Finite Moment Log-stable Model Research Articles

On the number of terms in the COS method for European option pricing

The Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.

Novel numerical techniques for the finite moment log stable computational model for European call option

AbstractOption pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second‐order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α‐stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi‐conjugate gradient stabilized method (Bi‐CGSTAB) and the fast Bi‐CGSTAB (FBi‐CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher‐order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.

Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model

We provide ready-to-use formulas for European options prices, risk sensitivities, and P&L calculations under Lévy-stable models with maximal negative asymmetry. Particular cases, efficiency testing, and some qualitative features of the model are also discussed.

A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases*

We study a generalized hyperbolic (GH) time-changed Levy process for option pricing and examine six three-parameter special cases: the variance gamma (VG) model of Madan, Carr, and Chang (1998), t, hyperbolic (H), normal inverse Gaussian (NIG), reciprocal hyperbolic (RH), and normal reciprocal inverse Gaussian (NRIG) option pricing models. We study the GH model’s moment properties of the associated risk-neutral distribution of logarithmic spot returns, and obtain an explicit pricing formula for European options facilitated by the time-change Levy process construction. Using S&P 500 Index European options during low and high volatility sample periods, we compare the GH model empirically with existing benchmark models such as the finite-moment log-stable model and the Black–Scholes model. The GH model offers the best in- and out-of-sample performance overall, and a proposed t model special case generally outperforms the existing VG special case. We also present a stochastic volatility extension of the GH model.

Analytic properties of American option prices under a modified Black–Scholes equation with spatial fractional derivatives

This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterized by a nonlinear fractional partial differential equation (FPDE) system. Using the technique of approximation, we prove that the American put price under the FMLS model is convex with respect to the underlying price, and specify the impact of the tail index on option prices. We also show numerical examples to support our analytic results.

Numerical simulation of a Finite Moment Log Stable model for a European call option

Compared to the classical Black-Scholes model for pricing options, the Finite Moment Log Stable (FMLS) model can more accurately capture the dynamics of the stock prices including large movements or jumps over small time steps. In this paper, the FMLS model is written as a fractional partial differential equation and we will present a new numerical scheme for solving this model. We construct an implicit numerical scheme with second order accuracy for the FMLS and consider the stability and convergence of the scheme. In order to reduce the storage space and computational cost, we use a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) to solve the discrete scheme. A numerical example is presented to show the efficiency of the numerical method and to demonstrate the order of convergence of the implicit numerical scheme. Finally, as an application, we use the above numerical technique to price a European call option. Furthermore, by comparing the FMLS model with the classical B-S model, the characteristics of the FMLS model are also analyzed.

General Smile Asymptotics with Bounded Maturity

We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [emphMath. Finance, 19 (2009), pp. 1--12]. We present applications to popular models, including the Carr--Wu finite moment logstable model, Merton's jump diffusion model, and Heston's model.

A predictor–corrector approach for pricing American options under the finite moment log-stable model

This paper investigates the pricing of American options under the finite moment log-stable (FMLS) model. Under the FMLS model, the price of American-style options is governed by a highly nonlinear fractional partial differential equation (FPDE) system, which is much more complicated to solve than the corresponding Black–Scholes (B–S) system, with difficulties arising from the semi-globalness of the fractional operator, in conjunction with the nonlinearity associated with the early exercise nature of American-style options. Albeit difficult, in this paper, we propose a new predictor–corrector scheme based on the spectral-collocation method to solve for the prices of American options under the FMLS model. In the current approach, the nonlinearity of the pricing system is successfully dealt with using the predictor–corrector framework, whereas the non-localness of the fractional operator is elegantly handled. We have also provided an elegant error analysis for the current approach. Various numerical experiments suggest that the current method is fast and efficient, and can be easily extended to price American-style options under other fractional diffusion models. Based on the numerical results, we have also examined quantitatively the influence of the tail index on American put options.

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.

ASYMPTOTIC EQUIVALENCE IN LEE'S MOMENT FORMULAS FOR THE IMPLIED VOLATILITY, ASSET PRICE MODELS WITHOUT MOMENT EXPLOSIONS, AND PITERBARG'S CONJECTURE

In this paper, we study the asymptotic behavior of the implied volatility in stochastic asset price models. We provide necessary and sufficient conditions for the validity of asymptotic equivalence in Lee's moment formulas, and obtain new asymptotic formulas for the implied volatility in asset price models without moment explosions. As an application, we prove a modified version of Piterbarg's conjecture. The asymptotic formula suggested by Piterbarg may be considered as a substitute for Lee's moment formula for the implied volatility at large strikes in the case of models without moment explosions. We also characterize the asymptotic behavior of the implied volatility in several special asset price models, e.g., the CEV model, the finite moment log-stable model of Carr and Wu, the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes, and SV1 and SV2 models of Rogers and Veraart.

First-passage probability, jump models, and intra-horizon risk

This paper proposes a risk measure, based on first-passage probability, which reflects intra-horizon risk in jump models with finite or infinite jump activity. Our empirical investigation shows, first, that the proposed risk measure consistently exceeds the benchmark value-at-risk (VaR). Second, jump risk tends to amplify intra-horizon risk. Third, we find large variation in our risk measure across jump models, indicative of model risk. Fourth, among the jump models we consider, the finite-moment log-stable model provides the most conservative risk estimates. Fifth, imposing more stringent VaR levels accentuates the impact of intra-horizon risk in jump models. Finally, using an alternative benchmark VaR does not dilute the role of intra-horizon risk. Overall, we contribute by showing that ignoring intra-horizon risk can lead to underestimation of risk exposures.