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

Abstract

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 by the 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

Turbo warrant first appears in Europe but is now available under various names in many markets including the UK, Germany, Switzerland, Italy, Australia, New Zealand, Singapore, South Africa, Taiwan, and Hong Kong. For instance, it is called turbo warrant in the Nordic Growth market, contract for difference (CFD) in the UK, and callable bull/bear contract (CBBC) in Hong Kong. According to a report by Hong Kong Exchanges and Clearing Limited (HKEx) in 2009 [1], the UK has a big OTC market for CFDs while Hong Kong has a big exchange market for CBBC. Typically, HKEx listed 1,525 CBBCs, constituting 28% of the total number of its listed securities, and the total issued amount reached HK$704 billion or US$95 billion at the end of May 2009 [1]. Hong Kong has two types of CBBC: N and R. The N-CBBC pays no rebate when a preselected barrier is crossed whereas the R-CBBC pays a lookback rebate. If the barrier is not breached by the underlying asset price during the life of the contract, then the turbo call (put) is equivalent to the standard call (put). Therefore, turbo warrant resembles a combination of barrier and lookback options.

A comprehensive mathematical treatment of turbo warrants is given in [2], which provides several model-free properties of turbo warrants. This leads to subsequent extensions to turbo warrant pricing with jump-diffusion model in [3] and with mean reversion in [4]. Although Wong and Chan [2] approximate turbo prices under a stochastic volatility (SV) model, their solution is restricted to the assumption that the mean-reverting speed of the stochastic volatility should either be close to zero or to infinity. Our goal is to derive an analytic solution to turbo prices by relaxing such an assumption using a homotopy analysis method.

Homotopy analysis method was introduced by Ortega and Rheinboldt [5] in 1970 and has been applied to solve many nonlinear problems since the work of Liao [6] in 1992. Zhu in [7] pioneers the use of homotopy analysis method in financial mathematics and derives the first analytic formula to American options. Zhao and Wong in [8] show that the approach of Zhu is also applicable to general diffusion models.

While general diffusion models belong to the class of complete market models, SV models are of incomplete market models because the number of Brownian motions driving the asset dynamics is larger than one. Park and Kim [9] apply homotopy analysis method to solve vanilla option and barrier option prices under SV models. Leung [10] extends the framework to lookback option pricing. However, the application of homotopy analysis method to turbo warrant pricing under SV models is yet to be considered.

This paper employs the homotopy analysis method to solve the PDE for the turbo warrant price under a SV model. As the price of turbo warrant under the Black-Scholes model is available, we construct a homotopy which deforms from the Black-Scholes solution to the desired solution under the SV model. We highlight the fundamental challenge in turbo warrant pricing. A turbo warrant consists of a barrier option and a lookback rebate. Although the barrier option pricing under SV models is investigated in [9], Park and Kim do not consider rebate in knockout options. Our solution should cover the case of state-dependent rebate under SV. Second, Leung [10] offers an analytic pricing formula for lookback options but his formula cannot be directly used in calculating the lookback rebate, which is an expectation on the discounted lookback option with a random starting time. Therefore, our homotopy analysis method has to simultaneously solve these two problems together.

The reminder of this paper is organized as follows. Section 2 presents the nature of turbo warrants under SV models and the corresponding pricing problem in a PDE approach. Section 3 contains the main result of this paper and solves the PDE using homotopy analysis method. Concluding remark is made in Section 4.

2. Problem Formulation

Let 𝑆_𝑡 be the underlying asset price at time 𝑡. A turbo call (put) warrant pays the contract holder (𝑆_𝑇−𝐾)⁺ at maturity 𝑇 if a specified barrier 𝐻≥𝐾 has not been passed by 𝑆_𝑡 at any time prior to the maturity. Denote 𝜏_𝐻 as the first passage time that the asset price crosses the barrier 𝐻, that is, 𝜏_𝐻=inf{𝑡∶𝑆_𝑡≤𝐻}. If 𝜏_𝐻≤𝑇; then the contract is void and a new contract starts. The new contract is a call option on 𝑚^𝑇0_𝜏𝐻=min_{𝜏𝐻≤𝑡≤𝜏𝐻+𝑇0}𝑆_𝑡, with the strike price 𝐾, and the time to maturity 𝑇₀. More precisely, the turbo call (TC) at time 𝑡 can be expressed as TC_𝑡=𝐸_𝑡𝑒^−𝑟𝑇𝑆_𝑇−𝐾⁺𝟏_{{𝜏𝐻>𝑇}}+𝑒^{−𝑟(𝜏𝐻+𝑇0)}𝑚^𝑇0_𝜏𝐻−𝐾⁺𝟏_{{𝜏𝐻≤𝑇}}, (1) where 𝑟 is the constant interest rate and 𝐸_𝑡 represents the risk-neutral expectation given information up to time 𝑡. Wong and Chan [2] explain the incentive of this security design and show a model-free representation that the turbo call warrant is sum of a down-and-out call (DOC) and an expectation of a nonstandard lookback option (LB) starting at 𝜏_𝐻.

Proposition 1 (see [2]). At 𝑡≤𝜏_𝐻, the model-free representation of the turbo call warrant is 𝑇𝐶(𝑡,𝑆)=𝐷𝑂𝐶(𝑡,𝑆)+𝐸_𝑡𝑒^{−𝑟𝜏𝐻}𝟏_{{𝜏𝐻≤𝑇}}𝐿𝐵𝜏_𝐻,𝑆_𝜏𝐻,𝑇₀, (2) where 𝐷𝑂𝐶(𝑡,𝑆) denotes the down-and-out call option price at 𝑡, 𝐿𝐵𝜏_𝐻,𝑆_𝜏𝐻,𝑇₀=𝐸_𝜏𝐻𝑒^−𝑟𝑇0𝑚^𝑇0_𝜏𝐻−𝐾⁺𝟏_{{𝜏𝐻≤𝑇}}=𝐿𝐶0,𝑆_𝜏𝐻,min𝑆_𝜏𝐻,𝐾,𝑣_𝜏𝐻,𝑇₀−𝐿𝐶0,𝑆_𝜏𝐻,𝑆_𝜏𝐻,𝑣_𝜏𝐻,𝑇₀, (3) and LC(𝑡,𝑆,𝑚,𝑣,𝑇₀) is the floating strike lookback call at time 𝑡 on 𝑆 with the realized minimum 𝑚, instantaneous volatility 𝑣, and time to maturity 𝑇₀.

In Proposition 1, the floating strike lookback call, LC(𝑡,𝑆,𝑚,𝑣,𝑇₀), has the payoff 𝑆_𝑇0−min_{0≤𝜏≤𝑇0}𝑆_𝜏. (4) Hence, its price has the representation, LC𝑡,𝑆,𝑚,𝑣,𝑇₀=𝑒^{−𝑟(𝑇0−𝑡)}𝐸_𝑡𝑆_𝑇0−min_{0≤𝜏≤𝑇0}𝑆_𝜏. (5) The term 𝐸_𝑡[𝑒^{−𝑟𝜏𝐻}𝟏_{{𝜏𝐻≤𝑇}}LB(𝜏_𝐻,𝑆_𝜏𝐻,𝑇₀)] is called the down-and-in lookback (DIL) option in [2] as a TC holder will knock in the lookback option LB, shown in (3), only when the underlying asset price hits the down-side barrier 𝐻.

In particular, if the asset price process follows the Black-Scholes (BS) model where the volatility is a constant, then the turbo call price at 𝑡∈[0,𝜏_𝐻] reduces to TC_BS(𝑡,𝑆)=DOC_BS()+𝑡,𝑆LC_BS0,𝐻,𝐾,𝑇₀−LC_BS0,𝐻,𝐻,𝑇₀DR_BS(𝑡,𝑆), (6) where the Black-Scholes pricing formulas of DOC_BS, LC_BS, and DR_BS=𝐸_𝑡[𝑒^{−𝑟𝜏𝐻}𝟏_{{𝜏𝐻≤𝑇}}] are classical results and can be found in [11]. However, the formulas of LC_BS and DR_BS also appear in some proofs of this paper. Specifically, LC_BS(𝑡,𝑆,𝐾,𝑇₀)=𝑆𝑈₀(𝑡,𝑦,𝑣,𝑇₀), where 𝑈₀ is given by (19) while DR_BS appears in (28).

2.1. Turbos under Stochastic Volatility

Although the explicit BS formula for the TC is known as in (6), its analytic pricing formula under SV model is yet to be considered. The Heston SV model assumes the following stochastic differential equation (SDE) for the underlying asset price 𝑑𝑆_𝑡𝑆_𝑡=𝑟−𝑞𝑑𝑡+√𝑣_𝑡𝑑𝑊^𝑆_𝑡, (7) where 𝑟 is the constant interest rate, 𝑞 is the constant dividend yield, and 𝑣_𝑡 is the stochastic instantaneous variance of the asset. The stochastic variance 𝑣_𝑡 follows the SDE: 𝑑𝑣_𝑡=𝜅𝜃−𝑣_𝑡𝑑𝑡+𝜎√𝑣_𝑡𝑑𝑊^𝑣_𝑡, (8) where 𝜅, 𝜃, and 𝜎 are constants. In (7) and (8), 𝑊^𝑆_𝑡 and 𝑊^𝑣_𝑡 are Wiener processes defined on a filtered complete probability space (Ω,ℱ,𝒬,ℱ_𝑡≥0) with 𝐸[𝑊^𝑆_𝑡𝑊^𝑣_𝑡]=𝜌𝑡, where 𝒬 denotes the risk-neutral probability measure.

Let 𝑉(𝑡,𝑆,𝑣) be the turbo warrant price at time 𝑡 under the stochastic volatility 𝑣. Using (1), we have 𝑉()𝑒𝑡,𝑆,𝑣=𝐸^−𝑟𝑇𝑆_𝑇−𝐾⁺𝟏_{{𝜏𝐻>𝑇}}+𝑒^{−𝑟(𝜏𝐻+𝑇0)}𝑚^𝑇0_𝜏𝐻−𝐾⁺𝟏_{{𝜏𝐻≤𝑇}}∣𝑆_𝑡=𝑆,𝑣_𝑡=𝑣. (9) By Proposition 1, the turbo price can be expressed as (2). As the SV model assumes a continuous process for the underlying asset price, 𝑆_𝜏𝐻=𝐻. Hence, we have 𝑉()𝑒𝑡,𝑆,𝑣=𝐸^−𝑟𝑇𝑆_𝑇−𝐾⁺𝟏_{{𝜏𝐻>𝑇}}+𝑒^{−𝑟𝜏𝐻}𝟏_{{𝜏𝐻≤𝑇}}LB𝜏_𝐻,𝐻,𝑣_𝜏𝐻,𝑇₀∣𝑆_𝑡=𝑆,𝑣_𝑡=𝑣, (10) where LB𝜏_𝐻,𝐻,𝑣_𝜏𝐻,𝑇₀=LC0,𝐻,𝐾,𝑣_𝜏𝐻,𝑇₀−LC0,𝐻,𝐻,𝑣_𝜏𝐻,𝑇₀. (11)

Let 𝑥_𝑡=ln(𝑆_𝑡/𝐻). By Ito’s lemma, we obtain the SDE for 𝑥_𝑡 as 𝑑𝑥_𝑡=𝑣𝑟−𝑞−^𝑡2+√𝑣_𝑡𝑑𝑊^𝑆_𝑡,𝑥₀𝑆=ln⁰𝐻. (12) Let 𝑉(𝑡,𝑥,𝑣)=𝑉(𝑡,𝑆,𝑣). Applying the Feynman-Kač formula to 𝑉 in (10) with respect to (12) and (8), we have ()𝑉()=ℒ+ℳ𝑉=0𝑇,𝑥,𝑣𝐻𝑒^𝑥−𝐾⁺𝑉(𝑡,𝑥,𝑣)|_𝑥=0=LB𝑡,𝐻,𝑣,𝑇₀, (13) where 𝜕ℒ=+𝑣𝜕𝑡𝑟−𝑞−2𝜕+1𝜕𝑥2𝑣𝜕²𝜕𝑥²()𝜕−𝑟,ℳ=𝜅𝜃−𝑣𝜕𝜕𝑣+𝜌𝜎𝑣²+1𝜕𝑥𝜕𝑣2𝜎²𝑣𝜕²𝜕𝑣². (14) In addition, the rebate LB is related to LC through (11). Using the transformation of variable LC(𝑡,𝑆,𝑚,𝑣,𝑇)=𝑆⋅𝑈(𝑡,log(𝑚/𝑆),𝑣,𝑇), the function 𝑈 is the solution of the following BVP [10]: ()𝑈()ℒ+ℳ𝑈=0𝑇,𝑥,𝑣,𝑇=𝑒^𝑥()−1𝜕𝑈𝑡,𝑥,𝑣,𝑇𝜕𝑥_𝑥=0=0. (15)

In principal, the exotic lookback option LB(𝑡,𝑆,𝑣,𝑇) should be solved from an alternative PDE governing lookback options. Fortunately, we know from (11) that this exotic lookback option is the difference between two floating strike lookback calls.

Specifically, LC(𝑡,𝑆,𝑚,𝑣,𝑇) has the following homotopy expression [10]: LC(𝑡,𝑆,𝑚,𝑣,𝑇)=𝑆⋅𝑈𝑡,𝑦,𝑣,𝑇,𝑝_𝑝=1, (16) where 𝑦=ln(𝑚/𝑆); 𝑈𝑡,𝑦,𝑣,𝑇,𝑝=^∞_𝑛=0𝑈_𝑛𝑡,𝑦,𝑣,𝑇𝑝^𝑛. (17) The functions 𝑈_𝑛(𝑡,𝑦,𝑣,𝑇),  𝑛=1,2,…, are obtained from the following iteration: 𝑈_𝑛𝑡,𝑦,𝑣,𝑇=𝜂𝑡,𝑦,𝑣𝑈_𝑛,𝜂1𝑡,𝑦,𝑣,𝑇𝜏,𝑦,𝑣=exp−𝑞−8𝑣(𝛼(𝑣)+1)²1𝜏+2(𝛼(𝑣))𝑥,𝛼(𝑣)=2+1𝑟−𝑞𝑣1,𝑘=2(𝛼(𝑣)),𝑈+1_𝑛𝑡,𝑦,𝑣,𝑇=^𝜏₀^∞₀𝜂(𝑠,𝜉,𝑣)𝔐𝑈_𝑛−1()×𝑇−𝑠,𝜉,𝑣,𝑇𝐺𝜕𝜏−𝑠,𝑦,𝜉,𝑣𝑑𝜉𝑑𝑠,𝔐=𝜌𝜎𝑣𝜕𝜕𝑣−𝜌𝜎𝑣()𝜕𝜕𝑦𝜕𝑣+𝜅𝜃−𝑣+1𝜕𝑣2𝜎²𝑣𝜕²𝜕𝑣²,𝐺=1𝑡,𝑦,𝜉,𝑣√𝑒2𝜋𝑣𝑡^{−(𝑦−𝜉)2/2𝑣𝑡}+𝑒^{−(𝑦+𝜉)2/2𝑣𝑡}+2𝑘^∞₀𝑒^{−((𝑦+𝜉+𝜂)2/2𝑣𝑡)+𝑘𝜂}𝑑𝜂, (18) where the iteration begins with the function 𝑈₀𝑡,𝑦,𝑣,𝑇=𝑒^{𝑦−𝑟(𝑇−𝑡)}𝑁−𝑑⁻_𝑚−𝑒^{−𝑞(𝑇−𝑡)}𝑁−𝑑⁺_𝑚+𝑣2𝑒𝑟−𝑞^{−𝑞(𝑇−𝑡)}𝑁𝑑⁺_𝑚−𝑒^{−𝑟(𝑇−𝑡)+(𝑦(𝑟−𝑞)/𝑣)}𝑁𝑑^𝑟_𝑚,𝑑^±_𝑚=−𝑦+𝑟−𝑞±(𝑣/2)(𝑇−𝑡)𝑣(),𝑑𝑇−𝑡_𝑟𝑚=𝑑⁺_𝑚−2𝑟−𝑞𝑇−𝑡𝑣, (19)(20) with 𝑁(⋅) being the standard normal cumulative distribution function.

3. The Homotopy Framework

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)=𝑉₀(𝑡,𝑥,𝑣) and 𝑉(𝑡,𝑥,𝑣,1)=𝑉(𝑡,𝑥,𝑣), the desired pricing formula. The function 𝑉₀ can be regarded as the initial guess of the solution. We use the BS formula of TC in (6) as the initial guess. Therefore, 𝑉₀(𝑡,𝑥,𝑣) is the solution of the following PDE: ℒ𝑉₀()𝑉𝑡,𝑥,𝑣=0₀(𝑇,𝑥,𝑣)=𝐻𝑒^𝑥−𝐾⁺𝑉₀(𝑡,𝑥,𝑣)_𝑥=0=𝐻𝑈₀0,0,𝑣,𝑇₀−𝑈₀𝐾0,ln𝐻,𝑣,𝑇₀, (21) where ℒ is defined in (14) and 𝑈₀ in (19).

Consider the following PDE for 𝑉(𝑡,𝑥,𝑣,𝑝): −𝑝(ℒ+ℳ)𝑉=1−𝑝ℒ𝑉−𝑉₀𝑉𝑇,𝑥,𝑣,𝑝=𝐻𝑒^𝑥−𝐾⁺𝑉𝑡,𝑥,𝑣,𝑝_𝑥=0=𝐻𝑈0,0,𝑣,𝑇₀,𝑝−𝑈𝐾0,ln𝐻,𝑣,𝑇₀,𝑝, (22) where 𝑥=ln(𝑆/𝐻), 𝑈 is defined in (17), and ℒ and ℳ are defined in (14).

If we set 𝑝=1 in (22), then we have (ℒ+ℳ)𝑉=0𝑉(𝑇,𝑥,𝑣,1)=𝐻𝑒^𝑥−𝐾⁺𝑉(𝑡,𝑥,𝑣,1)_𝑥=0=𝐻𝑈0,0,𝑣,𝑇₀,1−𝑈𝐾0,ln𝐻,𝑣,𝑇₀,1=LC𝐻,𝐻,𝑣,𝑇₀−LC𝐻,𝐾,𝑣,𝑇₀=LB𝑡,𝐻,𝑣,𝑇₀. (23) By comparing the systems (13) and (23), we confirm that 𝑉(𝑡,𝑥,𝑣,1)=𝑉(𝑡,𝑥,𝑣).

Alternatively, setting 𝑝=0 in (22) yields ℒ𝑉(𝑡,𝑥,𝑣,0)=ℒ𝑉₀(𝑡,𝑥,𝑣)𝑉(𝑇,𝑥,𝑣,0)=𝐻𝑒^𝑥−𝐾⁺𝑉(𝑡,𝑥,𝑣,0)_𝑥=0=𝐻𝑈0,0,𝑣,𝑇₀,0−𝑈𝐾0,ln𝐻,𝑣,𝑇₀𝑈,0=𝐻₀0,0,𝑣,𝑇₀−𝑈₀𝐾0,ln𝐻,𝑣,𝑇₀, (24) where 𝑈₀(0,𝑦,𝑣,𝑇₀) is obtained in (19). Therefore, 𝑉(𝑡,𝑥,𝑣,0)=𝑉₀(𝑡,𝑥,𝑣). Hence, the PDE (22) is desired construction for the homotopy solution.

Consider the Taylor expansion of 𝑉(𝑡,𝑥,𝑣,𝑝) with respect to 𝑝 as follows: 𝑉𝑡,𝑥,𝑣,𝑝=^∞_𝑛=0𝑉_𝑛(𝑡,𝑥,𝑣)𝑝^𝑛, (25) where 𝑉_𝑛(𝑡,𝑥,𝑣)=(1/𝑛!)(𝜕^𝑛/𝜕𝑝^𝑛)𝑉(𝑡,𝑥,𝑣,𝑝)|_𝑝=0. Substituting (25) into (22) yields ℒ𝑉_𝑛(𝑡,𝑥,𝑣)+ℳ𝑉_𝑛−1(𝑡,𝑥,𝑣)=0,𝑉_𝑛(𝑇,𝑥,𝑣)=0,𝑉_𝑛(𝑡,𝑥,𝑣)_𝑥=0=𝐻𝑈_𝑛0,0,𝑣,𝑇₀−𝑈_𝑛𝐾0,ln𝐻,𝑣,𝑇₀,for𝑛=1,2,…, (26) where 𝑈_𝑛(0,𝑦,𝑣,𝑇₀) is a known function given in (19). Let 𝜏=𝑇−𝑡. Then the PDE (26) becomes, for 𝑛=1,2,…, 𝜕𝑉_𝑛=1𝜕𝜏2𝑣𝜕²𝑉_𝑛𝜕𝑥²+𝑣𝑟−𝑞−2𝜕𝑉_𝑛𝜕𝑥−𝑟𝑉_𝑛+ℳ𝑉_𝑛−1𝑉_𝑛(𝑇,𝑥,𝑣)=0𝑉_𝑛(𝜏,𝑥,𝑣)_𝑥=0=𝐻𝑈_𝑛0,0,𝑣,𝑇₀−𝑈_𝑛𝐾0,ln𝐻,𝑣,𝑇₀∶=𝐻^∗_𝑛(𝑣). (27) This PDE can be transformed into a heat equation and its solution has a classical result. Specifically, the result is summarized as follows.

Theorem 2. Suppose the underlying asset price, 𝑆_𝑡, follows the SV model of (7) and (8). Then, the turbo call warrant price in (1) has an analytic formula derived from the iteration: for 𝑛=1,2,…, 𝑉_𝑛(𝜏,𝑥,𝑣)=𝐻^∗_𝑛(𝑣)𝐷𝑅_𝐵𝑆(𝜏,𝑥,𝑣)+^𝜏₀^∞₀ℳ𝑉_𝑛−1(𝑠,𝑥,𝑣)𝐺(𝜏−𝑠,𝑥,𝜉,𝑣)𝑑𝜉𝑑𝑠;𝐷𝑅_𝐵𝑆()=𝑣𝜏,𝑥,𝑣2^𝜏₀𝜕𝐺(𝜕𝜉𝜏−𝑠,𝑥,𝜉,𝑣)_𝜉=0𝐺()=1𝑑𝑠;𝑡,𝑥,𝜉,𝑣√𝑒2𝜋𝑣𝑡^{((𝑟−𝑞−(𝑣/2))(𝜉−𝑥)/𝑣)−(𝑟+((𝑟−𝑞−(𝑣/2))2/2𝑣))𝑡}×𝑒^{−(𝑥−𝜉)2/2𝑣𝑡}−𝑒^{−(𝑥+𝜉)2/2𝑣𝑡}. (28) An analytic pricing formula of TC under SV model is then given by 𝑉(𝑡,𝑥,𝑣)=𝑉(𝜏,𝑥,𝑣,1)=^∞_𝑛=0𝑉_𝑛(𝜏,𝑥,𝑣). (29)

3.1. DOC and DIL

As the TC price paying a lookback 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 DOC option price and DIL option price under SV. Hence, our solution also contributes to the valuation of barrier-type options under SV model. Note that the N-CBBC is actually a DOC option.

For DOC option with barrier level 𝐻, the corresponding PDE is the same as (13) except that the boundary condition is replaced by 𝑉(𝑡,𝑥,𝑣)|_𝑥=0=0. Using (28) and (29), we immediately obtain that DOC(𝑡,𝑥,𝑣,𝑇)=^∞_𝑛=0DOC_𝑛(𝜏,𝑥,𝑣), (30) where DOC_𝑛(𝜏,𝑥,𝑣)=^𝜏₀^∞₀ℳDOC_𝑛−1()()𝑠,𝑥,𝑣×𝐺𝜏−𝑠,𝑥,𝜉,𝑣𝑑𝜉𝑑𝑠,DOC₀(𝜏,𝑥,𝑣)=DOC_BS(𝑡,𝑥,𝑣,𝑇). (31) Here, we see that the DOC price under SV model from our approach is consistent with the result by Park and Kim [9].

For DIL option with barrier level 𝐻, the corresponding PDE is the same as (13) except that the terminal condition is replaced by 𝑉(𝑇,𝑥,𝑣)=0. Using (28) and (29), we obtain that DIL(𝑡,𝑥,𝑣,𝑇)=^∞_𝑛=0DIL_𝑛(𝜏,𝑥,𝑣), (32) where DIL_𝑛(𝜏,𝑥,𝑣)=𝐻^∗_𝑛(𝑣)DR_BS(𝜏,𝑥,𝑣)+^𝜏₀^∞₀ℳDIL_𝑛−1()()𝑠,𝑥,𝑣×𝐺𝜏−𝑠,𝑥,𝜉,𝑣𝑑𝜉𝑑𝑠,DIL₀(𝜏,𝑥,𝑣)=𝐻^∗₀(𝑣)DR_BS(𝜏,𝑥,𝑣). (33)

4. Conclusion

We use a PDE approach to solve the price of turbo warrant under a SV model. The PDE is solved by means of homotopy analysis method. The boundary condition of the PDE is simplified using the homotopy solution developed in [10]. 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.