The option-pricing problem has always been a hot research topic in finance, as options are important tools for investment and hedging risk. Among a variety of option products, American options have gained popularity in the market because they can be exercised before maturity. This feature makes them the predominant type of options sold in the market. Concurrently, stock loans have become commonplace in the current financial market, and determining how much money can be loaned against the current value of stocks has also emerged as a significant area of research. Given that both of these financial instruments can be exercised (or repaid) at any time, their prices (or loan amounts) can be determined by solving the following linear complementarity problem (LCP):
where
is a linear operator containing partial derivatives,
represents the value, and
signifies the constraint.
To solve the LCP (
1), it is necessary to understand the definition of the operator
, which depends on the assumed underlying asset price behavior. According to the well-known Black–Scholes model [
1], the underlying asset price is modeled as follows:
where
is the assets’ price,
represents the standard Brownian motion,
is the drift rate of the asset’s return, and:
with
r being the interest rate,
being the volatility, and
S being the asset’s price. However, this model cannot account for many empirical observations of asset prices, such as sudden price movements. Hence, new models have been proposed to handle these issues, including stochastic volatility models [
2,
3], jump diffusion models [
4], self-exciting jump models [
5], Hawkes jump diffusion models [
6], mixed fractional Brownian model [
7], two-factor non-affine stochastic volatility model [
8], and sub-fractional Brownian model [
9,
10]. The model based on the Lévy process, notable for its ability to model price jumps and transform into a fractional diffusion equation [
11], is among these. With different density functions, different models have been proposed, such as the KoBoL model [
12], the CGMY model [
13], and the FMLS model [
14]. The Lévy process with jumps has also been proposed to more accurately describe the price of underlying assets [
15,
16]. In this paper, this stochastic process is used to model the underlying assets, defined by
where
and
t denotes the current time, with
being the convexity adjustment. The variable
represents the Lévy-
-stable process with maximum skewness, where the tail index
falls within the range of (1, 2).
denotes a Poisson process characterized by the jump intensity
.
is a sequence of independent and identically distributed random variables, and
, where the parameters
and
represent the expectation and standard deviation of the jumps, respectively. Then, the operator
includes both a fractional derivative and an integral operator (for more details, refer to the following section). By solving the LCP in (
1) with this
under specific boundary and initial conditions, we can determine the price of American options or the value of stock loans.
To address the LCP arising from the valuation of American options and stock loans, various numerical algorithms have been developed. These can be categorized into five primary strategies. The first focuses on identifying the optimal execution boundary to resolve the challenges, as demonstrated by the approach presented in [
17]. The second strategy transforms the problem into a linear framework through either semi-implicit or implicit–explicit schemes, such as the L-stable method [
18], the linearly implicit predictor–corrector scheme [
19], and the IMEX BDF method [
20]. The third approach converts the problem into a nonlinear equation, utilizing iterative methods for solution acquisition, including the fixed-point method [
21], the preconditioned penalty method [
22], and the modulus-based matrix splitting iteration method [
23]. The fourth approach involves devising iterative methods for direct LCP resolution, exemplified by the projected SOR method [
24] and the projected algebraic multigrid method [
25]. The last category of algorithms consists of the recently popular deep learning algorithms, such as the neural network method [
26] and the physics-informed neural network method [
27]. For the LCP discussed in this paper, a Laplace transform method is introduced for rapid resolution by avoiding time marching [
15]. Moreover, a fast solution strategy that combines Newton’s method with the preconditioned conjugate gradient normal residual method is proposed in [
16], benefiting from fast Fourier transformation acceleration and achieving an operational cost of
per inner iteration. This paper introduces an alternative approach, translating the LCP into a Hamilton–Jacobi–Bellman (HJB) equation, and devising a fast algorithm for its resolution, complete with theoretical assurances.
This study aims to devise a fast algorithm for solving the HJB equation, which includes a fractional derivative and an integral operator derived from the LCP based on the Lévy--stable process with jumps model. To tackle this equation, a nonlinear finite difference method is developed, accompanied by stability and convergence analyses. Subsequently, the policy iteration method and the Krylov subspace method are applied to solve the resultant finite difference scheme. A banded preconditioner is also proposed to enhance the convergence speed of the internal iterative method, supported by theoretical analysis. These steps enable the development of a preconditioned policy–Krylov subspace method for solving the fractional partial integro-differential HJB equations in finance, ensuring an efficient and theoretically sound solution.
The rest of this article is organized as follows: The description of the equation considered in this paper is illustrated in
Section 2. A nonlinear finite difference scheme is proposed in
Section 3 to discretize the HJB equation with the unconditional stability guarantee and first-order convergence rate. In
Section 4, a fast algorithm framework incorporating the policy iteration method and the Krylov subspace method is introduced. In
Section 5, a banded preconditioner is proposed with theoretical analysis about the condition number of the preconditioned matrix. Numerical experiments are given in
Section 6, and conclusions are drawn in
Section 7.