Let us present an experiment concerning the use of the
Starting from (5.1) using Ito’s lemma and the equations (1.1), (1.2) of
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 Dt and D1,t, 0<t<T.
The portfolio whose value is Pt, 0<t<T, given in (5.1) is a hedged portfolio when Dt, D1,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 Ct, C1,t, 0<t<T, are prices
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 Ct, 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
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 Pt and the net values Nt and Nh,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 Ct, C1,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, K1=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 Nh,t,
(see formulae (5.5)
and (5.7)), and of the not hedged position
Nt, (see formula (5.4)) as a function of the time , where =, j=1,2,…,30, =April 2nd, 2012, =May 15th,
Figure 1: Hedged and not hedged net values versus time (Nh,t: dotted line, Nt: 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 (Nh,t) or not to hedge (Nt) 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 (Nh,t: dotted line, Nt: 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 2nd, 2012 - May 15th, 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 2nd, 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 Nt, , of the not-hedged portfolio made by a European call option having strike price and time to maturity and of the corresponding net value Nh,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 Rt and Rh,t of Nt and Nh,t , at maturity time (i.e. t = t) are given. That is the interactive application computes and shows Rt and Rh,t defined as:
In the interactive application a set of default parameters is given.