In this section, we are interested in comparing commonly used delta-hedging strategy with the discrete time hedging strategies introduced in Sect.22.2. The delta of a derivative is referred to as the rate of change in the price of a derivative security relative to the price of the underlying asset. Mathematically, the delta valuet at timet is defined as the partial derivative of the price of the derivative with respect to the price of the underlying, that is,t D@Vt=@St, whereVtandStare the prices of the derivative and the underlying asset at timet, respectively.
For example, considering a European call option with expiration dateT and strike priceK, the no-arbitrage option value at timet isCt D er.Tt/EtQf.ST K/Cg;wherer is the riskless interest rate. After simplifying the partial derivative
@Ct=@St and exploiting the following property,
h!0lim 1 h
Z log.K=St/
log.K=.StCh//f.StCh/eyKgdGt.y/D0;
where Gt.y/ is the conditional distribution of log.ST=St/ given Ft under the martingale measureQ, one can show that the delta of the European call option can be expressed as
t.c/D @Ct
@St
Der.Tt/EtQ
ST
StI.STK/
: (22.14)
And the delta value of the put option,t.p/, can also be derived from the following relationship based on the put-call parity:
t.c/t.p/D1: (22.15)
Since (22.15) is derived based on a simple arbitrage argument, the result is distribution-free, that is it does not depend on the distribution assumption of the underlying security. To calculate the delta value of a European call option, one can either approximate the conditional expectation,EtQ.STIfSTKg/, recursively by the DSA or approximate the partial derivative,t.c/D@Ct=@St, by the relative rate of changefCt.StCh/Ct.St/g= h, wherehis a small constant and the option price Ct’s can be obtained by the DSA.
22.3.1 LQRA-Hedging and Delta-Hedging Under Complete Markets
In a complete market every contingent claim is marketable, and the risk neutral probability measure is unique. There exists a self-financing trading strategy and the holding units of the stocks and bonds in the replicating portfolio are uniquely determined. This trading strategy is called the perfect hedging which attains the lower bound of the criteria (22.6) and (22.8). Thus we expect both the LQR- and LQRA-hedging strategies will coincide with the delta-hedging under the complete market models. In the following, we show directly the holding units of the stocks in an LQRA-hedging is the same as in the delta-hedging for the two complete market models – the binomial tree and the Black-Scholes models (Black and Scholes 1973).
For simplicity, let the bond priceBt D ert wherer represents a constant riskless interest rate. First, consider a binomial tree model. Assumes at each step that the underlying instrument will move up or down by a specific factor (u ord) per step of the tree, where.u; d /satisfies0 < d < er <u. For example, ifSt1 Ds, thenSt
will go up tosuDusor down tosd Ddsat timet, with the risk neutral probability qDP .St DsujSt1Ds/D erd
ud D1P .St DsdjSt1Ds/:
By straightforward computation, we have
CovQt1.Ft; St/Dq.1q/.susd/fFt.su/Ft.sd/g and
VarQt1.St/Dq.1q/.susd/2:
Thus by (22.10) the holding units of the stock in the-hedging is hO1t1D CovQt1.FQt;SQt/
VarQt1.SQt/ D Ft.su/Ft.sd/ susd ; which is consistent with the delta-hedging of the binomial tree model.
Next, consider the Black-Scholes model,
dSt DrStdtCStd Wt; (22.16) wherer and are constants andWt is the Wiener process. For a European call option with strike priceKand expiration dateT, the holding units of the stock in the delta-hedging of the Black-Scholes model ist D˚.d1.St//at timet, where
d1.St/D log.St=K/C.rC0:52/.T t/
p Tt
and˚./ is the cumulative distribution function of the standard normal random variable. We claim in the following that
h1t !˚.d1/
asdt !0, wheredtdenotes the length of the time periodŒt; tCdt. Denote the discounted stock price and option value at timet bySQt andFQt, respectively. Note that
CovQt .FQtCdt;SQtCdt/DEtQ.FQtCdtSQtCdt/ QFtSQt
DEtQ
er.tCdt/
n
StCdt˚.d1.StCdt//
Ker.Ttdt/˚.d2.StCdt//
oSQtCdt
QFtSQt
EtQ.SQtCdt2 /˚.d1.St// QStKerT˚.d2.St//
ertn
St˚.d1.St//Ker.Tt/˚.d2.St//
oSQt
D˚.d1.St//VarQt .SQtCdt/;
whered2.St/Dd1.St/p
T t and the approximation () is due to EtQf QStCdtk ˚.di.StCdt//g EtQ.SQtCdtk /˚.di.St//
for smalldt,i D 1; 2andk D 1; 2. Therefore, by (22.10) we haveh1t !˚.d1/ as dt ! 0. This result indicates that if practitioners are allowed to adjust the hedging portfolio continuously, then LQRA-hedging coincides with delta-hedging.
However, we should be aware that practitioners are not allowed to rebalance the hedging portfolio continuously and may want to reduce the number of the rebalancing time as less as possible due to the impact of transaction costs in practice.
22.3.2 LQRA-Hedging and Delta-Hedging Under Incomplete Markets
In this section, we consider that the log-return of the underlying assets follows a GARCH model such as
8<
:
Rt Dr1
2t2CtCt"t; "t D.0; 1/
t2D˛0C˛1t12 "2t1C˛2t12
; (22.17)
where the parameters are set the same as inDuan(1995)
D0:007452; ˛0 D0:00001524; ˛1D0:1883; ˛2D0:7162;
d Dq
˛0
1˛1ˇ1 D0:01263 .per day, i.e.0:2413per annum/;
KD40; rD0;
and the innovation"t is assumed to be normal or double exponential distributed with zero mean and unit variance. Suppose that a practitioner writes a European call option with strike priceK and expiration dateT, and set up a delta-hedging portfolio at the initial time, with the hedging capitalF0, that is,
F0Dh0C0S0;
whereF0denotes the risk-neutral price derived by the extended Girsanov principle and thus the cash position h0 can be obtained by F0 0S0. Similarly, we can construct the LQRA-hedging portfolio by
F0Dh0C0S0:
We simulatenD10;000random paths to generate the stock price,fST;igniD1, under the physical model (22.17), and then compute the ratio of the average variations of the delta hedging and LQRA-hedging portfolios
GT D Pn
iD1fh0erT C0ST;i.ST;iK/Cg2 Pn
iD1fh0erT C0ST;i.ST;iK/Cg2;
forT D 5; 10; 30(days). Table 22.1 shows the simulation results of GT, T D 5; 10; 30, of the GARCH-normal and GARCH-dexp models withK D 35; 40; 45 and several different parameter settings.
Table 22.1 The relative values of the average squared hedging costs of delta-hedging and LQRA- hedging in the GARCH(1,1) log-return model
GARCH-normal GARCH-dexp
kur. KD35 KD40 KD45 kur. KD35 KD40 KD45
Case 1:˛0D0:00001524; ˛1D0:1883; ˛2D0:7162; D0:007452
G5 4.10 1.01 1.00 1.01 6.67 1.02 1.01 1.03
G10 4.33 1.01 1.00 1.02 7.39 1.03 1.01 1.07
G30 4.29 1.01 1.01 1.05 9.24 1.04 1.03 1.16
Case 2:˛0D0:00002; ˛1D0:1; ˛2D0:8; D0:01
G5 3.50 1.00 1.01 1.01 4.91 1.02 1.01 1.05
G10 3.53 1.01 1.01 1.03 4.76 1.02 1.01 1.07
G30 3.42 1.01 1.03 1.04 4.28 1.00 1.03 1.08
Case 3:˛0D0:00002; ˛1D0:2; ˛2D0:7; D0:01
G5 4.18 1.01 1.01 1.03 6.95 1.04 1.01 1.09
G10 4.42 1.01 1.01 1.06 8.34 1.05 1.02 1.16
G30 4.39 1.00 1.03 1.08 9.21 1.01 1.06 1.21
Case 4:˛0D0:00002; ˛1D0:3; ˛2D0:6; D0:01
G5 5.09 1.02 1.01 1.06 10.52 1.06 1.02 1.15
G10 6.06 1.04 1.02 1.11 20.50 1.08 1.04 1.27
G30 8.87 1.01 1.06 1.20 53.67 1.04 1.25 1.79
Note that the values ofGT’s in Table22.1are all greater than 1, which means the average variation of the LQRA-hedging is smaller than the delta-hedging. Under the same parameter setting in both GARCH-normal and GARCH-dexp models, the kurtosis of the GARCH-dexp models is greater than the GARCH-normal model.
The results shows that GT tends to increase in the kurtosis of the log-returns, especially when the option is out-of-the-money.
In Fig.22.6, we plot the hedging strategies of delta- and LQRA-hedging for one period case, whereFQtis the discounted option value function at timet and the point .SQt;FQt/denotes the time-t discounted stock price and discounted hedging capital.
In the left-hand panel, the dash-line,.t1; t/, denotes the delta-hedging values, which is the tangent of the curveFQt at the point.SQt;FQt/. In the right-hand panel, the dot-line,.t1; t/, represents the LQRA-hedging, which is regression line of FQtC1 under the risk-neutral probability measure derived by the extended Girsanov principle (seeElliott and Madan 1998;Huang and Guo 2009d).
If the hedging period increases to`,` > 1, then the delta-hedging.t; tC`/
remains the same, that is,.t; tC`/ D .t; t C1/, (see the left-hand panel of Fig.22.7). However, the LQRA-hedging,.t; tC`/(see the red line in the right- hand panel of Fig.22.7), would be different from.t; tC1/since the hedging target is changed from FQt toFQtC`. This phenomenon states that the LQRA-hedging is capable of making adjustment to the hedging period`, which makes it more suitable for various time period hedging. This phenomenon also provides an explanation
S~ F~
~
1
Ft
~ Ft
~)
~, (StFt
S~
~
1
Ft
~ Ft
) 1 t (t,
~)
~, (StFt )
1 t (t,
F~
Fig. 22.6 One period delta- and LQRA-hedging
S~ F~
~ Ft
~ Ft
~)
~, (St Ft
S~ F~
~ Ft
~ Ft
) , (tt
~)
~, (StFt )
t (t,
) 1 t (t,
) 1 , (tt
Fig. 22.7 `-period delta- and LQRA-hedging
of why the GT’s in most cases of Table 22.1 tends to increase in the hedging period or in the kurtosis of the log-returns. Because when the hedging period or the kurtosis of the log-returns increases, the LQRA-hedging would reduce more variability between the hedging portfolio and the hedging target than delta-hedging.