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.