An optimal multi-step quadratic risk-adjusted hedging strategy

Journal of the Korean Statistical Society 42 (2013) 37–49

Contents lists available at SciVerse ScienceDirect

Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss

An optimal multi-step quadratic risk-adjusted hedging strategy Shih-Feng Huang a , Meihui Guo b,∗ a

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

b

Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

article

info

Article history: Received 4 July 2011 Accepted 27 April 2012 Available online 18 May 2012 AMS 2000 subject classifications: 62J05 62P05 91B28 Keywords: Discrete time hedging Extended Girsanov principle Multi-step hedging Quadratic risk minimization Risk-adjusted criterion

abstract An optimal multi-step hedging strategy is proposed to minimize one’s exposure to risk. The proposed strategy, called the QRA-hedging, is based on the minimization of the quadratic risk-adjusted hedging costs and extends the result of Elliott and Madan (1998) to the multistep case. The multi-step QRA-hedging cost is proved to be the same as the no-arbitrage price derived by the extended Girsanov principle. The QRA-hedging strategy is investigated under complete and incomplete market models. A regression-based method is proposed to estimate the QRA-hedging positions. And a dynamic programming is developed to facilitate computation of the QRA-hedging strategy. Simulation and empirical studies are performed to compare the QRA with other hedging strategies under complete and incomplete market models. © 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

1. Introduction A hedge is a financial strategy used to reduce the risk of adverse price movements in an asset by buying or selling others. Normally, a hedge consists of taking an offsetting position in a related security, such as a derivative. For example, when a trader writes a call option, she may set up a hedging portfolio consisting of the riskless bond and the underlying asset to hedge her short position of the option against future rise in the price of the underlying asset. If possible, traders will choose a perfect hedge to reduce their risk to nothing except the hedging capitals. In a complete market model, any derivative is attainable and thus admits a perfect hedge. And the cost of replication equals the price of the derivative, which is the expected discounted claim payoff under the unique equivalent martingale measure. For example, delta-hedging is a perfect-hedging strategy for the Binomial model as well as for the Black–Scholes model (Black & Scholes, 1973) if continuous rebalancing is allowed. However, completeness is only an idealization of a financial market. Relaxing the idealized assumption leads to incomplete market models, where financial products bear an intrinsic risk which cannot be hedged away completely. In addition, continuous hedging is practically impossible due to the consideration of transaction costs. If rebalancing can only be performed at discrete intervals of time, then delta-hedging is no longer optimal for the Black–Scholes model. Furthermore, many financial papers show that the hedging horizon plays a crucial role in financial hedging. For example, Cotter and Hanly (2009) examines the volatility and covariance dynamics of cash and futures contracts that underlie the optimal hedge ratio across different hedging time horizons. Their findings show that the volatility and covariance dynamics may differ considerably depending on the hedging horizon. Korn and Koziol (2009) investigates the price risk in foreign currency and

∗

Corresponding author. Tel.: +886 7 5253820; fax: +886 7 5253809. E-mail addresses: [email protected] (S.-F. Huang), [email protected] (M. Guo).

1226-3192/$ – see front matter © 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2012.04.008

38

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

shows that the hedge horizon might affect a firm’s usage of foreign exchange derivatives and lead to a term structure of optimal hedge ratios. Juhl, Kawaller, and Koch (2011) studies the effect of the hedge horizon on optimal hedge size and effectiveness when prices are cointegrated. The multi-step hedging in our study is corresponding to the hedging horizon longer than the unit time used in establishing models in these references. In practice, investors consider different hedging horizons for their own risk management (see for example Chen, Lee, and Shrestha (2004) and Cotter and Hanly (2009)). For example, traders may consider 1-day horizon while investors may consider 1-month or even 12 month horizon. In view of the importance of hedge horizon and the need for various horizons in practice, we are motivated to study multi-step hedging in this article and to search discrete hedging strategies which reduce risks especially unexpected risks as much as possible. In incomplete market models, since the equivalent martingale measure is not unique, various approaches were proposed to construct the martingale measures. Among others the minimal martingale law is a popular choice of risk-neutral measure for continuous time models (Colwell & Elliott, 1993; Duffie & Richardson, 1991; Föllmer & Schweizer, 1991). The minimal martingale law is related to the hedging approach designed to minimize the expected squared replication error, called the quadratic risk-minimizing (QR) strategy (Föllmer & Schweizer, 1991; Schäl, 1994; Schweizer, 1995). However, the minimal martingale law may not exist for discrete-time models. Elliott and Madan (1998) proposed an extended Girsanov principle to select an equivalent martingale measure to solve this problem. They also showed that the extended Girsanov principle is equivalent to the hedging strategy minimizing the variance of the discounted risk-adjusted hedging costs if the set of rebalancing times is equal to the time index set of the stochastic process. We call it the ‘‘single-step’’ quadratic risk-adjusted minimizing (QRA) hedging strategy throughout this paper. In this study, we consider the multi-step hedging case to enhance the application of the QRA-hedging in practice. The QRA-hedging can be viewed as a generalization of the QR-hedging by considering foreign exchange rates of differentcurrency assets, or risk premium. The economic consideration of the QRA-hedging is to minimize the squared adjusted risk of the additional capital (could be positive or negative) at the rebalancing time. Geometrically speaking, the QRA-hedging strategy finds the best approximation of the derivative value function with the shortest L2 distance with respect to the linear space spanned by the bond price function and the underlying security price function. The properties of the multi-step QRA hedging strategy are derived in Section 2 and a regression-based method is proposed to estimate the hedging positions. The initial QRA-hedging capital is proved to be invariant to the rebalancing period chosen and equals the no-arbitrage price by the extended Girsanov principle, which extends the result of Elliott and Madan (1998) to the multi-step case. In Section 3, we prove that the QRA-hedging is the perfect hedging in complete market models such as the Binomial model and the Black–Scholes model with continuous rebalancing. For discrete-time hedging in the Black–Scholes model, the hedging positions of the QRA and delta-hedging are different, although their hedging capitals are the same. Three risk measures are then employed to examine the performance of different hedging strategies in discrete-time hedging. The first one is the discounted risk-adjusted risk of the QRA-hedging. The other two are the Value-at-Risk (VaR) and the conditional Value-at-Risk (CVaR, or called the expected short-fall risk), which are commonly used in many financial papers (Chung, Shih, & Tsai, 2010; Cotter & Hanly, 2009). Simulation results show that the QRA-hedging provides better protection for hedgers than the delta-hedging in the Black–Scholes framework. The same conclusion holds for the incomplete GARCH models when the hedging horizon increases. Moreover, the additional cost of the QRA-hedging is less than the QR-hedging strategy when the investors make extra rebalancing to respond to the unexpected market events in most scenarios. This article is organized as follows. In Section 2, the QRA-hedging is introduced and its hedging positions are derived. In Section 3, the properties of the QRA-hedging in complete and incomplete models are studied. A dynamic programming procedure for establishing the QRA-hedging is proposed. In Section 4, simulation and empirical studies are performed to compare the QRA-hedging with the other hedging strategies. Conclusions are in Section 5. All the proofs are given in the Appendix. 2. A multi-step QRA-hedging strategy Consider a contingent claim involving m underlying assets with payoff function HT at maturity T . In order to hedge against this claim, we use a portfolio consisting of the riskless bond and the m securities. Let St = (S1,t , . . . , Sm,t ), denote the timet prices of the underlying securities and Bt denote the riskless bond price at time t which is assumed to be predictable, t = 0, 1, . . . , T . Both St and Bt are assumed to be positive stochastic processes. Suppose we have only a finite number of hedging times T = {0 = t0 < t1 < · · · < tR = T } ⊆ {0, 1, . . . , T }. At each rebalancing time tj ∈ T , the holding units of the bond and underlying assets are determined and remain unchanged in [tj , tj+1 ), then updated at time tj+1 . Let Vtj denote the value of the hedging portfolio at time tj ∈ T after rebalancing, that is Vtj = b0,j Btj + Stj bj ,

(1)

where b0,j and bj = (b1,j , . . . , bm,j ) are the holding units of Btj and Stj in the time interval [tj , tj+1 ), respectively, j = 0, . . . , R − 1, and set VT = HT at maturity. At time tj+1 , traders can relocate the hedging positions by the sale of the holding plus net additions of capital in the amount δj+1 = Vtj+1 − (b0,j Btj+1 + Stj+1 bj ). Denote the discounted cumulative cost by ′

Cj =

j  s =0

δ˜s ,

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

39

b

prices

a

t Fig. 1. (a) The discounted risk-adjusted asset value S˜t∗ = e−ut St /Bt . (b) St (the dashed line), S˜t (the solid line) and S˜t∗ (the dashed–dotted line), for t = 0, 10, 20, . . . , 60 (days), where the stock prices St , t = 1, . . . , 60, are generated from a Black–Scholes process with µ = 0.25, σ = 0.5, S0 = 40 and Bt = ert with riskless interest rate r = 0.05.

where

δ˜s = V˜ ts − (b0,s−1 + S˜ ts bs−1 ),

(2)

is the present value of the additional capital, S˜ ts = Sts /Bts , V˜ ts = Vts /Bts , j = 1, . . . , R, and set C0 = δ˜ 0 = V0 . In the literature, several risk-minimizing criteria based on the discounted cumulative hedging cost Cj were utilized to determine the hedging strategy, for instance, the total QR criterion (Duffie & Richardson, 1991; Schäl, 1994; Schweizer, 1995), the local QR criterion (Föllmer & Schweizer, 1991; Schäl, 1994), the local QRA criterion (Elliott & Madan, 1998), and the local piecewise linear risk minimization criterion (Coleman, Li, & Patron, 2003). If the trading strategy is selffinancing, then the additional discounted capital δ˜ j equals zero for j = 1, . . . , R − 1. Schweizer (1995) proved that there exists a self-financing strategy minimizing the total QR criterion, E(CT − C0 )2 , if the mean–variance tradeoff, [Etj (S˜tj+1 − S˜tj )]2 /Vartj (S˜tj+1 − S˜tj ), is P-a.s. uniformly bounded, where P denotes the dynamic (or physical) probability measure and the conditional expectations Etj are taken under P given the information generated by {Bt , St , t ≤ tj }. If δ˜ j is nonzero, then additional money needs to be placed into the hedging portfolio. The local QR strategy is the optimal hedging scheme minimizing the quadratic incremental discounted cost, Etj (δ˜ j2+1 ), at each rebalancing time tj ∈ T \ {T }. Yet the discounted costs δ˜ t as well as Ct considered in the QR criteria, by their definitions only adjusts values to the time effect not to the cross-sectional factors. It is a well known fact that assets differ cross-sectionally in their risk characteristics. To accommodate this Elliott and Madan (1998) considered the hedging criterion based on the risk adjusted cost (called abbreviately as QRA-hedging strategy). After the hedging position was set at time t, the discounted asset value S˜i,t is multiplied by the factor e−ui,t , where ui,t denotes the single-step risk premium of the ith asset such that Es−1 (S˜i,s ) = S˜i,s−1 eui,s for i = 1, . . . , m. The multiplicative rates are referred to as differences in asset risk quality or exchange rate between asset i and the reference asset (bond) if one thinks of different assets as different currencies. See Fig. 1(a) for illustration of the discounted risk adjusted asset value. Fig. 1(b) gives an example of random paths of St (the dashed line), S˜t (the solid line) and S˜t∗ (the dashed–dotted line), for

t = 0, 10, 20, . . . , 60 (days), that is, S˜t ’s are 10-step (or 10 days) discounted stock prices and S˜t∗ ’s are the corresponding discounted risk adjusted stock prices, where the daily stock prices St , t = 1, . . . , 60, are generated from a Black–Scholes process with µ = 0.25, σ = 0.5, S0 = 40 and Bt = ert with riskless interest rate r = 0.05. Elliott and Madan (1998) showed that the initial cost of the ‘‘single-step’’ QRA-hedging, that is tj − tj−1 = 1, is the same as the no-arbitrage price of the contingent claim derived by the extended Girsanov principle. Brief introduction of the extended Girsanov principle is given in the Appendix. In this study, we extend the QRA hedging strategy to the multi-step case, that is, tj − tj−1 > 1, and compare the QR and QRA-hedging strategies when investors make extra rebalancing to respond to unexpected market events. Define the multi-step discounted risk-adjusted hedging cost by

δ˜j∗ = V˜ t∗j − (b0,j−1 + S˜ ∗tj bj−1 ),

j = 1, . . . , R,

(3)

where −U1,tj −U ∗ S˜ ∗tj = S∗tj /Btj = (S1∗,tj , . . . , Sm , . . . , Sm,tj e m,tj )/Btj . ,tj )/Btj = (S1,tj e

Ui,tj =

tj

s=tj−1 +1

ui,s denotes the multi-step risk premium of the ith underlying asset in [tj−1 , tj ] and V˜ t∗j is the abbreviation

of V˜ tj (S∗tj ). A multi-step QRA-hedging is the trading strategy minimizing the multi-step conditional expected discounted

40

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

quadratic risk-adjusted hedging costs. The objective is to find the holding units bi,j ’s minimizing Etj (δ˜ j∗+1 )2 for the hedging portfolio Vtj defined in (1). In the following theorem, we derive the explicit representation of the QRA-hedging holding units with a given rebalancing time set T . Theorem 1. Consider the rebalancing time set T = {tj }Rj=0 and assume that S˜i∗,tj+1 |Ftj are nondegenerate random variables, ∗ where Ftj = σ {Bt , St , t ≤ tj }, for i = 1, . . . , m and j = 1, . . . , R − 1. Let Σ = (σuv )(m+1)×(m+1) with σuv = Etj (S˜u∗,tj+1 S˜v, tj+1 ), for u, v = 0, 1, 2, . . . , m and j = 1, . . . , R − 1, where S˜0∗,tj+1 ≡ 1. Assume that Σ −1 exists. Then we have the following. (i) The holding units bi,j , i = 0, . . . , m, at time tj ∈ T of the QRA-hedging strategy are solved recursively from j = R − 1, R − 2, . . . , 0 by setting VT = HT and the following equations,

(bˆ 0,j , . . . , bˆ m,j )′ = Σ −1 v, ∗ ˜∗ ′ where v = (Etj (V˜ t∗j+1 ), Etj (S˜1∗,tj+1 V˜ t∗j+1 ), . . . , Etj (S˜m ,tj+1 Vtj+1 )) . In particular, if there is only one asset (m = 1), then the

hedging positions are given by bˆ 1,j =

Covtj (V˜ t∗j+1 , S˜1∗,tj+1 ) Vartj (S˜1∗,tj+1 )

and

bˆ 0,j = V˜ tj − bˆ 1,j S˜1,tj ,

(4)

where Covtj and Var tj denote the conditional covariance and variance given Ftj under the dynamic measure P, respectively. Q (ii) V˜ tj = Etj (HT /BT ), where Q is the risk-neutral measure derived by the extended Girsanov change of measure process defined in (7).

Hedging position from a regression perspective The criterion of minimizing Etj (δ˜ j∗+1 )2 is equivalent to find the best approximation of V˜ t∗j+1 , which is a nonlinear function of S˜ ∗tj+1 , in the linear space spanned by S˜ ∗tj+1 with shortest L2 -distance defined by the conditional dynamic probability measure given Stj . Statistically speaking, this is corresponding to regress the discounted risk-adjusted hedging capital V˜ t∗j+1 on the security price variables S˜ ∗tj+1 . The regression line will pass through the mean point (Etj (S˜ ∗tj+1 ), Etj (V˜ t∗j+1 )) = (S˜ tj , V˜ tj ), where the equality is due to Lemma A.1 and Eq. (8) in Appendix. In addition, Theorem 1(ii) extends the result of Theorem 4.1 in Elliott and Madan (1998) to the multi-step hedging case, that is the multi-step discounted hedging capital equals the no arbitrage derivative price derived under the extended Girsanov principle and it is invariant to the length of the hedging period, h = tj+1 − tj . Moreover the optimal QRA holding units bˆ i,j ’s are adapted to the hedging length h to minimize the criterion Etj (δ˜ j∗+1 )2 . Advantages of the QRA-hedging

• The risk adjusted factor used in the QRA-hedging strategy is useful to handle the exchange rate problem for differentcurrency underlying assets or the risk premium for assets of different risk characteristics.

• In Theorem 1(ii), we prove that the discounted value of the hedging portfolio at time t, V˜ t , is exactly equal to the discounted no-arbitrage price computed from the risk-neutral measure Q . Therefore, the component V˜ t∗ in (4) can j+1

be obtained by computing Etj (HT /BT ) given St∗j directly. Instead of deriving the hedging positions iteratively backward from the maturity in the QR-hedging, this property helps us to save on the computational effort in deriving the hedging positions in the QRA-hedging. Q

In particular, we give the minimum value of the criterion Etj (δ˜ j∗+1 )2 when there is only one underlying asset in the following. Corollary 1. Consider the single underlying asset case, m = 1, we have min Etj (δ˜ j∗+1 )2 = Vartj (V˜ t∗j+1 )(1 − Corr2tj (V˜ t∗j+1 , S˜1∗,tj+1 )), where Corrtj (V˜ t∗j+1 , S˜1∗,tj+1 ) is the conditional correlation of V˜ t∗j+1 and S˜1∗,tj+1 given Ftj . For deep in-the-money (ITM) and deep out-of-money (OTM) European options, we have min Etj (δ˜ j∗+1 )2 ≈ 0. 3. The QRA hedging in complete and incomplete market models In a complete market, every contingent claim is marketable and the risk neutral probability measure is unique. Hence, traders can construct a self-financing trading strategy where the holding units of the replicating portfolio are uniquely determined and the additional capitals are zero at all rebalancing times, that is Etj (δ˜ j∗+1 )2 = 0, for all tj ∈ T . This trading strategy is called the ‘‘perfect hedging’’ and it reaches the minimal bounds of the criterion of the QRA strategy. Therefore, we expect that the QRA strategy coincides with the perfect hedging in the complete market models. In the following, we consider two popular complete market models such as the Binomial model and the Black–Scholes model in

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

41

Sections 3.1 and 3.2, respectively. On the other hand, if the market model is incomplete, then the hedging problem becomes complicated and usually the hedging positions of the QRA-hedging do not have closed-form representation. We therefore propose a dynamic programming to solve this problem and employ the GARCH model (Bollerslev, 1986; Engle, 1982) as an example for illustration in Section 3.3. 3.1. The Binomial market model In a Binomial model, assume that the underlying instrument will move up or down by a specific factor (u or d) per step, that is, St +1 = uSt or St +1 = dSt for t = 0, . . . , T − 1, where (u, d) satisfies 0 < d < er < u and r is the continuous compounded riskless interest rate. For a contingent claim with payoff HT at the expiration date T , the no-arbitrage price of Q the contingent claim can be evaluated by Et (e−r (T −t ) HT ) at time t, where Q is the unique risk-neutral probability measure Q er −d such that Et (I{St +1 =uSt } ) = u−d and I{·} is an indicator function, for t = 0, . . . , T − 1. If we assume the rebalancing time set to be T = {0, 1, . . . , T }, then the delta-hedging is a self-financing trading strategy consisting of the riskless bond and the underlying instrument and is capable of replicating the price process of the contingent claim. That is, the delta-hedging strategy is the perfect hedging and is derived recursively from time T to time 0 by VT = HT and Vt = ∆0,t + ∆1,t St , for t = 0, 1, . . . , T − 1, where

∆0,t = e−r

uVt +1 (dSt ) − dVt +1 (uSt )

and ∆1,t =

u−d

Vt +1 (uSt ) − Vt +1 (dSt )

(u − d)St

.

In the following proposition, we demonstrate that the QRA-hedging coincides with the perfect hedging as well by proving the equivalence of the QRA and delta-hedging in a Binomial model. Proposition 1. Suppose that the price of the underlying security, denoted by St , follows a Binomial model, t = 0, 1, . . . , T . If the rebalancing time set is assumed to be T = {0, 1, . . . , T }, then the QRA-hedging position of the underlying asset is bˆ 1,t = ∆1,t =

Vt +1 (uSt ) − Vt +1 (dSt )

(u − d)St

,

for all t ∈ T . 3.2. The Black–Scholes model Assume that the underlying asset process {St } satisfies the following Black–Scholes model, dSt = µSt dt + σ St dWt ,

(5)

where µ is the annualized constant drift rate of the underlying asset, σ is the volatility and Wt is the Wiener process. If continuous rebalancing is allowed, then the delta-hedging is a perfect hedging of a European option and the hedging position of the underlying asset for the European call option is Φ (d1 (St )) at time t ∈ [√ 0, T ], where Φ (·) is the standard normal cumulative distribution function and d1 (St ) = [ln(St /K ) + (r + 0.5σ 2 )(T − t )]/σ T − t. In the following theorem, we show that the QRA-hedging also coincides with the perfect hedging under Model (5). Theorem 2. Suppose that the price of the underlying security follows Model (5). If continuous rebalancing is allowed, then the QRA-hedging position of the underlying asset for the European call option satisfies lim bˆ 1,t (dt ) = Φ (d1 (St )),

dt →0

where dt denotes the length of the time period [t , t + dt ]. If continuous rebalancing is not available, then neither the delta-hedging nor the QRA-hedging is perfect. In Section 4, simulation and empirical studies are performed to investigate the hedging performances of these two hedging strategies in discrete rebalancing. 3.3. The GARCH model Assume the log return process {Rt = ln(St /St −1 )} under the dynamic measure P satisfies the following:



Rt = µt − γt + σt εt , εt ∼ i.i.d. D(0, 1) σt2 = fθ (σk , εk , k ≤ t − 1),

(6)

where µt is the Ft −1 -measurable continuously compounded risk interest rate of the asset in the time period [t − 1, t ), σt2 is the conditional variance of the log return in [t − 1, t ), the mean correction factor γt = ln Et −1 (eσt εt ) is included to ensure Et −1 (St ) = St −1 eµt , D(0, 1) denotes the distribution of εt which has zero mean and unit variance, and the function fθ (·)

42

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

governs the volatility dynamic where θ is a vector parameter. Model (6) includes the cases considered in Duan (1995) and Duan and Simonato (2001) when µt = r + λσt , and in Heston and Nandi (2000) when µt = r + λσt2 + 12 σt2 , and εt is Gaussian distributed, where λ denotes the constant unit risk premium. We first consider the Gaussian innovation case in Model (6). Suppose that the innovation εt is N (0, 1) distributed, µt = r + λσt and fθ (σk , εk , k ≤ t − 1) = α0 + α1 σt2−1 εt2−1 + α2 σt2−1 , then γt = 21 σt2 . The risk-neutral model derived by the extended Girsanov change of measure process (7) is



Rt = r −

1 2

σt2 + σt ξt ,

σ = α0 + α σ 2 t

2 1 t −1

ξt ∼ i.i.d. N (0, 1)

(ξt −1 − λ)2 + β1 σt2−1 ,

which is the same as the result in Duan (1995). For risk-neutral GARCH models with heavy-tailed innovations εt , we refer to Huang and Guo (2011). In general, the hedging positions of the QRA-hedging have no closed-form representation in GARCH models. Based on the viewpoint of regression discussed in Section 2, we propose the following dynamic programming to compute bˆ i,j ’s and use the case of the hedging short position of a European option underlying single asset (m = 1) for illustration. Dynamic programming of the QRA-hedging 1. For a given stock price Stj at time tj , generate n stock prices {Stj+1 ,ℓ }nℓ=1 at time tj+1 conditional on Stj from the dynamic model. 2. Compute the corresponding European option prices, Vtj+1 (Stj+1 ,ℓ ), for each Stj+1 ,ℓ by either the dynamic semi-parametric approach (DSA) (Huang & Guo, 2009) or the empirical martingale simulation (EMS) method (Duan & Simonato, 1998). If j = R − 1, then VT (ST ,ℓ ) = HT (ST ,ℓ ). 3. By (4), the hedging positions bˆ i,j , i = 0, 1, are approximated by n 

ˆ bˆ 1,j =

ℓ=1

V˜ t∗j+1 ,ℓ S˜t∗j+1 ,ℓ − n 

1 n

(S˜t∗j+1 ,ℓ )2 −

ℓ=1

n 

ℓ=1 1 n



V˜ t∗j+1 ,ℓ

n 

ℓ=1

n 

ℓ=1

S˜t∗j+1 ,ℓ

S˜t∗j+1 ,ℓ

2

ˆ and bˆ 0,j = V˜ tj − bˆ 1,j S˜tj , where V˜ t∗j+1 ,ℓ is short for V˜ tj+1 (S˜t∗j+1 ,ℓ ). In Section 4, the comparison between the hedging performances of the QRA hedging and other hedging strategies in GARCH models are investigated via simulation studies. 4. Simulation and empirical studies In this section, several simulation and empirical studies are performed to investigate the hedging performances of the QRA-, QR- and delta-hedging strategies. Three risk measurements, the quadratic discounted risk-adjusted hedging error (denoted by MQR ), the Value-at-Risk (VaR) and the conditional VaR (CVaR), are employed to compare the hedging performance in the simulation studies presented in Sections 4.1, 4.2 and 4.4. Section 4.3 demonstrates the empirical hedging performance of the QRA-hedging and the delta-hedging by computing the differences between the two hedging portfolios. 4.1. QRA- vs. delta-hedging in the Black–Scholes model in static hedging Suppose that a trader decides to set up a static hedging strategy at time 0 and holds the positions till the expiration date T to hedge her short position of a European call option with strike price K and payoff HT = VT . Denote the time-0 deltaQ hedging portfolio by C0 = ∆0,0 + ∆1,0 S0 , where ∆1,0 = Φ (d1 (S0 )), C0 = E0 (V˜ T ) is the no-arbitrage price of the call option. Further let V0 = b0,0 + b1,0 S0 denote the hedging portfolio of the QRA trading strategy at time 0. By Theorem 1(ii), the initial Q hedging capitals of QRA satisfy V0 = E0 (V˜ T ) and hence delta-hedging and the QRA-hedging have the same initial hedging capital, that is, V0 = C0 . However, the hedging positions of the QRA and delta-hedging strategies are in general different. For comparing the hedging performances of the QRA and delta-hedging strategies, three risk measures are employed to evaluate the loss distributions. Let η˜ T∗ = V˜ T∗ − (∆0,0 + ∆1,0 S˜T∗ ) and δ˜ T∗ defined in (3) be the discounted risk-adjusted loss functions of the delta- and QRA-hedging, respectively. The first risk measure is MQR . For example, the MQR of delta-hedging is computed by E(η˜ T∗ )2 . In addition, denote the ratio of E(η˜ T∗ )2 and E(δ˜ T∗ )2 by R(QR). The second one is a widely used risk measure in practice, the 100 α % VaR, defined by VaR = inf{z ∈ R; P (HE ≥ z ) ≤ α}, where HE = VT − FT is the hedging error with VT being the value of hedging target and FT being the value of a hedging portfolio at time T . The third risk measure is the CVaR (or called the expected shortfall), which quantifies the mean loss beyond the VaR, that is, CVaR = E(HE | HE ≥ VaR).

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

43

a

b

Fig. 2. (a) The values of R(QR) of the delta-hedging and the QRA-hedging with T = 5, 10, 30 and K /S0 = 0.5, 0.6, . . . , 1.4. (b) The values of R(QR), R(VaR) and R(CVaR) of the delta-hedging and the QRA-hedging with T = 5, 10, 30 and K /S0 = 0.90, 0.95, . . . , 1.10 in the Black–Scholes model.

In practice, α is set to be 0.05 or 0.01. In the following, we perform the results of the ratios of VaRs and CVaRs with α = 0.01 of delta-hedging over QRA-hedging, and denote the ratios by R(VaR) and R(CVaR), respectively. Fig. 2(a) plots R(QR) versus K /S0 with S0 = 40, µ = r = 0.05, σ = 0.5, K /S0 = 0.5, 0.6, . . . , 1.4 and T = 5, 10, 30 (days). It is not surprising to see that all the ratios are greater than 1 since the QRA-hedging is designed to attend the minimal QRA error. However, the magnitude of the ratios shows that the maximum ratios occur in the ITM zone (that is, K /S0 < 1 in Fig. 2)(a), which is the case when the seller demands more to hedge her short position because the option has higher chance to be excised at the expiration date. Furthermore, most of the ratios are greater than 1.05 when T ≥ 10, which means that the QRA-hedging reduces at least 5% hedging risks of its delta counterpart in most cases. In practice, the strike prices K are between 0.9S0 and 1.1S0 for most of the liquid European options when T ≤ 30 (days) (Duan & Simonato, 1998; Huang & Guo, 2009, 2011; Yung & Zhang, 2003). Fig. 2(b) plots R(QR), R(VaR) and R(CVaR), respectively, with T = 5, 10, 30 and K /S0 = 0.90, 0.95, . . . , 1.10. All the ratios are greater than 1 no matter which risk measure is used. Therefore, the QRA-hedging is capable of providing the option seller significant risk reduction and achieves better economic benefits than the delta-hedging in the Black–Scholes framework. 4.2. QRA- vs. delta-hedging in GARCH models Suppose that the log returns follow Model (6) with µt = r + λσt and volatility equation σt2 = α0 + α1 σt2−1 εt2−1 + β1 σt2−1 , −5 where the parameters √ are set to be the same as in Duan (1995), that is, λ = 0.007452, α0 = 1.524 × 10 , α1 = 0.1883, α2 = 0.7162, σd = α0 /(1 − α1 − β1 ) = 0.01263 (per day, i.e., 0.2413 per annum), r = 0, and the innovations εt ’s are assumed to be either N (0, 1) or double exponential distribution (dexp) with zero mean and unit variance. Consider hedging short position of a European option at time t = T − h. The hedging positions of the QRA-hedging strategy are derived by the dynamic programming scheme and the time-t delta-hedging position is computed by

∂ Ct = e−r (T −t ) EQt ∂ St



ST St

 I{ST ≥K }

,

which can be computed by the EMS method under the risk-neutral model, where Ct denotes the European call option prices at time t (see Corollary 2.4 in Duan (1995)). Fig. 3 plots R(QR), R(VaR) and R(CVaR) (based on n = 5 × 104 sample paths) of the delta-hedging and the QRA-hedging versus T − h, h = 5, 10, 30 days for strike prices K = 36, 40, 40. The initial

44

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

a

c

b

Fig. 3. The values of (a) R(QR), (b) R(VaR) and (c) R(CVaR) of the delta-hedging and the QRA-hedging in the GARCH-normal (G-N) and GARCH-dexp (G-D) models.

Table 1 The S&P500 index call options traded in 4 different categories of moneyness and for 3 expiration dates during the period from March, 2003 to September, 2008 after the filtration procedure in Section 4.3. T (days)

5 10 30

Moneyness (K /S0 ) 0.9∼0.97

0.97∼1

1∼1.03

1.03∼1.1

76 100 64

392 375 187

220 488 392

134 419 473

stock price is set to be 40 for all cases in the GARCH-normal and GARCH-dexp models. The phenomenon shown in Fig. 3(a) is similar to Fig. 2(a) and (b) in the Black–Scholes model. However, the VaRs of the QRA-hedging are less than those of the delta-hedging when the time to maturity is 30 days. The CVaRs of the QRA-hedging performs better than delta-hedging when the time to maturity is 10 or 30 days. According to these results, the QRA-hedging performs better than the deltahedging when the hedging horizon increases in the GARCH framework in terms of the MQR , VaR and CVaR risk measures. In the case of short term hedging horizon, delta-hedging might have less VaR and CVaR than the QRA-hedging if the option is not near-the-money. 4.3. Empirical studies In this empirical example, we compare the hedging performances of the QRA- and delta-hedging for S&P500 index call options during the period from March, 2003 to September, 2008. The data set contains the Trading Date, Expiration Date, Spot Price, Strike Price, Best Bid and Offer Prices, Trading Volume, Open Interest, BS Implied Volatility and other Greeks. The QRA- and delta-hedging are set up 5, 10, 30 trading days before each expiration date. At each trading date, we classify the call options into 4 categories with the moneyness (K /S0 ) in 0.9∼0.97, 0.97∼1, 1∼1.03 and 1.03∼1.1 (cf. Li, Favero, & Ortu, 2010; Yung & Zhang, 2003). The riskless interest rate is set up to be the Treasury Bills rate with 4-week maturity for each option. In order to ensure that options are liquid enough, we rule out options with trading volume less than 100. Furthermore, we compute the option prices implied by the BS implied volatility and exclude those options if the implied option price less than 0.9 times the best bid price or greater than 1.1 times the best offer price. Table 1 illustrates the options traded in 4 different

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

45

Table 2 The means and the corresponding standard deviations (in the parentheses) of the values of the QRA-hedging portfolio minus the values of the delta-hedging portfolio at expiration dates. The boldface denotes the case of the mean value greater than zero significantly. T (days)

5 10 30

Moneyness (K /S0 ) 0.9∼0.97

0.97∼1

1∼1.03

0.0255 (0.0165) 0.0066 (0.0189) 0.1243 (0.0537)

0.0259 (0.0096) 0.0002 (0.0120) 0.0411 (0.0061)

0.0186 (0.0123) 0.0051 (0.0263) 0.0779 (0.0371)

1.03∼1.1 0.0108 (0.0091) −0.0028 (0.0532) 0.0281 (0.1367)

Table 3 The criteria and properties of the QR- and QRA-hedging strategies.

Criterion Risk-neutral measure Hedging capitals a b

QR-hedging

QRA-hedging

min Etj (δ˜ t2j+1 )

min Etj (δ˜ t∗j+1 )2

Minimal martingale lawa Depend on T b

Extended Girsanov principle Independent of T

The minimal martingale measure may not exist in discrete time models. T is the set of the rebalancing time points.

categories of moneyness and for 3 expiration dates. Because we only observe one realization of the S&P500 indices, the MQR , VaR and CVaR are not proper to use here. Instead, we compute the difference of the QRA- and delta-hedging portfolios for each traded option. Table 2 gives the means and the corresponding standard deviations of the values of the QRA-hedging portfolio minus the values of the delta-hedging portfolio at expiration dates. Note that the means are greater than zero in most cases and are significantly positive in the first 3 categories when the hedging period is 30 days. As a result, the QRA-hedging provides option sellers more protection than the delta-hedging, especially when investor decides to hold her static hedging portfolio for longer time horizon. 4.4. QRA- vs. QR-hedging strategies The last simulation study is to investigate the risk-adjusted effect on the hedging performance. In general, for any prechosen rebalancing time intervals, the hedging capitals and hedging positions of the QRA and QR strategies are not equal to each other since their optimal criteria are different. The only exception is when the risk premium is zero since the criteria of these two trading strategies are the same if µ = r. Table 3 summarizes the criteria and properties of the QRA and QRhedging strategies. As mentioned above, the QRA and QR-hedging portfolios are set up iteratively from the expiration date to the initial time for any pre-chosen rebalancing time intervals based on the corresponding optimal criterion. However, in practice the option seller may make additional rebalancing to reduce her risk caused by unexpected market events. The conduct of this extra rebalancing enforces the trader to reconstruct the optimal hedging portfolio during the remaining hedging period and leads to additional hedging costs. In the following, we use a simulation study to investigate the impact of the extra rebalancing on the QRA and QR-hedging strategies by comparing their additional quadratic hedging costs. Consider a European option with payoff HT with prechosen rebalancing time set T = {0, T }. Denote the initial hedging portfolio for the European option by V0T = a0T + b0T S0 , where a0T and b0T are the hedging positions of the QRA (or QR) strategy, V0T is the corresponding initial hedging capital, and the subscripts of a0T , b0T and V0T are used to highlight that the hedging portfolio begins at time 0 and ends at time T . If an extra rebalancing occurs at time t during the prechosen time interval [0, T ], then the optimal (QR/QRA) time-t trading strategy, denoted by VtT = atT + btT St , will be reconstructed for the remaining hedging period [t , T ]. Generate n stock price paths from the dynamic model, and compute the values of R(QR), R(VaR) and R(CVaR) of the QR- and QRA-hedging at time t. Fig. 4 performs the values of R(QR), R(VaR) and R(CVaR) versus the time t = T /4, T /2, 3T /4 for µ = 0.10 and 0.15 under a Black–Scholes framework. The parameters in the Black–Scholes model are S0 = 40, r = 0.05, σ = 0.5, T = 30, 60 (days) and K = 36, 40, 44. We summarized the findings in the following. 1. All R(QR) and most of R(VaR) and R(CVaR) are greater than 1. Thus the QRA-hedging pays less additional hedging costs than the QR-hedging for the reconstruction in most cases, especially when the option is in-the-money (K = 36). 2. The ratio decreases as t increases and increases as T increases. This implies that the QR-hedging pays (in percentage) more for earlier extra rebalancing and longer initial hedging period. 3. The ratios increase in µ for fixed K , T and the extra rebalancing time t. Consequently, difference between QR and QRA becomes significant when the risk premium is large.

46

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

Fig. 4. The values of R(QR), R(VaR) and R(CVaR) of the QR-hedging over the QRA-hedging in the Black–Scholes model with µ = 0.10, 0.15, K = 36, 40, 44, T = 30, 60 and the extra rebalancing occurring at t = T /4, T /2, 3T /4, where the cases of the case of (K , T ) = (36, 30), (40, 30), (44, 30), (36, 60), (40, 60) and (44, 60) are denoted by •, ×, +, , ♦ and ⃝, respectively.

5. Conclusion In this study, we introduce the multi-step QRA-hedging strategy based on minimizing the multi-step conditional expected quadratic risk-adjusted costs. The QRA-hedging is capable of making adjustment to the rebalancing period and its hedging cost is shown to be invariant to the hedging time period. Also, the hedging capital of the QRA strategy is proved to be equal to the no-arbitrage price derived by the extended Girsanov principle. In complete market models, the QRAhedging is the perfect hedging strategy. In incomplete market models, a dynamic programming of the QRA-hedging is proposed for practical implementation. To examine the hedging performance in discrete-time hedging, three risk measures, MQR , VaR and CVaR, are employed. Simulation results show that the QRA-hedging achieves better economic benefits than the delta-hedging in the Black–Scholes framework. The same conclusion holds for the incomplete GARCH models when the hedging horizon increases. In addition, when the investors make extra rebalancing to respond to unexpected market events, the QRA strategy pays less additional hedging costs than the QR-hedging in most cases. In the future, we will extend the QRA-hedging strategy to American and exotic options and study the proper selection of the rebalancing period. Acknowledgments The authors acknowledge helpful comments by the AE and two anonymous reviewers. The research of the first author was supported by the grant NSC 100-2118-M-390-001 from the National Science Council of Taiwan. The research of the second author was supported by the grant NSC 100-2118-M-110-003 from the National Science Council of Taiwan. Appendix A.1. The extended Girsanov principle In the following, we introduce the extended Girsanov principle briefly. The extended Girsanov principle is a discretized version of the Girsanov change of measure. Consider the multiplicative decomposition of the ith discounted asset price process S˜i,t = S˜i,0 Ai,t Mi,t , where Ai,t ≡

t  k=1

Ek−1 (S˜i,k /S˜i,k−1 )

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

47

is a predictable process and Mi,t = S˜i,t /(S˜i,0 Ai,t ) is a positive martingale, i = 1, . . . , m, under the dynamic measure P. Define the change of measure density process {Λt , t = 0, 1, . . . , T } by

Λt =

t  φk (S˜1,k /S˜1,k−1 , . . . , S˜m,k /S˜m,k−1 )eu1,k +···+um,k k =1

φk (e−u1,k S˜1,k /S˜1,k−1 , . . . , e−um,k S˜m,k /S˜m,k−1 )

,

(7)

which is also a P-martingale, where ui,k = ln Ek−1 (S˜i,k /S˜i,k−1 ) denotes the risk-premium of the ith underlying asset for M1,k

M

, . . . , Mmm,k,−k 1 ) given Fk−1 under P, where Fk−1 is the information set generated by the riskless bonds and the underlying assets before time k − 1. The extended Girsanov equivalent martingale measure Q is defined by dQ = ΛT dP, and the positive process Λt is the i = 1, . . . , m, Ek−1 (·) is the conditional expectation and φk (·) is the conditional density of ( M

1,k−1

Radon–Nikodym derivative of Q with respect to P restricted to Ft −1 . Furthermore, Elliott and Madan (1998) showed the law of S˜i,t /S˜i,t −1 given Ft −1 under measure Q is equal to the law of Mi,t /Mi,t −1 given Ft −1 under measure P for i = 1, . . . , m. In order to prove Theorem 1, we give the following Lemma first. Lemma A.1. For any given t ∈ {0, 1, . . . , T } and for all h ≥ 1, we have Et (S˜i∗,t +h ) = S˜i,t , where i = 1, . . . , m. Proof. By the tower property, we have Et (S˜i,t +h e−Ui,t +h ) = Et · · · Et +h−2 Et +h−1 (S˜i,t +h e−Ui,t +h )

= Et · · · Et +h−2 (S˜i,t +h−1 e−Ui,t +h−1 ) = Et (S˜i,t +1 e−ui,t +1 ) = S˜i,t , for i = 1, . . . , m, where the second equality is due to the definition of ui,s = ln Es−1 (S˜i,s /S˜i,s−1 ) and ui,s is Fs−1 -measurable, s = t + 1, . . . , t + h.  Proof of Theorem 1. (i) Let f (b0,j , . . . , bm,j ) = Etj [V˜ t∗j+1 − (b0,j + S˜ ∗tj+1 bj )]2 . By solving the normal equations ∂ b∂ f (b0,j , . . . , bm,j ) = 0 for i,j i = 0, . . . , m, we have the desired formula of (bˆ 0,j , . . . , bˆ m,j ).

∗ (ii) Notice that the normal equation ∂ b∂ f (b0,j , . . . , bm,j ) = 0 is equivalent to bˆ 0,j = Etj [V˜ t∗j+1 −(bˆ 1,j S˜1∗,tj+1 +· · ·+ bˆ m,j S˜m ,tj+1 )]. 0,j

By substituting bˆ i,j ’s into (1), we obtain V˜ tj = Etj (V˜ t∗j+1 ),

(8)

for tj ∈ T . In the following, we show that V˜ tj = Etj (HT /BT ) recursively from j = R − 1 to 0. At time tR−1 , for a fixed StR−1 we have Q

V˜ tR−1 (StR−1 ) = EtR−1 [V˜ T (S∗T )] = EtR−1 · · · ET −2 ET −1 [V˜ T (S∗T )]

= EQtR−1 · · · EQT−2 EQT−1 [V˜ T (ST )] = EQtR−1 (HT /BT ),

(9)

where the first equality holds by (8), the second and fourth equalities hold by the tower property, and the third equality is due to the results of Elliott and Madan (1998) for the single-step case since Si∗,tR−1 +1 /Si,tR−1 = e

−ui,tR−1 +1

Si,tR−1 +1 /Si,tR−1 +1

and Si∗,k+1 /Si∗,k = e−ui,k+1 Si,k+1 /Si,k , for k = tR−1 , . . . , T − 1 and i = 1, . . . , m. At time tR−2 , for a given StR−2 we have V˜ tR−2 (StR−2 ) = EtR−2 [V˜ (S∗tR−1 )] = EtR−2 [V˜ (StR−1 )] Q

= EQtR−2 [EQtR−1 (HT /BT )] = EQtR−2 (HT /BT ), where the first equality is due to (8), the second equality holds by similar arguments of the third equality in (9), and the third equality follows by the result of (9). Finally, the desired result can be obtained by backward induction.  Proof of Corollary 1. By (3), (4), Lemma A.1, (8), we have min Etj (δ˜ j∗+1 )2 = Vartj (V˜ t∗j+1 ) − 2bˆ 1,j Covtj (V˜ t∗j+1 , S˜1∗,tj+1 ) + bˆ 21,j Vartj (S˜1∗,tj+1 )

= Vartj (V˜ t∗j+1 ) − Cov2tj (V˜ t∗j+1 , S˜1∗,tj+1 )/Vartj (S˜1∗,tj+1 ) = Vartj (V˜ t∗j+1 )(1 − Corr2tj (V˜ t∗j+1 , S˜1∗,tj+1 )).

48

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

Consider the deep ITM European call options when Stj+1 ≫ K , since the option price Vtj+1 ≈ Stj+1 − Ke−r (T −t ) , thus V˜ t∗j+1 ≈ S˜t∗j+1 − Ke−rT and Corrtj (V˜ t∗j+1 , S˜t∗j+1 ) ≈ 1. As for the deep OTM cases, we have Vartj (V˜ t∗j+1 ) ≈ 0 and Corrtj (V˜ t∗j+1 , S˜t∗j+1 ) ≈ 0 since V˜ tj+1 ≈ 0 when Stj+1 ≪ K . Therefore, the minimal average squared hedging costs are almost zero for both deep ITM and OTM options.  Proof of Proposition 1. Since the delta-hedging is a self-financing trading strategy and is capable of replicating the price process of the contingent claim, thus it is a perfect hedging and our objective is to prove that the hedging positions of the QRA strategy are equal to those of the delta-hedging at each rebalancing time. According to Theorem 1, the holding unit of the underlying asset in the QRA-hedging at time t can be derived backward from time t + 1 by bˆ 1,t =

Covt (V˜ t∗+1 , S˜t∗+1 ) Vart (S˜t∗+1 )

Covt (V˜ t +1 , S˜t +1 ) Q

=

Vart (S˜t +1 ) Q

Covt (Vt +1 , St +1 ) Q

=

Vart (St +1 ) Q

, Q

Q

where the second equality can be obtained by the extended Girsanov change of measure process (7), Covt and Vart are the conditional covariance and variance given Ft under the probability measure Q , respectively, and Vt denotes the time-t Q hedging capital of the QRA-hedging defined in (1). By Theorem 1(ii), we know that Vt = Et (e−r (T −t ) HT ) is the no-arbitrage r Q Q price of the derivative at time t. Further by utilizing the equation of Et (I{St +1 =uSt } ) = eu−−dd = 1 − Et (I{St +1 =dSt } ), we have bˆ 1,t =

Vt +1 (uSt ) − Vt +1 (dSt )

(u − d)St

and bˆ 0,t ert = Vt − bˆ 1,t St = ∆0,t ,

which is the same as the hedging positions of the delta-hedging at time t.



Proof of Theorem 2. By Theorem 1, the time-t hedging position of the underlying asset of the QRA-hedging for a European call option with strike price K and maturity date T is bˆ 1,t (dt ) =

Covt (V˜ t∗+dt , S˜t∗+dt ) Vart (S˜t∗+dt )

Covt (V˜ t +dt , S˜t +dt ) Q

=

Q Vart (S˜t +dt )

,

where the second equality can be obtained by the extended Girsanov change of measure process (7), and V˜ t +dt Q Et +dt (e−rT HT ) is the discounted no-arbitrage price of the derivative at time t + dt. Note that

=

Covt (V˜ t +dt , S˜t +dt ) = Et (V˜ t +dt S˜t +dt ) − V˜ t S˜t Q

Q

= EQt {[S˜t +dt Φ (d1 (S˜t +dt )) − K˜ Φ (d2 (S˜t +dt ))]S˜t +dt } − V˜ t S˜t ≈ EQt (S˜t2+dt )Φ (d1 (St )) − S˜t K˜ Φ (d2 (St )) − [S˜t Φ (d1 (St )) − K˜ Φ (d2 (St ))]S˜t = Φ (d1 (St ))VarQt (S˜t +dt ), √ Q where d2 (St ) = d1 (St ) − σ T − t and the approximation (≈) is due to the property that Et [S˜tk+dt Φ (di (S˜t +dt ))] ≈ Q Et (S˜tk+dt )Φ (di (S˜t )) for small dt, i = 1, 2 and k = 1, 2. Therefore, as dt → 0, we have Covt (V˜ t +1 , S˜t +1 ) Q

bˆ 1,t (dt ) =

Vart (S˜t +1 ) Q

→ Φ (d1 (St )),

which indicates that if investors are allowed to rebalance the hedging portfolio continuously, then the QRA-hedging coincides with the delta-hedging.  References Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy, 81, 637–654. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Economet., 31, 307–327. Chen, S. S., Lee, C. F., & Shrestha, K. (2004). An empirical analysis of the relationship between the hedge ratio and hedging horizon: a simultaneous estimation of the short- and long-run hedge ratios. J. Futures Markets, 24, 359–386. Chung, S. L., Shih, P. T., & Tsai, W. C. (2010). A modified static hedging method for continuous barrier options. J. Futures Markets, 30, 1150–1166. Coleman, T. F., Li, Y., & Patron, M. C. (2003). Discrete hedging under piecewise linear risk minimization. J. Risk, 5, 39–65. Colwell, D. B., & Elliott, R. J. (1993). Discontinuous asset prices and non-attainable contingent claims. Math. Finance, 3, 295–308. Cotter, J., & Hanly, J. (2009). Hedging: scaling and the investor horizon. J. Risk, 12, 49–77. Duan, J. C. (1995). The GARCH option pricing model. Math. Finance, 5, 13–32. Duan, J. C., & Simonato, J. G. (1998). Empirical martingale simulation for asset prices. Manage. Sci., 44, 1218–1233. Duan, J. C., & Simonato, J. G. (2001). American option pricing under GARCH by a Markov chain approximation. J. Econ. Dyn. Control, 25, 1689–1718. Duffie, D., & Richardson, H. R. (1991). Mean-variance hedging in continuous time. Ann. Appl. Prob., 1, 1–15. Elliott, R. J., & Madan, D. B. (1998). A discrete time equivalent martingale measure. Math. Finance, 8, 127–152. Engle, R. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica, 50, 987–1007. Föllmer, H., & Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In M. H. A. Davis, & R. J. Elliott (Eds.), Applied stochastic analysis. New York: Gordon and Breach.

S.-F. Huang, M. Guo / Journal of the Korean Statistical Society 42 (2013) 37–49

49

Heston, S. L., & Nandi, S. (2000). A closed-form GARCH option valuation model. Rev. Financial Stud., 13, 585–625. Huang, S. F., & Guo, M. H. (2009). Financial derivative valuation — a dynamic semiparametric approach. Stat. Sin., 19, 1037–1054. Huang, S. F., & Guo, M. H. (2011). Model risk of implied conditional heteroscedastic models with Gaussian innovation in option pricing and hedging. Quantitative Finance, http://dx.doi.org/10.1080/14697688.2011.630323. Juhl, T., Kawaller, I. G., & Koch, P. D. (2011). The effect of the hedge horizon on optimal hedge size and effectiveness when prices are cointegrated. J. Futures Markets., http://dx.doi.org/10.1002/fut.20544. Korn, O., & Koziol, P. (2009). The term structure of currency hedge ratios. CFR-Working paper No. 09-01. Li, J., Favero, C., & Ortu, F. (2010). A spectral estimation of tempered stable stochastic volatility models and option pricing. Comput. Stat. Data Anal., http://dx.doi.org/10.1016/j.csda.2010.11.013. Schäl, M. (1994). On quadratic cost criteria for option hedging. Math. Oper. Res., 19, 121–131. Schweizer, M. (1995). Variance-optimal hedging in discrete time. Math. Oper. Res., 20, 1–32. Yung, H. M., & Zhang, H. (2003). An empirical investigation of the GARCH option pricing model: hedging performance. J. Futures Markets, 23, 1191–1207.