^{1}

^{2}

^{1}

^{2}

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.

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 [

A comprehensive mathematical treatment of turbo warrants is given in [

Homotopy analysis method was introduced by Ortega and Rheinboldt [

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 [

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 [

The reminder of this paper is organized as follows. Section

Let

At

In Proposition

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

Although the explicit BS formula for the TC is known as in (

Let

Let

In principal, the exotic lookback option

Specifically,

We aim to construct a homotopy solution

Consider the following PDE for

If we set

Alternatively, setting

Consider the Taylor expansion of

Suppose the underlying asset price,

As the TC price paying a lookback rebate is a sum of the DOC and DIL prices, the solution of

For DOC option with barrier level

For DIL option with barrier level

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 [

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.