Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 682524, 5 pages http://dx.doi.org/10.1155/2013/682524

Research Article Homotopy Analysis Method for Boundary-Value Problem of Turbo Warrant Pricing under Stochastic Volatility

Hoi Ying Wong1 and Mei Choi Chiu2

1 Department of Statistics, The Chinese University of Hong Kong, Hong Kong 2 Department of Mathematics and Information Technology, Hong Kong Institute of Education, Hong Kong

Correspondence should be addressed to Hoi Ying Wong; [email protected]

Received 13 December 2012; Accepted 30 January 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 H. Y. Wong and M. C. Chiu. 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.

Turbo warrants are liquidly traded financial derivative securities in over-the-counter and exchange markets in Asia and Europe. The structure of turbo warrants is similar to barrier options, but a lookback rebate will be paid if the barrier is crossed bythe underlying asset price. Therefore, the turbo warrant price satisfies a partial differential equation (PDE) with a boundary condition that depends on another boundary-value problem (BVP) of PDE. Due to the highly complicated structure of turbo warrants, their valuation presents a challenging problem in the field of financial mathematics. This paper applies the homotopy analysis method to construct an analytic pricing formula for turbo warrants under stochastic volatility in a PDE framework.

1. Introduction properties of turbo warrants. This leads to subsequent exten- sions to turbo warrant pricing with jump-diffusion model in Turbo warrant first appears in Europe but is now available [3] and with mean reversion in [4]. Although Wong and Chan under various names in many markets including the UK, [2] approximate turbo prices under a stochastic volatility (SV) Germany, Switzerland, Italy, Australia, New Zealand, Singa- model, their solution is restricted to the assumption that the pore,SouthAfrica,Taiwan,andHongKong.Forinstance,itis mean-reverting speed of the stochastic volatility should either called turbo warrant in the Nordic Growth market, contract beclosetozeroortoinfinity.Ourgoalistoderiveananalytic for difference (CFD) in the UK, and callable bull/bear solution to turbo prices by relaxing such an assumption using contract (CBBC) in Hong Kong. According to a report by a homotopy analysis method. Hong Kong Exchanges and Clearing Limited (HKEx) in 2009 Homotopy analysis method was introduced by Ortega [1],theUKhasabigOTCmarketforCFDswhileHong and Rheinboldt [5] in 1970 and has been applied to solve Kong has a big exchange market for CBBC. Typically, HKEx many nonlinear problems since the work of Liao [6]in1992. listed 1,525 CBBCs, constituting 28% of the total number of its Zhu in [7] pioneers the use of homotopy analysis method in listed securities, and the total issued amount reached HK$704 financial mathematics and derives the first analytic formula billion or US$95 billion at the end of May 2009 [1]. Hong to American options. Zhao and Wong in [8]showthat Kong has two types of CBBC: N and R. The N-CBBC pays the approach of Zhu is also applicable to general diffusion norebatewhenapreselectedbarrieriscrossedwhereastheR- models. CBBC pays a lookback rebate. If the barrier is not breached by While general diffusion models belong to the class of the underlying asset price during the life of the contract, then complete market models, SV models are of incomplete the turbo call (put) is equivalent to the standard call (put). market models because the number of Brownian motions Therefore, turbo warrant resembles a combination of barrier driving the asset dynamics is larger than one. Park and and lookback options. Kim [9] apply homotopy analysis method to solve vanilla A comprehensive mathematical treatment of turbo war- option and barrier option prices under SV models. Leung [10] rants is given in [2], which provides several model-free extends the framework to lookback option pricing. However, 2 Abstract and Applied Analysis the application of homotopy analysis method to turbo war- where DOC(𝑡, 𝑆) denotes the down-and-out call option price at rant pricing under SV models is yet to be considered. 𝑡,

This paper employs the homotopy analysis method to −𝑟𝑇 𝑇 + (𝜏 ,𝑆 ,𝑇 )=𝐸 [𝑒 0 (𝑚 0 −𝐾) 1 ] solve the PDE for the turbo warrant price under a SV model. LB 𝐻 𝜏𝐻 0 𝜏𝐻 𝜏𝐻 {𝜏𝐻≤𝑇} As the price of turbo warrant under the Black-Scholes model = (0, 𝑆 , (𝑆 ,𝐾),V ,𝑇 ) (3) is available, we construct a homotopy which deforms from LC 𝜏𝐻 min 𝜏𝐻 𝜏𝐻 0 the Black-Scholes solution to the desired solution under − (0, 𝑆 ,𝑆 , V ,𝑇 ), theSVmodel.Wehighlightthefundamentalchallengein LC 𝜏𝐻 𝜏𝐻 𝜏𝐻 0 turbo warrant pricing. A turbo warrant consists of a barrier option and a lookback rebate. Although the barrier option and LC(𝑡,𝑆,𝑚,V,𝑇0) is the floating strike lookback call at time pricing under SV models is investigated in [9], Park and Kim 𝑡 on 𝑆 with the realized minimum 𝑚,instantaneousvolatility do not consider rebate in knockout options. Our solution V, and time to maturity 𝑇0. should cover the case of state-dependent rebate under SV. Second, Leung [10] offers an analytic pricing formula for In Proposition 1, the floating strike lookback call, (𝑡,𝑆,𝑚,V,𝑇 ) lookback options but his formula cannot be directly used in LC 0 ,hasthepayoff calculating the lookback rebate, which is an expectation on 𝑆 − {𝑆 }. 𝑇0 min 𝜏 the discounted lookback option with a random starting time. 0≤𝜏≤𝑇0 (4) Therefore, our homotopy analysis method has to simultane- ously solve these two problems together. Hence, its price has the representation, The reminder of this paper is organized as follows. −𝑟(𝑇0−𝑡) Section 2 presents the nature of turbo warrants under SV LC (𝑡, 𝑆, 𝑚, V,𝑇0)=𝑒 𝐸𝑡 [𝑆𝑇 − min {𝑆𝜏}] . (5) 0 0≤𝜏≤𝑇 models and the corresponding pricing problem in a PDE 0 −𝑟𝜏 approach. Section 3 contains the main result of this paper and 𝐸 [𝑒 𝐻 1 (𝜏 ,𝑆 ,𝑇 )] The term 𝑡 {𝜏𝐻≤𝑇}LB 𝐻 𝜏𝐻 0 is called the down- solves the PDE using homotopy analysis method. Concluding and-in lookback (DIL) option in [2] as a TC holder will remark is made in Section 4. knock in the lookback option LB, shown in (3), only when the underlying asset price hits the down-side barrier 𝐻. 2. Problem Formulation In particular, if the asset price process follows the Black- Scholes (BS) model where the volatility is a constant, then the 𝑡∈[0,𝜏 ] Let 𝑆𝑡 be the underlying asset price at time 𝑡.Aturbocall turbo call price at 𝐻 reduces to + (put) warrant pays the contract holder (𝑆𝑇 −𝐾) at maturity TC (𝑡,) 𝑆 = DOC (𝑡,) 𝑆 𝑇 if a specified barrier 𝐻≥𝐾has not been passed by 𝑆𝑡 at BS BS any time prior to the maturity. Denote 𝜏𝐻 as the first passage +[ (0,𝐻,𝐾,𝑇 ) LCBS 0 (6) time that the asset price crosses the barrier 𝐻,thatis,𝜏𝐻 = {𝑡:𝑆 ≤𝐻} 𝜏 ≤𝑇 − (0,𝐻,𝐻,𝑇 )] (𝑡,) 𝑆 , inf 𝑡 .If 𝐻 ; then the contract is void and LCBS 0 DRBS a new contract starts. The new contract is a call option on 𝑇0 𝑚𝜏 = min𝜏 ≤𝑡≤𝜏 +𝑇 𝑆𝑡,withthestrikeprice𝐾,andthetime where the Black-Scholes pricing formulas of DOCBS,LCBS, 𝐻 𝐻 𝐻 0 −𝑟𝜏𝐻 =𝐸𝑡[𝑒 1{𝜏 ≤𝑇}] to maturity 𝑇0.Moreprecisely,theturbocall(TC)attime𝑡 and DRBS 𝐻 are classical results and canbeexpressedas can be found in [11]. However, the formulas of LCBS and DRBS also appear in some proofs of this paper. Specifically, LC (𝑡,𝑆,𝐾,𝑇0)=𝑆𝑈0(𝑡, 𝑦, V,𝑇0),where𝑈0 is given by (19) + BS =𝐸[𝑒−𝑟𝑇(𝑆 −𝐾) 1 while DR appears in (28). TC𝑡 𝑡 𝑇 {𝜏𝐻>𝑇} BS (1) −𝑟(𝜏 +𝑇 ) 𝑇 + +𝑒 𝐻 0 (𝑚 0 −𝐾) 1 ], 2.1. Turbos under Stochastic Volatility. Although the explicit 𝜏𝐻 {𝜏𝐻≤𝑇} BS formula for the TC is known as in (6), its analytic pricing formula under SV model is yet to be considered. where 𝑟 is the constant interest rate and 𝐸𝑡 represents the risk- The Heston SV model assumes the following stochastic neutral expectation given information up to time 𝑡.Wong differential equation (SDE) for the underlying asset price andChan[2] explain the incentive of this security design and 𝑑𝑆𝑡 𝑆 show a model-free representation that the turbo call warrant =(𝑟−𝑞)𝑑𝑡+√V𝑡 𝑑𝑊𝑡 , (7) is sum of a down-and-out call (DOC) and an expectation of 𝑆𝑡 a nonstandard lookback option (LB) starting at 𝜏𝐻. where 𝑟 is the constant interest rate, 𝑞 is the constant dividend Proposition 1 𝑡≤𝜏 yield, and V𝑡 is the stochastic instantaneous variance of the (see [2]). At 𝐻, the model-free representa- V tion of the turbo call warrant is asset. The stochastic variance 𝑡 follows the SDE: V 𝑑V𝑡 =𝜅(𝜃−V𝑡)𝑑𝑡+𝜎√V𝑡 𝑑𝑊𝑡 , (8)

−𝑟𝜏𝐻 TC (𝑡,) 𝑆 = DOC (𝑡,) 𝑆 +𝐸𝑡 [𝑒 1{𝜏 ≤𝑇}LB (𝜏𝐻,𝑆𝜏 ,𝑇0)] , 𝑆 V 𝐻 𝐻 where 𝜅, 𝜃,and𝜎 are constants. In (7)and(8), 𝑊𝑡 and 𝑊𝑡 are (2) Wiener processes defined on a filtered complete probability Abstract and Applied Analysis 3

𝑆 V space (Ω, F, Q, F𝑡≥0) with 𝐸[𝑊𝑡 𝑊𝑡 ]=𝜌𝑡,whereQ denotes In principal, the exotic lookback option LB(𝑡, 𝑆, V,𝑇) the risk-neutral probability measure. should be solved from an alternative PDE governing lookback Let 𝑉(𝑡,̃ 𝑆, V) be the turbo warrant price at time 𝑡 under options. Fortunately, we know from (11)thatthisexotic the stochastic volatility V.Using(1), we have lookback option is the difference between two floating strike lookback calls. ̃ 𝑉 (𝑡, 𝑆, V) Specifically, LC(𝑡,𝑆,𝑚,V,𝑇)has the following homotopy

−𝑟𝑇 + expression [10]: =𝐸[𝑒 (𝑆𝑇 −𝐾) 1{𝜏 >𝑇} 𝐻 󵄨 LC (𝑡, 𝑆, 𝑚, V,𝑇) =𝑆⋅𝑈(𝑡,𝑦,V,𝑇,𝑝)󵄨 , −𝑟(𝜏 +𝑇 ) 𝑇 + 󵄨𝑝=1 (16) +𝑒 𝐻 0 (𝑚 0 −𝐾) 1 |𝑆 =𝑆,V = V]. 𝜏𝐻 {𝜏𝐻≤𝑇} 𝑡 𝑡 (9) where 𝑦=ln(𝑚/𝑆);

By Proposition 1, the turbo price can be expressed as (2). ∞ 𝑛 As the SV model assumes a continuous process for the 𝑈(𝑡, 𝑦, V,𝑇,𝑝) = ∑𝑈𝑛 (𝑡, 𝑦, V,𝑇) 𝑝 . (17) 𝑆 =𝐻 𝑛=0 underlying asset price, 𝜏𝐻 .Hence,wehave 𝑉̃ (𝑡, 𝑆, V) The functions 𝑈𝑛(𝑡, 𝑦, V,𝑇), 𝑛 = 1, 2, .,areobtainedfrom . the following iteration: =𝐸[𝑒−𝑟𝑇(𝑆 −𝐾)+1 𝑇 {𝜏𝐻>𝑇} ̂ −𝑟𝜏 𝑈 (𝑡, 𝑦, V,𝑇)=𝜂(𝑡,𝑦,V) 𝑈 (𝑡, 𝑦, V,𝑇), +𝑒 𝐻 1 (𝜏 ,𝐻,V ,𝑇)|𝑆 =𝑆,V = V], 𝑛 𝑛 {𝜏𝐻≤𝑇}LB 𝐻 𝜏𝐻 0 𝑡 𝑡 1 1 (10) 𝜂(𝜏,𝑦,V)= {(−𝑞− V(𝛼 (V) +1)2)𝜏+ (𝛼 (V) +1) 𝑥} , exp 8 2 where 2(𝑟−𝑞) 1 LB (𝜏𝐻,𝐻,V𝜏 ,𝑇0)=LC (0,𝐻,𝐾,V𝜏 ,𝑇0) 𝛼 (V) = ,𝑘=(𝛼 (V) +1) , 𝐻 𝐻 V 2 (11) − (0, 𝐻, 𝐻, V ,𝑇 ). 𝜏 ∞ LC 𝜏𝐻 0 ̂ 𝑈𝑛 (𝑡, 𝑦, V,𝑇)= ∫ ∫ 𝜂 (𝑠, 𝜉, V) M𝑈𝑛−1 (𝑇−𝑠,𝜉,V,𝑇) 0 0 Let 𝑥𝑡 = ln(𝑆𝑡/𝐻). By Ito’s lemma, we obtain the SDE for 𝑥𝑡 as × 𝐺 (𝜏 − 𝑠, 𝑦,𝜉, V)𝑑𝜉𝑑𝑠,

V𝑡 𝑆 𝑆0 2 𝑑𝑥 = (𝑟−𝑞− ) + V 𝑑𝑊 ,𝑥= ( ) . 𝜕 𝜕 𝜕 1 2 𝜕 𝑡 √ 𝑡 𝑡 0 ln (12) M =𝜌𝜎V −𝜌𝜎V +𝜅(𝜃−V) + 𝜎 V , 2 𝐻 𝜕V 𝜕𝑦𝜕V 𝜕V 2 𝜕V2 Let 𝑉(𝑡, 𝑥, V)=𝑉(𝑡,̃ 𝑆, V). Applying the Feynman-Kacˇ 1 −(𝑦−𝜉)2/2V𝑡 −(𝑦+𝜉)2/2V𝑡 formula to 𝑉 in (10)withrespectto(12)and(8), we have 𝐺 (𝑡, 𝑦, 𝜉, V)= {𝑒 +𝑒 √2𝜋V𝑡 (L + M) 𝑉=0 ∞ −((𝑦+𝜉+𝜂)2/2V𝑡)+𝑘𝜂 𝑥 + +2𝑘 ∫ 𝑒 𝑑𝜂} , 𝑉 (𝑇, 𝑥, V) =(𝐻𝑒 −𝐾) (13) 0 (18) 𝑉 (𝑡, 𝑥, V)|𝑥=0 = LB (𝑡, 𝐻, V,𝑇0), where the iteration begins with the function where 𝜕 V 𝜕 1 𝜕2 𝑈 (𝑡, 𝑦, V,𝑇)=𝑦−𝑟(𝑇−𝑡) 𝑒 𝑁(−𝑑− )−𝑒−𝑞(𝑇−𝑡)𝑁(−𝑑+ ) L = +(𝑟−𝑞− ) + V −𝑟, 0 𝑚 𝑚 2 𝜕𝑡 2 𝜕𝑥 2 𝜕𝑥 V (14) + [𝑒−𝑞(𝑇−𝑡)𝑁(𝑑+ ) 𝜕 𝜕2 1 𝜕2 2(𝑟−𝑞) 𝑚 M =𝜅(𝜃−V) +𝜌𝜎V + 𝜎2V . 𝜕V 𝜕𝑥𝜕V 2 𝜕V2 −𝑟(𝑇−𝑡)+(𝑦(𝑟−𝑞)/V) 𝑟 −𝑒 𝑁(𝑑𝑚)] , In addition, the rebate LB is related to LC through (11). (19) Using the transformation of variable LC(𝑡, 𝑆, 𝑚, V,𝑇) = 𝑆 ⋅ 𝑈(𝑡, log(𝑚/𝑆), V,𝑇),thefunction𝑈 is the solution of the ± −𝑦 + (𝑟 − 𝑞± (V/2)) (𝑇−𝑡) following BVP [10]: 𝑑 = , 𝑚 √V (𝑇−𝑡) (L + M) 𝑈=0 (20) 𝑟 + 𝑇−𝑡 𝑥 𝑑 =𝑑 −2(𝑟−𝑞)√ , 𝑈 (𝑇, 𝑥, V,𝑇) =𝑒 −1 𝑚 𝑚 V (15) 󵄨 𝜕𝑈 (𝑡, 𝑥, V,𝑇)󵄨 󵄨 =0. with 𝑁(⋅) being the standard normal cumulative distribution 󵄨 𝜕𝑥 󵄨𝑥=0 function. 4 Abstract and Applied Analysis

3. The Homotopy Framework Consider the Taylor expansion of 𝑉(𝑡, 𝑥, V,𝑝)with respect to 𝑝 as follows: We aim to construct a homotopy solution 𝑉(𝑡, 𝑥, V,𝑝) with ∞ auxiliary parameter 𝑝∈[0,1]such that 𝑉(𝑡, 𝑥, V,0) = 𝑛 𝑉(𝑡,𝑥,V,𝑝)= ∑𝑉𝑛 (𝑡, 𝑥, V) 𝑝 , (25) 𝑉0(𝑡, 𝑥, V) and 𝑉(𝑡, 𝑥, V, 1) = 𝑉(𝑡, 𝑥, V), the desired pricing 𝑛=0 formula. The function 𝑉0 can be regarded as the initial guess 𝑛 𝑛 ofthesolution.WeusetheBSformulaofTCin(6)as where 𝑉𝑛(𝑡, 𝑥, V) = (1/𝑛!)(𝜕 /𝜕𝑝 )𝑉(𝑡, 𝑥, V,𝑝)|𝑝=0.Substitut- the initial guess. Therefore, 𝑉0(𝑡, 𝑥, V) is the solution of the ing (25)into(22) yields following PDE: L𝑉𝑛 (𝑡, 𝑥, V) + M𝑉𝑛−1 (𝑡, 𝑥, V) =0, L𝑉0 (𝑡, 𝑥, V) =0 𝑉 (𝑇, 𝑥, V) =0, 𝑥 + 𝑛 𝑉0 (𝑇, 𝑥, V) =(𝐻𝑒 −𝐾) 󵄨 𝐾 󵄨 󵄨 𝐾 𝑉𝑛 (𝑡, 𝑥, V)󵄨 =𝐻[𝑈𝑛 (0, 0, V,𝑇0)−𝑈𝑛 (0, ln , V,𝑇0)] , 𝑉 (𝑡, 𝑥, V)󵄨 =𝐻[𝑈 (0, 0, V,𝑇 )−𝑈 (0, , V,𝑇 )] , 󵄨𝑥=0 𝐻 0 󵄨𝑥=0 0 0 0 ln 𝐻 0 (21) for 𝑛 = 1, 2, . . ., (26) where L is defined in (14)and𝑈0 in (19). 𝑉(𝑡, 𝑥, V,𝑝) Consider the following PDE for : where 𝑈𝑛(0, 𝑦, V,𝑇0) is a known function given in (19). Let 𝜏=𝑇−𝑡.ThenthePDE(26) becomes, for 𝑛 = 1, 2, ., . −𝑝 (L + M) 𝑉=(1−𝑝)L (𝑉−𝑉0) 𝜕𝑉 1 𝜕2𝑉 V 𝜕𝑉 𝑥 + 𝑛 = V 𝑛 +(𝑟−𝑞− ) 𝑛 −𝑟𝑉 + M𝑉 𝑉(𝑇,𝑥,V,𝑝)=(𝐻𝑒 −𝐾) 𝜕𝜏 2 𝜕𝑥2 2 𝜕𝑥 𝑛 𝑛−1 󵄨 𝐾 𝑉 (𝑇, 𝑥, V) =0 𝑉(𝑡,𝑥,V,𝑝)󵄨 =𝐻[𝑈 (0,0, V,𝑇 ,𝑝)−𝑈(0, , V,𝑇 ,𝑝)], 𝑛 󵄨𝑥=0 0 ln𝐻 0 󵄨 (22) 𝑉 (𝜏, 𝑥, V)󵄨 =𝐻[𝑈 (0, 0, V,𝑇 ) 𝑛 󵄨𝑥=0 𝑛 0 where 𝑥=ln(𝑆/𝐻), 𝑈 is defined in (17), and L and M are 𝐾 ∗ defined in (14). −𝑈𝑛 (0, ln , V,𝑇0)] := 𝐻𝑛 (V) . If we set 𝑝=1in (22), then we have 𝐻 (27) (L + M) 𝑉=0 This PDE can be transformed into a heat equation and 𝑉 (𝑇, 𝑥, V,1) =(𝐻𝑒𝑥 −𝐾)+ its solution has a classical result. Specifically, the result is summarized as follows. 󵄨 𝐾 𝑉 (𝑡, 𝑥, V,1)󵄨 =𝐻[𝑈 (0, 0, V,𝑇0,1)−𝑈(0,ln , V,𝑇0,1)] 󵄨𝑥=0 𝐻 Theorem 2. Suppose the underlying asset price, 𝑆𝑡,followsthe SV model of (7) and (8).Then,theturbocallwarrantprice = LC (𝐻, 𝐻, V,𝑇0)−LC (𝐻, 𝐾, V,𝑇0) in (1) has an analytic formula derived from the iteration: for 𝑛 = 1, 2, ., . = LB (𝑡, 𝐻, V,𝑇0). 𝑉 (𝜏, 𝑥, V) =𝐻∗ (V) (𝜏, 𝑥, V) (23) 𝑛 𝑛 DRBS By comparing the systems (13)and(23), we confirm that 𝜏 ∞ + ∫ ∫ M𝑉𝑛−1 (𝑠, 𝑥, V) 𝐺 (𝜏−𝑠,𝑥,𝜉,V) 𝑑𝜉 𝑑𝑠; 𝑉(𝑡, 𝑥, V, 1) = 𝑉(𝑡, 𝑥, V). 0 0 Alternatively, setting 𝑝=0in (22)yields 󵄨 V 𝜏 𝜕 󵄨 (𝜏, 𝑥, V) = ∫ 𝐺 (𝜏−𝑠,𝑥,𝜉,V)󵄨 𝑑𝑠; DRBS 󵄨 L𝑉 (𝑡, 𝑥, V,0) = L𝑉0 (𝑡, 𝑥, V) 2 0 𝜕𝜉 󵄨𝜉=0

𝑥 + 1 2 𝑉 (𝑇, 𝑥, V,0) =(𝐻𝑒 −𝐾) 𝐺 (𝑡, 𝑥, 𝜉, V) = 𝑒((𝑟−𝑞−(V/2))(𝜉−𝑥)/V)−(𝑟+((𝑟−𝑞−(V/2)) /2V))𝑡 √2𝜋V𝑡 󵄨 𝐾 𝑉 (𝑡, 𝑥, V,0)󵄨 =𝐻[𝑈 (0, 0, V,𝑇0,0)−𝑈(0,ln , V,𝑇0,0)] 2 2 󵄨𝑥=0 𝐻 ×(𝑒−(𝑥−𝜉) /2V𝑡 −𝑒−(𝑥+𝜉) /2V𝑡). 𝐾 =𝐻[𝑈 (0, 0, V,𝑇 )−𝑈 (0, , V,𝑇 )] , (28) 0 0 0 ln𝐻 0 (24) An analytic pricing formula of TC under SV model is then given by 𝑈 (0, 𝑦, V,𝑇 ) where 0 0 is obtained in (19). Therefore, ∞ 𝑉(𝑡, 𝑥, V,0) = 𝑉0(𝑡, 𝑥, V).Hence,thePDE(22) is desired 𝑉 (𝑡, 𝑥, V) = 𝑉 (𝜏, 𝑥, V,1) = ∑𝑉𝑛 (𝜏, 𝑥, V) . (29) construction for the homotopy solution. 𝑛=0 Abstract and Applied Analysis 5

3.1. DOC and DIL. As the TC price paying a lookback References rebate is a sum of the DOC and DIL prices, the solution of 𝑉(𝑡, 𝑥, V,𝑝) from (25)and(28) is useful to determine the [1] The HKEx callable bull/bear contract (CBBC) market, Report by Research & Corporate Development Department of Hong DOC option price and DIL option price under SV. Hence, Kong Exchanges and Clearing Limited, 2009, http://www.hkex our solution also contributes to the valuation of barrier-type .com.hk/eng/stat/research/rpaper/Documents/cbbc2009.pdf. options under SV model. Note that the N-CBBC is actually a [2]H.Y.WongandC.M.Chan,“Turbowarrantsunderstochastic DOC option. volatility,” Quantitative Finance,vol.8,no.7,pp.739–751,2008. For DOC option with barrier level 𝐻,thecorresponding [3] H. Y. Wong and K. Y. Lau, “Analytical valuation of turbo PDE is the same as (13)exceptthattheboundarycondition 𝑉(𝑡, 𝑥, V)| =0 warrants under double exponential jump diffusion,” Journal of is replaced by 𝑥=0 .Using(28)and(29), we Derivatives,pp.61–73,2008. immediately obtain that [4] H. Y. Wong and K. Y. Lau, “Path-dependent currency options ∞ with mean reversion,” The Journal of Futures Markets,vol.28, DOC (𝑡, 𝑥, V,𝑇) = ∑DOC𝑛 (𝜏, 𝑥, V) , (30) pp. 275–293, 2008. 𝑛=0 [5]J.M.OrtegaandW.C.Rheinboldt,IterativeSolutionofNonlin- where ear Equations in Several Variables, Academic Press, New York, 𝜏 ∞ NY, USA, 1970. DOC𝑛 (𝜏, 𝑥, V) = ∫ ∫ MDOC𝑛−1 (𝑠, 𝑥, V) [6] S. J. Liao, The proposed homotopy analysis technique for the 0 0 solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong ×𝐺(𝜏−𝑠,𝑥,𝜉,V) 𝑑𝜉 𝑑𝑠, (31) University, 1992. [7] S.-P. Zhu, “An exact and explicit solution for the valuation of (𝜏, 𝑥, V) = (𝑡, 𝑥, V,𝑇) . American put options,” Quantitative Finance,vol.6,no.3,pp. DOC0 DOCBS 229–242, 2006. Here, we see that the DOC price under SV model from our [8]J.ZhaoandH.Y.Wong,“Aclosed-formsolutiontoAmer- approach is consistent with the result by Park and Kim [9]. ican options under general diffusion processes,” Quantitative For DIL option with barrier level 𝐻,thecorresponding Finance,vol.12,no.5,pp.725–737,2012. PDE is the same as (13) except that the terminal condition is [9] S.-H. Park and J.-H. Kim, “Homotopy analysis method for replaced by 𝑉(𝑇, 𝑥, V)=0.Using(28)and(29), we obtain that option pricing under stochastic volatility,” Applied Mathematics Letters,vol.24,no.10,pp.1740–1744,2011. ∞ [10] K. S. Leung, “An analytic pricing formula to lookback options DIL (𝑡, 𝑥, V,𝑇) = ∑DIL𝑛 (𝜏, 𝑥, V) , (32) 𝑛=0 under stochastic volatility,” Applied Mathematics Letters,vol.26, pp.145–149,2013. where [11] Y.-K. Kwok, Mathematical Models of Financial Derivatives, (𝜏, 𝑥, V) =𝐻∗ (V) (𝜏, 𝑥, V) Springer, Berlin, Germany, 2008. DIL𝑛 𝑛 DRBS 𝜏 ∞ + ∫ ∫ MDIL𝑛−1 (𝑠, 𝑥, V) 0 0 (33) ×𝐺(𝜏−𝑠,𝑥,𝜉,V) 𝑑𝜉 𝑑𝑠,

(𝜏, 𝑥, V) =𝐻∗ (V) (𝜏, 𝑥, V) . DIL0 0 DRBS

4. Conclusion WeuseaPDEapproachtosolvethepriceofturbowarrant underaSVmodel.ThePDEissolvedbymeansofhomotopy analysis method. The boundary condition of the PDE is simplified using the homotopy solution developed in10 [ ]. As byproducts, we offer analytic pricing formulas for DOC and DIL options under SV model. Future research can apply this solution to investigate the impact of volatility to the prices of turbo warrants empirically.

Acknowledgments H. Y. Wong acknowledges the support by GRF of Research Grant Council of Hong Kong with Project n. 403511. M. C. Chiu acknowledges the Start-up Research Grant RG44/2012- 2013R by Department of MIT, Hong Kong Institute of Education. Copyright of Abstract & Applied Analysis is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.