Discretely Monitored Research Articles

Pricing Discretely Monitored Asian Options Under Regime-Switching and Stochastic Volatility Models with Jumps

Pricing Discretely Monitored Asian Options Under Regime-Switching and Stochastic Volatility Models with Jumps

Pricing Discretely Monitored Barrier Options under Markov Processes through Markov Chain Approximation

The authors propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options with an underlying asset that evolves according to a one-dimensional Markov process, which includes diffusion and jump-diffusion processes. The prices and Greeks of a discretely monitored double barrier option are explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice—for example, time-dependent barriers and nonuniform monitoring time intervals. They provide numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature in a unified framework. TOPICS:Derivatives, options Key Findings ▪ Markov chain approximations to stochastic processes are used to develop a simple closed-form pricing formula for discretely monitored barrier options. ▪ This article explores price and Greek computations within the constant elasticity of a variance model and Merton’s and Kou’s jump-diffusions models. ▪ Numerical examples are provided to demonstrate the robustness and accuracy of the pricing technique, both to recreate specialized results in related barrier option literature and to extend to multiple novel settings.

A Fourier-Cosine Method for Pricing Discretely Monitored Barrier Options under Stochastic Volatility and Double Exponential Jump

In this paper, the valuation of the discrete barrier options on the condition that the underlying asset price process follows the GARCH volatility and double exponential jump is studied. We derived an analytical approximation of the characteristic function for the underlying log-asset price. Then, a quasianalytical approximate formula of the price of the discrete barrier option is obtained based the on Fourier-cosine method. Numerical examples show that the Fourier-cosine method is fast and efficient for pricing discrete barrier options compared with the Monte Carlo simulation method. Finally, the influences of some important parameters on the prices of discrete barrier options are studied to further illustrate the rationality of the model.

A General Stochastic Degradation Modeling Approach for Prognostics of Degrading Systems With Surviving and Uncertain Measurements

This paper is concerned with estimating remaining useful life (RUL) for a class of stochastic degrading systems with surviving degradation paths and uncertain measurements. The motivation comes from two engineering facts: the system's degradation state is discretely monitored and the measured degradation signals are taken from a survival degradation path, i.e., its lifetime is greater than the latest observation time, and the underlying degradation state cannot be perfectly observed due to the noise, disturbance, nonideal instruments, etc. Thus, the measured degradation signals are uncertain, but related to the underlying degradation state. Toward this end, we first present a general stochastic degradation modeling approach to account for the abovementioned facts. Then, a non-Gaussian degradation state transition equation is derived considering the constraint of the survival path and thus the particle filtering algorithm is used to estimate the underlying degradation state in real time from uncertain measurements. Furthermore, we derive the RUL distribution based on the first-passage time concept which incorporates the uncertainty of the estimation for the degradation state and can be real-time updated based on the available surviving yet uncertain measurements. The novelty of this paper is to allow us to exclude the probability of failure between discrete monitoring times and to account for the impacts of surviving and uncertain measurements on estimating both the degradation state and the RUL. To apply the presented methodology, a maximum likelihood estimation framework is provided to determine the model parameters based on the expectation maximization algorithm together with particle filtering and smoothing methods. As a special case fallen into the presented framework, a linear degradation modeling and prognostic realization is discussed. Finally, we demonstrate the proposed approach by a case study.

Pricing Discretely Monitored Barrier Options Under Markov Processes Using a Markov Chain Approximation

We propose an explicit closed-form approximation formula for the price of discretely monitored single or double barrier options whose underlying asset evolves according to a generic one-dimensional Markov process. This set of stochastic processes includes, but is not limited to, diffusion and jump diffusion processes commonly used in derivative pricing applications. The formula’s derivation combines the integral equation method, the Z−transform technique, and a continuous-time Markov chain approximation of the underlying Markov process. It does not require one to perform an intermediate numerical quadrature or related potentially runtime-intensive, error-prone, or otherwise complicated numerical procedure that may require a high degree of tuning to ensure appropriate accuracy (e.g. an inverse Laplace transform or inverse Z−transform). Rather, the price and Greeks of a discretely-monitored double barrier option may be explicitly expressed in terms of rudimentary matrix operations. In addition, this framework may be extended to include additional features of barrier options often encountered in practice. Examples including time-dependent barriers and non-uniform monitoring time intervals, may be seamlessly incorporated into this framework. In addition, by limiting the monitoring frequency to a large value, we also obtain an accurate closed-form formula for the price and Greeks of continuously-monitored double barrier options with time-dependent barriers under general Markov processes. Finally, we provide many numerical examples to demonstrate the accuracy and efficiency of the proposed formula as well as its ability to reproduce existing benchmark results in the relevant literature.

Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Lévy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener–Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-z domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

SWIFT Valuation of Discretely Monitored Arithmetic Asian Options

In this work, we propose an efficient and robust valuation of discretely monitored arithmetic Asian options based on Shannon wavelets. We employ the so-called SWIFT method, a Fourier inversion numerical technique with several important advantages with respect to the existing related methods. Particularly interesting is that SWIFT provides mechanisms to determine all the free-parameters in the method, based on a prescribed precision in the density approximation. The method is applied to two general classes of dynamics: exponential Levy models and square-root diffusions. Through the numerical experiments, we show that SWIFT outperforms state-of-the-art methods in terms of accuracy and robustness, and shows an impressive speed in execution time.

Fluctuation Identities with Continuous Monitoring and Their Application to Price Barrier Options

We present a numerical scheme to calculate fluctuation identities for exponential L'evy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential L'evy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener-Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-$z$ domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme.

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Levy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Levy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

Asian Options with Jumps

We derive a closed-form formula for the fair value of call and put options written on the arithmetic average of security prices driven by jump diffusion processes displaying (possibly periodical) trend, time varying volatility, and mean reversion. The model allows one for jointly fitting quoted futures curve and the time structure of spot price volatility. Our result extends the no-jump case put forward in [Fusai, G., Marena, M., Roncoroni, A. 2008. Analytical Pricing of Discretely Monitored Asian-Style Options: Theory and Application to Commodity Markets. Journal of Banking and Finance 32 (10), 2033-2045]. A few tests based on commodity price data assess the importance of introducing a jump component on the resulting option prices.

A Closed-Form Expansion Approach for Pricing Discretely Monitored Variance Swaps - Online Supplementary Material

This supplementary material for A Closed-form Expansion Approach for Pricing Discretely Monitored Variance Swaps contains (1) more details on the computational results (Section 1), (2) an illustration of expansion formulas (Section 2), and (3) an interpretation of our expansions for the examples (Section 3).

Pricing Path-Dependent Options with Discrete Monitoring under Time-Changed Lévy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

In this paper we propose a closed-form asymptotic expansion approach to pricing discretely monitored Asian options in general one-dimensional diffusion models. Our expansion is a small-time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval. This expansion method is distinguished from many other pricing-oriented expansion algorithms in the literature because of two appealing features. First, we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way. Second, the convergence of the expansion is proved rigorously under some regularity conditions. Numerical experiments suggest that the closed-form expansion formula with only a few terms (e.g., four terms up to the third order) is accurate, fast, and easy to implement for a broad range of diffusion models, even including those violating the regularity conditions.

Pricing Path-Dependent Options with Discrete Monitoring Under Time-Changed Levy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Levy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic functions of the intertemporal joint distribution of time-changed Levy processes. Using the derived representation of the characteristic function we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader, and lookback options, all of which are discretely monitored.

Pricing Discretely Monitored Asian Options by Maturity Randomization

We present a new methodology based on maturity randomization to price discretely monitored arithmetic Asian options when the underlying asset evolves according to a generic Levy process. Our randomization technique considers the option expiry to be a random variable distributed according to a geometric distribution of a parameter independent of the underlying process. This allows one to transform the pricing backward procedure into a set of independent integral equations. Numerical procedures for a fast and accurate solution of the pricing problem are provided.

The Hedging Costs of Discrete Monitoring of FX Barrier Options

The effects of the discrete monitoring of barrier options is analysed. We determine the costs that a market-maker should charge when making prices for such a kind of exotic options. The costs are related to the Delta-hedging activity around the barrier level between two subsequent monitoring dates, by assuming a reasonable hedging rule followed by the trader. We explain how our result is related to an apparently similar result by Becker and Wystup (2005) and finally we present a practical application showing that the inclusion of the Delta-hedging costs has a relevant impact on the discretely monitored barrier options prices.

Pricing Discretely Monitored Barrier Options by a Markov Chain

Barrier options have become commonplace in the option market, and a variety of other financial contracts may also be thought of in terms of barrier options. But the existence of a price barrier can significantly complicate the option valuation problem when volatility is time-varying, or the barrier itself moves over time, or the barrier is only monitored at discrete intervals. In this article, Duan et al. present a new Markov chain technology for pricing barrier options that readily handles all of these problems. Out-and-in options can be valued within their framework even when volatility follows a GARCH process and a discretely monitored time-varying barrier is present.