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

and Applied Analysis 3 space (Ω,F,Q,F t≥0 ) with E[WS t W V t ] = ρt, where Q denotes the risk-neutral probability measure. Let ?̃?(t, S, V) be the turbo warrant price at time t under the stochastic volatility V. Using (1), we have


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.

Problem Formulation
Let   be the underlying asset price at time .A turbo call (put) warrant pays the contract holder (  − ) + at maturity  if a specified barrier  ≥  has not been passed by   at any time prior to the maturity.Denote   as the first passage time that the asset price crosses the barrier , that is,   = inf{ :   ≤ }.If   ≤ ; then the contract is void and a new contract starts.The new contract is a call option on   0   = min   ≤≤  + 0   , with the strike price , and the time to maturity  0 .More precisely, the turbo call (TC) at time  can be expressed as where  is the constant interest rate and   represents the riskneutral expectation given information up to time .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   .
Proposition 1 (see [2]).At  ≤   , the model-free representation of the turbo call warrant is where DOC(, ) denotes the down-and-out call option price at , and LC(, , , V,  0 ) is the floating strike lookback call at time  on  with the realized minimum , instantaneous volatility V, and time to maturity  0 .
In Proposition 1, the floating strike lookback call, LC(, , , V,  0 ), has the payoff Hence, its price has the representation, The term   [ −  1 {  ≤} LB(  ,    ,  0 )] is called the downand-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 .

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 where  is the constant interest rate,  is the constant dividend yield, and V  is the stochastic instantaneous variance of the asset.The stochastic variance V  follows the SDE: where , , and  are constants.In ( 7) and ( 8),    and  V  are Wiener processes defined on a filtered complete probability space (Ω, F, Q, F ≥0 ) with [    V  ] = , where Q denotes the risk-neutral probability measure.
Let Ṽ(, , V) be the turbo warrant price at time  under the stochastic volatility V. Using (1), we have By Proposition 1, the turbo price can be expressed as (2).
As the SV model assumes a continuous process for the underlying asset price,    = .Hence, we have where Let   = ln(  /).By Ito's lemma, we obtain the SDE for   as Let (, , V) = Ṽ(, , V).Applying the Feynman-Kač formula to  in (10) with respect to (12) and ( 8), we have where In addition, the rebate LB is related to LC through (11).
Using the transformation of variable LC(, , , V, ) =  ⋅ (, log(/), V, ), the function  is the solution of the following BVP [10]: In principal, the exotic lookback option LB(, , V, ) 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.
Theorem 2. Suppose the underlying asset price,   , 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  = 1, 2, . .., An analytic pricing formula of TC under SV model is then given by 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, ) 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 , the corresponding PDE is the same as (13) except that the boundary condition is replaced by (, , V)| =0 = 0. Using (28) and ( 29 (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 , the corresponding PDE is the same as (13) except that the terminal condition is replaced by (, , V) = 0. Using (28) and ( 29 (33)

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.