Vulnerable Options Research Articles

Vulnerable options with regime switching and stochastic liquidity

Investigating default risk in pricing options holds significant practical importance, as nearly all market participants and institutions face credit risk. Additionally, economic cycles and asset liquidity are crucial factors that should be incorporated. This paper considers these factors and derives an analytical pricing formula. Specifically, we model the economic cycles through switching volatility driven by a continuous-time Markov chain, while we adopt a discounting factor based on market liquidity levels to model the asset liquidity. We establish a risk-neutral measure after embracing a regime-switching Esscher transform, and formulate a price representation to value vulnerable options analytically despite the complexity of the developed model. We conduct several numerical experiments to validate the model’s efficacy and flexibility.

Pricing Vulnerable Options With Variance Gamma Systematic and Idiosyncratic Factors by Laplace Transform Inversion

Pricing Vulnerable Options With Variance Gamma Systematic and Idiosyncratic Factors by Laplace Transform Inversion

European vulnerable options pricing under sub-mixed fractional jump-diffusion model with stochastic interest rate

With the development of financial markets, the trading of financial derivatives has become more flexible. To meet the demands of investors, over-the-counter (OTC) trading of options has also become increasingly frequent, raising concerns about accurately pricing vulnerable options. This study investigates the pricing problem of European vulnerable options under the sub-mixed fractional jump-diffusion (SMFJ) model. Firstly, the underlying asset price model and the counterparty company value model are established under the SMFJ model with a sub-mixed fractional Vasicek stochastic interest rate. Then, analytical solutions for the European vulnerable options are obtained using the quasi-martingale measure transformation and the risk-neutral pricing method. Additionally, the accuracy of the pricing formula is verified through the Monte Carlo simulation method. Finally, the rationality and validity of the established model are confirmed by numerical simulations, and the influence of key parameters on the pricing models is analyzed.

Pricing vulnerable options under cross-asset markov-modulated jump-diffusion dynamics

In this study, the dynamics of the underlying asset price and the counterparty's asset price are governed by a cross-asset Markov-modulated jump-diffusion model that captures the time-inhomogeneity and systematic cojumps. Additionally, the forward interest rate processes are driven by a Markov-modulated Heath–Jarrow–Morton model to depict stochastic volatilities. Under an incomplete-market setting, we apply the Markov-modulated Esscher transform technique to determine a risk-neutral martingale measure. After determining the Markov-modulated Esscher parameters, we obtain an integral expression on the prices of vulnerable European-style Black–Scholes options. The numerical illustrations indicate that the findings are consistent with Klein (1996) and contribute to the extant literature on cojumping impacts on vulnerable option prices.

Pricing Vulnerable Options in Fractional Brownian Markets: a Partial Differential Equations Approach

Pricing Vulnerable Options in Fractional Brownian Markets: a Partial Differential Equations Approach

A Simplified Approach to the Pricing of Vulnerable Options with Two Underlying Assets in an Intensity-Based Model

In this paper, we study a simplified approach to determine the pricing formula for vulnerable options involving two correlated underlying assets. We utilize an intensity-based model to describe the credit risk associated with these vulnerable options. Without the change of measure technique, we derive pricing formulas for vulnerable options involving two underlying assets based on the probabilistic approach. We provide closed-form pricing formulas for two specific types of options: the vulnerable exchange option and the vulnerable foreign equity option. Finally, we present numerical results to demonstrate the accuracy of our formulas using the Monte-Carlo method and the effect of various parameters on the price of options.

Pricing European Vulnerable Options with Jumps and Stochastic Default Obstacles Barrier under Regime Switching

In this paper, we propose an enhanced model for pricing vulnerable options. Specifically, our model assumes that parameters such as interest rates, jump intensity, and asset value volatility are governed by an observable continuous-time finite-state Markov chain. We take into account European vulnerable options that are exposed to both default risk and rare shocks from underlying and counterparty assets. We also consider stochastic default barriers driven by a regime-switching model and geometric Brownian motion, thus improving upon the assumption of fixed default barriers. The risky assets follow a related jump-diffusion process, whereas the default barriers are influenced by a geometric Brownian motion correlated with the risky assets. Within the framework of our model, we derive an explicit pricing formula for European vulnerable options. Furthermore, we conduct numerical simulations to examine the effects of default barriers and other related parameters on option prices. Our findings indicate that stochastic default barriers increase credit risk, resulting in a decrease in option prices. By considering the aforementioned factors, our research contributes to a better understanding of pricing vulnerable options in the context of counterparty credit risk in over-the-counter trading.

Valuation of vulnerable options with stochastic corporate liabilities in a mixed fractional Brownian motion environment

Valuation of vulnerable options with stochastic corporate liabilities in a mixed fractional Brownian motion environment

Pricing of vulnerable options based on an uncertain CIR interest rate model

<abstract><p>The traditional Cox-Ingersoll-Ross (CIR) interest rate model follows a stochastic differential equation that cannot obtain the closed solution while the uncertain CIR interest rate model is an uncertain differential equation. First, this paper gives the solution in terms of the distribution of the uncertain CIR interest rate model based on uncertainty theory. Second, the pricing formulas of vulnerable European call option and vulnerable European put option are obtained by using the uncertain CIR interest rate model. Finally, according to the proposed pricing formula, the corresponding numerical algorithms are designed and several numerical examples are given to verify the effectiveness of the algorithm. Our results not only enrich the option pricing theory, but they also have a certain guiding significance for the derivatives market.</p></abstract>

Valuation of vulnerable European options with market liquidity risk

AbstractIn this paper, we investigate the pricing of vulnerable European options in a market where the underlying stocks are not perfectly liquid. A liquidity discount factor is used to model the effect of liquidity risk in the market, and the default risk of the option issuer is incorporated into the model using a reduced-form model, where the default intensity process is correlated with the liquidity risk. We obtain a semiclosed-form pricing formula of vulnerable options through the inverse Fourier transform. Finally, we illustrate the effects of default risk and liquidity risk on option prices numerically.

Valuation of European-style vulnerable options under the non-affine stochastic volatility and double exponential jump

The pricing of European-style vulnerable option when the price process of the underlying asset follows non-affine stochastic volatility and double exponential jump is investigated. An approximate expression for the joint characteristic function of the log-price of underlying asset and the log-value of counterparty asset is derived. An analytical approximate price of European-style vulnerable option is also obtained by means of Fourier-cosine method. Numerical experiments are given to confirm the accuracy and efficiency of the proposed result for pricing the European-style vulnerable option compared with Monte Carlo simulation. Finally, sensitivity analysis is presented to further explain the theoretical results.

Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate

This paper considers pricing of European-style vulnerable options under the Heston stochastic volatility and stochastic interest rate model in which the mean-reversion levels of both variance and interest rate processes are modulated by a continuous-time Markov process with a finite state space. An analytical pricing formula is derived by using the Esscher transform, joint characteristic function and multivariate Fourier transform technique, where the closed-form solution of the characteristic function is obtained by the law of iterated expectation. Then we provide the efficient approximation to calculate the analytical pricing formula of option using the FFT approach and examine the accuracy of the approximation by Monte Carlo simulation. Finally, the sensitivity analysis of different parameters in the proposed model on the vulnerable call option price and its Delta value are provided, and the difference between the proposed model and the Heston and stochastic interest rate model with non-Markov regime-switching are presented by some numerical experiments, which shows the influence of introducing regime-switching into Heston model with stochastic interest rate.

Valuing vulnerable Asian options with liquidity risk under Lévy processes

AbstractIn this paper, we study the pricing of vulnerable Asian options with liquidity risk. We employ general Lévy processes to capture the changes in the liquidity discount factors and the information processes of all assets. In the proposed pricing model, we obtain the closed-form pricing formula of vulnerable Asian options using the Fourier transform methods. Finally, the derived pricing formula is used to illustrate the effects of asymmetric jump risk, and the effects are relatively stable on (vulnerable) Asian options with different moneynesses.

Pricing vulnerable options under correlated skew Brownian motions

AbstractIn this paper, we consider vulnerable options in a pricing model with correlated skew Brownian motions. In the proposed pricing model, both the underlying asset and option issuer's assets are exposed to endogenous and exogenous risks. We deduct a new pricing formula of vulnerable European options, and then use it to illustrate the effect of the skewness parameters on vulnerable option prices. An interesting finding is that vulnerable option prices are higher when the variance of the logarithm of option issuer's assets is larger in some cases.

Pricing vulnerable options with stochastic liquidity risk

In this paper, we consider vulnerable options with stochastic liquidity risk. We employ liquidity-adjusted pricing models to describe the underlying stock price and option issuer’s assets. In addition, the correlation between these assets is stochastic, depending on the market liquidity measures. In the proposed framework, we derive closed forms of vulnerable European options with stochastic liquidity risk and then use them to illustrate the effects of stochastic liquidity risk on vulnerable option prices. Numerical results show that the effects of liquidity risk on the prices of out-of-the-money options or the options with a short maturity are not negligible.

Pricing Vulnerable Options in a Mixed Fractional Brownian Motion with Jumps

A new framework for pricing European vulnerable options is developed in the case where the underlying stock price and firm value follow the mixed fractional Brownian motion with jumps, respectively. This research uses the actuarial approach to study the pricing problem of European vulnerable options. An analytic closed-form pricing formula for vulnerable options with jumps is obtained. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in the paper. Finally, we compare and analyze the pricing results of different pricing models and discuss the influences of basic parameters on the pricing results of our proposed model by using numerical simulations, and the corresponding economic analyses about these influences are given.

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Pricing of vulnerable options under hybrid stochastic and local volatility

In this study, considering the paradoxical stochastic characteristics of over-the-counter markets during a financial crisis, we examine the price of vulnerable options under the constant-elasticity-of-variance-with-stochastic-volatility (SVCEV) model. This model describes the market situation better than the stochastic volatility model as well as the constant-elasticity-of-variance model. We provide the corrected option price derived by asymptotic analysis, which is an approximation to the price of a vulnerable option under the SVCEV model. Furthermore, we numerically verify the accuracy of the price of a vulnerable option (as obtained using the SVCEV model) by comparing the approximate option price with the option price obtained by Monte Carlo simulation.

Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility

<p style='text-indent:20px;'>In this paper, we propose a model to price vulnerable European options where the dynamics of the underlying asset value and the counter-party's asset value follow two jump-diffusion processes with fast mean-reverting stochastic volatility. First, we derive an equivalent risk-neutral measure and transfer the pricing problem into solving a partial differential equation (PDE) by the Feynman-Kac formula. We then approximate the solution of the PDE by pricing formulas with constant volatility via multi-scale asymptotic method. The pricing formula for vulnerable European options is obtained by applying a two-dimensional Laplace transform when the dynamics of the underlying asset value and the counter-party's asset value follow two correlated jump-diffusion processes with constant volatilities. Thus, an analytic approximation formula for the vulnerable European options is derived in our setting. Numerical experiments are given to demonstrate our method by using Laplace inversion.</p>

Pricing vulnerable options with jump risk and liquidity risk

In this paper, we consider vulnerable options with jump risk and liquidity risk. In the proposed framework, we allow discontinuous changes in the information processes and the liquidity discount factors as well, and default risk is taken into consideration. Specially, we investigate the effect of jumps in the liquidity discount factors and find that the effects of jumps in the liquidity discount factors are stable for different maturities and alternative moneynesses. Further, option prices behave differently with respect to alternative intensities of common jumps, depending on whether there are jumps in the liquidity discount factors or not.