Lorella Fatone,
Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli
.
In [1] the
SABR stochastic volatility model with bvolatility b Î (0,1) and an absorbing barrier in zero imposed to
the forward prices/rates stochastic process is studied. The model is considered
in presence of (possibly) nonzero correlation between the stochastic
differentials that appear on the right hand side of the dynamic equations. A
series expansion of the probability density function of the model in powers of
the correlation coefficient of these stochastic differentials is presented. Explicit
formulae for the first three terms of this expansion are derived. These
formulae are integrals of known integrands. The zeroth order term of the
expansion is a new integral formula containing only elementary functions of the
transition probability density function of the SABR model when the correlation
coefficient is zero. The expansion is deduced from the final value problem for
the backward Kolmogorov equation satisfied by the transition probability
density function. Each term of the expansion is defined as the solution of a
final value problem for a partial differential equation. The integral formulae
that give the solutions of these final value problems are based on the Hankel
and on the KontorovichLebedev transforms. From the series expansion of the
probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover
we deduce closed form formulae for the moments of the forward prices/rates
variable. The moment formulae do not contain integrals or series expansions and
are expressed using only elementary functions. The option pricing formulae are
used to study synthetic and real data. In particular we study a time series of
futures prices of the EUR/USD currency's exchange rate and of some of the
corresponding option prices. A general
reference to the work of the authors and of their coauthors in mathematical
finance is the website: http://www.econ.univpm.it/recchioni/finance.
The SABR model is a stochastic volatility model widely used in
mathematical finance and in the practice of the financial markets to describe
the dynamics of the forward prices/rates variable and of its stochastic
volatility. The variables of the SABR model x_{t} , v_{t}, t>0, satisfy
the following system of stochastic differential equations:
_{}
_{}
and the
initial conditions:
_{}
_{}
The variable x_{t} represents the forward
prices/rates at time t, t>0, and v_{t} is the stochastic volatility variable
associated to x_{t} at time t, t>0. The parameters b and e are called
respectively bvolatility of the forward
prices/rates variable and volatility of volatility. We assume 0<b<1 and e>0. The
stochastic processes W_{t}, Z_{t}, t>0, are standard Wiener
processes, such that W_{0}=Z_{0}=0 and dW_{t}, dZ_{t},
t>0, are their stochastic differentials. The stochastic differentials dW_{t},
dZ_{t}, t >0, are assumed to satisfy the condition _{}, where <^{.}> denotes the expected value of ^{.}
and rÎ(1,1). That is the
stochastic differentials of dW_{t}, dZ_{t}, t>0, are assumed to be correlated with correlation
coefficient rÎ(1,1). The initial
conditions _{} are random variables
that are assumed to be concentrated in a point with probability one and, for
simplicity, we identify them with the points where they are concentrated.
Moreover we assume _{}>
For m = 1,2,¼ let M_{m}
be the mth moment with respect to zero of x_{t}, t >0, we have:
_{}
where p_{S} is the
transition probability density function associated to the stochastic processes x_{t},
v_{t}, t >0, implicitly
defined by (2.1), (2.2), (2.3), (2.4) and by the absorbing barrier in zero
imposed to x_{t}, t>0. Let_{}, in [1] it has been shown that:
where [^{.}] denotes the integer part of^{
.} and the functions b_{j,m}, j=0,1,2,…,n*, m=1,2,…, are defined
by the following recursive relation:
_{}
_{}
_{}
_{}
The integrals that appear in (2.10) are elementary integrals that can be
computed explicitly using the following formula:
_{}
Formula (2.6) that gives the
moments _{}, m=1,2,..., of the
forward prices/rates variable _{}, t>0, is an
explicit formula expressed using only elementary functions that does not
contain integrals or series expansions. Formula (2.6) is a new formula that can be used in many
circumstances. For example it can be used in the formulation of calibration
problems for the SABR model. In fact formula (2.6), thank to its closed form
character and to the fact that contains only elementary functions can be
evaluated very efficiently. In [2], [3]
formulae analogous to (2.6) have been used in the study of calibration problems
for the normal and the lognormal SABR models.
We provide an interactive
application and an app that given the values of the parameters of the SABR
model, a time value t >0 and a moment order m (0<m£5) give the mth
order moment M_{m }at time t computed using formula (2.6) and using the
Monte Carlo method. These two values of the same quantity can be compared. In the interactive
application and in the app the user can choose the size of the sample used in
the Monte Carlo computation of the moment. A set of default values of the parameters of
the SABR model needed in the interactive application and in the app is given.
The app is a simple translation of the interactive application in the Android
environment. The app is freely downloadable
and runs on devices based on the Android software system. For example it can be used on Android smart phones and tablets.
Let us
give an expression of the transition probability density function of the variables
x_{t}, v_{t}, t>0, implicitly defined by (2.1), (2.2),
(2.3), (2.4) when the correlation coefficient r is zero. Let us define the
stochastic process:

(4.1) 
From Ito's lemma and equations (2.1), (2.2) it follows that the
stochastic processes x_{t}, v_{t}, t > 0, satisfy the following
system of stochastic differential equations:


The initial conditions (2.3), (2.4) become:
_{}
_{}
An absorbing barrier in zero is imposed to the
stochastic process x_{t}, t > 0. The barrier imposed to x_{t}, t > 0, comes from the barrier
imposed to x_{t}, t > 0.
Let p_{S}(t,x,v,t¢,x¢,v¢), x, x¢, v, v¢, t, t¢ > 0, t¢t > 0, be the transition
probability density function of model (2.1), (2.2), (2.3), (2.4) with the
absorbing barrier in zero imposed to x_{t} , t>0, that is let p_{S}(t,x,v,t¢,x¢,v¢), x, x¢, v, v¢, t, t¢ > 0, t¢t > 0, be the probability density
function of having x_{t}_{¢} = x¢, v_{t}_{¢} = v¢ given the fact
that we have x_{t} = x, v_{t} = v when t¢t > 0. Moreover let p(t,x,v,t¢,x¢,v¢), x, x¢, v, v¢, t, t¢ > 0, t¢t > 0, be the transition
probability density function of model (4.2), (4.3), (4.4), (4.5) with the
absorbing barrier in zero imposed to x_{t} , t>0, that is let p(t,x,v,t¢,x¢,v¢), x, x¢, v, v¢, t, t¢ > 0, t¢t > 0, be the probability density
function of having x_{t}_{¢} = x¢, v_{t}_{¢} = v¢ given the fact that we have x_{t} = x, v_{t} = v when t¢t > 0. We have:


The
function p is the solution of the backward Kolmogorov equation associated to (4.2),
(4.3), that is:
_{}



(4.9) 
where d is the Dirac's
delta. The Dirichlet boundary condition (4.9) imposed to the function p is the
condition that translates to p the condition imposed prescribing
the absorbing barrier in zero to the
stochastic process x_{t}, t>0. Note that p does not depend from t
and t¢ separately, it depends
only from s = t¢t > 0. Let us introduce the
function p^{*}(s,x,v,x¢,v¢) = p(t,x,v,t¢,x¢,v¢), where s = t¢t, x, x¢, v, v¢, t, t¢ > 0. From (4.7), (4.8), (4.9) it
follows that p^{*} is the solution of the partial differential
equation:
_{}
with initial condition:
_{}
Let _{} be the transition probability density function
p* when r=0. From (4.10), (4.11), (4.12) it follows that the function p*_{0}
satisfies the partial differential equation:
_{}
with initial condition:
_{}
and boundary condition:
In [1] it is shown that the solution of problem (4.13), (4.14),
(4.15) is given by:
_{}
where
_{}
_{}
and
_{}
It is easy to see that the
function q(u,x,v, x’,v’)^{2}4e^{4}x^{2}x’^{2} is
positive for u>0, x, x’, v, v’ > 0. Note that the function p^{*}_{0}
defined in (4.16) satisfies the boundary condition (4.9). In fact when x = 0 the term x^{2}^{n} is zero and the
functions q and m are “well behaved” in v, v’ for v, v’ > 0, x’, u >0.
Let s=t’t>0, t=0 and let “year” be the time unit on the t
axis. In MOVIE
1 we show the integral on [0,+¥)´[0,+¥) of the
probability density function p^{*}_{0} with respect to the
future variables x’, v’ as a function
of the past variables x, v, for two values of b=0.1, 0.5 when e=0.6, r=0 and the time
varies from s=0 years to s=10 years. That is for the choices of the parameters
specified previously we show the quantity:
_{}
In the models where the probability is conserved the function _{} defined in (4.19) is equal to one for every x, v, s > 0. This
is not the case in the SABR model studied here. Note that we have chosen t=0 so
that the past variables x, v corresponds to x_{0}, v_{0} assigned
in (4.4), (4.5). MOVIE 1 shows that when time goes from s=0 to s=10
years there is an increasing loss of probability. Furthermore it is shown that the loss of probability observed for b=0.5 is larger than
the loss of probability observed for b=0.1.
Let us present a numerical
experiment. In the experiment we consider the daily values of the futures price
of the EUR/USD currency's exchange rate having maturity September 16th, 2011,
(the third Friday of September 2011) (ticker YTU1 Curncy of Figure
1) and the daily prices of the corresponding European call and put options
with expiry date September 9th, 2011 and strike prices E_{i} = 1.375+0.005*(i1), i = 1,2,¼,18. The strike prices E_{i},
i = 1,2,¼, 18, are expressed
in USD. These prices have been observed in the time period that goes from
September 27th, 2010, to July 19th, 2011. The observations are made daily and
the prices considered are the closing prices of the day at the New York Stock
Exchange. Recall that a year is made of about 250255 trading days and that a
month is made of about 2022 trading days. Figure1
shows the futures price (ticker YTU1 Curncy) (blue line) and the EUR/USD
currency's exchange rate (pink line) as a function of time. Figures
2 and 3 show respectively the prices (in USD) of
the corresponding call and put options with maturity time September 9th, 2011
and strike price E_{i}, i = 1,2,¼,18, as a function of time.
Figure 1: YTU1 (blue line) and EUR/USD currency's exchange rate (pink
line) versus time.
Figure 2: Call option prices on YTU1 with
strike price E_{i} = 1.375+0.005*(i1), i = 1,2,¼,18, and expiry date T= September 9th, 2011 versus time.
Figure 3: Put option prices
on YTU1 with strike price E_{i} = 1.375+0.005*(i1), i = 1,2,¼,18, and expiry date T= September 9th, 2011 versus time.
We use the SABR model to study the data shown
in Figures 1, 2, 3. Let R^{5} be the fivedimensional real Euclidean space, we
introduce the vector Q = (r, e, b, _{}, r) Î R^{5} and the set MÌ R^{5} defined as
follows:
_{}
The parameter r is the risk
free interest rate that appears in the option pricing formulae derived in [1] and used in the
calibration problem. We consider r as an unknown of the calibration problem. Note
that since _{} cannot be observed in
the financial markets it must be regarded as a parameter to be estimated in the
calibration problem. The inequalities contained in (5.1) that define M are the
natural constraints implied by the meaning in the model equations of the components
of the vector Q. We use as data of
the calibration problem the option prices observed at a given time (i.e. the
option prices of Figures 2, 3 at a given date and we formulate the calibration
problem as a nonlinear constrained least squares problem.
In the calibration problem
studied the vector Q Î R^{5} is the unknown that
is determined from the data (i.e. the futures EUR/USD prices and the option
prices shown in Figures 1,2,3) and M is the set of the “feasible”
choices of Q. This means that the
solution of the calibration problem consists in finding the choice of Q ÎM that gives the
best fit in the least squares sense between the observed option prices (i.e.
the data) and the option prices obtained evaluating the formulae deduced in [1].
MOVIE 2 shows six animated windows. The upper left
corner window shows the value of the futures prices of Figure 1 and the current
time. The remaining five windows show the observed and forecast prices of the European
call and put options for five values of the
strike price (that is: E=E_{3} = 1.3850 (window (a)), E=E_{5} = 1.3950 (window (b)), E= E_{7}
= 1.4050 (window (c)), E= E_{9} = 1.415 (window (d)), E= E_{11}
= 1.4250 (window (e))) are shown as a function of time. The forecasts shown in MOVIE 2 are the
forecasts one dayahead of the day of observation of the option prices used to
calibrate the SABR model.
[2] Fatone, L., Mariani, F., Recchioni,
M.C., Zirilli, F.: The use of statistical tests to calibrate the normal SABR
model, Journal of Inverse
and Ill Posed Problems, 21(1),
(2013), 5984, http://www.econ.univpm.it/recchioni/finance/w15/
.
[3] Fatone, L., Mariani, F., Recchioni, M.C.,
Zirilli, F.: Closed form moment formulae for
the lognormal SABR model and applications
to calibration problems, Open Journal of Applied Sciences, 3, (2013) , 345359, http://www.econ.univpm.it/recchioni/finance/w16/
.