Double Exponential Jump-diffusion Research Articles

Pricing airbag option via first passage time approach

Airbag option is a new structured product that has been highly favored by investors in the over-the-counter derivatives market. It provides protections against the downside loss to investors like the ‘airbag’ in a car crash. In contrast with its popularity, there is little literature on the pricing of airbag options, especially from the mathematical finance point of view. In this paper, we study the pricing of airbag options using the first passage time approach. We propose two innovative new types of airbag options, namely airbag-TB and airbag-ER, which enhance the downside loss protections by integrating a time-varying barrier and early redemption features respectively in the payoffs. Explicit and closed-form pricing formulas for the original and the two new types of airbag options are derived under the mixed-exponential jump diffusion (MEJD) model, double exponential jump diffusion (DEJD) model and Black-Scholes (BS) model. In doing so, we significantly improve the Hybrid model proposed by Jaimungal and Sigloch (Incorporating risk and ambiguity aversion into a hybrid model of default. Math. Finance, 2012, 22, 57–81) by incorporating a stochastic redemption intensity model and more general underlying asset dynamics. We further demonstrate the performance of our pricing formulas and the two new types of airbag options with numerical examples based on Monte Carlo simulation, which verifies the accuracy and efficiency of our pricing formulas.

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.

Option pricing under a double-exponential jump-diffusion model with varying severity of jumps

This paper extends the standard double-exponential jump-diffusion (DEJD) model to allow for successive jumps to bring about different effects on the asset price process. The double-exponentially distributed jump sizes are no longer assumed to have the same parameters; instead, we assume that these parameters may take a series of different values to reflect growing or diminishing effects from these jumps. The mathematical analysis of the stock price requires an introduction of a number of distributions that are extended from the hypoexponential (HE) distribution. Under such a generalized setting, the European option price is derived in closed-form which ensures its computational convenience. Through our numerical examples, we examine the effects on the return distributions from the growing and diminishing severity of the upcoming jumps expected in the near future, and investigate how the option prices and the shapes of the implied volatility smiles are influenced by the varying severity of jumps. These results demonstrate the benefits of the modeling flexibility provided by our extension.

An Algorithm for the Pricing and Timing of the Option to make a Two-Stage Investment with Credit Guarantees

We develop a jump-diffusion model for a guarantee-investment combination financing mode (G-I mode) that is recently popular in financial practice. We assume that a borrower has exclusively an option to invest in a project in two stages. The project’s cash flow follows a double exponential jump-diffusion process and it is increased by a growth factor once the second-stage investment is exercised. The first-stage investment cost is financed by a bank loan with the guarantee provided by an insurer, who promises to provide the second-stage investment cost as well as take the lender’s all default losses. In return for the guarantee and investment, the borrower pays a guarantee fee upon first investment and grants a fraction of equity upon second investment to the insurer. In sharp contrast to prior papers on guarantee, the guarantee costs are contracted prior to investment. We provide closed-form solutions and produce a numerical algorithm for the timing and pricing of the two investment options.

Option valuation under double exponential jump with stochastic intensity, stochastic interest rates and Markov regime-switching stochastic volatility

In this paper, we present a double exponential jump-diffusion option pricing model with stochastic interest rates, stochastic volatility, and stochastic jump intensity. In addition, Markov regime-switching is introduced to modulate the mean-reverting level of the squared volatility. We obtain the analytical pricing formulae for European options under this model. Finally, we use numerical examples to explore the effects of the regime-switching, stochastic jump intensity and the distribution of jump size on the option price or (and) the implied volatility.

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.

An algorithm for the pricing and timing of the option to make a two-stage investment with loan guarantees

We develop a jump-diffution model for a guarantee-investment combination financing mode (G-I mode) that is recently popular in financial practice. We assume that a borrower has exclusively an option to invest in a project in two stages. The project's cash flow follows a double exponential jump-diffusion process and it is increased by a growth factor once the second-stage investment is exercised. The first-stage investment cost is financed by a bank loan with the guarantee provided by an insurer, who promises to provide the second-stage investment cost as well as take the lender's all default losses. In return for the guarantee and investment, the borrower pays a guarantee fee upon first investment and grants a fraction of equity upon second investment to the insurer. In sharp contrast to prior papers on guarantee, the guarantee costs are contracted prior to investment. We provide closed-form solutions and produce a numerical algorithm for the timing and pricing of the two investment options.

A Model for Small Basket Equities Financing

Small basket equities financing is the bread and butter of hedge fund financing businesses. The collapse of Archegos Capital Management and the losses of its lending banks incurred in early 2021 are timely reminders of how challenging it is for banks to price and risk manage hedge fund transactions. This article develops an equity portfolio financing haircut model that gives credit analysts flexibility when assessing borrower credit quality given hedge funds’ operational opacity and lack of market traded credit products. Recent historical data are used to estimate the Black-Karasinski credit model and double-exponential jump-diffusion asset price model per stock. Computational results for a basket of four Archegos holding stocks show sensible haircuts and haircut diversification effect. A bank opting to trade at less favorable haircut levels could price in higher economic capital charge to compensate and use that to incentivize its funding unit.

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.

Specification Analysis of Structural Credit Risk Models

In this paper we conduct a specification analysis of structural credit risk models, using term structure of credit default swap (CDS) spreads and equity volatility from high-frequency return data. 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) barrier model, and the Longstaff and Schwartz (1995) model with stochastic interest rates. The double exponential jump-diffusion barrier model (Huang and Huang, 2003) improves significantly over the three models. The best one among the five models considered is the stationary leverage model of Collin-Dufresne and Goldstein (2001), which we cannot reject in more than half of our sample firms. 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 investment grade names. This points to a potential role of time-varying asset volatility, a feature that is missing in the standard structural models.

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.

Dynamic asset allocation with asymmetric jump distribution

PurposeThe purpose of this paper is to solve the optimal dynamic portfolio problem under the double-exponential jump diffusion (DEJD) distribution, which can allow the asset returns to jump asymmetrically.Design/methodology/approachThe authors solve the problem by solving the HJB equation. Meanwhile, in the presence of jump component in the asset returns, the investor may suffer a large loss due to high leveraged position, so the authors impose the short-sale and borrowing constraints when solving the optimization problem.FindingsThe authors provide sufficient conditions such that the optimal solution exists and show theoretically that the optimal risky asset weight is an increasing function of jump-up probability and average jump-up size and a decreasing function of average jump-down size.Research limitations/implicationsIn this study, the authors assume that the jump-up and jump-down intensities are constant. In the future, the authors will relax the assumption and allows the jump intensities to be time varying.Practical implicationsEmpirical studies based on Chinese Shanghai stock index data show that the jump distribution of Shanghai index returns is asymmetric, and the DEJD model can fit the data better than the log-normal jump-diffusion model. The numerical results are consistent with the theoretical prediction, and the authors find that the less risk-averse investor will suffer more economic cost if ignoring asymmetric jump distribution.Originality/valueThis study first examines how asymmetric jumps affect the investor’s portfolio allocation.

CDS Option Valuation with Double-Exponential Jumps

We demonstrate how the Double-Exponential Jump-Diffusion (DEJD) process can be used to value iTraxx CDS options based on historical returns of the underlying CDS index. In the first step we find Maximum Likelihood estimates for the volatility of the normal component of returns and the Poisson frequencies and mean sizes of upward and downward exponential jumps. In results similar to Ramezani and Zeng’s (2006) application of DEJD to equities, we find that the DEJD provides a better fit than either a normal or single-jump specification. We take the additional step of using parameter estimates as inputs to the semi-closed European option pricing formula proposed by Kou (2002) under DEJD. We compare the model and market option prices across strikes and find that both the level and shape of the implied volatility smile match closely. Our findings suggest that the DEJD provides a realistic description of the joint role of positive and negative economic surprises in credit markets. In addition, the use of Maximum Likelihood with the option pricing model is a practical way to analyse divergence between realised and market-implied distributions of credit returns, and can be used to check pricing.

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.

Haircutting Non-Cash Collateral

Haircutting non-cash collateral has become a key element of the post-crisis regulatory reform of the shadow banking system and OTC derivatives markets. This article develops a haircut model by expanding haircut definitions beyond the traditional value-at-risk measure and employing a double-exponential jump-diffusion model for collateral price. Haircuts are solved to target traditional credit risk measurements, such as probability of default, expected loss or unexpected loss given certain internal or external rating criteria. Computational results are presented for main equities, securitization, and corporate bonds.

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.

Pricing currency options in the Heston/CIR double exponential jump-diffusion model

We examine currency options in the double exponential jump-diffusion version of the Heston stochastic volatility model for the exchange rate. We assume, in addition, that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semi-analytical formula for the price of the European currency call option in the hybrid foreign exchange/interest rates model.

A Laplace Space Approach to American Options

In this paper, we extend the lower-upper bound approximation (LUBA) idea of Broadie and Detemple [Broadie, M., Detemple, J., (1996) American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies. 9(4): 1211-1250] to the Laplace space. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, for which there exist closed-form expressions for the Laplace transforms of the corresponding “capped (barrier) option prices”. The method is applied to a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures similar as Broadie and Detemple in the Laplace space. Finally we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option, and further utilize the early exercise premium (EEP) representation in the Laplace space to obtain the upper bound. We obtain explicit expressions in the case of the constant elasticity of variance (CEV) model (Wong and Zhao) and the double-exponential jump diffusion (DEJD) model (Leippold and Vasiljevic). Numerical examples show that our lower and upper bounds are accurate and efficient compared to results in the literature. To the best of authors’ knowledge, it is the first time that the LUBA idea of Broadie and Detemple is applied to a model with jumps, and this solves an open question stated on page 1181 of Kou and Wang.