Trinomial Tree Method Research Articles

Convergence Rate of the High-Order Finite Difference Method for Option Pricing in a Markov Regime-Switching Jump-Diffusion Model

The high-order finite difference method for option pricing is one of the most popular numerical algorithms. Therefore, it is of great significance to study its convergence rate. Based on the relationship between this algorithm and the trinomial tree method, as well as the definition of local remainder estimation, a strict mathematical proof is derived for the convergence rate of the high-order finite difference method for option pricing in a Markov regime-switching jump-diffusion model. The theoretical result shows that the convergence rate of this algorithm is O(Δτ) . Moreover, the results also hold in the case of Brownian motion and jump-diffusion models that are specialized forms of the given model.

Optimal Selling Strategies under Regime-Switching Market Environment with Finite Expiry

This paper addresses an optimal stock liquidation problem over a finite-time horizon; to that end, we model it as an optimal stopping problem in a regime-switching market. The optimal stopping time is written as a solution to a system of Volterra type integral equations. Moreover, it reveals that when the risk-free interest rate is always lower than the return rate of the stock, it is never optimal to sell the stock early; otherwise, one should sell the stock in bear market if the stock price reaches a critical value and hold the stock in bull market until the maturity date. Finally, we present a trinomial tree method for numerical implementation. The numerical results are consistent with the theoretical findings.

Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

Calculation of call option using trinomial tree method and black-scholes method case study of Microsoft Corporation

Options are one of the most commonly known financial instrument used in financial market. Options are expected to minimize risk or even increase profits in stock trading. Therefore, it is important for options traders to take into account the contract price of options. This study discusses the determination of call option prices using the Trinomial Tree method and the Black-Scholes method with the data of Microsoft Corporation’s stock price and strike price. From the result of the calculations, a conclusion will be drawn from the comparison of the use of the two methods. In conclusion, the call option price using the trinomial tree method is close to the call option price using the Black-Scholes method. Therefore, the Trinomial tree method and the Black-Scholes method is worth using for the base calculation of option pricing.

Analytic valuation of European continuous-installment barrier options

This study examines the valuation of European continuous-installment barrier options with dividend payments, under the classical Black–Scholes model. By using Mellin transform techniques, we derive the analytic integral solutions of European continuous-installment options with two types of barriers, so that the integral representations are decomposed into the value of the European barrier option and the premium of the early exercise installment. In the similar manner, the pricing of European continuous-installment barrier option which includes a rebate provision is investigated. Additionally, we investigate the effect of the free boundaries of European installment barrier options against barrier levels. Finally, by comparing the recursive integration method with the existing finite difference method – which is, in turn, based on the trinomial tree method as a benchmark – we analyze the accuracy and efficiency of the recursive integration method with respect to continuous-installment barrier options.

A lattice-based approach to option and bond valuation under mean-reverting regime-switching diffusion processes

Nowadays, the pricing of financial instruments under continuous-time Markov switching models have received a widespread attention from researchers and practitioners in the finance industry. Lattice-based approaches are amongst the most widely used approaches to solve the pricing problem in this context. Recently, Yuen and Yang (2010) have proposed a simple and fast recombining trinomial tree method to handle the case of regime-switching geometric Brownian motion processes. In this paper, we generalize their approach to the regime-switching (exponential) mean-reverting case with state-dependent switching rates and derive the necessary conditions for the positivity of conditional branching probabilities. We employ the Hull and White’s tree-building procedure to limit tree growth away from the long-run mean of the process. We use the proposed lattice framework to price contingent claims of European, American and barrier type, and demonstrate its applicability in pricing default-free zero-coupon bonds. In the proposed tree structure, the number of nodes is substantially decreased and the computational cost is effectively reduced compared to the usual approaches. Extensive numerical experiments illustrate the efficiency and flexibility of the proposed scheme.

Non-recombining trinomial tree pricing model and calibration for the volatility smile

Abstract In this paper, we consider the non-recombining trinomial tree pricing model under the volatility, which is a function of time, establish the option pricing model and give the convergence rates of the non-recombining trinomial tree method. In addition, we research the calibration problem of volatility and adopt an exterior penalty method to transform this problem into a nonlinear unconstrained optimization problem. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S&P 500 index. The results show the effectiveness of the non-recombining trinomial tree pricing model.

Efficient Lattice Method for Valuing of Options with Barrier in a Regime Switching Model

We propose an efficient lattice method for valuation of options with barrier in a regime switching model. Specifically, we extend the trinomial tree method of Yuen and Yang (2010) by calculating the local average of prices near a node of the lattice. The proposed method reduces oscillations of the lattice method for pricing barrier options and improves the convergence speed. Finally, computational results for the valuation of options with barrier show that the proposed method with interpolation is more efficient than the other tree methods.

On pricing options with stressed-beta in a reduced form model

We consider the valuation of options with stressed-beta in a reduced form model. Under this two-state beta model, we provide the analytic pricing formulae for the European options and American options as the integral forms. Specifically, we provide the integral representation of the early exercise premium of an American put option. We use the quadrature method to evaluate the integral forms and we measure the performance of our pricing framework comparing the benchmarks set by the trinomial tree method. It turns out that our pricing framework with the quadrature methods are computationally efficient and accurate. We also calibrate the market data successfully.

A COST-EFFECTIVE MODIFICATION OF THE TRINOMIAL METHOD FOR OPTION PRICING

A new method for option pricing based on the trinomial tree method is introduced. The new method calculates the local average of option prices around a node at each time, instead of computing prices at each node of the trinomial tree. Local averaging has a smoothing effect to reduce oscillations of the tree method and to speed up the convergence. The option price and the hedging parameters are then obtained by the compact scheme and the Richardson extrapolation. Computational results for the valuation of European and American vanilla and barrier options show superiority of the proposed scheme to several existing tree methods.

“Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010

(2010). “Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method”, Fei Lung Yuen and Hailiang Yang, April, 2010. North American Actuarial Journal: Vol. 14, No. 2, pp. 272-277.

Pricing Asian Options and Equity-Indexed Annuities with Regime Switching by the Trinomial Tree Method

Equity-indexed annuities (EIAs) provide investors with a minimum rate of return and at the same time the opportunity of gaining a profit that is linked to the performance of an equity index. These properties make EIAs a popular product in the market. For modeling the equity index process and calculating the price of EIAs, as the maturity of EIAs usually is long, it is more reasonable to assume that the interest rate and the volatility of the equity index are stochastic processes. One simple way is to apply the regime-switching model, which allows these parameters depending on the market situation. However, the valuation of derivatives in such models is challenging, especially for the strong path-dependent options such as Asian options. A trinomial tree model is introduced to provide an efficient way to solve this problem. The valuation of Asian options is studied and extended to Asian-option-related EIAs.

Option Pricing in a Jump-Diffusion Model with Regime Switching

AbstractNowadays, the regime switching model has become a popular model in mathematical finance and actuarial science. The market is not complete when the model has regime switching. Thus, pricing the regime switching risk is an important issue. In Naik (1993), a jump diffusion model with two regimes is studied. In this paper, we extend the model of Naik (1993) to a multi-regime case. We present a trinomial tree method to price options in the extended model. Our results show that the trinomial tree method in this paper is an effective method; it is very fast and easy to implement. Compared with the existing methodologies, the proposed method has an obvious advantage when one needs to price exotic options and the number of regime states is large. Various numerical examples are presented to illustrate the ideas and methodologies.

Option pricing with regime switching by trinomial tree method

We present a fast and simple tree model to price simple and exotic options in Markov Regime Switching Model (MRSM) with multi-regime. We modify the trinomial tree model of Boyle (1986) [12] by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure. In MRSM, the market might not be complete, therefore we provide some ideas and discussions on managing the regime switching risk in support of our results.

Convergence of the trinomial tree method for pricing European/American options

In this paper, we present a trinomial tree method for pricing options and show its equivalence to certain explicit difference scheme. Using the numerical analysis and the notion of viscosity solution, we prove the uniform convergence of the trinomial tree method for European/American options. The accuracy and efficiency are shown through numerical simulations.

Selective On-line Parameter Tuning of Adaptive PSS by Trinomial Tree Method in Wide-Area Power System

In this study, an adaptive PSS using data measured at its generator bus is developed. The low-order linear models are utilized for the eigenvalues analysis of the large-scale power system and for the PSS on-line tuning. A generator with the PSS is modeled as the 5th order model, and the models are identified by the least squares method using measured variables at each generator. Control parameters of the phase compensator in the PSS are updated online by using the trinomial tree method based on the eigenvalue analysis. The 4th order model is also developed to obtain the dominant oscillation mode of the whole system. The PSS parameters are updated only when the dominant eigenvalues detected at the 4th and 5th order models are converged for a given period, and the identification is considered to be accurate enough to update the parameters. Dynamic digital simulation is carried out for IEEJ WEST 10-machine system model to show the effectiveness of the proposed tuning method.