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.
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 R5 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  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 KC,i = KP,i = Ki = 1075+25(i-1), i = 1,2,¼,4, KC,5 = KP,5 = K5 = 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 Ki, 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) KC,1 = KP,1 = K1 = 1075, (b) KC,2 = KP,2 = K2 = 1100, (c) KC,3 = KP,3 = K3 = 1125, (d) KC,4 = KP,4 = K4 = 1150, (e) KC,5 = KP,5 = K5 = 1170) versus time to maturity expressed in days.