Let us present an experiment concerning the use of the _{t} , 0<t<T. We want to hedge this position
during the time period 0<t<T using a quantity D_{t} of the
underlying asset whose price is S_{t}, 0<t<T, and a quantity D_{1,t} of
another call option whose value in the Hull and White model is C_{1,t},
0<t<T, having a strike price different from the strike price of the
option to be hedged and having the same expiry date T. That is we consider a
second portfolio made of the short position on the call option whose value is C_{t} and a long position of D_{t} unit of the
underlying asset whose price is S_{t}, and a long position of D_{1,t} unit
of the call option whose value is C_{1,t},
0<t<T. The value of the second portfolio
P_{t} ,
0<t<T, is:

_{}

Starting from (5.1) using Ito’s lemma and the equations (1.1), (1.2) of
the _{t}, 0<t<T. It is easy to see that the randomness in the
value of the portfolio P_{t}, 0<t<T, that comes from
to the stochastic processes that describe the asset price S_{t},
0<t<T, and the associated stochastic
volatility v_{t}, 0<t<T, can be eliminated imposing respectively the
conditions:

_{}

and

_{}

where S, v>0 are real variables that denote respectively the
underlying asset price and the associated stochastic volatility.

The equations (5.2), (5.3) are a system of two linear equations depending
on the time variable t in the two unknowns D_{t} and D_{1,t}, 0<t<T.

The portfolio whose value is P_{t}, 0<t<T, given in (5.1) is a hedged portfolio when D_{t}, D_{1,t},
0<t<T, are chosen as solution of (5.2), (5.3).

Let us consider a financial institution that writes at
time t=_{}³0 a call option
whose price at the time t=_{} is _{} and receives at the
time t=_{} _{}dollars as premium to write this option (for simplicity we
suppose _{}). The financial institution can choose to hedge or not to
hedge this position. The need to hedge this position is due to the possible
movement of the call value induced by the movement of the asset price and/or by
the changes of the market condition (described in the

Should the financial institution not hedge its
obligation, no hedging strategy is applied (to the not hedged portfolio) and
the net gain or loss at time t, _{}<t<T, is given by:

_{}, _{}<t<T,
(5.4)

where r* is the risk free interest rate. Note that, as
usual when no hedging strategy is applied, we have assumed that the premium
received to write the option by the financial institution at time t=_{} is invested at the risk free interest rate. Should the financial institution hedge its
obligation, the hedging strategy at t=_{} consists in buying _{}units of the underlying asset whose price is _{} and _{}units of the call option whose price is _{}where _{}, _{} are the solution of
(5.2), (5.3) when t=_{}. Note we have denoted with _{} the realization of the
random variable _{}_{ }, actually observed, 0<t<T. At
this point the prices C_{t}, C_{1,t}, 0<t<T, are prices
in the _{} received to write the option and, when this money is not enough for this
purpose, it is borrowed from the market at the interest rate r*. Hence at the
inception of the contract, that is at t=_{}, the net value of the hedged position is:

_{}

where _{} is the money borrowed from the market at time t=_{}, that is:

_{}

Note that _{} could be negative. In
this case no money has been borrowed
from the market, and instead there is some liquidity available, this liquidity
is invested at the risk free interest rate. After one (trading) day, that is at
time _{} (i.e. _{}=one trading day), the position is re-hedged selling _{} and buying _{} units of the asset
whose price is _{}and selling _{}and buying _{} units of the option
whose price is _{} and eventually borrowing
money from the market when necessary. Applying this strategy the net value of
the hedged position _{} at the time _{} is:

_{}

where _{} is the total amount of
money borrowed from the market until time_{} given by:

_{}

The quantity _{}, 0<t<T, is the value of the so called replicating
portfolio. The value of this portfolio replicates the option price C_{t},
0<t<T. In the previous analysis we have considered the value of the replicating
portfolio at time _{} and _{}.

Note that the variation in time of _{} is approximately the
variation of _{}, _{}<t<T. This last
variation is _{}, _{}<t<T.

This hedging strategy that has been illustrated in the
days _{} and _{} can be applied to an
arbitrary time period rehedging the
portfolio each day of hedging period.

Let us discuss an example of the hedging strategy
described above applied to the _{}, j=1,2,…,30. The values of
_{}, j=1,2,…,30, are those used in Section 4, that is _{}= April 2nd, 2012, _{} _{}one trading day, j = 1,2,¼,29, and we have _{}= May 15th, 2012.

In Section 4 we have calibrated the Hull and White
model using as data the U.S.A. S&P
500 and the prices of its European options when _{}, j=1,2,…,30. In order
to test the reliability of the parameters obtained in Section
4 calibrating the Hull and White model we generate the portfolio whose
value is P_{t} and the net values N_{t} and N_{h,t}
of the not hedged and of the hedged portfolios when _{}, j=1,2,…,30, using _{}, j = 1,2,¼,30, where _{}, j=1,2,…,30, are the values of the U.S.A. S&P 500 index observed
in the period _{}= April 2nd, 2012 -_{}= May 15th, 2012 and choosing the option prices C_{t},
C_{1,t} used to evaluate the portfolio value to be the values (closing
prices of the day) observed at the New York Stock Exchange (N.Y.S.E.) of
the European call options on the U.S.A.
S&P 500 index having expiry date T= March 16th, 2013 and strike prices respectively K=1075, K_{1}=1100
respectively. The quantities _{}, _{}, _{}, _{} needed to determine _{}, _{}, when t=_{}, j=1,2,…,30, are
obtained evaluating numerically the derivatives of the option price formulae
deduced in Section 3. Note that the option price
formulae needed to evaluate numerically these derivatives are computed using as
model parameters those obtained in the calibration discussed in Section 4 and the S&P 500 index (closing value of
the day) actually observed. The abuse of notation in the option prices is due to
the desire of keeping the notation simple.

The hedged portfolio is rehedged at the end of every trading day in the
period _{}.

Let us consider the net value of the not hedged and of the hedged
portfolio when _{}.

In Figure 1 we show the net value of the hedged position N_{h,t},
_{} (see formulae (5.5)
and (5.7)), and of the not hedged position
N_{t}, _{}(see formula (5.4)) as a function of the time _{}, where _{}=_{}, j=1,2,…,30, _{}=April 2nd, 2012, _{}=May 15th, _{}=_{} (i.e. time to maturity T-t, T-_{}=237 days) and the last index value 29 (i.e. day 29) corresponds
to t=_{} (30 days) (i.e. to
time to maturity T-_{}=208 days).

Figure 1: Hedged and not hedged net values
versus time (N_{h,t}: dotted
line, N_{t}: dashed line)

Figure 2 shows the same quantities of Figure 1 when we
choose _{}=_{}. Note that _{}=April 10th, 2012. That is Figure 2 shows the same
quantities of Figure 1 when the call
option is written at time t=_{}=_{} = April 10th, 2012, instead of being written when t=_{}=_{} = April 2nd, 2012. For _{}<t<_{} Figure 2 shows the net value of the position of the
financial institution that wrote the call option at t=_{}=_{} and chooses to hedge (N_{h,t}) or not to hedge (N_{t})
the corresponding position. Note that in Figures 1 and 2 we have maintained the
same index scale on the horizontal axis. This choice is done to make easy the
comparison by inspection between Figure 1 and Figure 2.

Figure 2: Hedged and not hedged net values
versus time (N_{h,t}: dotted
line, N_{t}: dashed line)

Figure 1
and Figure 2 show how the value _{} of the call option
(observed at the N.Y.S.E.) the day _{} when the call is written influences the gain/loss
obtained when the position is closed at maturity time. In fact when the call is
written at t =_{}(i.e. index value equal zero in Figures 1, 2, 3) the hedging strategy may be omitted without
incurring in losses. In fact the price of the call option (observed at the
N.Y.S.E.) _{} for t >_{} (see Figure 3) is smaller than _{}, in this case the writer without hedging his position will
make a profit. On the contrary when the
call is written at t =_{} (index value equal seven in Figures 1, 2, 3) the hedging
strategy plays a relevant role in protecting the writer from losses. In fact
when t>_{} the price of the call option (observed) _{} is greater than _{}most of the days (see Figure 3) and the writer loses money most of the days if he
does not apply a hedging strategy.

_{}

Figure 3: Call option prices _{}observed in the period April 2^{nd}, 2012 - May 15^{th},
2012

In the example shown in Figure 1 the hedged position
when _{} has a net value _{}= - 0.026 USD. This means that the hedging strategy has
produced a loss of 0.026 USD on a value of 344 USD. This loss is due the fact
that the asset price at _{} (i.e. _{}) is smaller than its value at _{} (i.e._{}). That is, hedging a
position of 344 USD when_{}=April 2nd, 2012 and re-hedging the position every trading
day in the period _{}April 2^{nd}, 2012 - _{}May 15th, 2012 has produced a loss of 0.026/344=0.008% of the
position value. The net value of the not-hedged portfolio _{} is _{}=_{}= (344.4-289.9) USD = 54.5 USD, that is the not hedged
portfolio has a gain of 54/344=15.8% of the position value. Note that to
evaluate _{} we have used r*=1%(year^{-1}).

In the example shown in Figure 2 we can see that if
the not hedged position is closed in a date t with _{}£ t £_{} the financial institution has an average loss of about 26 USD, that is an average loss of 26/344=7.5% of the position value. That is 26
USD is the value of the sum _{}.

The total
cost of the hedging shown in Figure 1 is _{}and it is about -20 USD,
that is no money has been borrowed from the market, and on the contrary, the
hedging strategy has generated 20 USD.
Note that in this experiment we neglect the transaction costs associated to the
hedging. The transaction costs are very important in real financial markets. In
this sense the hedging experiment presented here is an ideal experiment.

We conclude this Section presenting an interactive
application that implements a numerical experiment. The numerical experiment is
concerned with the hedging of a short position on a European call option. The
hedged position is obtained proceeding as explained previously. The user must
provide as input parameters of the application the values of the parameters of the Hull and
White model (i.e.: risk free interest rate r*,
drift of the variance m, volatility of
volatility e, initial volatility
_{}), the time to
maturity _{} (0< t <T), where the
current time _{} is assumed to be zero
(_{}=0), the initial value _{} of the asset price and
the size of the sample used in the statistical simulation to compute the option
prices and the derivatives of the option prices with respect to the asset price
and to the volatility in the Hull and White model. Note that the interactive
application, given the values of the parameters of the

The interactive application shows the evolution of the
net value N_{t, }_{}, of the not-hedged portfolio made by a European call option
having strike price _{} and time to maturity _{}and of the corresponding net value N_{h,t} , _{} of the hedged
portfolio. Note that in the interactive application the time _{} corresponds to t=0 and
the maturity time corresponds to t = t. The European call
option used to hedge the portfolio has the same expiry date of the call option
that must be hedged and has strike price _{}. Furthermore the
realized variances R_{t }and R_{h,t} of N_{t } and N_{h,t} , _{} at maturity time (i.e.
t = t) are given. That
is the interactive application computes and
shows R_{t}_{ }and R_{h,}_{t}_{ } defined as:

_{}_{}

In the interactive application a set of default
parameters is given.