Exponential Jump-diffusion Model Research Articles

SPECIFICATION ANALYSIS OF STRUCTURAL CREDIT RISK MODEL WITH STOCHASTIC VOLATILITY

In this research, we conduct specification analysis of structural credit risk models with stochastic volatility using term structure of Credit Default Swap (CDS) spreads and equity volatility form high-frequency return data. We also test five representatives of structural credit risk models using samples of 93 single name CDS contracts from January 2010 -2022. The model we consider are; the standard Merton(1974) model, the Black & Cox (1976) model with flat barrier, the Longstaff and Schwartz(1995)model with stochastic interest rates, the Collin- Dufresne and Goldstein (2001) model with stationary leverage, and the double exponential jump diffusion model used in Huang & Huang(2003). Our study provides consistent econometric estimation of the pricing model parameters and specification tests based on the joint behavior of time-series asset dynamics and cross-sectional pricing errors. Our empirical tests reject strongly the standard Merton (1974) model, the Black and Cox (1976) with flat barrier model, the Longstaff, Schwartz (1995) model with stochastic interest rates. The double exponential jump-diffusion barrier model used in (Huang and Huang, 2003) improves significantly over the five models. The best model is the stationary leverage model of Collin-Dufresne and Goldstein (2001), which we cannot reject at 0.5 level of significance in our sample firm. However, our empirical results document the inability of the existing structural models to capture the dynamic behavior of CDS spreads and equity volatility, especially for high investment grade names derivatives. These points to a potential role of time-varying asset volatility, a feature that is missing in the standard structural models.

An Analysis of Asymptotic Properties and Error Control under the Exponential Jump-Diffusion Model for American Option Pricing

Our work is aimed at modeling the American option price by combining the dynamic programming and the optimal stopping time under two asset price models. In doing so, we attempt to control the theoretical error and illustrate the asymptotic characteristics of each model; thus, using a numerical illustration of the convergence of the option price to an equilibrium price, we can notice its behavior when the number of paths tends to be a large number; therefore, we construct a simple estimator on each slice of the number of paths according to an upper and lower bound to control our error. Finally, to highlight our approach, we test it on different asset pricing models, in particular, the exponential Lévy model compared to the simple Black and Scholes model, and we will show how the latter outperforms the former in the real market (Microsoft “MSFT” put option as an example).

Pricing Bitcoin under Double Exponential Jump-Diffusion Model with Asymmetric Jumps Stochastic Volatility

The objective of this study is, to show the importance of incorporating jumps in both returns and volatility dynamics for Bitcoin. For that purpose, we introduce the Double Exponential Jump-Diffusion model with Stochastic Volatility (DEJDSVJ) that contains asymmetric jumps. The use of the Markov Chain Monte Carlo methods for estimation has proved the meaningful presence of jumps in Bitcoin price and volatility. Moreover, based on the Bitcoin options market, a comparison between the underlying model, the Double Exponential Jump Diffusion model (DEJD) with Stochastic Volatility (no Jumps) and the Stochastic Volatility (SV) shows the goodness of the DEJDSVJ model’s calibration over others for pricing Bitcoin options.

Pricing and hedging defaultable participating contracts with regime switching and jump risk

This paper develops a transform-based approach for the pricing of participating life insurance contracts with a constant or floating guaranteed rate. Our analysis incorporates credit, market (jump), and economic (regime switching) risks, where the evolution of the reference portfolio is described by a regime switching double exponential jump-diffusion model. We provide semi-analytical formulas for the contract value by using a Laplace or Laplace–Fourier transform that involves matrix Wiener–Hopf factors. Then, the price is obtained by implementing the matrix Wiener–Hopf factorization and by performing a numerical Laplace and Fourier inversion. By comparing the results with Monte Carlo simulations, we show that our pricing method is easy to implement and accurate. We also show that the contract with a floating guaranteed rate is riskier but more profitable than the contract with a constant guaranteed rate. Two hedging strategies are introduced to hedge jump and regime switching risks in the participating contracts.

Option pricing on sesame price using jump diffusion models

In this paper, we aim at developing a model for option pricing to reduce the risks associated with Ethiopian sesame price fluctuations. The White Humera Gondar Sesame Grade 3 (WHGS3) price, which is recorded from 5 November 2010 to 30 March 2018 at Ethiopia Commodity Exchange (ECX) market, is used to analyze the price fluctuation. The nature of log-returns of the price is asymmetric (positively skewed) and exhibits high kurtosis. We used jump diffusion models for modeling and option pricing of sesame price. The method of maximum likelihood is applied to estimate the parameters of the models. We used the Root Mean Square Error (RMSE) to test the goodness of fitting for the two models to the data. This test indicates that the models fit the data well. The techniques of analytical and Monte Carlo simulation are used to find the call option pricing of WHGS3 sesame price. From the results, we concluded that Double Exponential Jump Diffusion (DEJD) model is more efficient than Merton’s model for modeling and option pricing of this sesame price.

Option Pricing for Coffee Price Using Jump Diffusion Models

In this study, we aim at developing a model for option pricing to reduce the risks associated with Ethiopian coffee price fluctuations. We used daily closed Washed Sidama class A Grade3 (WSDA3) coffee price recorded in the period 31 May 2011 to 30 March 2018 obtained from Ethiopia commodity exchange (ECX) market to analyse the price fluctuation. The nature of log-returns of the price is asymmetric (negatively skewed) and exhibits high kurtosis. We used jump diffusion models for modeling and option pricing the coffee price. The method of maximum likelihood is applied to estimate the parameters of the models. We used the root mean square error (RMSE) to test the validation of the models. The values of RMSE for Merton's and double exponential jump diffusion models are 0.1093 and 0.0783, respectively. These results indicate that the models fit the data very well. We used analytical and Monte Carlo technique to find the call option pricing of WSDA3 price. Based on the empirical results, we concluded that double exponential jump diffusion model is more efficient than Merton's model for modeling and option pricing of this coffee price.

Evaluation of energy forward curves with jumps under the general Levy process

In this study, we evaluate the relationship between the forward rates and the future delivery period with the consideration of the Levy process for a time-inhomogeneous exponential jump-diffusion process and model the forward curve. This is a large variety of stylized features observed in the Samuelson effect of increasing volatilities close to maturity. However, a new method based on characteristic functions is used to estimate the jump component in a finite-activity Levy process, which includes the jump frequency and the jump size distribution which enables ´ the further investigation of the properties of estimators without the presence of high frequency data ∆. Numerical implementation of the approach was applied on sample electricity data of about 10,000 observations between the period of 2 years and then a seasonalized forecast for an extra year was implemented to normalize the volatility in forwards contracts.

A combined compact difference scheme for option pricing in the exponential jump-diffusion models

In the present paper, starting with the Black–Scholes equations, whose solutions are the values of European options, we describe the exponential jump-diffusion model of Levy process type. Here, a jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general “partial integro-differential equation” (PIDE). With a combined compact difference (CCD) scheme for the spatial discretization, a high-order method is proposed for solving exponential jump-diffusion models. The method is sixth-order accurate in space and second-order accurate in time. A known analytical solution to the model is used to evaluate the performance of the numerical scheme.

Pricing and Hedging Defaultable Participating Contracts with Regime Switching and Jump Risk

This paper develops a transform-based approach for the pricing of participating life insurance contracts with a constant or floating guaranteed rate. Our analysis incorporates credit, market (jump), and economic (regime switching) risks, where the evolution of the reference portfolio is described by a regime switching double exponential jump diffusion model. We provide semi-analytical formulas for the contract value by using a Laplace or Laplace-Fourier transform that involves matrix Wiener-Hopf factors. Then, the price is obtained by implementing the matrix Wiener-Hopf factorization and by performing a numerical Laplace and Fourier inversion. By comparing the results with Monte-Carlo simulations, we show that our pricing method is easy to implement and accurate. We also show that the contract with a floating guaranteed rate is riskier but more profitable than the contract with a constant guaranteed rate. Two hedging strategies are introduced to hedge jump and regime switching risks in the participating contracts.

Defined Contribution Pension Plans: Who Has Seen the Risk?

The trend towards eliminating defined benefit (DB) pension plans in favour of defined contribution (DC) plans implies that increasing numbers of pension plan participants will bear the risk that final realized portfolio values may be insufficient to fund desired retirement cash flows. We compare the outcomes of various asset allocation strategies for a typical DC plan investor. The strategies considered include constant proportion, linear glide path, and optimal dynamic (multi-period) time consistent quadratic shortfall approaches. The last of these is based on a double exponential jump diffusion model. We determine the parameters of the model using monthly US data over a 90-year sample period. We carry out tests in a synthetic market which is based on the same jump diffusion model and also using bootstrap resampling of historical data. The probability that portfolio values at retirement will be insufficient to provide adequate retirement incomes is relatively high, unless DC investors adopt optimal allocation strategies and raise typical contribution rates. This suggests there is a looming crisis in DC plans, which requires educating DC plan holders in terms of realistic expectations, required contributions, and optimal asset allocation strategies.

Defined Contribution Pension Plans: Who Has Seen the Risk?

The trend towards eliminating defined benefit (DB) pension plans in favor of defined contribution (DC) plans implies that increasing numbers of pension plan participants will bear the risk that final realized portfolio values may be insufficient to fund desired retirement cash flows. We compare the outcomes of various asset allocation strategies for a typical DC plan investor. The strategies considered include constant proportion, linear glide path, and optimal dynamic (multi-period) time consistent quadratic shortfall approaches. The last of these is based on a double exponential jump diffusion model. We determine the parameters of the model using monthly US data over a 90 year sample period. We carry out tests in a synthetic market which is based on the same jump diffusion model and also using bootstrap resampling of historical data. The probability that portfolio values at retirement will be insufficient to provide adequate retirement incomes is relatively high, unless DC investors adopt optimal allocation strategies and raise typical contribution rates. This suggests there is a looming crisis in DC plans, which requires educating DC plan holders in terms of realistic expectations, required contributions, and optimal asset allocation strategies.

Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space [Formula: see text] or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases, we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods.

Evaluation of the Kou-Modified Lee-Carter Model in Mortality Forecasting: Evidence from French Male Mortality Data

Mortality forecasting has always been a target of study by academics and practitioners. Since the introduction and rising significance of securitization of risk in mortality and longevity, more in-depth studies regarding mortality have been carried out to enable the fair pricing of such derivatives. In this article, a comparative analysis is performed on the mortality forecasting accuracy of four mortality models. The methodology employs the Age-Period-Cohort model, the Cairns-Blake-Dowd model, the classical Lee-Carter model and the Kou-Modified Lee-Carter model. The Kou-Modified Lee-Carter model combines the classical Lee-Carter with the Double Exponential Jump Diffusion model. This paper is the first study to employ the Kou model to forecast French mortality data. The dataset comprises death data of French males from age 0 to age 90, available for the years 1900–2015. The paper differentiates between two periods: the 1900–1960 period where extreme mortality events occurred for French males and the 1961–2015 period where no significant jump is observed. The Kou-modified Lee-Carter model turns out to give the best mortality forecasts based on the RMSE, MAE, MPE and MAPE metrics for the period 1900–1960 during which the two World Wars occurred. This confirms that the consideration of jumps and leptokurtic features conveys important information for mortality forecasting.

Introducing Fuzziness in CDS Pricing under a Structural Model

OTC credit derivatives are nonstandardized financial derivatives which have the following characteristics. (1) Information on trades is not public. (2) There is no performance guarantee from the stock exchange. (3) The bigger the risk in performance, the bigger the price floating. These result in an asymmetry of market information flow and this asymmetry acts as a decisive factor in the credit risk pricing of financial instruments. The asymmetry of market information flows will lead to obvious fuzziness in how counterparty risks are characterized, as in the process of valuing assets when discontinuous jumping takes place. Accurately measuring the amplitude and frequency of asset values when influenced by information asymmetry cannot be arrived at just by analyzing the random values of historical data. With this in mind, this paper hypothesizes both asset value jump amplitude and frequency of random parameters as a triangular fuzzy interval, i.e., a new double exponential jump diffusion model with fuzzy analysis. It then gives a credit default swap pricing formula in the form of fuzziness. Through the introduction of fuzzy information, this model has the advantage of being able to arrive at results in the form of triangular fuzziness and, consequently, being able to solve some inherent problems in a world characterized by asymmetry in the flow of market information and, to a certain extent, the inadequate disclosure of information.

Option Pricing under the Double Exponential Jump-Diffusion Model with Stochastic Volatility and Interest Rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump-diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump-diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models.

Compound option pricing under a double exponential Jump-diffusion model

A compound option, an option on another option, plays an important role in financial field since it can be used to price American option and corporate debt with discrete coupons. In the real options literature, compound options are most suitable to be employed to investment problems involving sequential decision making. Most compound option and real options formulae are based on log-normal distribution while the empirical evidence shows that the return distribution in real market exhibits asymmetric leptokurtic feature, higher peak and two heavier tails. This paper introduces the jump-diffusion process into pricing compound options and derives the related valuation formulas. We assume that the dynamic of the underlying asset return process consists of a drift component, a continuous Wiener process and discontinuous jump-diffusion processes which have jump times that follow the compound Poisson process and the logarithm of jump size follows the double exponential distribution proposed by Kou (2002). Numerical results indicate that the advantage of combining the double exponential distribution and normal distribution is that it can capture the phenomena of both the asymmetric leptokurtic features and the volatility smile. Furthermore, the compound options under the double exponential jump diffusion model which we derived are more generalized than those proposed by Gukhal (2004) and Geske (1979), and thus have wider application.

Real options under a double exponential jump-diffusion model with regime switching and partial information

We consider the irreversible investment in a project which generates a cash flow following a double exponential jump-diffusion process and its expected return is governed by a continuous-time two-state Markov chain. If the expected return is observable, we present explicit expressions for the pricing and timing of the option to invest. With partial information, i.e. if the expected return is unobservable, we provide an explicit project value and an integral-differential equation for the pricing and timing of the option. We provide a method to measure the information value, i.e. the difference between the option values under the two different cases. We present numerical solutions by finite difference methods. By numerical analysis, we find that: (i) the higher the jump intensity, the later the option to invest is exercised, but its effect on the option value is ambiguous; (ii) the option value increases with the belief in a boom economy; (iii) if investors are more uncertain about the economic environment, information is more valuable; (iv) the more likely the transition from boom to recession, the lower the value of the option; (v) the bigger the dispersion of the expected return, the higher the information value; (vi) a higher cash flow volatility induces a lower information value.

The Total Return Swap Pricing Model under Fuzzy Random Environments

This paper models the jump amplitude and frequency of random parameters of asset value as a triangular fuzzy interval. In other words, we put forward a new double exponential jump diffusion model with fuzziness, express the parameters in terms of total return swap pricing, and derive a fuzzy form pricing formula for the total return swap. Following simulation, we find that the more the fuzziness in financial markets, the more the possibility of fuzzy credit spreads enlarging. On the other hand, when investors exhibit stronger subjective beliefs, fuzzy credit spreads diminish. Using fuzzy information and random analysis, one can consider more uncertain sources to explain how the asset price jump process works and the subjective judgment of investors in financial markets under a variety of fuzzy conditions. An appropriate price range will give investors more flexibility in making a choice.

On Two Transform Methods for the Valuation of Contingent Claims

This paper presents two transform methods for pricing contingent claims namely the fast Fourier transform method and the fast Hilbert transform method. The fast Fourier transform method utilizes the characteristic function of the underlying instrument’s price process. The fast Hilbert transform method is obtained by multiplying a square integrable function f by an indicator function associated with the barrier feature in the real domain. This is also obtained by taking the Hilbert transform in the Fourier domain. We derived closed-form solutions for European call options in a double exponential jump-diffusion model with stochastic volatility. We developed fast and accurate numerical solutions by means of the Fourier transform method. The comparison of the probability densities of the double exponential jump-diffusion model with stochastic volatility, the Black-Scholes model and the double exponential jump-diffusion model without stochastic volatility showed that the double exponential jump-diffusion model with stochastic volatility has better performance than the two other models with respect to pricing long term options. An analysis of the fast Fourier transform method revealed that the volatility of volatility σ and the correlation coefficient ρ have significant impact on option values. It was also observed that these parameters σ and ρ have effect on long-term option, stock returns and they are also negatively correlated with volatility. These negative correlations are important for contingent claims valuation. The fast Fourier transform method is useful for empirical analysis of the underlying asset price. This method can also be used for pricing contingent claims when the characteristic function of the return is known analytically. We considered the performance of the fast Hilbert transform method and Heston model for pricing finite-maturity discrete barrier style options under stochastic volatility and observed that the fast Hilbert transform method gives more accurate results than the Heston model as shown in Table 3.

Sequential Modeling of Dependent Jump Processes

Two multivariate models for asset returns are introduced. Both generalize popular univariate models without altering their marginal laws; a very convenient feature, e.g., for a sequential calibration of the model's parameters to market quotes. The first is a double exponential jump-diffusion model with constant volatility as presented in the univariate case by Kou (2002). The second is a generalization of the stochastic volatility model of Gamma-Ornstein-Uhlenbeck type as first presented by Barndorff-Nielsen and Shephard (2001). Univariate processes of these kinds are driven by a Brownian motion as well as a compound Poisson process. Unlike other existing multivariate extensions of these univariate models, one important advantage of the presented approach is to keep the parameters that govern the dependence structure between the assets separate from the parameters that determine the marginal distributions of the assets. This is achieved by introducing dependence to the univariate compound Poisson processes by means of a bespoke common stochastic time change.