Journal of Mathematical Finance, 2011, 1, 120-124
doi:10.4236/jmf.2011.13015 Published Online November 2011 (http://www.SciRP.org/journal/jmf)
Copyright © 2011 SciRes. JMF
Analysis of Hedging Profits under Two Stock
Pricing Models
Lingyan Cao1, Zheng-Feng Guo2
1Department of Mat hem at i cs, University of Maryland, Maryland, USA
2Department of Economics, Vanderbilt University, Nashville, USA
E-mail: {lingyancao, gzfsadie}@gmail.com
Received August 29, 2011; revised October 13, 2011; acce pted October 22, 2011
Abstract
In this paper, we employ two stock pricing models: a Black-Scholes (BS) model and a Variance Gamma
(VG) model, and apply the maximum likelihood method (MLE) to estimate corresponding parameters in
each model. With the estimated parameters, we conduct Monte Carlo simulations to simulate spot prices and
deltas of the European call option at different time spots over different sample paths. We focus on calculate-
ing the deltas for the two models by the method of the innitesimal perturbation analysis (IPA). Then, we
employ dynamic delta hedging using the simulated spot prices and deltas for different hedging periods. Fi-
nally, hedging profits under these two models are calculated and analyzed.
Keywords: Likelihood Ratio, Variance Gamma, Geometric Brownian Motion, Delta Hedge
1. Introduction
Delta hedging is a particular type of hedging strategy based
on the Greek called Delta, which measures the sensitivity
of option prices with respect to spot prices. A general in-
troduction of delta hedging was provided in [1]; a detailed
explanation of dynamic delta hedging and the process of
how to replicate portfolios with delta hedging and achie-
ve a delta-neutral position are studied in [2].
In order to employ delta hedging, we need to first esti-
mate deltas. Generally, there are two types of techniques:
direct and indirect methods. The former includes the in-
nitesimal perturbation analysis method (IPA) and the like-
lihood ratio method (LR). The latter includes finite diffe-
rence methods, such as forward difference method, central
difference method and backward difference method. How-
ever, the indirect methods require additional simulations
and are sensitive to the cho ice of scalars. Thus, the direct
methods are more popular in estimating Greeks. There is a
vast literature in discussing the direct methods. The IPA
was first applied to option pricin g for both European and
American options in [3,4] reviews various Monte Carlo
methods for financial engineering; see also [5] reviews
various methods of gradient estimation in stochastic simula-
tion, including both direct and indirect methods, as well
as [6]. The LR method was first applied to option pricing
in [7] for European and Asian option. However, we only
focus on estimating deltas by IPA method in this paper.
In this paper, we assume two stock pricing models, one
is the classical Black-Scholes (BS) model, in which the
stock prices follows a geometric Brownian motion (GBM)
process; another is the Variance Gamma (VG) model, in
which stock prices follow a VG model. The BS m odel was
first introduced by [8] and [9] to the finance community
as a model for stock prices and option pricing; while the
VG process was introduced for log-price returns and op-
tion pricing by [10], and further developed in [11,12]. A
general review of the VG process in the context of sto-
chastic Monte Carlo simulation is shown in [13]. The es-
timations and comparisons of Greeks under different mod-
els by different methods are studied in [14-16]. The com-
parisons of profits f rom delta hedging under differ ent mod-
els and through different methods are studied in [17,18].
In this paper, we focus on developing gradient estimates,
i.e. deltas, for a European call option of which stock prices
follow a GBM process and a VG process, respectively.
We apply the MLE to the historical data of some assets
to estimate corresponding parameters, price European call
options, and calculate the deltas according to different
stochastic processes through IPA. The remainder of the
paper is organized as follows. Section 2 provides an in-
troduction of geometric Brownian motion and VG proc-
ess and shows how to price stock prices under these two
models. Section 3 reviews several methods of gradient
121
L. Y. CAO ET AL.
estimation. Section 4 explains the delta hedging strategy
and shows how to em ploy dynam ic delt a hedging to obtai n
a risk-neutral position . Section 5 provides an example of
delta hedging, of which data are from real market.
2. Geometric Brownian Motion and
Variance Gamma
2.1. Geometric Brownian Motion Process
A stochastic process St follows a geometric Brownian
motion if log(St) is a Brownian motion with initial va-
lue log(S0). In the BS model, the price of the underlying
stock St following a geometric Brownian motion satisfies
ddd


tt
t
StW
S
where t is a standard Brownian motion. With dividend
yield q, spot 0, volatility
WS
and drift rq
, we
can obtain the stock price as:

2
0exp
t
SSrqt W


t
The stock price can be simulated through
t
S


2
0exp Z
t
SSrqt t


(1)
where is a random variable following a standard nor-
mal distribution.
Z
2.2. Variance Gamma Process
The VG process is a Levy process of independent and sta-
tionary increments. Let

,
t
be a Gamma process with
drift parameter
and vari ance para me ter v. There are two
ways to define the VG process:
1) The VG process is defined as a Gamma-time-
changed Brownian motion subordinated by a Gamma
process. The representation of VG process (say GVG) is:

 

1, 1,
,1,
tt
tt
XB W
 
 

t
(2)
where t
BtW
 represents a brownian motion with
constant drift rate
and volatility
.
2) The VG process is expressed as a difference of two
Gamma processes. The representation of the VG process
as a difference of two gamma processes (say DVG) is

,
tt t
X
,


 

(3)
where
22
22

 , and 2


.
Under the risk-neutral measure, with no dividends and a
constant risk-free interest rate r, the stock price i s given by


0exp
tt
SSr tX

2
ln 12
 

(4)
where .
nitesimal Perturbation Analysis
seful in
nancial engineering applications. To employ delta hedg-
3. In
Simulation and gradient estimation have been u
fi
ing, we first need to estimate deltas, one of gradient esti-
mates of the price of a European call option. Delta is one
of the Greeks, which are quantities representing the sen-
sitivities of derivatives, such as options. Each Greek let-
ter measures a different dimension to the risk in an option
position and the aim of a trader is to manage the Greeks
so that all risks are acceptable. More precisely, delta
is
defined as the rate of change of option prices with re-
spect to the underlying asset price. Therefore, deltas can
be calculated by taking the derivative of the option prices
with respect to the spot price.
Assuming the objective function

V
depends on the
parameter
, we focus on calculating:
d
d
V
Suppose the objective function xpectation of the
sample performance measure L, that is:
is an e

;VELX



(5)
12
,,,
n
X
XX X
where are dependent on
, and n
. The expectation is the f variables
ritten as:
ixed number of random
can be w

d,
X
ELXLXF X


(6)
where FX is the distribution of the input rando
X. If we use inverse transform method to sim
where the parameter
m variables
ulate the
input random variable X, we can re-write Equation (6) as
 

1
0;d,ELXLXu u


(7)
dependence is in t
tion F . IPA method originally comes from
(
he distribu-
taking the
X
derivative of Equation7). Moreover, IPA estimates re-
quire the integrability condition which is easily satisfied
when the performance function is continuous with respect
to the given parameter. Assume we can interchange the
expectation and differentiation. The IPA estimate is:

1
0
dd
dd,
ELX X
Lu
X



dd
d
and the IPA estimator is:

d
d
dXd
X
L
(8)
From Lebesgue dominated con
co
vergence theorem, the
ndition of uniform integrability of
d
d
dd
L
X
X
must
Copyright © 2011 SciRes. JMF
L. Y. CAO ET AL.
122
be satisfied to make the interchangeability. We only ap-
te in the f
which attains to main-
ically by offsetting the
ply IPA to calculate the gradient estimaollow-
ing paper.
4. Delta Hedging Strategy
Delta hedging is a trading strategy
ain a delta-neutral portfolio dynamt
delta of the option position through the delta of the stock
position. Delta measures the sensitivity of the option price
f with respect to the stock price S. In other words, when
the stock price changes by small amoun S, the option
price would change by the discounted amount of S
 .
The investor could hedge the risk by adjusting (long or
short) the shares of stocks to achieve a delta-neutral port-
folio position. However, as delta changes, the investor’s
risk-neutral position will change accordingly. Thus, we
have to adjust the hedging position periodically, which is
called rebalancing. If we could rebalance immediately
when the stock price changes, perfect hedge is achieved.
However, perfec t he d ge is alw a ys difficult to achieve. What
we can do is to employ a dynamic hedging and make the
hedging period as short as possible. In the following, we
show the process of delta hedging by setting the portfo-
lio’s delta zero. Details of the procedure to conduct delta
hedging can refer to [4,5].
5. An Example of Delta Hedging
European call option gives the buyer the rig
igation to buy certain amount of financial inht, not the ob-
strument from
process.
ws a GBM process, we
ay:
l
the seller at the maturity time for a certain strike price.
Let St be the stock price at t, T be the maturity time, K be
the strike price, and r be the risk-free interest rate. The price
(value) of the European call option at t is

rT
TT
Ve SK

where ST can follow a GBM process or a VG
5.1. Estimating under GBM
Assuming the stock price St follo
an simulate St in the following wc


2
0exp Z
t
SSrqt t


where Z represents the standard normal random variable.
The estimator of delta fo r the European call opon through ti
IPA is

00
where
dd
1,
dd
T
rT
TT
SK
VS
e
SS
(14)


Z .t
2
00
dexp
TT
SS rq t
vS S
 
llows a VG process, we
5.2. Estimating under VG
Assuming the stock price St fo
an write St as: c
0exp
tt
SSr tX

where Xt follows a VG process. We have two different
ways to represent the VG process Xt as iEquations (2) n
and (3). Thus we estimate the deltas in the corresponding
two ways. The estimators for deltas of European call
option through IPA are:
the estimate of delta (under GVG):

00
1,
d
T
SK
vS S
d
T
V
ed
rT T
S
(15)
where


00
dexp
dTT T
X
.
SS rT
SS
 
te of delta (under DVG): the estima

00
1,
dd
T
SK
e
SS
dd
rT
TT
VS
(16)
where


00
dexp
dTT T
X
.
SS rT
SS
 
ical Experiment
al data in WRDS of the
ar.10, 2008 to Sept.10,
5.3. Numer
In this paper, we use the historic
tock price of Apple Ltd. from Ms
2008. Assuming stock price follows a GBM and VG
process respectively, we apply the MLE to estimate the
corresponding parameters. Assuming the maturity time
for the option is 38 days, i.e., 38 365T
, the risk free
interest rate minus the dividend rate be 0.02451rq ,
we get the variance parameter 0.38204
for GBM;
and get0.35873
,0.00000262
, 0.01011
 fo
.61 ar
VG process. The spot price is 151S
t t = t0 and
the strike price is 155K0
.
We simulate all the spot pricat all different spot
times and the deltas on all hedgots. The algo-
rit
es Si
ing time sp
ˆ
hms to estimate spot prices and deltas in one sample
path refer to [4]. We then conduct the simulation on 10,000
sample paths. Moreover, after having the corresponding
spot prices and deltas on each sample path, we employ
delta hedging technique to calculate net gains on each
sample path.
Table 1 shows the summary statistics for the net prof-
its by delta hedging only once initially, i.e., 38 365tT 
by the methods above. Table 2 shows the summary sta-
tistics for the net profits by delta hedging 7 days, i.e.,
7 365t
in all the methods above is shown. Table 3
shows the summary statistics for the net profits by delta
hedging ev ery 3 days i.e., 3 365t
in all th e meth ods
Copyright © 2011 SciRes. JMF
L. Y. CAO ET AL. 123
Table 1. Summary statistics of net profits by delta hedging
once.
summary stat GBM IPA GVG IPA DVG IPA
mean 3659.8 3944.2 4116.7
std 5353.3 5580.7 4530.9
m
ian
sis
ess
in –21004 –25285 –31987
max 7562.9 7962.3 7148.1
med7058.7 7575.3 6686.3
kurto4.1588 4.4942 4.5388
skewn 1.3766 v1.4405 –1.9523
Table 2. Summartics of nts by deing
ays. y statiset profilta hedg
every 7 d
summary stat GBM IPA GVG IPA DVG IPA
mean 4457.2 4763.0 4592.4
std 2558.5 2985.4 2857.1
m
ian
sis
ess
in –8498 –8183 –8074
max 7330.7 9420.3 9527.4
med5051.4 4857.9 4644.6
kurto3.7302 2.5653 4.0837
skewn–1.0192 –0.3517 –0.5850
Table 3. Summartics of nts by deing
ays. y statiset profilta hedg
every 3 d
summary stat GBM IPA GVG IPA DVG IPA
mean 4875 4993.4 4992.6
std 1594 2476 2227
m
ian
sis
ess –
in –342035904025
max 6828 9157 9901
med5253 4854 4205
kurto3.9906 2.1933 2.2340
skewn 1.07070.0631 0.05913
above. Figures 1 the grempiricula-
ibutionns of trofits
e methods above.
y comparing the numerical results we obtained so far,
lowing conclusions:
The mean value of net gain from delta hedg ing every
n value of net gain from VG is larger than
DVG are
close.
-3 showaph of al cum
tive distr functiohe net pthrough all
th
6. Conclusions
B
we can make the fol
7 days is larger than hedging only once in the initial
time.
The mean value of net gain from delta hedg ing every
3 days is larger than hedging every 7 days.
The mea
the mean from GBM.
The mean value of net gain from GVG and
Figure 1. Empirical cdf of net prots of delta hedging once
initially.
Figure 2. Empirical cdf of net prot of delta hedging every
7 days.
Figure 3. Empirical cdf of net prot of delta hedging every
3 days.
Copyright © 2011 SciRes. JMF
L. Y. CAO ET AL.
Copyright © 2011 SciRes. JMF
124
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