 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 inﬁnitesimal 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 ; 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 . 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 inﬁ- 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  reviews various methods of gradient estimation in stochastic simula- tion, including both direct and indirect methods, as well as . The LR method was first applied to option pricing in  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  and  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 , 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 . 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 121L. 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tttStWS 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: 20exptSSrqt Wt The stock price can be simulated through tS20exp ZtSSrqt 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,ttttXB W  t (2) where tBtW 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 tX,  (3) where2222 , 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 0expttSSr tX 2ln 12  (4) where . nitesimal Perturbation Analysis seful in nancial engineering applications. To employ delta hedg- 3. Inﬁ Simulation and gradient estimation have been ufiing, 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: ddV Suppose the objective function xpectation of the sample performance measure L, that is: is an e;VELX (5) 12,,,nXXX Xwhere are dependent on , and n . The expectation is the f variablesritten as: ixed number of randomcan be wd,XELXLXF X (6) where FX is the distribution of the input randoX. If we use inverse transform method to simwhere the parameter m variables ulate the input random variable X, we can re-write Equation (6) as  10;d,ELXLXu u (7)  dependence is in ttion F . IPA method originally comes from (he distribu- taking the Xderivative 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: 10dddd,ELX XLuX dddand the IPA estimator is: dddXdXL (8) From Lebesgue dominated concovergence theorem, the ndition of uniform integrability of ddddLXX 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 strategyain a delta-neutral portfolio dynamtdelta 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 rigigation to buy certain amount of financial inht, not the ob- strument from process. ws a GBM process, we ay: lthe 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 rTTTVe SK where ST can follow a GBM process or a VG 5.1. Estimating under GBM Assuming the stock price St folloan simulate St in the following wc20exp ZtSSrqt t where Z represents the standard normal random variable. The estimator of delta fo r the European call opon through tiIPA is 00where dd1,ddTrTTTSKVSeSS (14) Z .t 200dexpTTSS rq tvS S llows a VG process, we 5.2. Estimating under VG Assuming the stock price St foan write St as: c0expttSSr 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): 001,dTSKvS SdTVedrT TS (15) where 00dexpdTT TX. SS rTSS te of delta (under DVG):  the estima001,ddTSKeSSddrTTTVS (16) where 00dexpdTT TX. SS rTSS ical Experiment al data in WRDS of the ar.10, 2008 to Sept.10, 5.3. Numer In this paper, we use the historictock price of Apple Ltd. from Ms2008. 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- rites Si ing time spˆhms to estimate spot prices and deltas in one sample path refer to . 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. 123Table 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 mian 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 hedgevery 7 dsummary stat GBM IPA GVG IPA DVG IPA mean 4457.2 4763.0 4592.4 std 2558.5 2985.4 2857.1 mian 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 hedgevery 3 dsummary stat GBM IPA GVG IPA DVG IPA mean 4875 4993.4 4992.6 std 1594 2476 2227 mian sis ess – – – in –3420 –3590 –4025 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 cumtive distr functiohe net pthrough allth 6. Conclusions Bwe 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 meathe mean from GBM.  The mean value of net gain from GVG and Figure 1. Empirical cdf of net proﬁts of delta hedging once initially. Figure 2. Empirical cdf of net proﬁt of delta hedging every 7 days. 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