1 INTRODUCTION
Exotic options traded in the over-the-counter market has become increasingly important since the early 1980s and is larger than the exchange-traded market. An advantage of exotic options is that they can be tailored by a financial institution to meet the particular needs of a corporate treasurer or fund manager. One of them is called shout options [1] given the holder to shout to the writer once or more times according to specified rules during its life. At the end of the life of the option, the option holder receives either the usual payoff from a European option or the intrinsic value at the time of the shout, whichever is greater. The valuation and hedging of shout options is more complicated than that of standard options because there is an element of uncertainty in the investor’s actions.
Up to present, academic research on shout options has not been extensive. Thomas [1] described the simple type of shout options in 1993. Cheuk and Vorst [2] considered shout options with multiple exercise opportunities and presented explicit type methods for pricing these types of derivatives under the assumption that the underlying asset followed geometric Brownian motion in 1997. Boyle et al [9] used Monte Carlo simulation to deal with Greek function integrals and dynamic programming to price 11*Corresponding author:Jun Liu E-mail address: junliu7903@126.com complicated shout options in 1999.In his paper Boyle presented when the number of factors exceeds four, the complication of this method far exceeds that of numerical partial differential method. Windcliff et al. [4] solved a system of interdependent linear complementarity problem to value the shout options and considered numerical issues related to interpolation and choice of time stepping method in detail in 2001.Dai and Kwok [6] developed a linear complementarity problem to analyze shout options and provided shouting boundaries using the binomial scheme and recursive integration approach in 2004. Goard [8] derived exact solutions for both the price of the shout call option and the strike reset put option where they each have a single shout right during the life of the contract in 2012. Ballestra and Cecere [21] presents a new numerical method for solving the linear complementarity problem controlled by partial integro-differential equations in 2016. Mallier and Goard [5] used an integral equation method to value shout options and found the behavior of the optimal exercise boundary for one and two shout options close to maturity in 2018.
More complicated shout options were embedded in other financial products, such as segregated funds sold by Canadian life insurance companies. These products provided a guarantee for the holders to permit to reset shout times, up to some limit during the life of the contract. It is also worth noting that some energy derivative contracts have included a feature called swing options [3], which is similar in many respects to complicated shout options. About these contracts embedded with shout options Windcliff et al. [7] explored the valuation using an approach based on the numerical solution of a set of linear complementarity problems in 2001.In his paper he indicated the shout option components of many of these contracts may be underpriced.
During last decade there have been some literature to propose and extend high-order difference approach for solving the partial integro-differential equation (PDE) arising from option pricing. Düring and Fournié [12] derived a high-order difference scheme for option pricing in Heston model in 2012 .They extended this method to non-uniform grids in 2014[13] and multiple space dimensions in 2015[14]. In 2019 Düring and Pitkin [15] applied this approach to extend to stochastic volatility jump modes for option pricing. The advantage of this approach is that it is very parsimonious in terms of memory requirements and computational effort and is more efficient than finite element approaches for option pricing [15].
About Howard algorithm (also called policy iteration [17]), Howard [16] proposed this technique for the solution of the Hamilton-Jacobi-Bellman(HJB)equations in finance in 1960.Thakoor et al. [18] developed a new procedure for the linear complementarity formulation and used Howard’s algorithm to solve the discrete problem obtained through a higher-order Crandall-Douglas discretization in 2019.The advantage of this algorithm is to ensure convergence for solution of the discretized equations under sufficient conditions[17].
The purpose of this paper is to evaluate complicated shout options more accurately. So far research on shout options remains only on the assumption that the price of the underlying asset follows either Brownian motion or geometric Brownian motion. Now, Compared with geometric Brownian motion, jump-diffusion model can describe the underlying asset more accurately. Therefore, this paper derives a new partial integro-differential inequality (PIDI) for shout options pricing on the assumption that the price of the underlying asset follows the jump-diffusion model and constructs the mathematical model by combining specific features and terminal conditions. Another innovation is that this paper proposes a new competitive algorithm to choose two aspects for this mathematical model. One is employing high-order difference for integral and partial derivative terms, the other is using Howard algorithm (also called policy iteration) for the complementarity problem. The advantage of this action is to make full use of the advantages of Howard algorithm and the high-order difference algorithm, that is, to ensure the convergence and achieve valuation result more accurately than the traditional finite element method for the shout options.
The rest of the paper is organized as follows. In Section 2 we derive a new partial integro-differential inequality (PIDI) for shout options pricing on the assumption that the price of the underlying asset follows the jump-diffusion model and constructs the mathematical model by combining specific features and terminal conditions. In Section 3 we propose a new competitive algorithm by combining high-order difference and Howard algorithm. In Section 4 we present numerical examples to compare the convergence and efficiency of the scheme to traditional methods. Section 5 concludes.