Research paper
Nonlinear PDE model for European options with transaction costs under Heston stochastic volatility

https://doi.org/10.1016/j.cnsns.2021.105986Get rights and content

Highlights

  • European options prices with transaction costs under Heston stochastic volatility.

  • Non-linear pricing partial differential equation is solved by the finite difference method.

  • Bid and ask prices for the buyer and the writer in the presence of transaction costs.

  • Model can be extended to price American options with appropriate boundary conditions.

Abstract

In this work, we formulate a pricing model for European options with transaction costs under Heston-type stochastic volatility. The resulting pricing partial differential equations (PDEs) are a pair of nonlinear convection-diffusion-reaction equations with mixed derivative terms, for the writing and holding prices, respectively. The equations are solved numerically by the explicit Euler method. Numerical experiments are presented to illustrate the order of convergence and the effect of the transaction costs on option prices.

Introduction

Options are an important form of financial instrument in which two parties agree to buy or sell an asset at a specified price on or before a future date depending on the type of the contract. Since its introduction in 1973, the Black-Scholes model [1] has been widely used for pricing options. The Black-Scholes model provides the theoretical base for other more complicated pricing models, and its formula is powerful for pricing options. However, market data shows some limitations of the model. The most quoted phenomenon is the so called “volatility smile”, which indicates that the assumption of constant volatility of the Black-Scholes model is unrealistic in practice. Many studies have been carried out to overcome this problem, pricing models with non-constant volatility have been developed, such as the local volatility model [2], regime-switching in volatility [3], [4], [5] and the stochastic volatility models [6], [7]. The Heston model [6], which assumes the stochastic volatility following a mean reverting process (CIR process) [8], has been widely adopted in the financial market.

Transaction costs in trading are important to an investor, as they affect the net returns. However, in the presence of transaction costs, the costs associated with continuous hedging of the underlying stock could become significant. Investors would not hedge the portfolio very frequently as the transaction costs could be too much, resulting in an extremely small or negative option holding price and a very large option writing price. Hence, perfect hedging is impossible and the absence of arbitrage argument is no longer valid. There have been two main groups of approaches dealing with transaction costs in option pricing: one group is based on hedging strategies and the other based on the utility maximization theory. Among the former, Leland [9] proposed a pricing model with transaction costs, deriving a Black-Scholes type PDE with the volatility term adjusted by the hedging interval and the transaction costs rate. There are other variations of the Leland model, such as the works in [10], [11], [12], [13], [14], to name a few. The hedging strategies are easy to formulate and implement, but the only choice an investor has on this type of models is the hedging interval. On the other hand, the utility-based approaches, such as Davis et al. [15], Hodges [16], and Soner et al. [17] among others, adopt optimal strategies by taking investors’ preferences into consideration, but require lengthy computation, as the utility-based option price is usually computed by solving two three-dimensional Hamilton-Jacobi-Bellman (HJB) equations numerically. As a result, many follow up studies focused on finding efficient numerical techniques or approximations as in [18], [19], [20]. However, hedging strategies are preferred over utility approaches, when stochastic volatility is considered [21], [22], [24], [23], because stochastic volatility presents an additional dimensionality, making it impractical using utility approaches.

Under hedging strategies, the main difficulty in pricing options with transaction costs under stochastic volatility results from the need to hedge the extra volatility risk, which affects the total number of underlying traded in the hedging process, in turn, affects the costs involved in trading. Mariani and SenGupta [22] proposed a pricing model with transaction costs under the modified Hull and White stochastic volatility model [7]. The authors assume that the ratio of the vega of the unknown option and that of the additional option brought in to hedge the volatility risk, is a constant by arguing that these options are from the same option chain, and thus form their pricing equation system in terms of a linear combination of the unknown option value and the known option value. They then prove analytically the existence of a strong solution to the problem and solve it numerically in [24]. However, a major drawback of the model is that their final solution depends on an unknown constant ratio and the additional option which needs to be carefully chosen. This makes their approach almost impractical, as there is no suggestion on how to choose the option or the constant vega ratio. Florescu et al. [21] used a traded proxy for the volatility as well as the underlying to form a portfolio, and derived a nonlinear PDE whose solution provides the option price under general volatility in the presence of transaction costs, which are the sum of the transaction costs for the traded stocks and volatility. The authors proved the existence of strong solutions to the problem, the numerical results for the modified Hull and White model are presented in Mariani et al. [23]. While the suitability of this hedging approach is beyond the scope of this paper, we would like to point it out that the pricing PDE in [21], [23] does not reduce to the pricing PDE under either the modified Hull and White or the Heston volatility model when the transaction costs rates are set to zero.

In this paper, we propose a new approach for the price of European options with transaction costs under the widely market-adopted Heston’s volatility [6]. We derive our formulation by using a dynamic hedging strategy and utilizing a known option whose price does not include transaction costs. A nice feature of our model is that the final pricing PDE does not involve the known option at all, and the pricing equation reduces to the original Heston PDE when the transaction costs rate is set to zero. Although our approach is formulated under the Heston framework, the approach could be easily applied to price options whose volatility follows other stochastic models, such as the modified Hull and White model. Due to the complexity of the governing PDEs, which are convection-diffusion-reaction equations with mixed derivative terms, an explicit finite difference method is used to obtain the numerical values of options. However, the value of the model should not be discounted, as obtaining the numerical solution of a PDE, in general, is still easier and faster than the purely numerical approaches, such as, the Monte Carlo simulations. As the aim of this paper is to establish a model for the price of a European option under stochastic volatility with transaction costs, our emphasis is establishing the pricing PDEs. For the application of this model, more advanced numerical techniques could be explored, such as the alternating-direction implicit (ADI) finite difference method, to improve both accuracy and efficiency of the numerical solution.

The rest of the paper is organized as follows. In Section 2, we provide the formulation and some financial interpretation of the valuation problem of European options under the Heston volatility model with transaction costs. Numerical results and discussions are detailed in Section 3, and concluding remarks in Section 4.

Section snippets

Formulation of the model

In this section, we formulate a pricing model for European option prices under the Heston volatility model [6] assuming the trading of the underlying stocks is subject to proportional transaction costs. The price of the underlying asset S and the variance u satisfy the following stochastic differential equations1{dS=μSdt+uSdW1,du=κH(θu)dt+σudW2,where μ is the drift, θ

Numerical experiments and discussions

In this section, Eqs. (2.12) and (2.13) are solved numerically by using the finite-difference method. To validate our formulations, the numerical solutions with zero transaction costs are compared with results from the analytical solution of the Heston PDE [6]. The orders of convergence along the time and space directions are also discussed in this section.

Unless otherwise mentioned, all of the calculations are carried out for the following parameters: κH=2.5, θ=0.16, r=0.1, σ=0.45, ρ=0.1, the

Conclusion

In this paper, we formulated the pricing PDEs for European calls with transaction costs under the Heston volatility framework. As the market is no longer complete when transaction costs are taken into consideration, there is no unique option price, instead the price falls within an upper bond (the writer price) and a lower bond (the holder price). Due to the highly nonlinear term associated with the cost of trading the underlying stock, the PDEs had to be solved numerically. Nevertheless, the

CRediT authorship contribution statement

Xiaoping Lu: Formal analysis, Methodology, Supervision, Writing – original draft, Writing – review & editing. Song-Ping Zhu: Conceptualization, Methodology, Supervision, Writing – original draft. Dong Yan: Software, Investigation, Methodology, Writing – original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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