Semi-analytical Pricing Research Articles

Implied roughness in the term structure of oil market volatility

This paper analyses the attributes and the significance of the roughness of oil market volatility. We employ unspanned stochastic volatility models driven by rough Brownian motions that yield semi-analytical prices for future options entailing efficient calibration applications. By performing a Monte Carlo simulation study, we show that the semi-analytical pricing performs well thus establishing its efficiency for calibration applications. Thus we calibrate option prices written on oil futures and provide empirical evidence of the roughness in oil volatility. Introducing just one additional parameter, the Hurst parameter, indicating the roughness of the volatility improves the calibration by almost a factor of 10. The calibrated option-implied Hurst parameter varies over time, but the entire set of parameters becomes more stable than in the non-rough case corresponding to a fixed Hurst parameter 1/2. These results underscore the importance to model the time dependency of the roughness of oil market volatility.

Semi-analytical pricing of options written on SOFR futures

Semi-analytical pricing of options written on SOFR futures

American Options in Time-Dependent One-Factor Models: Semi-Analytic Pricing, Numerical Methods, and ML Support

American Options in Time-Dependent One-Factor Models: Semi-Analytic Pricing, Numerical Methods, and ML Support

A novel stochastic modeling framework for coal production and logistics through options pricing analysis

We propose a novel stochastic modeling framework for coal production and logistics using option pricing theory. The problem of valuing the inherent real optionality a coal producer has when mining and processing thermal coal is modelled as pricing spread options of three assets under the stochastic volatility model. We derive a three-dimensional Fast Fourier Transform (“FFT”) lower bound approximation to value the inherent real optionality and for robustness check, we compare the semi-analytical pricing accuracy with the Monte Carlo simulation. Model parameters are estimated from the historical monthly data, and stochastic volatility parameters are obtained by matching the Kurtosis of the low-ash diff data to the Kurtosis of the stochastic volatility process which is assumed to follow Cox–Ingersoll–Ross (“CIR”) model.

Semi-Analytical Pricing for Generalized Short Rate Models

Semi-Analytical Pricing for Generalized Short Rate Models

Semi-Analytical Pricing of Barrier Options in the Time-Dependent λ-SABR Model: Uncorrelated Case

We consider semi-analytical pricing of barrier options for the time-dependent SABR stochastic volatility model (with drift in the instantaneous volatility) with zero correlation between spot and stochastic volatility. In doing so, we modify the general integral transform method (see Itkin et al. 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the article. Numerical examples illustrate the speed and accuracy of our method, which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.

Barrier Option Pricing with Trading and Non-Trading Hours

We introduce a semi-analytical pricing approach, which takes non-trading hours into account.

Pricing defaultable bonds under Hawkes jump-diffusion processes

In this paper, we propose a reduced-form model for the embedded credit risk in corporate bonds. We specify the default hazard rate as an affine function of a series of influential variables. To capture the clustering property in some extreme situations, Hawkes jump-diffusion processes are adopted to model the variables. We derive the semi-analytical pricing formula for defaultable bonds. The empirical results from U.S. bond market illustrate the significance of jump clustering when pricing low credit-rating bonds.

Pricing and hedging for correlation options with regime switching and common jump risk

This article investigates the pricing of the correlation options under regime switching models with common jump risk. We assume that the values of model parameters are modulated by a continuous-time, finite-state, observable Markov chain which is used to describe the states of an economy. In addition, the common jump reflects the correlated jump risk between the underlying assets. According to the Fourier transform method, we first derive the semi-analytical pricing formula for the correlation options. Then, some hedging strategies such as Greek letters and minimum variance hedging strategies are offered. Finally, we provide numerical examples to illustrate the effects of the regime switching and the common jump risk on the correlation option price by using the fast Fourier transform (FFT) algorithm.

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

<p style='text-indent:20px;'>The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.</p>

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

<p style='text-indent:20px;'>We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.</p>

Option Pricing under Double Heston Model with Approximative Fractional Stochastic Volatility

We establish double Heston model with approximative fractional stochastic volatility in this article. Since approximative fractional Brownian motion is a better choice compared with Brownian motion in financial studies, we introduce it to double Heston model by modeling the dynamics of the stock price and one factor of the variance with approximative fractional process and it is our contribution to the article. We use the technique of Radon–Nikodym derivative to obtain the semianalytical pricing formula for the call options and derive the characteristic functions. We do the calibration to estimate the parameters. The calibration demonstrates that the model provides the best performance among the three models. The numerical result demonstrates that the model has better performance than the double Heston model in fitting with the market implied volatilities for different maturities. The model has a better fit to the market implied volatilities for long-term options than for short-term options. We also examine the impact of the positive approximation factor and the long-memory parameter on the call option prices.

A semi-analytical pricing formula for European options under the rough Heston-CIR model

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.
 
 doi:10.1017/S1446181120000024

Pricing vulnerable European options with dynamic correlation between market risk and credit risk

In this paper, we study the valuation of vulnerable European options incorporating the reduced-form approach, which models the credit default of the counterparty. We provide an analytical pricing model in which the components of the state processes, including the dynamics of the underlying asset value and the intensity process corresponding to the default event, are cross-exciting and they could facilitate the description of complex structure of events dependence. To illustrate how our model works, we present an application when the state variables follow specific affine jump-diffusion processes. Semi-analytical pricing formulae are obtained through a system of matrix Riccati equations. The derived formula can be implemented numerically, and we give numerical analysis to investigate the impact of the dynamic correlation between jump risk of the underlying asset value and default risk of the counterparty.

Geometric Step Options with Jumps: Parity Relations, PIDEs, and Semi-Analytical Pricing

The present article studies geometric step options in exponential Levy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and parity relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Semi-analytic pricing formulas for basket credit-linked notes with and without counterparty risks

This paper discusses the pricing of basket credit linked notes (BCLN) under the reduced model. Three types of BCLNs are discussed: the first, the second and the mth-to-default BCLNs with and without counterparty risks. Based on conditional independence, the joint distribution of default times can be calculated and thus the price problems are transformed into partial differential equation (PDE) forms. Then the semi-analytical pricing formulas for the above BCLNs are obtained with PDE method and the calculation of credit valuation adjustments are also obtained. Finally, numerical test results show the high precision of the derived pricing formulas.

A semianalytical formula for European options under a hybrid Heston–Cox–Ingersoll–Ross model with regime switching

AbstractIn this paper, we consider the pricing of European options under a regime‐switching Heston–Cox–Ingersoll–Ross (CIR) hybrid model, where the mean‐reversion levels of both the stochastic volatility and interest rate are assumed to change among different states. Albeit difficult, we have still managed to derive an semianalytical pricing formula for European options after the generalized moment generating function of this particular model is worked out. Numerical experiments are also carried out to demonstrate the accuracy of the newly derived formula as well as the influence of the introduction of the regime‐switching mechanics on option prices. Finally, through a preliminary empirical study, our model is shown to be superior to the Heston‐CIR model, which demonstrates the importance of introducing the regime‐switching mechanics.

A SEMI-ANALYTICAL PRICING FORMULA FOR EUROPEAN OPTIONS UNDER THE ROUGH HESTON-CIR MODEL

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

Generalized affine transform on pricing quanto range accrual note

This paper was to price and hedge a quanto floating range accrual note (QFRAN) by an affine term structure model with affine-jump processes. We first generalized the affine transform proposed by Duffie et al. (2000) under both the domestic and foreign risk-neutral measures with a change of measure, which provides a flexible structure to value quanto derivatives. Then, we provided semi-analytic pricing and hedging solutions for QFRAN under a four-factor affine-jump model with the stochastic mean, stochastic volatility, and jumps. The numerical results demonstrated that both the common and local factors significantly affect the value and hedging strategy of QFRAN. Notably, the factor of stochastic mean plays the most important role in either valuation or hedging. This study suggested that ignorance of these factors in a term-structure model will result in significant pricing and hedging errors in QFRAN. In summary, this study provided flexible and easily implementable solutions in valuing quanto derivatives.

Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales

In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numeraire technique, proposed by Geman et al. [ 14 ], is employed to obtain a semi-analytical pricing formula for an vulnerable European option in the presence of regime switching effect. The method is numerically implemented using the multinomial approach in Costabile et al. [ 6 ]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.