Wen-Ting Chen

A predictor–corrector scheme based on the ADI method for pricing American puts with stochastic volatility

Abstract In this paper, we introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options with a stochastic volatility model. Upon applying a front-fixing transformation to transform the unknown free boundary into a known and fixed boundary in the transformed space, a predictor–corrector finite difference scheme is then developed to solve for the optimal exercise price and the option values simultaneously. Based on the local von Neumann stability analysis, a stability requirement is theoretically obtained first and then tested numerically. It is shown that the instability introduced by the predictor can be damped, to some extent, by the ADI method that is used in the corrector. The results of various numerical experiments show that this new approach is fast and accurate, and can be easily extended to other types of financial derivatives with an American-style exercise. Another key contribution of this paper is the proposition of a set of appropriate boundary conditions, particularly in the volatility direction, upon realizing that appropriate boundary conditions in the volatility direction for stochastic volatility models appear to be controversial in the literature. A sound justification is also provided for the proposed boundary conditions mathematically as well as financially.

Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative

This paper investigates the option pricing under the FMLS (finite moment log stable) model, which can effectively capture the leptokurtic feature observed in many financial markets. However, under the FMLS model, the option price is governed by a modified Black-Scholes equation with a spatial-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. This has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Despite this difficulty, we have managed to find an explicit closed-form analytical solution for European-style options after successfully solving the FPDE (fractional partial differential equation) derived from the FMLS model. After the validity of the put-call parity under the FMLS model is verified both financially and mathematically, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice. Received August 2, 2012 and, in revised form, June 24, 2013. 2010 Mathematics Subject Classification. Primary 97M30, 35R11.

A new analytical approximation for European puts with stochastic volatility

Abstract In this paper, we apply singular perturbation techniques to price European puts with a stochastic volatility model, and derive a simple and elegant analytical formula as an approximation for the value of European put options. In contrast to the existing Heston’s semi-analytical formula, this approximation has the following unique feature: the latter only involves the standard normal distribution function, which is as fast and easy to implement as the Black–Scholes formula; whereas the former requires the evaluation of a logarithm with a complex argument during the involved Fourier inverse transform, which may sometimes result in numerical instability. Various numerical experiments suggest that our new formula can achieve a high order of accuracy for a large class of option derivatives with relatively short tenor.

A spectral-collocation method for pricing perpetual American puts with stochastic volatility

Abstract Based on the Legendre pseudospectral method, we propose a numerical treatment for pricing perpetual American put option with stochastic volatility. In this simple approach, a nonlinear algebraic equation system is first derived, and then solved by the Gauss–Newton algorithm. The convergence of the current scheme is ensured by constructing a test example similar to the original problem, and comparing the numerical option prices with those produced by the classical Projected SOR (PSOR) method. The results of our numerical experiments suggest that the proposed scheme is both accurate and efficient, since the spectral accuracy can be easily achieved within a small number of iterations. Moreover, based on the numerical results, we also discuss the impact of stochastic volatility term on the prices of perpetual American puts.

Pricing perpetual American options under a stochastic-volatility model with fast mean reversion

Abstract In this paper, we present a “correction” to Merton’s (1973) well-known classical case of pricing perpetual American puts by considering the same pricing problem under a general fast mean-reverting SV (stochastic-volatility) model. By using the perturbation method, two analytic formulae are derived for the option price and the optimal exercise price, respectively. Based on the newly obtained formulae, we conduct a quantitative analysis of the impact of the SV term on the price of a perpetual American put option as well as its early exercise strategies. It shows that the presence of a fast mean-reverting SV tends to universally increase the put option price and to defer the optimal time to exercise the option contract, had the underlying been assumed to be falling. It is also noted that such an effect could be quite significant when the option is near the money.

Stock loan valuation under a stochastic interest rate model

Stock loans are loans collateralized by stocks. They are modern financial products designed for investors with large equity positions. Mathematically, stock loans can be regarded as American call options with a time-dependent strike price. This study is the first in the literature that considers the valuation of stock loans in a stochastic interest rate framework. Based on portfolio analysis, a partial differential equation (PDE) governing the value of stock loans is derived. A set of appropriate boundary conditions, particularly in the interest rate direction, are also proposed to close the pricing system. A sound justification is provided for the proposed boundary conditions mathematically as well as financially. To solve the proposed nonlinear PDE system, a predictor-corrector finite difference method is adopted. Moreover, an alternating direction implicit (ADI) method is used to improve the computational efficiency. Numerical results suggest that the current method is reliable and the stochastic interest rate leads to a higher optimal exercise price of the stock loan in comparison with that calculated from the Black-Scholes model.

Pricing perpetual American puts under multi-scale stochastic volatility

In this paper, we consider the problem of pricing perpetual American put options with volatility driven by two other processes. By using a perturbation approach, we obtain approximate but explicit closed-form pricing formulae for the option and optimal exercise prices, respectively, under a general multi-scale SV (stochastic volatility) model. A key feature of the expansion methodology employed here is to balance the two SV processes, while dealing with the free boundary conditions properly. It turns out that in the current formulae, the fast volatility factor does not play an explicit role, while the slow factor is quite crucial, a phenomenon that is shown to be quite reasonable through our discussions.

Optimal exercise price of American options near expiry

This paper investigates American puts on a dividend-paying underlying whose volatility is a function of both time and underlying asset price. The asymptotic behavior of the critical price near expiry is deduced by means of singular perturbation methods. It turns out that if the underlying dividend is greater than the risk-free interest rate, the behavior of the critical price is parabolic, otherwise an extra logarithmic factor appears, which is similar to the constant volatility case. The results of this paper complement numerical approaches used to calculate the option values and the optimal exercise price at times that are not close to expiry. doi:10.1017/S1446181110000052