Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 407145, 10 pages http://dx.doi.org/10.1155/2014/407145

Research Article Turbo Warrants under Hybrid Stochastic and Local

Min-Ku Lee,1 Ji-Hun Yoon,2 Jeong-Hoon Kim,3 and Sun-Hwa Cho3

1 Department of Mathematics, Sungkyunkwan University, Suwon-si, Gyeonggi-do 440-746, Republic of Korea 2 Department of Mathematical Science, Seoul National University, Seoul 151-747, Republic of Korea 3 Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea

Correspondence should be addressed to Jeong-Hoon Kim; [email protected]

Received 28 September 2013; Revised 12 December 2013; Accepted 14 December 2013; Published 8 January 2014

Academic Editor: Sergio Polidoro

Copyright © 2014 Min-Ku Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the pricing of turbo warrants under a hybrid stochastic and local volatility model. The model consists of the constant elasticity of variance model incorporated by a fast fluctuating Ornstein-Uhlenbeck process for stochastic volatility. The sensitive structure of the turbo price is revealed by asymptotic analysis and numerical computation based onthe observation that the elasticity of variance controls leverage effects and plays an important role in characterizing various phases of volatile markets.

1. Introduction security price. It has been proposed by Cox [3]andCox and Ross [4]. Another version of volatility models has been Turbo warrants, which appeared first in Germany in late 2001, developed by assuming security’s volatility to be a random have experienced a considerable growth in Northern Europe process governed by another state variables such as the and Hong Kong. They are special types of knockout barrier tendency of volatility to revert to some long-run mean value, optionsinwhichtherebateiscalculatedasanotherexotic the variance of the volatility process itself, and so forth. . For one thing, this contract has a low vega so that Thistypeofmodelsiscalled(pure)stochasticvolatility theoptionpriceislesssensitivetothechangeoftheimplied models. The models developed by Heston [5] and Fouque volatility of the security market, and for another, it is highly et al. [6] are popular ones among others in this category. geared owing to the possibility of knockout. Closed form expression for the price has been presented by Eriksson [1] Also, there is another type of generalized models for the under geometric Brownian motion (GBM) for the underlying underlying security which have been developed in terms of security. Levy processes which may have discontinuous paths. The It is well-known that the assumption of the GBM for the model by Carr et al. [7] is a well-known one of this kind underlying security price in the Black-Scholes model [2]does among others. not capture many empirical lines of evidence appeared in Evenifthosemodelshavebeendevelopedtoovercome financial markets. Maybe, the two most significant shortcom- the major shortcomings of the Black-Scholes model, there ings of the model’s assumption lie in flat , still remain limitations. For instance, volatility and underly- whereas the volatility fluctuates depending upon market con- ing risky asset price changes are perfectly correlated either ditions and the underestimation of extreme moves, yielding positively or negatively in the CEV model, whereas empirical tail risk. So, there have been a lot of alternative underlying studies indicate no definite correlation all the time. Also, the models developed to extend the GBM and overcome these dynamics of the implied volatility surface created by the CEV problems. Local volatility models are one type of them, where model may have an opposite direction to the observed market the volatility depends on the price level of the underlying dynamics as shown by Hagan et al. [8]. Stochastic volatility security itself. The most well-known local volatility model models usually do not take into account the relationship is the constant elasticity of variance (CEV) model in which between volatility and underlying price (the leverage or the volatility is given by a power function of the underlying inverse leverage effects) directly. Much of computing time 2 Abstract and Applied Analysis

+ tends to be required for the implementation of Levy type of time 𝑡 is given by 𝑆𝑡,paystheoptionpossessor(𝑆𝑇 −𝐾) at models. So, recently, one has begun to see a trend of develop- maturity 𝑇 on the understanding that a specified barrier 𝐻≥ ing some sort of hybrid formulation in asset pricing. A hybrid 𝐾 has not been passed by 𝑆𝑡 atanytimepriortothematurity. stochastic and local volatility model created by Choi et al. [9] Let us define 𝜏𝐻 as the first time that the asset price hits the is among those alternative hybrid models for pricing options. barrier 𝐻,thatis,𝜏𝐻 = inf{𝑡 |𝑡 𝑆 ≤𝐻}.Thecontractisuseless Thisnewhybridmodelisequippedwiththestochasticvolatil- andanewcontractbeginsif𝜏𝐻 ≤𝑇.Thenewcontractisa ityandCEVtermsinthevolatilitypart.Fromthehybrid whose payoff is given by the difference between 𝑇 nature of the volatility, one can remove the hedging instability 𝑚 0 := 𝑆 𝐾 𝜏𝐻 min𝜏𝐻≤𝑡≤𝜏𝐻+𝑇0 𝑡 and the with time to created by the CEV model as well as capture the leverage effect maturity 𝑇0. Thus the turbo call contract is given by fit to the corresponding market. To date, there have been a number of applications of this hybrid model to the pricing −𝑟(𝑇−𝑡) + TC (𝑡, 𝑠) =𝐸[𝑒 (𝑆𝑇 −𝐾) 1{𝜏 >𝑇} |𝑆𝑡 =𝑠] of derivatives. Refer to Choi et al. [9]andBocketal.[10]for 𝐻 + European options and Lee et al. [11]forAsianoptions. −𝑟(𝜏𝐻+𝑇0−𝑡) 𝑇0 +𝐸[𝑒 (𝑚𝜏 −𝐾) 1{𝜏 ≤𝑇} |𝑆𝑡 =𝑠], This paper is concerned with the pricing of turbo warrants 𝐻 𝐻 which has been studied by several authors for different (1) models of security price in recent years. The pricing of 𝐸 turbo warrants has been studied by Eriksson [1] under the where denotes the expectation with respect to a risk- Black-Scholes model, Wong and Chan [12] under a stochastic neural probability. The turbo call warrant denoted by (1)is volatility model, Domingues [13] under the CEV model, and comprised of two parts. The first part looks like a down-and- Wong and Lau [14]underajumpdiffusionmodel.Thiswork out barrier (DOC) call option with a zero rebate and the studies the pricing of a turbo warrant under the hybrid setting secondpartisadown-and-inlookback(DIL)calloption.So, of stochastic and local volatility provided by [9]. In particular, the price of each part is expressed as follows: the main concern is given to the sensitivity of the turbo −𝑟(𝑇−𝑡) + (𝑡, 𝑠) =𝐸[𝑒 (𝑆𝑇 −𝐾) 1{𝜏 >𝑇} |𝑆𝑡 =𝑠], warrant to the elasticity parameter and parameters driving DOC 𝐻 the stochastic volatility. The choice to price a turbo warrant −𝑟(𝜏 +𝑇 −𝑡) 𝑇 + (𝑡, 𝑠, 𝑇 )=𝐸[𝑒 𝐻 0 (𝑚 0 −𝐾) 1 |𝑆 =𝑠]. with this kind of hybrid model is justified as follows. Turbo DIL 0 𝜏𝐻 {𝜏𝐻≤𝑇} 𝑡 warrants are barrier options with the rebate whose value is (2) computed by another path dependent option. They can be −𝑟𝑇 𝑇 + very sensitive to the change in volatility under stochastic (𝜏 ,𝑆 ,𝑇 ):=𝐸 [𝑒 0 (𝑚 0 −𝐾) ] If LB 𝐻 𝜏𝐻 0 𝜏𝐻 𝜏𝐻 (nonstandard volatility models as pointed out by [12], which is contrary ) and LC (𝑡,𝑠,𝑚,𝑇)denotes the price of the to the case of the Black-Scholes model. However, a frequent fl floating strike lookback call on 𝑆𝑡 =𝑠with realized minimum criticism of stochastic volatility models for path dependent 𝑚 and time to maturity 𝑇,thenonecanobtainthefollowing options is that they do not produce deltas precise enough for result. hedging purposes. So, relevant industry experts recommend using a hybrid stochastic local volatility model of their own Theorem 1. At 𝑡<𝜏𝐻, the model-free representation of the development for the best pricing of option products. Refer to turbo call warrant is given by Tataru and Travis [15]. Since the elasticity of variance controls leverage effects and plays an important role in characterizing 𝑇𝐶 (𝑡, 𝑠) various phases of volatile markets as described in Kim et al. −𝑟(𝜏 −𝑡) [16], it is worth investigating the sensitivity of the price to =𝐷𝑂𝐶(𝑡, 𝑠)+𝐸[𝑒 𝐻 1 𝐿𝐵 (𝜏 ,𝑆 ,𝑇 )|𝑆 =𝑠] {𝜏𝐻≤𝑇} 𝐻 𝜏𝐻 0 𝑡 elasticity parameter under a hybrid structure of the stochastic −𝑟(𝜏 −𝑡) volatility and the constant elasticity of variance. =𝐷𝑂𝐶(𝑡, 𝑠)+𝐸[𝑒 𝐻 1 𝐿𝐵 (𝜏 ,𝐻,𝑇 )|𝑆 =𝑠] {𝜏𝐻≤𝑇} 𝐻 0 𝑡 This paper is structured as follows. Section 2 provides a −𝑟(𝜏 −𝑡) =𝐷𝑂𝐶(𝑡, 𝑠)+𝐿𝐵(𝜏 ,𝑆 ,𝑇 )𝐸 [𝑒 𝐻 1 |𝑆 =𝑠], review on the decomposition of the price of turbo warrants 𝐻 𝜏𝐻 0 {𝜏𝐻≤𝑇} 𝑡 call option and establish a partial differential equation (PDE) (3) forthepricebasedonthehybridmodel(calledtheSVCEV model) of [9]. In Section 3, asymptotic analysis is given where for the PDE under the assumption of fast mean reverting 𝐿𝐵 (𝜏 ,𝑆 ,𝑇 )=𝐿𝐶 (𝜏 ,𝑆 , (𝑆 ,𝐾),𝑇 ) stochastic volatility. Section 4 solves the approximate price 𝐻 𝜏𝐻 0 𝑓𝑙 𝐻 𝜏𝐻 min 𝜏𝐻 0 by using the finite difference method and investigates the (4) price behavior and compares it with the ones corresponding −𝐿𝐶 (𝜏 ,𝑆 ,𝑆 ,𝑇 ). 𝑓𝑙 𝐻 𝜏𝐻 𝜏𝐻 0 to a stochastic volatility model as well as the CEV model. Section 5 concludes. In particular, if 𝑆𝑡 =𝐻(i.e., 𝑡=𝜏𝐻), then

+ −𝑟𝑇0 𝑇0 𝑇𝐶 (𝑡, 𝐻) =𝐸𝜏 [𝑒 (𝑚𝜏 −𝐾) ]=𝐿𝐵(𝜏𝐻,𝑆𝜏 ,𝑇0). 2. Model Formulation 𝐻 𝐻 𝐻 (5) The contract concerned in this paper is defined as follows. A turbo call warrant for an underlying asset, whose price at Proof. Refer to Domingues [13]. Abstract and Applied Analysis 3

This section constructs a pricing problem for turbo Note that the operator (1/𝜖)L0 is the infinitesimal generator warrants based upon a hybrid stochastic and local volatility (market probability measure version) of the OU process 𝑌𝑡 model given by [9]. Let 𝑆𝑡 be the underlying asset price at time andactsonlyonthe𝑦 variable. Since the operator L2 ∗ ∗ 𝑡 and 𝑊𝑡 and 𝑍𝑡 independent Brownian motions. Then the corresponds to the CEV operator for the turbo warrant, it is 𝑆 L dynamics of 𝑡 are given by denoted by CEV also. 𝑑𝑆 = (𝑟−𝑞) 𝑆 𝑑𝑡 +𝑓 (𝑌 ) 𝑆𝜃/2𝑑𝑊∗, 𝑡 𝑡 𝑡 𝑡 𝑡 (6) 3. Approximation 1 1 1 ̂∗ 𝑑𝑌𝑡 = [ (𝑚−𝑌𝑡) − Λ (𝑌𝑡)] 𝑑𝑡 + 𝑑𝑍𝑡 (7) Solution of the PDE (9) is not analytically available. So, this 𝜖 √𝜖 √𝜖 section is devoted to value the turbo warrant by transforming 𝜖 𝑄𝛾 (9) into a system of PDEs (under the assumption that is under an equivalent martingale measure with an arbitrary sufficiently small but positive) and solving the asymptotic adapted process 𝛾 to be determined, where 𝑟 (interest rate), 𝑞 𝜃 𝜖 PDEs numerically. (dividend yield rate), (elasticity parameter), (reciprocal of Under the assumption of the fast mean reversion of the 𝑚 𝑍̂∗ =𝜌𝑊∗ + rate of mean reversion), and are constants and 𝑡 𝑡 process 𝑌𝑡, this paper takes the asymptotic expansion, 2 ∗ √1−𝜌 𝑍𝑡 , −1≤𝜌≤1,andΛ (market price of volatility 𝑃=𝑃0 + √𝜖𝑃1 +𝜖𝑃2 +𝜖√𝜖𝑃3 +⋅⋅⋅, (11) risk) is independent of 𝑆𝑡,andthefunction𝑓 is assumed to 0<𝑐 <𝑓<𝑐 <∞ 𝑐 satisfy 1 2 forsomepositiveconstants 1 and is interested in the first order approximation 𝑃0 + √𝜖𝑃1. 𝑐 𝑌 and 2.NotethattheOrnstein-Uhlenbeck(OU)process 𝑡 is a This work assumes a growth condition for each 𝑃𝑖, 𝑖= Gaussian process which has an invariant distribution given by 0, 1, 2, . . ., such that the 𝜕𝑃𝑖/𝜕𝑦 does not grow N(𝑚, 1/2).Wecallthemodel(6)and(7)theSVCEVmodel 𝑦2/2 exponentially as much as (𝜕𝑃𝑖/𝜕𝑦) ∼𝑒 as 𝑦→∞. as in [9]. First, a boundary condition on 𝑃𝑖, 𝑖=0,1,isprovided Now, under the SVCEV model above, the dependence of as follows. Note that the first order approximation for the the value 𝑌𝑡 =𝑦is added to the price of the turbo warrant call floating lookback call option LC is given by option which yields fl 0 1 LC (𝜏𝐻,𝐻,𝑚,𝑦,𝑇0)≈LC (𝐻,0 𝑇 )+√𝜖LC (𝐻,0 𝑇 ), TC (𝑡, 𝑠, 𝑦) = DOC (𝑡, 𝑠, 𝑦) + DIL (𝑡,𝑠,𝑦,𝑇0), (8) fl fl fl (12) (𝑡, 𝑠, 𝑦) (𝑡,𝑠,𝑦,𝑇 ) where DOC and DIL 0 are defined by (2) 0(𝐻, 𝑇 ) 𝑦 where LCfl 0 istheCEVpriceofLCfl with effec- with the additional dependence of ,respectively.Also, 2 2 𝑦 (𝐻,𝑦,𝑇 )= tive volatility 𝜎 =⟨𝑓⟩,where𝑓 is a function in (6) -dependence is added to LB and LCfl.LB 0 1 (𝜏 ,𝑆 ,𝑦,𝑇 ) (𝐻,𝑦,𝑇 )= (𝜏 ,𝑆 ,𝑚,𝑦,𝑇 ) and LC (𝐻,0 𝑇 ) isthefirstcorrectedpriceofLC.Here, LB 𝐻 𝜏𝐻 0 and LCfl 0 LCfl 𝐻 𝜏𝐻 0 fl fl 0(𝐻, 𝑇 ) 1(𝐻, 𝑇 ) 𝑦 areusedforbriefness. LCfl 0 and LCfl 0 are independent of as in Wong From now on, let 𝑃(𝑡, 𝑠, 𝑦) := TC(𝑡, 𝑠, 𝑦) for conve- and Chan [18]. By plugging (12)intoLB(𝜏𝐻,𝑆𝜏 ,𝑦,𝑇0) = (𝑆 , (𝑆 , 𝐾), 𝑦, 𝑇 )− (𝑆 ,𝑆 ,𝑦,𝑇 )𝐻 nience. Then, by the well-known Feynman-Kac formula (cf. LCfl 𝜏𝐻 min 𝜏𝐻 0 LCfl 𝜏𝐻 𝜏𝐻 0 ,onecan Øksendal [17]) on (8), 𝑃(𝑡, 𝑠, 𝑦) satisfies a PDE problem given obtain the first order approximation: by 𝑃(𝜏𝐻,𝐻,𝑦)=LB (𝜏𝐻,𝐻,𝑦,𝑇0)≈LB0 (𝐻,0 𝑇 ) L𝜖𝑃(𝑡,𝑠,𝑦)=0, 0≤𝑡<𝑇,𝑥>0, (13) +√𝜖LB1 (𝐻,0 𝑇 ) , 𝑃(𝑇,𝑠,𝑦)=𝑠−𝐾, (9) where LB0 and LB1 are independent of 𝑦 and time 𝑡. 𝑃(𝜏𝐻,𝐻,𝑦)=LB (𝜏𝐻,𝐻,𝑦,𝑇0). Consequently, the boundary conditions 𝑃 (𝑇,𝑠,𝑦)=𝑠−𝐾, Here, if 𝜖 is positive and small, which corresponds to the 0 assumption of the fast mean reversion of the process 𝑌𝑡,then 𝜖 𝑃1 (𝑇, 𝑠, y)=0, the operator L can be expressed by the following well- (14) ordered form: 𝑃𝑖 (𝜏𝐻,𝐻)=LB𝑖 (𝐻,0 𝑇 ) 1 1 L𝜖 = L + L + L , 𝑃 𝑖=0,1 𝜖 0 √𝜖 1 2 are obtained for 𝑖, . The following lemma is extremely useful for the analysis 1 of interest in this work. L := 𝜕2 +(𝑚−𝑦)𝜕 , 0 2 𝑦𝑦 𝑦 Lemma 2. Let L0 be the operator given by (10).Ifsolutionto 𝜃/2 2 (10) L1 := 𝜌𝑓(𝑦)𝑠 𝜕𝑠𝑦 −Λ(𝑦)𝜕𝑦, the Poisson equation

1 2 L0𝜒(𝑦)+𝜙(𝑦)=0, (15) L := 𝜕 + 𝑓(𝑦) 𝑠𝜃𝜕2 2 𝑡 2 𝑠𝑠 exists, then condition ⟨𝜙⟩ = 0 must be satisfied, where +(𝑟−𝑞)𝑠𝜕 −𝑟:=L (𝑓(𝑦)) . ⟨⋅⟩ 𝑠 CEV denotes the expectation with respect to the invariant 4 Abstract and Applied Analysis

5.5 0.1

5 0.05

4.5 0

4 −0.05 1 0 𝜖P P

3.5 √ −0.1

3 −0.15

2.5 −0.2

2 −0.25 13 13 1 1 12 12

S 11 0.5 S 11 0.5 t t 10 0 10 0

(a) 𝑃0 (b) √𝜖𝑃1

Figure 1: The leading term and the first correction term of the turbo warrant call price is drawn under the SVCEV model with respect to time to maturity and stock price. The parameters used here are 𝑆0 =10, 𝜃 = 1.8, 𝑟 = 0.05, 𝑞 = 0.03, 𝜎 = 0.125, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 𝑊2 = −0.01,and𝑊3 = 0.0004.

5 5

4.5 4.5

4 4

3.5 3.5

Price 3 Price 3

2.5 2.5

2 2

1.5 1.5 10.5 11 11.5 12 12.5 10.5 11 11.5 12 12.5 S S

SVCEV: W2 = 0.02 SVCEV: W2 =−0.04 SVCEV: W3 =−0.002 SVCEV: W3 = 0.001 SVCEV: W2 =−0.01 CEV SVCEV: W3 = −0.001 CEV

(a) 𝑊3 = 0.0004 (b) 𝑊2 = −0.01

Figure 2: The turbo warrant call prices for different values of 𝑊2 and 𝑊3 under the SVCEV model are drawn. The parameters used here are 𝑆0 =10, 𝜃 = 1.8, 𝑟 = 0.05, 𝑞 = 0.03, 𝜎 = 0.125, 𝑇=1, 𝑇0 = 0.2, 𝐻=9,and𝐾=8.

distribution of 𝑌𝑡.Further,solutionsof (15) are given by the Theorem 3. Under the assumed growth condition on 𝑃𝑖,the form leading term 𝑃0 is independent of 𝑦 and is the CEV price given 𝑡 by the solution of the PDE, 𝑦 𝜒 (𝑦) = ∫ 𝐸 [𝜙 (𝑌𝑡)] 𝑑𝑡 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. (16) 0 Proof. Refer to Fouque et al. [19]. 1 2 𝜃 2 𝜕 𝑃 + 𝜎 𝑠 𝜕 𝑃 +(𝑟−𝑞)𝑠𝜕𝑃 −𝑟𝑃 =0, (17) First, the leading term 𝑃0 is obtained as follows. 𝑡 0 2 𝑠𝑠 0 𝑠 0 0 Abstract and Applied Analysis 5

5.5 0

5 −0.05

4.5 −0.1 4 −0.15 1

0 3.5 𝜖P P

√ −0.2 3 −0.25 2.5

2 −0.3

1.5 −0.35 10 10.5 11 11.5 12 12.5 13 10 10.5 11 11.5 12 12.5 13 S S

𝜃 = 1.6 𝜃 = 2.0 𝜃 = 1.6 𝜃 = 2.0 𝜃 = 1.8 𝜃 = 2.2 𝜃 = 1.8 𝜃 = 2.2

(a) 𝑃0 (b) √𝜖𝑃1

Figure 3: The turbo warrant call prices under the CEV model and the first correction term of the SVCEV model are drawn. The parameters used here are given by 𝑆0 =10, 𝑟 = 0.05, 𝑞 = 0.03, 𝜎 = 0.125, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 𝑊2 = −0.01,and𝑊3 = 0.0004.

with the boundary condition: 𝑦-derivative and 𝑃1 is 𝑦-independent. Then by Lemma 2 one has ⟨L2⟩𝑃0 =0,where 𝑃0 (𝑇, 𝑠) =𝑠−𝐾, 1 2 𝜃 2 ⟨L2⟩=𝜕𝑡 + ⟨𝑓 ⟩𝑠 𝜕𝑠𝑠 +(𝑟−𝑞)𝑠𝜕𝑠 −𝑟𝐼, (21) 𝑃0 (𝜏𝐻,𝐻)=𝐿𝐵0 (𝐻,0 𝑇 ) (18) 2 0 0 𝐼 𝑃 (𝑡, 𝑠) =𝐿𝐶𝑓𝑙 (𝐻,𝐾,𝑇0)−𝐿𝐶𝑓𝑙 (𝐻,𝐻,𝑇0), where is the identity operator. Hence, 0 is equal to the CEV option price with the effective volatility 𝜎. 2 where 𝜎=√⟨𝑓 ⟩. Next, the term 𝑃1 is obtained as follows.

Proof. By substituting the expansion (11)intothePDE(9), we Theorem 4. Under the assumed growth condition on 𝑃𝑖,the have correction term 𝑃1 is independent of 𝑦 and given by the solution of the PDE 1 1 L 𝑃 + (L 𝑃 + L 𝑃 ) 0 0 0 1 1 0 1 𝜖 √𝜖 𝜕 𝑃 + 𝜎2𝑠𝜃𝜕2 𝑃 +(𝑟−𝑞)𝑠𝜕𝑃 −𝑟𝑃 𝑡 1 2 𝑠𝑠 1 𝑠 1 1 +(L 𝑃 + L 𝑃 + L 𝑃 ) (19) 0 2 1 1 2 0 2 2 (22) 𝜃/2 𝜕 𝜃 𝜕 𝑃0 𝜃 𝜕 𝑃0 =𝑊3𝑠 (𝑠 )+𝑊2𝑠 + √𝜖(L0𝑃3 + L1𝑃2 + L2𝑃1)+⋅⋅⋅=0, 𝜕𝑠 𝜕𝑠2 𝜕𝑠2 and thus the following equations are satisfied. with the boundary condition:

L0𝑃0 =0, L0𝑃1 + L1𝑃0 =0, 𝑃1 (𝑇, 𝑠) =0, L 𝑃 + L 𝑃 + L 𝑃 =0, 0 2 1 1 2 0 (20) 𝑃1 (𝜏𝐻,𝐻)= 𝐿𝐵1 (𝐻,0 𝑇 ) (23) 1 1 L0𝑃3 + L1𝑃2 + L2𝑃1 =0. =𝐿𝐶𝑓𝑙 (𝐻,𝐾,𝑇0)−𝐿𝐶𝑓𝑙 (𝐻,𝐻,𝑇0),

Then, from the growth condition applied to L0𝑃0 =0, 𝑃0 where 𝑊3 and 𝑊2 are given by becomes a function of only 𝑡 and 𝑠.So,𝑃0 =𝑃0(𝑡, 𝑠). Since each term of the operator L1 contains 𝑦-derivative 𝜌] 󸀠 ] 󸀠 𝑊3 = ⟨𝑓𝜓 ⟩, 𝑊2 =− ⟨Λ𝜓 ⟩, (24) and 𝑃0 is 𝑦-independent, L0𝑃1 +L1𝑃0 =0leads to L0𝑃1 =0 √2 √2 and so 𝑃1 also is a function of 𝑡 and 𝑠 only; 𝑃1 =𝑃1(𝑡, 𝑠). Next, L0𝑃2 +L1𝑃1 +L2𝑃0 =0in (20)reducestoL0𝑃2 + respectively. Here, 𝜓 is defined as the solution of the Poisson 2 2 L2𝑃0 =0becauseeachtermoftheoperatorL1 contains equation L0𝜓=𝑓 −⟨𝑓 ⟩. 6 Abstract and Applied Analysis

3.2 3.2

3 3

2.8 2.8

2.6 2.6

2.4 2.4 1 1 𝜖P 𝜖P

√ 2.2 √ 2.2 + + 0 0

P 2 P 2

1.8 1.8

1.6 1.6

1.4 1.4

1.2 0.2 0.25 0.3 0.35 0.4 0.2 0.25 0.3 0.35 0.4 𝜎 𝜎

Turbo warrant under SVCEV Turbo warrant under SV Turbo warrant under CEV Turbo warrant under BS European vanilla under SVCEV European vanilla under SV European vanilla under CEV European vanilla under BS

(a) 𝜃=1.6 (b) 𝜃=2 3.2

3

2.8

2.6

2.4 1 𝜖P

√ 2.2 + 0

P 2

1.8

1.6

1.4

1.2 0.2 0.25 0.3 0.35 0.4 𝜎

Turbo warrant under SVCEV Turbo warrant under CEV European vanilla under SVCEV European vanilla under CEV

(c) 𝜃=2.2

Figure 4: The prices of the turbo warrant call and the European vanilla call are drawn against 𝜎 under the CEV and SVCEV models. The parameters used here are 𝑆0 =10, 𝑟 = 0.05, 𝑞 = 0.03, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 𝑊2 = −0.01,and𝑊3 = 0.0004.

Proof. The 𝑦-independence of 𝑃1 has been obtained in the theresultofDavydovandLinetsky[20], we obtain 𝑃0 = proof of Theorem 3. Applying Lemma 2 to the equation TCCEV analytically as follows: L0𝑃3 +L1𝑃2 +L2𝑃1 =0in (20)leadsto⟨L1𝑃2 +L2𝑃1⟩=0. Combining this one with the result L0𝑃2+L2𝑃0 =0obtained 𝑃 (𝑡, 𝑠) = (𝑡, 𝑠) in the middle of proof of Theorem 3 yields the PDE (22)for 0 DOCCEV 𝑃1(𝑡, 𝑠). +[ CEV (𝐻, 𝐾, 𝑇 )− CEV (𝐻,𝐻,𝑇 )] LCfl 0 LCfl 0 (25) Now, we need to solve the PDEs (17)and(22)for𝑃0 and 𝑃 × (𝑡, 𝑠) , 1, respectively, for the approximation of interest. By using DRCEV Abstract and Applied Analysis 7

2.5 Here, the functions 𝑀𝑘,𝑚(𝑥) and 𝑊𝑘,𝑚(𝑥) are the well-known Whittaker functions and 𝐼](𝑥) and 𝐾](𝑥) are the modified 2.4 Bessel functions. √ 2.3 Next, the first correction term 𝜖𝑃1 is required to be solved from the PDE (22). Apparently, it is difficult to solve 2.2 (22) analytically. It requires the use of a numerical computing method. Here, we use the finite difference scheme of the 2.1 Crank-Nicolson method to compute the first correction term 2 √𝜖𝑃1. The truncation error for the solution is O((Δ𝑡) ) + 2 2 O((Δ𝑠) ),whereΔ𝑡 = 0.01 and Δ𝑠 = 0.18. Turbo warrant price warrant Turbo 1.9 Figure 1 shows the behavior of both the leading term and the first correction term. One can notice from the 1.8 correction term that the stochastic volatility of the SVCEV 1.7 model tends to lower the CEV turbo warrant price as the 1.5 2 2.5 stock price approaches the barrier. However, the impact of the 𝜃 stochastic volatility decreases as time to maturity decreases. More detailed implications of the SVCEV model for the turbo SVCEV warrant price are described in the following section. CEV From the results of Theorems 3 and 4,weformallyobtain Figure 5: The turbo warrant call price against 𝜃 is drawn under the the first order approximation: CEV and SVCEV models. The parameters used here are 𝑆0 =10, 𝑃 (𝑡, 𝑠, 𝑦) ≈𝑃 (𝑡, 𝑠) + √𝜖𝑃 (𝑡, 𝑠) , 𝑟 = 0.05, 𝑞 = 0.03, 𝜎 = 0.125, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 0 1 (29) 𝑊2 = −0.01,and𝑊3 = 0.0004. where 𝑃0 and 𝑃1 are independent of the unobservable variable 𝑦. In terms of accuracy of the approximation, we note that the where DOCCEV is the well-known CEV pricing formula (cf. error estimates for barrier and lookback options have been CEV obtained by Park and Kim [21]fortheCEVmodel.Essentially, [20]) and LC and DRCEV are given by fl therequiredestimateforthepresentSVCEVmodelcanbe −𝑞𝑇 −𝑟𝑇 (𝑠, 𝑚, 𝑇 )=𝑒 0 𝑠−𝑒 0 𝑚 given by a combination of the result of [21]fortheCEVmodel LCfl 0 and the error estimate obtained by Fouque et al. [22]forthe 𝑚 −𝑟𝑇0 𝑇0 stochastic volatility model. The detailed proof is omitted here. +𝑒 ∫ Q (𝑚𝜏 ≤𝜉)𝑑𝜉, (26) 0 𝐻

−𝑟(𝜏 −𝑡) (𝑡, 𝑠) =𝐸[𝑒 𝐻 1 ], 4. Implications DRCEV {𝜏𝐻≤𝑇} The pricing of turbo warrants under the SVCEV model is respectively. Here, under the CEV model with the volatility 𝑇0 −𝑟(𝜏𝐻−𝑡) more interesting than either the CEV or stochastic volatility coefficient 𝜎, Q(𝑚 ≤𝜉)and 𝐸[𝑒 1{𝜏 ≤𝑇}] are given by 𝜏𝐻 𝐻 models. In this section, we analyze the sensitivity of the 1 𝜙 (𝐻) turbo warrant call option with respect to several involved 𝑇0 −1 𝜆 𝜃 Q (𝑚𝑡 ≤𝜉)=L𝑇 ,𝜆 { }, parameters, in particular, the elasticity parameter , under the 0 𝜆 𝜙 (𝜉) 𝜆 SVCEVmodel.WenotethatthecaseofSVCEVwith𝜃=2 (27) 1 𝜙 (𝑠) corresponds to the stochastic volatility (SV) model of [6]and −𝑟(𝜏𝐻−𝑡) −1 𝑟+𝜆 𝐸[𝑒 1{𝜏 ≤𝑇}]=L { }, the case of CEV with 𝜃=2is the same as the Black-Scholes 𝐻 𝑇,𝜆 𝜆 𝜙 (𝐻) 𝑟+𝜆 (BS) model in the following arguments. respectively, with the Laplace transform L,where𝜆>0and Figure 2 plots the turbo warrant call price with respect to the parameters 𝑊2 and 𝑊3 under the SVCEV and the CEV 𝜃 = 1.8 𝑊 𝜙𝜆 (𝑠) price for .Asseenfrom(24), 2 is a parameter related to the market price of volatility risk and 𝑊3 is a parameter ̃ {𝑠𝛼+(1/2)𝑒(𝜃/2)𝑥̃𝑊 (𝑥̃) ,𝛼<0,𝑟−𝑞=0,̸ related to the correlation of the stock price and the fast-mean { 𝑘,𝑚̃ { 𝛼+(1/2) (𝜃/2)̃ 𝑥̃ reverting process. It can be seen from Figure 2 that the price 𝑠 𝑒 𝑀𝑘,𝑚̃ (𝑥̃) ,𝛼>0,𝑟−𝑞=0,̸ = { of the turbo warrant call increases as 𝑊2 increases, whereas {𝑠(1/2)𝐾 (√2𝜆𝑧),̃ 𝛼<0,𝑟−𝑞=0, { ̃] the price decreases as 𝑊3 increases. This behavior appears (1/2) √ {𝑠 𝐼̃] ( 2𝜆𝑧),̃ 𝛼>0,𝑟−𝑞=0, more sensitively near the barrier. 󵄨 󵄨 Figure 3 shows that the turbo warrant call price under the 󵄨𝑟−𝑞󵄨 −𝛼 󵄨 󵄨 −2𝛼 𝑠 ̃ CEV model and the first correction term under the SVCEV 𝑥=̃ 𝑠 , 𝑧=̃ , 𝜃=sign (𝛼 (𝑟 − 𝑞)), 𝜎2 |𝛼| 𝜎 |𝛼| model are drawn with respect to the stock price for different values 𝜃.ThestochasticvolatilityimpactontheCEVprice 1 1 1 𝜆 1 ̃ ̃ ̃ becomes more apparent as 𝜃 increases. It is interesting to note 𝑚= ,𝑘=𝜃( + )− 󵄨 󵄨, ] = . 4 |𝛼| 2 4𝛼 2 󵄨𝛼(𝑟−𝑞)󵄨 2 |𝛼| that the impact begins to show a nonmonotonic behavior (28) with respect to the stock price as 𝜃 becomes larger than 2. 8 Abstract and Applied Analysis

2.4 2.4

2.2 2.2 1 1 𝜖P 𝜖P

√ 2 √ 2 + + 0 0 P P 1.8 1.8

1.6 1.6

1.4 1.4 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −3 −3 W2 ×10 W2 ×10 Turbo warrant under SVCEV Turbo warrant under SV European vanilla under SVCEV European vanilla under SV

(a) 𝜃=1.8 (b) 𝜃=2 2.4 2.6

2.3 2.4 2.2

2.1 2.2 1 2 1 𝜖P 𝜖P √ √ 2 + 1.9 + 0 0 P P 1.8 1.8

1.7 1.6 1.6

1.5 1.4 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −3 −3 W2 ×10 W2 ×10 Turbo warrant under SVCEV Turbo warrant under SVCEV European vanilla under SVCEV European vanilla under SVCEV

(c) 𝜃=2.2 (d) 𝜃=2.4

Figure 6: The sensitivity of the turbo warrant call and the European vanilla call to 𝑊2 is drawn under the SVCEV model for different values of 𝜃. The parameters used here are 𝑆0 =10, 𝑟 = 0.05, 𝑞 = 0.03, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 𝜎 = 0.125,and𝑊3 = 0.0004.

Figure 4 plots the functional behavior of the European 𝜃.AsshowninFigure 4, the lower 𝜃 is, the more sensitively vanillacallandtheturbowarrantcallpriceswithrespect the turbo warrant price behaves with respect to the change in totheeffectivevolatility𝜎. It is interesting to note that the volatility. Europeanvanillaoptionisoverpriced,whereastheturbo Figure 5 shows the sensitivity difference between the warrant is underpriced in the SVCEV model with respect to CEVandSVCEVmodelsfortheturbowarrantpriceto theCEVone.Ingeneral,theEuropeanvanillapriceincreases the change of the elasticity parameter 𝜃. Apparently, the astheeffectivevolatility𝜎 rises. However, it becomes opposite stochastic volatility of the SVCEV model lowers the price for the turbo warrant case as shown in Figure 4.Theturbo level of the CEV turbo warrant regardless of the 𝜃 level. As call price under the Black-Scholes model may not be much 𝜃 increases, this impact tends to become larger. less sensitive to the change in volatility than the European Figures 6 and 7 show the sensitivity of the turbo warrant vanilla call option as shown in [14]. But the degree of the call and the European vanilla call against 𝑊2 and 𝑊3 under sensitivity depends on the value of the elasticity parameter the SVCEV Model. The turbo warrant call is always cheaper Abstract and Applied Analysis 9

2.4 2.4

2.3 2.3

2.2 2.2

2.1 2.1

1 2 1 2 𝜖P 𝜖P √ √

+ 1.9 + 1.9 0 0 P P 1.8 1.8

1.7 1.7

1.6 1.6

1.5 1.5 3 3.5 4 4.5 5 3 3.5 4 4.5 5 −4 −4 W3 ×10 W3 ×10 Turbo warrant under SVCEV Turbo warrant under SVCEV European vanilla under SVCEV European vanilla under SVCEV

(a) 𝜃=1.6 (b) 𝜃=1.8 2.4 2.4

2.3 2.3

2.2 2.2

2.1 2.1 1 2 1 2 𝜖P 𝜖P √ √ + 1.9 + 1.9 0 0 P P 1.8 1.8

1.7 1.7

1.6 1.6

1.5 1.5 3 3.5 4 4.5 5 3 3.5 4 4.5 5 −4 −4 W3 ×10 W3 ×10 Turbo warrant under SV Turbo warrant under SVCEV European vanilla under SV European vanilla under SVCEV

(c) 𝜃=2 (d) 𝜃=2.2

Figure 7: The sensitivity of the turbo warrant call and the European vanilla call to 𝑊3 is drawn under the SVCEV model. The parameters used here are 𝑆0 =10, 𝑟 = 0.05, 𝑞 = 0.03, 𝑇=1, 𝑇0 = 0.2, 𝐻=9, 𝐾=8, 𝜎 = 0.125,and𝑊2 = −0.01. than the European vanilla call as expected. Figure 6 shows (effectivevolatility,elasticity,themarketpriceofvolatility that as 𝑊2 increases, the price of the turbo warrant call risk, and the correlation between stock price and volatility) increases, whereas the price of the vanilla call decreases. The under the SVCEV model. The hybrid nature of the model European vanilla call is more sensitive to 𝑊2 than the turbo enables us to find the rich sensitivity structure of the turbo warrant as 𝜃 increases. Figure 7 showsthatthepriceofthe warrant price in various ways depending upon the stochastic turbo warrant call drops moderately while the price of the volatility as well as the elasticity of variance. This paper not vanilla call rises very slowly as 𝑊3 increases. only shows the stochastic volatility effect on the CEV price oftheturbowarrantbutalsotheimpactoftheelasticity 5. Conclusion of variance on the stochastic volatility price. Moreover, a comparison analysis of the turbo warrant contract and the In this paper, we have examined the price change behavior European vanilla option has been performed effectively based of a turbo warrant with respect to the involved parameters onthehybridmodel. 10 Abstract and Applied Analysis

Conflict of Interests [15] G. Tataru and F. Travis, Stochastic Local Volatility,Quantitative Development Group, Bloomberg, 2010, version 1. The authors declare that there is no conflict of interests [16] J.-H. Kim, J. Lee, S.-P. Zhu, and S.-H. Yu, “Amultiscale correc- regarding the publication of this paper. tion to the Black-Scholes formula,” Applied Stochastic Models in Business and Industry.Inpress. Acknowledgments [17] B. Øksendal, Stochastic Differential Equations, Springer, New York, NY, USA, 6th edition, 2003. The authors would like to thank the anonymous referees [18] H. Y. Wong and C. M. Chan, “Lookback options and dynamic for their valuable comments and suggestions to improve fund protection under multiscale stochastic volatility,” Insur- thepaper.TheresearchofJeong-HoonKimwassup- ance,vol.40,no.3,pp.357–385,2007. ported by the National Research Foundation of Korea NRF- [19] J.-P.Fouque, G. Papanicolaou, R. Sircar, and K. Sølna, Multiscale 2013R1A1A2A10006693. The research of Min-Ku Lee was Stochastic Volatility for Equity, Interest Rate, and Credit Deriva- supportedbyBK21PLUSSungkyunkwanUniversityMath- tives, Cambridge University Press, Cambridge, UK, 2011. ematics Division and the research of Ji-Hun Yoon was sup- [20] D. Davydov and V. Linetsky, “Pricing and hedging path- ported by BK21 PLUS SNU Mathematical Sciences Division. dependent options under the CEV process,” Management Sci- ence,vol.47,no.7,pp.949–965,2001. [21] S.-H. Park and J.-H. Kim, “Asymptotic option pricing under References the CEV diffusion,” Journal of Mathematical Analysis and Applications,vol.375,no.2,pp.490–501,2011. [1] J. Eriksson, On the pricing equations of some path-dependent options [Ph.D. dissertation], Uppsala University, Uppsala, Swe- [22] J.-P.Fouque, G. Papanicolaou, R. Sircar, and K. Solna, “Singular den, 2006. perturbations in option pricing,” SIAM Journal on Applied Mathematics,vol.63,no.5,pp.1648–1665,2003. [2] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy,vol.81,pp.637–654, 1973. [3] J. Cox, “Notes on option pricing I: constant elasticity of variance diffusions,” Working Paper, Stanford University, 1975, reprinted in The Journal of Portfolio Management, vol. 22, pp. 15–17, 1996. [4] J.C.CoxandS.A.Ross,“Thevaluationofoptionsforalternative stochastic processes,” Journal of Financial Economics,vol.3,no. 1-2, pp. 145–166, 1976. [5] S. L. Heston, “Closed-form solution for options with stochastic volatility with applications to bond and currency options,” The Review of Financial Studies,vol.6,pp.327–343,1993. [6] J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, “Asymptotics of a two-scale stochastic volatility model,” in Equations Aux Derivees Partielles et Applications, Articles Dedies a Jacques- Louis Lions, pp. 517–525, Gauthier-Villars, Paris, France, 1998. [7] P. Carr, H. Geman, D. B. Madan, and M. Yor, “Stochastic volatility for Levy´ processes,” Mathematical Finance,vol.13,no. 3, pp. 345–382, 2003. [8]P.S.Hagan,D.Kumar,A.S.Lesniewski,andD.E.Woodward, Managing Smile Risk, Wilmott Magazine, 2002. [9] S.-Y. Choi, J.-P. Fouque, and J.-H. Kim, “Option pricing under hybrid stochastic and local volatility,” Quantitative Finance,vol. 13, no. 8, pp. 1157–1165, 2013. [10] B.-H. Bock, S.-Y. Choi, and J.-H. Kim, “The pricing of European options under the constant elasticity of variance with stochastic volatility,” Fluctuation and Noise Letters,vol.12,no.1,ArticleID 135004,13pages,2013. [11] M. -K. Lee, J. -H. Kim, and K. -H. Jang, “Pricing arithmetic Asian options under hybrid stochastic and local volatility,” Journal of Applied Mathematics,ArticleID784386,2014. [12] H. Y. Wong and C. M. Chan, “Turbo warrants under stochastic volatility,” Quantitative Finance,vol.8,no.7,pp.739–751,2008. [13] A. Domingues, The valuation of turbo warrants under the CEV Model [M.S. thesis], ISCTE Business School, University Institute of Lisbon, Lisbon, Portugal, 2012. [14] H. Y. Wong and K. Y. Lau, “Analytical valuation of turbo warrants under double exponential jump diffusion,” Journal of Derivatives,vol.15,no.4,pp.61–73,2008. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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