Heston Á2
and the characteristic function of
23
-
the jump process, ÁJ . This is given in Schoutens et al.
(2005).
ÁHJ2 (u, t) = exp(A)× exp(D)× exp(E)× exp(F )× exp(G).
(3.16)
where
F = −¸¹J iut+ ¸t((1 + ¹J)iu.
G = ¾2J(iu/2)((iu− 1))− 1).
We can see that
ÁJ = exp(F )× exp(G).
3.4 Pricing the FX Quanto option
Let Xa,bt denote the exchange rate which follows the asset
process in the Heston with jumps
equation. Here a denotes the foreign currency and b denotes the
domestic currency. The price of
a plain vanilla European option with the exchange rate as
underlying,strike K, time to maturity
K under an EMM Q is given by
Cvanilla0 = e−raTEQ[Xa,bT −K]+ (3.17)
Here ra denotes the risk free rate in the foreign currency i.
The price of an FX quanto option
is given by
Cquanto0 = e−rcTEQ[Xa,bT −K]+ (3.18)
Here rc denotes the risk free rate in the quanto currency c.
This is the formulae we will be
using while pricing the FX plain vanilla and the quanto options
without a copula. Later we
24
-
also incorporate a copula. This is discussed in section 5.3.
Once the prices of the plain vanilla
options are evaluated we back out the Black Scholes implied
volatilities which are then used for
the calibration process. The parameters we get after the
calibration of the plain vanilla options
are then used to price the quanto with the copula.
25
-
Chapter 4
Pricing of FX Quanto under the
GARCH Option Pricing Model
4.1 Duan’s GARCH Option Pricing Model
In this section we will discuss the methodology adopted in
pricing the quanto option using the
GARCH Option Pricing Model. This has been developed by Duan
(1995). Duan and Wei (1999)
have further extended the GARCH Option Pricing Model for
valuation of Foreign Exchange
Options. The conditional variance ¾t follows a non linear
asymmetric GARCH(1,1) model. As
before our currency set is given by A in section 2.1. Let us
denote by Xi,jt the exchange rate
between currency i and currency j where i,j ∈ A. The asset
process of Xi,jt is governed by the
probability law ℙ with respect to information filtration Ft. We
take the measure ℙ with respect
to the domestic currency.
26
-
We have the following process for the exchange rate Xi,jt .
ln
[Xi,jt+Δ
Xi,jt
]= (rf − rd) + ¸¾t+Δ − 1
2¾2t+Δ + ¾t+Δ²t+Δ (4.1)
²t+Δ ∼ N(0, 1)
under ℙ
¾2t+Δ = ¯0 + ¯1¾2t + ¯2¾
2t (²t − µ)2 (4.2)
Here rf denotes the foreign risk free rate and rd denotes the
domestic risk free rate. The
parameter ¸ represents the unit risk premium for the exchange
rate.The parameter µ represents
a leverage parameter. The leverage effect in GARCH models comes
from the empirical evidence
that increase in volatility is larger when the returns are
negative than when they are positive.
Duan (1995) comes to the conclusion that if µ is positive there
is a negative correlation between
the innovations of the asset return and its conditional
volatility. As we can see ¾2t+Δ is expressed
in terms of Ft measurable random variables. The volatility
process is hence predictable or known
at time t.
Under the locally risk neutralized probability measure ℚ the
exchange rate dynamics given by
Duan(1995) is
ln
[Xi,jt+Δ
Xi,jt
]= (rf − rd)− 1
2¾2t+Δ + ¾t+Δ²̃t+1 (4.3)
²̃t+1 = ²t+1 + ¸ ∼ N(0, 1)
under ℚ
¾2t+Δ = ¯0 + ¯1¾2t + ¯2¾
2t (²̃t − µ − ¸)2
The condition for first order stationarity is ¯1 + ¯2[1+ (µ+¸)2]
< 1. Under the risk neutralized
system the volatility remains as an NGARCH process. The leverage
parameter here is µ +
27
-
¸. When we try to fit the prices from the model to market prices
we have to calibrate four
parameters. They are ¯0, ¯1, ¯2 and µ + ¸.
4.2 Analytical Approximation of the GARCH Option Pricing
Model
Duan et al. (1999) has derived an analytical approximation to
the option pricing problem under
GARCH. We give a review of this method in this section. The
option price is determined in
terms of the moments of the cummulative asset return ½T = ln
(Xi,j0Xi,jT
). We denote the standard
normal distribution function as N and the density function as
n(). The formulae for the price
of a European FX call option with spot price Xi,j0 , strike K,
time to maturity T is given as
Capprox = C + ·3A3 + (·4 − 3)A4 (4.4)
where
C = Xi,j0 Á(d̃)−Ke−(rd−rf )TN(d̃− ¾½T )
The constants A3, A4 are given as
A3 =1
3!Xi,j0 ¾½T
[(2¾½T − d̃)n(d̃)− ¾2½TN(d̃)
]
A4 =1
4!Xi,j0 ¾½T
{[d̃2 − 1− 3¾½T (d̃− ¾½T )]n(d̃) + ¾3½TN(d̃)
}
with
d̃ = d+ ±
where
d =ln(Xi,j0 /K
)+((rd − rf )T + 12¾2½T
)
¾½T
28
-
and
± =¹½T − (rd − rf )T + 12¾2½T
¾½T
Here
¹½T = Eℚ0
[½T
]
and
¾½T =
√V arℚ0
[½T
]
We use Eℚ0 to denote conditional expectation under ℚ with
respect to F0.
·3 and ·4 represent the third and fourth cumulants of the
normalized cummulative return zT
which is given by
zT =½T − ¹½T
¾½T
We note that
Xi,jT = Xi,j0 exp(¹½T + ¾½T zT ) (4.5)
The price of an European FX Call option with strike K is given
by
exp(−(rd − rf )T )Eℚ0[max(Xi,jT −K, 0)
]
Hence
Xi,jT ≥ K ⇔ −zT ≤lnXi,j0
K + ¹½T¾½T
= K̃ (4.6)
Now
Eℚ0[max(Xi,jT −K, 0)
]= (4.7)
∫ K̃−∞
[Xi,j0 exp(¹½T − ¾½T z)−K
]g(z)dz
29
-
To evaluate the above expression we observe from equation (4.5)
that we need the true density
function g(z) of zT . Duan et al. (1999) follow the method
adopted by Jarrow and Rudd (1982). It
involves the approximation of a probability distribution by an
arbitrary distribution.The density
function is expressed in terms of an expansion which involves
second and higher order moments
of the arbitrary distribution. In this case the approximation
was given as
g(z) = n(z)
[1− ·3
3!(z3 − 3z) + ·4 − 3
4!(z4 − 6z2 + 3)
](4.8)
With this approximation Duan et al. (1999) uses equation (4.7)
to evaluate the price of the
option. Equation (4.4) gives the approximated price of the
option after the approximation.
Duan et al. (1999) have also provided explicit expressions for
¹½T ,¾½T , ·3 and ·4 in terms of the
parameters involved in the NGARCH volatility process which
enables the computation of the
option prices.
30
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Chapter 5
Modelling the Dependence Structure
Using a Copula
The previous two sections spoke about pricing the FX quanto
option using the Heston Model
and the GARCH Option Pricing Model. Let us denote by Xi,jt the
exchange rate with i as the
foreign currency and j as the domestic currency. By principle of
no arbitrage we can say
Xi,jt =Xk,jt
Xk,it(5.1)
The payoff of the FX quanto option with Xi,jt as underlying,
maturity T , strike K is given by
P quantoT = max
[Xk,jt
Xk,it−K, 0
](5.2)
Such an approach could be useful if for instance Xi,jt was a
less frequently traded, illiquid asset.
And both Xk,jt and Xk,it were more liquid in the market as
compared to the original currency. In
the context of our pricing problem there is another motivation
for doing so. In the Black model
31
-
the implied volatility quote of a quanto on an exchange rate,
take for instance the USD/JPY
exchange rate depends on the implied volatilities of the EUR/USD
and the EUR/JPY rates.
Here EUR is the quanto currency. Since the forward rate and the
exchange rate process is linked
only by a scaler factor we can say that the volatility of the
USD/JPY rate is influenced by the
EUR/USD and the EUR/JPY rates. We have not accounted for this in
the GARCH or Heston
model with jumps to price the FX quanto option. We need to model
the dependence structure
of the EUR/USD and the EUR/JPY rates. One way of doing this
could be by using a copula.
The marginals are free to be chosen depending on the calibration
of plain vanilla quotes.
We can now view the FX quanto option as a multivariate
contingent claim. In such a situation
it is valid to think that the payoff of the option would depend
on the co-movement between the
two exchange rate returns. We will use a copula to measure the
association between the two
assets.We can use a wide choice of dependent structures using a
copula, for instance linear, non
linear or tail dependent.
We will formalise the definition of a copula. This is given in
Frees and Valdez (1977).
Let us consider p uniform random variables u1, u2, ...., up. We
need not assume that they are inde-
pendent. The dependence relationship is given by a joint
distribution function C(u1, u2, ..., up) =
Prob(U1 ≤ u1, U2 ≤ u2, ...., Up ≤ up))
We select arbitrary marginal distribution functions F1(x1),
F2(x2), ...., Fp(xp). Then the function
C[F1(x1), F2(x2), ...., Fp(xp)
]= F
(x1, x2, ...., xp
)(5.3)
defines a multivariate distribution function evaluated at x1,
x2, ...., xp with marginal distributions
F1, F2, F3..., Fp. We call the function C a copula.
32
-
In our case of modelling the dependence between the exchange
rates we will use the T-copula
from the Elliptic distribution and a copula from the Archimedean
family. Archimedean copulas
are simpler to apply. It allows us to reduce the study of a
multivariate copula to a single
univariate family.
5.1 Archimedean Copulas
We first formalise the definition of Archimedean copulas.
Definition 5.1. Let Á be a convex, decreasing function with with
domain (0, 1] and range [0,∞)
such that Á(1) = 0. Then the function CÁ(u, v) = Á−1(Á(u) +
Á(v)) for u,v ∈ (0, 1] is said to be
an Archimedean Copula. Á is called the generator of the
copula.
It is worthwhile to mention that a generator uniquely determines
an Archimedean Copula. We
give below a table of Archimedean Copulas and their generators
in table 5.1.
Family Generator Á(t) Domain(®) CÁ(u, v)
Clayton t−® − 1 ® > 1 (u−® + v−® − 1)−1/®
Gumbel(− ln t)® ® ≥ 1 exp{-[(-lnu)® + (− ln v)®]1/®}
Frank ln e®t−1e®−1 −∞ < ® < ∞ 1® ln
(1 + (e
®u−1)(e®v−1)e®−1
)
Table 5.1: Archimedean Copulas and their Generators
5.1.1 Archimedean Copulas and their Measures of Dependence
There is an important result due to Schweizer and Wolf (1981).
They have established that if
f1 and f2 were strictly increasing but arbitrary functions over
the range of the random variables
33
-
X1, X2 then f1(X1) and f2(X2) have the same copula as X1 and X2.
This ensures that the
co-movement between the two random variables X1, X2 is captured
by the copula. Schweizer
and Wolf (1981)have also shown that two standard non parametric
correlation measures, the
Spearman’s Rank Correlation and the Kendall’s correlation
coefficient can be expressed in terms
of the copula function.
The first one is the Spearman’s Rank Correlation Co-efficient
defined as
½(X1, X2) = 12
∫ ∫ {C(u, v)− uv}dudv
The second one is the Kendall’s correlation coefficient defined
by
¿(X1, X2) = 4
∫ ∫C(u, v)dC(u, v)− 1
For the Archimedean copula family both the Spearman’s Rho and
Kendall Tau correlation can
be expressed in terms of the parameter ®. This is summarised in
the table 5.2.
Family Kendall’s ¿ Spearman’s ½
Clayton ®®+2 No closed form
Gumbel 1− ®−1 No closed form
Frank 1− 4®{D1(−®)− 1} 1− 12® {D2(−®)−D1(−®)}
Table 5.2: Archimedean Copulas and Measures of Dependence
where for the Frank Copula we have
Dk(x) =k
xk
∫ x0
tk
et − 1dt
Dk(−x) = Dk(x) + kxx+ 1
34
-
5.1.2 Identifying the Right Copula from the Archimedean Family
to Model
the Dependence
We will use the proceedure used by Genest and Rivest (1993) to
identify the right copula to
model the dependence structure between the two exchange rates.
We start with a bivariate set of
observations{(X1i, Y2i)
}ni=1
which in our case are the log returns of exchange rate. We
assume
that the distribution function F of the bivariate data has an
associated Archimedean Copula
CÁ. Let Zi = F (X1i, X2i) be a random variable which has the
distribution function K(z) =
Prob(Zi ≤ z). It can be proved that the distribution function is
related to the generator of an
Archimedean copula through the expression
K = t− Á(t)Á′(t)
The following steps are taken to find Á. This is the method
followed by Genest and Rivest
(1993).
1. The Kendall correlation coefficient is estimated using the
historical bivariate exchange rate
data. This is given as
¿n =∑
i
-
3. We now construct a parametric estimate of K using
KÁ(z) = t− Á(z)Á′(z)
We select Á and hence the particular copula from the Archimedean
family such that the para-
metric estimate most closely resembles the non parametric
estimate. We choose the copula for
what the value of d is the lowest where d is given by
d =
∫ [KÁn(z)−Kn(z)
]dKn(z) (5.5)
Once we know the copula to choose we will proceed by first
finding the Kendall Tau Rank
Correlation from the historical data. The parameter of the
copula ® is then estimated from the
Kendall Tau Rank Correlation from Table 5.2.
Our copula fititng technique is based on the historical spot
exchange rate data. The use of a
member of the Archimedean Copula Family simplifies the fitting
procedure a lot. Apart from
the independent copula we have three classes of members of the
Archimedean family which have
distinct forms of the generator function. We have covered all
three families by picking one from
each. The Frank, Gumbel and Clayton copulas belong to three
distinct Archimedean families.
These three copulas can model various kinds of dependencies. The
Clayton Copula exhibits
greater negative tail dependence than positive dependency. The
Gumbel Copula on the other
hand exhibits more positive dependence than negative. The Frank
Copula is symmetric. It has
lighter tails than a Normal Distribution. We generate a copula
from the Archimedean family
with the proceedure given by Nelsen (2006). This is shown in
Figures 5.1,5.2 and 5.3.
36
-
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
u
v
Random sample from Frank Copula
Figure 5.1: Frank Copula with ® = 1.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
u
v
Random sample from Clayton Copula
Figure 5.2: Clayton Copula with ® = 1.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
u
v
Random Sample from Gumbel Copula
Figure 5.3: Gumbel Copula with ® = 1.5
5.2 The T Copula
We begin by formalising the definition of the T Copula
Definition 5.2. Let R be a symmetric, positive definite matrix
with diag (R)= 1. Let TR,º be
the standardized multivariate Student’s t distribution with
correlation matrix R and º degrees
of freedom. Then the multivariate Student’s t copula is defined
as C(u1, u2, ......, un;R, º) =
TR,º(t−1º (u1), t−1º (u2), ....., t−1º (un)
)Here t−1º (u) denotes the inverse of the Student’s t
cumulative
distribution function.
37
-
5.2.1 Calibration of the T Copula
Mashal and Zeevi(2002) have shown that the empirical fit of a T
Copula is better than a
Gaussian copula. Financial data sometimes have heavy tails. Log
returns of exchange rates we
have taken have high kurtosis of 9. Mashal and Zeevi(2002) say
that a T Copula captures better
dependent extreme values. In the copula function the choice of
marginals and the dependency
structure is independent. In the T Copula we estimate the
correlation matrix R and the degrees
of freedom º. The methodology we will adopt to calibrate the T
Copula to the log returns of
the spot exchange rates is due to Mashal and Zeevi (2002). In
this method there is no prior
assumption on the distribution form of the marginals. It is a
Canonical Maximum Likelihood
Estimate(CML) method. We take the observation vector (of
historical log returns of exchange
rates)X = (X1t, X2t, .....XNt)Tt=1 and and approximate the
parametric marginals
ˆFn(.) as
ˆFn(x) =1
T
T∑
t=1
1{Xnt≤x}
We first give a result due to Lindskog et al. (2001).
Theorem 5.3. Let X ∼ EN (º,∑
) where for i,j ²{1, 2, ...., N}, Xi and Xj are continuous.
Then,
Γ(Xi, Xj) =2
¼arcsinRi,j (5.6)
where EN (º,∑
) denotes the N-dimensional elliptical distribution with
parameters (º,∑
). Γ(Xi, Xj)
and Ri,j denote the Kendall’s Tau and Pearson’s linear
correlation coefficient for the random
variables (Xi, Xj).
The procedure we adopt to calibrate the T Copula is as
follows
38
-
1. We start with a historical sample X of the exchange rate
data. We then transform the initial
data set into a set of uniform variates U using an empirical
marginal transformation. Given the
data set we use the empirical marginal distribution. For t = 1,
......, T let
ût = (ût1, ût2, û
tN ) =
[ ˆF1(X1t), ...., ˆFN (XNt)]
2. We use equation 5.6 to estimate the correlation matrix
RCML.
3. We then find the estimate º̃ of the degrees of freedom by
maximizing the log likelihood
function of the Student’s t-copula density.
º̃ = argmaxº∈Θ
ΣTt=1 ln cstudent(ût1, û
t2, ...., û
tN ;R
CML, º)
5.3 Pricing using the copula
We use a Monte Carlo simulation to price options with the
copula. Let us denote the number
of samples as M and the number of timesteps as m.We generate
pairs (ui, vi) from the copula
where i = 1, 2, ....,m. We take the marginals which are exchange
rates to follow the Heston
with jumps process as discussed in Section 3.0.6. We denote Δt =
Tm where T denotes the
time to maturity of the option.The Brownian motions (dW 1t ,
dW2t ) driving each of the marginal
processes are constructed as follows
dW 1t = N−1(ui)
√Δt
dW 2t = N−1(vi)
√Δt
39
-
Here N denotes the distribution function of a standard normal
random variable. We also know
that the Brownian motions driving the variance process and asset
process for the first exchange
rate have a correlation ½ (which we obtain from the calibration
of the marginals). We construct
the Brownian motion for the variance process as
dW̃ 1t = xi√Δt
where
xi = N−1(ui)½+
√1− ½2N−1(ai)
Here ai is the third uniform variate which is chosen
independently from the first two. The first
two ui and vi are generated from the copula.
We follow the same procedure for the second exchange rate as
well. In this way we simulate the
asset process and the variance process. The price of the FX
Quanto option is evaluated by the
usual procedure followed in a Monte Carlo.
40
-
Chapter 6
Data
6.1 Discount Factors, Implied Volatility quotes
In this section we will present the data which will be used to
price the FX quanto options. We
have obtained the data from Bennett and Kennedy (2003) The
source of data in the paper is
Meryll Lynch United Kingdom. The data comprises of
a. The discount factors of three currencies - United States
Dollar (USD), Euros(EUR) and
Japanese Yen(JPY) pertaining to maturities of 1 month, 3 months,
6 months, 1 year and 2 year
as on 7 July 2001. Hence we price our options on 7 July
2001.
b. The implied volatilities of plain vanilla FX options written
on EUR/USD, EUR/JPY and
USD/JPY exchange rates on 7 July 2001. The implied volatilities
are given in terms of the
Black Delta (Δ) of the option.
41
-
Maturity EUR USD JPY
1/12 0.996243 0.996661 0.999948
1/4 0.988734 0.990054 0.999805
1/2 0.978343 0.980503 0.999566
1 0.958074 0.959798 0.998965
2 0.916243 0.911070 0.997404
Table 6.1: Discount factors Di0,T corresponding to currency i
and maturity T
Maturity 90%(Δ) 80%(Δ) 50%(Δ) 20%(Δ) 10%(Δ)
1/12 0.1289 0.123 0.1215 0.126 0.134
1/4 0.1277 0.121 0.119 0.1235 0.1319
1/2 0.1272 0.1203 0.118 0.1223 0.1305
1 0.1269 0.1196 0.117 0.1216 0.1303
2 0.1249 0.1176 0.115 0.1196 0.1283
Table 6.2: EUR/USD implied volatility quotes of call options
corresponding to standard values
of Black Delta and maturity T
6.2 Recovering Strikes and Prices of FX vanilla options
For calibration of plain vanilla FX options we need to price
options using a particular model.
For this purpose we need to recover strikes (K)for FX call
options from the implied volatility
quotes. We use the formulae given by Carr and Wu (2004).
K = F exp
[1
2¾2T − ¾
√TÁ−1(erfTΔ)
]
42
-
Maturity 90%(Δ) 80%(Δ) 50%(Δ) 20%(Δ) 10%(Δ)
1/12 0.1467 0.1441 0.148 0.1606 0.1747
1/4 0.1466 0.1372 0.1385 0.1487 0.1667
1/2 0.1444 0.1329 0.132 0.1414 0.1597
1 0.1443 0.132 0.13 0.138 0.1557
2 0.1508 0.1358 0.1308 0.1358 0.1508
Table 6.3: EUR/JPY implied volatility quotes of call options
corresponding to standard values
of Black Delta and maturity T
Maturity 90%(Δ) 80% (Δ) 50%(Δ) 20%(Δ) 10%(Δ)
1/12 0.1109 0.1015 0.095 0.0945 0.099
1/4 0.1156 0.1045 0.0985 0.0995 0.1066
1/2 0.1178 0.1055 0.1 0.1024 0.1122
1 0.1186 0.1065 0.101 0.1048 0.1166
2 0.1235 0.1105 0.106 0.1105 0.1235
Table 6.4: USD/JPY implied volatility quotes of call options
corresponding to standard values
of Black Delta and maturity T
where
F = Xi,j0 e(rd−rf )T
Here Á denote the distribution function of a standard normal
variable, Xi,j0 denotes the spot
exchange rate,rd the domestic risk free interest rate, rf the
foreign risk free interest rate, T the
time to maturity and ¾ the implied volatility quote.
43
-
We price an FX plain vanilla FX call option using the following
formulae used by Carr and Wu
(2004).We will use this when we have to calculate implied
volatility using the bisection method.
This will be discussed in Chapter 7.
Cvanilla0 = erfTXi,j0 Á(d+)− erdTKÁ(d−) (6.1)
where
d+ =ln(F/K)
¾√T
+1
2¾√T
d− =ln(F/K)
¾√T
− 12¾√T
44
-
Chapter 7
Implementation of Pricing Methods
In this section we will look at pricing the FX quanto option
under two different models-the
Heston model with jumps (HJ) and the GARCH Option Pricing Model
with the analytical
approximation.
7.1 Implementation of the Heston Stochastic Volatility Model
with Jumps
7.1.1 Calibration of the Model
Before pricing the FX quanto option we would like to fit our
pricing model to the market implied
volatilities. Carr and Wu (2004) have carried out an analysis on
the suitability of the Heston
model to fit the market smile. They find that for extremely
short and longer maturities the
Heston model does not give a good fit. They say that the Heston
model with jumps fits the
45
-
market smile better. We will look to model the dynamics of the
USD/JPY exchange rate with
the Heston with jumps model. As we mentioned in the data section
we intend to price the
options as on 7 July 2001. Before proceeding to applying a model
we take a look at the pattern
of the historical spot USD/JPY exchange rate. We download data
of the spot rates from 1
January 1998 to 7 July 2001 (from Datastream)
0 200 400 600 800 1000 12006.5
7
7.5
8
8.5
9
9.5
10x 10
−3
Observations
USD/
JPY
Exch
ange
Rat
e
Time series plot of USD/JPY Exchange Rate
Figure 7.1: Time series observations of USD/JPY spot exchange
rate
Before we price a FX quanto option on an exchange rate we will
look to calibrate plain vanilla FX
options. The purpose of our calibration is to obtain values of
parameters of the Heston Model
with jumps that accurately describes current market implied
volatilities. There is a reason why
we choose to calibrate with market implied volatilities (or
market prices of options)of the option
instead of the asset prices of the underlying. If we calibrate
to asset prices we obtain parameters
which correspond to the true process of the asset and not the
risk neutral process. We would
the need to calculate the market risk premium associated with
exposure to volatility changes by
estimating returns on options that are being used to hedge
against volatility.
We take up the pricing of vanilla FX options on the USD/JPY rate
(please refer to Data for all
46
-
values).In the calibration process we minimize the least squares
error function given by
Min(Error) = Σni=1(Xi − X̃i)2P
where X and X̃ denote market and model implied volatilities and
P denotes the parameter
vector. The parameters involved in the Heston model with jumps
are are ¾0, the initial volatility,
·, the mean reversion rate of volatility, ´ the long run
variance,¸ the volatility of volatility, µ,the
frequency of jumps in a year, ¹ the percentage size of the jump
and ¾J the variance of the jump
process and ½, the correlation between the Brownian motions
driving the asset price process and
the variance process. The constraint
2·´ > µ2
is imposed to ensure that the variance is always positive. We
see that the calibration problem for
the Heston model with jumps is thus a general non linear
optimization problem. We calibrate to
all maturities using an in-built function called fminsesarch in
MATLAB. This function uses the
Nelder and Mead (1965) Algorithm which is an unconstrained non
linear optimization algorithm.
Here we add the constraints externally. We set the error
function to a very high number if the
constraints are violated. Since fminsearch is a local
optimization algorithm we have to take care
while choosing the initial parameters We choose the same values
of parameters as by Schoutens
et al. (2005) while pricing options using the Heston with jumps
model.
(a) ¾0 = 0.05
(b) · = 0.5
(c) ´ = 0.06
47
-
(d) ¸ = 0.22
(e) ½ = -0.9
(f) µ = 0.13
(g) ¹ = 0.17
(h) ¾J = 0.13
After the calibration we obtain the following results of the
optimum parameters.
Parameter Value
¾0 0.1182
· 1.5288
´ 0.0481
¸ 0.3705
½ -0.8102
µ 0.120
¹ 0.005
¾J 0.01137
Table 7.1: Parameters of Heston with jumps calibration on
USD/JPY rates
The Carr Madan formula with the FFT pricer is most effective for
ATM options. Hence we
would expect that the relative error between OTM model and
market volatilities is higher. For
options which are in the money region we expect more accurate
results.We have checked our
pricer using the Carr Madan formula with the results given by
Schoutens et al. (2005).
48
-
In the calibration excercise we vary the number of iterations
from 50 to 2000. We observe that
the parameters converge to the optimum in approximately 1000
iterations.
We would like our model results to be coherent with the market.
This can be examined by
comparing the market smile with the smile obtained from our
model. We obtained our prices
from the Heston with Jumps model and then inverted them to get
the implied volatilities during
the calibration proceedure.
7.1.2 Finding Implied Volatility
During our calibration process we obtain call prices from our
Heston with jumps model. To find
the corresponding implied volatility we need to invert the Black
Scholes equation. In the Black
Scholes case there is no analytic solution for the Implied
Volatility. Instead one has to use a
numerical proceedure or approximation.In our case we use the
bisection method to back out the
implied volatilities.The bisection method relies on the fact
that option prices increase when the
volatility increases.
The fits to the market smile are given in Figures 7.2 - 7.6 for
varying maturities.
We do a piecewise linear interpolation to construct both the
volatility surface of the Heston
model with jumps and the market volatility surface for USD/JPY
quotes.We observe from the
surface of Heston with jumps (Figure 7.8) that as the delta of
the option increases from 0
to 0.50 (roughly in the money region) the implied volatility
decreases. As we go from at the
money region to the out of money regions the implied volatility
increases. This is similar to the
behaviour of the market implied volatilities (Figure 7.7).
49
-
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08
0.09
0.1
0.11
0.12
0.13
Delta
Vola
tilit
y
Market and model volatilities for USD/JPY FX options with
maturity 1 month
Heston with jumps volatilityMarket volatility
Figure 7.2: Market and model volatility(in %) for USD/YEN call
options with maturity 1
months
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08
0.09
0.1
0.11
0.12
0.13
Delta
Vola
tilit
y
Market and model volatilities for USD/JPY FX options with
maturity 3 months
Heston with jumps VolatilityMarket Volatility
Figure 7.3: Market and model volatility(in %) for USD/YEN call
options with maturity 3
months
7.2 Implementation of the GARCH Option Pricing Model
Our aim is to price the FX quanto options with USD/JPY exchange
rate as underlying using
the GARCH Option Pricing Model with a Monte Carlo simulation. We
first calibrate to the
implied volatility quotes on the USD/JPY rates. In the
calibration proceedure we use the same
50
-
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08
0.09
0.1
0.11
0.12
0.13
Delta
Vola
tilit
y
Market and model volatilities for USD/JPY FX options with
maturity 6 months
Heston with jumps VolatilityMarket Volatility
Figure 7.4: Market and model volatility(in %) for USD/YEN call
options with maturity 6
months
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08
0.09
0.1
0.11
0.12
0.13
Delta
Vola
tilit
y
Market and model volatilities for USD/JPY FX options with
maturity 1 year
Heston with jumps VolatilityMarket Volatility
Figure 7.5: Market and model volatility(in %) for USD/YEN call
options with maturity 12
months
set of innovations during each optimisation step. We calibrate
to all maturities. The use of
the GARCH Option Pricing Model in pricing the quanto would be
justifiable if we can fit the
market smile using this model.
We follow the same optimisation technique as in the calibration
of the Heston model with
51
-
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.08
0.09
0.1
0.11
0.12
0.13
Delta
Vola
tilit
y
Market and model volatilities for USD/JPY FX options with
maturity 2 years
Heston with jumps VolatilityMarket Volatility
Figure 7.6: Market and model volatility(in %) for USD/YEN call
options with maturity 24
months
jumps.The parameters we have to calibrate are ¯0, ¯1, ¯2 and µ +
¸.
Duan (1996) had fit this model to the market smile with the help
of a Monte Carlo simula-
tion.Duan et al. (1999) have shown that the difference between
the Monte Carlo price and the
analytical approximation is very low. The analytical
approximation is considered to be precise
by the authors. Our Monte Carlo was computationally very
extensive. It would have been better
to use the analytical approximation. Our choice of initial
parameters is based on a parameter
set used by Duan et al. (1999) to price options.
(a) ¯0 = 1E-5.
(b) ¯1 = 0.1.
(c) ¯2 = 0.7
(d) µ + ¸ =0.5
After the calibration we obtain the following results of the
optimum parameters as shown in
52
-
0.1 0.20.3 0.4
0.5 0.60.7 0.8
0
0.5
1
1.5
20.09
0.1
0.11
0.12
0.13
0.14
Delta
Market Volatility Surface
Time to Matutity
Impl
ied
Vol
atili
ty
Figure 7.7: Market Implied Volatility Surface
0.10.2
0.30.4
0.50.6
0.7
0
0.5
1
1.5
20.08
0.1
0.12
0.14
Delta
Heston with Jumps Volatility Surface
Time to Matutity
Impl
ied
Vol
atili
ty
Figure 7.8: Heston with Jumps Implied
Volatility Surface
Table 7.2.
Parameter Value
¯0 1.23E-5
¯1 0.231
¯2 0.859
µ + ¸ 0.935
Table 7.2: Parameters of GARCH calibration on USD/JPY rates
7.2.1 Calibration Results
Finally the values of the implied volatilities obtained from the
GARCH Model, Heston Model
with Jumps (HJ) as compared to the market volatilities are given
in Tables 7.3- 7.7. We observe
that the volatility we obtain from the GARCH model does not
behave in similar way to the
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market. Actually the behaviour is somewhat opposite to the
market. Duan (1996) has fit the
same model to FTSE 100 index options. It could be the case that
for this particular set of data
the GARCH model was not well applicable. However our Heston with
jumps model fits the
market behaviour a lot better.
Delta Market volatility GARCH volatility HJ volatiltiy
0.10 0.0990 0.0959 0.1044
0.20 0.0945 0.1078 0.0934
0.5 0.095 0.1221 0.0950
0.8 0.1015 0.1286 0.1004
0.9 0.1109 0.1242 0.1036
Table 7.3: Implied volatilities for maturity of 1 month
Delta Market volatility GARCH volatility HJ volatiltiy
0.10 0.1066 0.0907 0.1088
0.20 0.0995 0.1068 0.0997
0.50 0.0985 0.1248 0.1015
0.80 0.1045 0.1306 0.1069
0.90 0.1156 0.1185 0.1086
Table 7.4: Implied volatilities for maturity of 3 months
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Delta Market volatility GARCH volatility HJ volatiltiy
0.10 0.1122 0.0989 0.1153
0.20 0.1024 0.1153 0.1098
0.50 0.1 0.1320 0.1123
0.80 0.1055 0.1343 0.1168
0.90 0.1178 0.1102 0.1191
Table 7.5: Implied volatilities for maturity of 6 months
Delta Market volatility GARCH volatility HJ volatiltiy
0.10 0.1166 0.1124 0.1168
0.20 0.1048 0.1266 0.1126
0.50 0.101 0.1380 0.1162
0.80 0.1065 0.1299 0.1209
Table 7.6: Implied volatilities for maturity of 1 year
Delta Market volatility GARCH volatility HJ volatiltiy
0.10 0.1235 0.1211 0.1082
0.20 0.1105 0.1295 0.0971
0.50 0.106 0.1288 0.102
0.80 0.1105 0.1310 0.1114
0.90 0.1235 0.1342 0.1243
Table 7.7: Implied volatilities for maturity of 2 years
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7.3 Choosing a model to price the Quanto option
We obtain the Root Mean Squared Errors (RMSE) for the GARCH and
the HJ model for the
implied volatility fits.The RSME is defined as
RSME =
√∑(Market volatility−Model volatility)2
Number of options
Method RMSE
GARCH 0.31
HJ 0.089
Table 7.8: Root mean squared errors for the implied volatility
fits on USD/JPY FX rate
We can observe from table 7.8 that the error for the HJ model is
substantially lower. We can
say that the HJ model would suit our purpose of pricing the FX
quanto option better. However
we price the quanto option using both the models.
7.4 Results of pricing the quanto option
We now proceed to price the FX quanto option on the USD/JPY
underlying with EUR as the
quanto currency. We denote the Heston model with jumps price as
HJ Quanto,the GARCH
option pricing model price as the GARCH price and the Black
Scholes price as BS Quanto.The
prices are in tables 7.9, 7.10, 7.11,7.12 and 7.13. Relative
difference (RD) of the HJ model as
compared to the Black Scholes model is calculated as
RD = (Black Scholes Price-Model Price)/(Black Scholes
Price).
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A negative value of the relative difference would mean that the
price under stochastic volatility
is more than under constant volatility and vice versa.
Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ)
Rel.difference(GARCH)
0.00864 1.12E-05 1.65E-05 9.88E-06 -47.95% 11.81%
0.00852 2.52E-05 3.58E-05 3.45E-05 -42.31% -40%
0.00833 9.01E-05 1.05E-04 1.15E-04 -16.49% -28.5%
0.00812 2.28E-04 2.30E-04 2.48E-04 -1.57% -8.39%
0.00798 3.47E-04 3.39E-04 3.53E-04 2.18% -1.7%
Table 7.9: Quanto prices for maturity of 1 month. Spot price is
0.0083 dollars. All the prices
are in the Euro currency.Strike prices in dollars.
Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ)
Rel.difference(GARCH)
0.0089 2.09E-05 2.105E-05 1.08E-05 -0.56% 48.44%
0.00868 4.60E-05 5.20E-05 5.48E-05 -12.72% -19.17%
0.00832 1.61E-04 1.75E-04 2.05E-04 -8.158% -27.03%
0.00796 4.07E-04 4.04E-04 4.39E-04 0.79% -8.01%
0.00771 6.2E-04 6.04E-04 6.30E-04 3.71% -7.36%
Table 7.10: Quanto prices for maturity of 3 months.Spot price is
0.0083 dollars. All the prices
are in the Euro currency.Strike prices in dollars.
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Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ)
Rel.difference(GARCH)
0.00923 3.13E-05 2.61E-05 1.88E-05 16.653% 39.78%
0.00885 6.73E-05 7.09E-05 8.98E-05 -5.17% -33.52%
0.00832 2.33E-04 2.49E-04 3.08E-04 -6.81% -32.39%
0.00780 5.84E-04 5.83E-04 6.35E-04 0.74% -8.7%
0.00743 9.12E-04 8.86E-04 9.1 E-04 3.5% -0.10%
Table 7.11: Quanto prices for maturity of 6 months. Spot price
is 0.0083 dollars. All the
prices are in the Euro currency.Strike prices in dollars.
Strikes BS Quanto HJ Quanto GARCH Quanto Rel.difference(HJ) Rel.
difference(HN)
0.00970 4.68E-05 3.51E-05 4.07E-05 25% 12.88%
0.00911 9.92E-05 1.05E-04 1.55E-04 -6.51% -56.40%
0.00832 3.40E-04 3.7E-04 4.62E-04 -8.46% -36.30%
0.00755 8.59E-04 8.67E-04 9.14E-04 -0.03% -6.4%
0.00698 1.3E-03 1.34E-03 1.32E-03 1.72% 3.3%
Table 7.12: Quanto prices for maturity of 1 year. Spot price is
0.0083 dollars. All the call
prices are in the Euro currency.Strike prices in dollars.
7.5 Implementing a copula to price the FX quanto option
In sections 5.1.2 we discussed the methodology by which we would
choose our copula from the
Archimedean family. We follow it to obtain the Frank Copula to
model the dependence. The
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Strikes BS Quanto HJ Quanto GARCH Quanto Rel. difference(HJ)
Rel.difference(GARCH)
0.01047 7.43E-05 4.47E-05 6.85E-05 40% 7.8%
0.00950 1.57E-04 1.54E-04 2.26E-04 1.9% -44.20%
0.00826 5.39E-04 5.57E-04 6.40E-04 -3.3% -18.84%
0.00702 1.4E-03 1.38E-03 1.29E-03 1.85% 8.28%
0.00570 2.65E-03 2.67E-03 2.12E-03 -0.57% 20%
Table 7.13: Quanto prices for maturity of 2 year. Spot price is
0.0083 dollars. All the prices
are in the Euro currency.Strike prices in dollars.
Copulas will be used to model the dependence between the EUR/USD
and EUR/JPY rates.
7.5.1 Parameter estimation for the T Copula
We used the Mashal and Zeevi (2002) approach discussed in
Chapter 5 to find the optimal
degrees of freedom º, the correlation ½T and the log likelihood
value L for the T Copula. We
get the following results as in Table 7.14. We get the plot for
the log likelihood function as in
Parameter Value
º 14
½T 0.6
L 246.52
Table 7.14: Parameters for the T copula
Figure7.9.
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10 12 14 16 18 20 22 24 26 28 30232
234
236
238
240
242
244
246
248Cannonical Maximum Likelihood Estimation (Mashal and
Zeevi)
Degrees of Freedom
Va
lue
of L
og
−L
ike
liho
od
Fu
nct
ion
Figure 7.9: Canonical Log likelihood function (Mashal and
Zeevi)
7.5.2 Parameter estimation for the Frank Copula
For the Frank Copula we get the Kendall Tau Rank Correlation (Γ)
and ®