Let us consider option prices under a risk neutral measure, that is let
us consider the model:
_{}
_{}
together with the initial conditions:
_{}
_{}
where W^{Q}_{t}, Z^{Q}_{t}, t Î R^{+}, are standard
Wiener processes such that W^{Q}_{0} = Z^{Q}_{0}
= 0, and dW^{Q}_{t}, dZ^{Q}_{t}, t Î R^{+}, are their
stochastic differentials. The correlation structure of the model is assumed to
be:


where r Î (1,1). The
model (4.1), (4.2), (4.3), (4.4), (4.5) is parametrized by five real
parameters, that is: r^{*}, e, _{}, _{}, r. Due to presence
of r^{*}, _{} these parameters are
called risk neutral parameters.
Figure 1: The U.S.A. S&P 500
index versus time.
Figure 2: Prices of the call options
on the U.S.A. S&P 500 index with strike prices K_{i} =
1075+25(i1), i = 1,2,¼,4 and K_{5} = 1170, and
expiry date T= March 16th, 2013 versus time.
Figure 3: Prices of the put options
on the U.S.A. S&P 500 index with strike prices K_{i} =
1075+25(i1), i = 1,2,¼,4 and K_{5} = 1170, and
expiry date T= March 16th, 2013 versus time.
Let R^{5} be the fivedimensional real
Euclidean space, we introduce the vector Q Î R^{5}
given by Q = (r^{*}, e, m^{*}, _{}, r) Î R^{5} and the set M^{*}
Ì R^{5}
defined as follows:
_{}
The inequalities that define M^{*}
are dictated by the “meaning” of the parameters r^{*}, e,_{}, r in the model
equations. In the calibration problem that we study the vector Q Î R^{5} is the unknown that must be determined
from the data and M^{*} is the set of the “feasible” choices of Q. We use as data of the
calibration problem a set of option prices observed at a given time and we
formulate the calibration problem as a nonlinear constrained least squares
problem. This means that the solution of the calibration problem consists in
fitting in the least squares sense, under the constraints defined in (4.6), the
observed option prices (i.e. the data) with the option prices obtained
evaluating the formulae deduced in Section 3
adapted to the circumstances (see [7] for
further details). For example one of these circumstances is the fact that the option prices must be evaluated at the
time when the option prices used as data are observed, that is at a given time _{}, not necessarily at time t=0 as done in Section 3.
In the numerical experiment presented we consider as
data the closing value of the day of the U.S.A. S&P 500 index and the
closing prices of the day of the European call and put options on the U.S.A.
S&P 500 index with expiry date March 16th, 2013 and strike prices of the call
and put options K_{C,i} = K_{P,i}
= K_{i} = 1075+25(i1), i = 1,2,¼,4, K_{C,5} = K_{P,5} = K_{5}
= 1170. These prices are observed in the time period that goes from April 2nd,
2012, to July 25th, 2012. Note that the observations are daily observations.
Recall that in the study of financial data time series a year is made of about
252 trading days and a month is made of about 21 trading days. Figure
1 shows the value of the U.S.A. S&P 500 index as a function of time
during the period of interest. Figure 2 and Figure
3 show respectively the prices of the European call and put options on the U.S.A.
S&P 500 index with maturity time March 16th, 2013 and strike price K_{i},
i = 1,2,¼,5, specified
previously as a function of time during the same time period. To download these
data click here.
Let _{}= April 2nd,2012, _{}one trading day, j = 1,2,¼,29, we have that _{}= May 15th,2012. We calibrate the Hull and White model (4.1),
(4.2), (4.3), (4.4) every (trading) day
during the period that goes from t =_{} = April 2nd,2012 to t = _{}= May 15th, 2012 using the prices of the European call
and put options shown in Figure
2 and Figure 3 when _{} , j = 1,2,¼,30. That is we consider a
rolling window of data made of the data of a day that covers the period April
2nd, 2012, May 15th, 2012, that is thirty trading days, and we solve the
corresponding thirty calibration problems.
Figure 4 shows the risk neutral
parameters obtained using the calibration procedure on the data described above.
We can see that the parameter values as functions of time do not change
significantly. That is the values of the parameters of the model (4.1), (4.2),
(4.3), (4.4) obtained solving the calibration problems are “stable” during the
observation period. The values of the parameters shown in Figure
4 are used to forecast the option prices one day ahead. That is we use the
parameter values obtained calibrating the model with the data of _{} to compute the option prices at _{}, obtained using as asset price (in the option pricing
formulae) _{}, j = 1,2,¼,29. The forecasts of the option prices are obtained evaluating the
European call option with formula (3.15) (g=0.5) and, given
the call price, evaluating the European put option with the put call parity
relation (3.16) . As already said formulae (3.15), (3.16) are adapted to the
circumstances. Figure 5
shows the observed and forecast values of the European call and put option
prices. The average relative errors on the forecast values of the European call
and put option prices when compared with the corresponding prices observed in
the financial market are respectively 5% and 7%. Note that if we remove from
the constraints contained in the definition of M^{*} the request that r^{*}
must be non negative the solution of the calibration procedure shows a negative
risk free interest rate of about 2% with an average of the relative errors between
forecast and observed call and put prices respectively of approximately 2% and
3% . That is if we allow negative risk free interest rates in the calibration
problem we improve the forecast of the prices of the European call and put
options. This unexpected finding may be a consequence of the anomalous market
conditions registered in the spring 2012. The following movie gives an idea of the
difference in the forecast option prices obtained with and without the
positivity constraint on the risk free interest rate. The movie shows the
forecast call (window on the left) and put (window on the right) option prices
obtained with the positivity constraint on the risk free interest rate and
those obtained without the positivity constraint.
Figure 4: Parameter values estimated in the
period April 2nd, 2012, May 15th, 2012 versus time to maturity expressed in
days. The unit of measure of e, _{} is years^{1/2}
and the unit of r^{*} is years^{1}. The parameters r and m^{*} are dimensionless.
Figure 5: Observed and one day ahead forecast
call and put option prices (in USD) for five different strike prices: ((a) K_{C,1}
= K_{P,1} = K_{1} = 1075, (b) K_{C,2} = K_{P,2}
= K_{2} = 1100, (c) K_{C,3} = K_{P,3} = K_{3} =
1125, (d) K_{C,4} = K_{P,4} = K_{4} = 1150, (e) K_{C,5}
= K_{P,5} = K_{5} = 1170) versus time to maturity expressed in
days.