Pricing Of Vulnerable Options Research Articles

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.

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 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>

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 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.

An asymptotic expansion approach to the valuation of vulnerable options under a multiscale stochastic volatility model

Abstract In this study, we examine the pricing of vulnerable options under a stochastic volatility model based on the partial differential equation approach. Specifically, we consider a multiscale stochastic volatility model that is assumed to be driven by two diffusions (fast-scale and slow-scale) and use an asymptotic expansion approach to drive the approximate pricing formulas of vulnerable options, which allows the counterparty credit risk at maturity. Furthermore, we provide the Greek Delta of vulnerable options for the dynamic hedge and present the numerical results to examine the effect of the multiscale stochastic volatility model and to show the accuracy of our formula.

Valuing vulnerable options with bond collateral

ABSTRACT This paper investigates vulnerable European options with bond collateral. Working with stochastic interest rate, we evaluate vulnerable options under four cases of collateral according to the length of time to maturity of the government bonds, including cash, short term bonds, medium term bonds, and long term bonds. The impacts of collateral and credit risk on vulnerable option prices are illustrated numerically.

Vulnerable options pricing under uncertain volatility model

In this paper, we consider the pricing problem of options with counterparty default risks. We study the asymptotic behavior of vulnerable option prices in the worst case scenario under an uncertain volatility model which contains both corporate assets and underlying assets. We propose a method to estimate the price of vulnerable options when the volatility of the underlying assets is within a small interval. By imposing additional conditions on the boundary condition and cutting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximate method for solving the fully nonlinear partial differential equation satisfied by the price of vulnerable options under the uncertain volatility model.

Pricing Vulnerable Options with Stochastic Volatility and Stochastic Interest Rate

This paper considers the pricing issue of vulnerable European options when the price process of the underlying asset follows the GARCH diffusion model with stochastic interest rate. Based on the proposed model, we obtain an approximate solution for the vulnerable European option price via means of Fourier transform. In addition, the Greeks of vulnerable option price are derived explicitly. Besides, the approximate solution of vulnerable option price can be quickly computed by using the fast Fourier transform (FFT) algorithm. The results of Monte Carlo simulations indicate that FFT is accurate, fast and easy to implement. More important, the pricing model also reveals that: (i) a negative correlation of volatility with the spot return creates a fat left tail and thin right tail in the distribution of continuously compounded spot returns. Thus, for in-the-money options, the vulnerable option prices of the proposed model are higher than those of Klein (J Bank Finance 20(7):1211–1229, 1996). While for deep-out-of-the-money options, the vulnerable option prices of the proposed model are smaller; (ii) the higher long-run mean of the underlying asset price’s instantaneous variance, the higher vulnerable option price; (iii) the long-run mean of the stochastic interest rate exerts a positive effect on the value of vulnerable European option.

The continuity and estimates of a solution to mixed fractional constant elasticity of variance system with stochastic volatility and the pricing of vulnerable options

Stochastic volatility models play an important role in finance modeling. Under a mixed fractional Brownian motion environment, we study the continuity and estimates of a solution to a kind of stochastic differential equations with double volatility terms. Besides, we propose to price the vulnerable option with the discretization method and present the results using a Monte Carlo simulation.

Pricing vulnerable options with variable default boundary under jump-diffusion processes

For the pricing of vulnerable options, we improve the results of Klein and Inglis [Journal of Banking and Finance] and Tian et al. [The Journal of Futures and Markets], considering the circumstances in which the writers of options face financial crisis. Our pricing model faces the risks of default and the occasional impact experienced by the underlying assets and counterparty’s assets. The correlation between the option’s underlying assets and the option writer’s assets is clearly modeled. Asset prices are driven by the jump-diffusion processes of two related assets. Furthermore, we consider a variable default boundary (VDB) based on the option’s potential debt and the option writer’s other liabilities. In case financial distress happens, the payout rate is connected to the option writer’s assets. Through the Taylor expansion, we derive an approximate explicit valuation for vulnerable options.

Pricing of vulnerable options with early counterparty credit risk

The counterparty credit risk should be considered when valuing options traded in the over-the-counter (OTC) markets because the OTC markets have rapidly grown, and the credit risk in the OTC markets has become an important issue since the global financial crisis. In this paper, we propose two types of vulnerable options whose payoffs allow for the counterparty credit risk prior to maturity of the options. We use Mellin transforms to solve the partial differential equations for the vulnerable option prices with early counterparty credit risk. In addition, we present numerical experiment results with the Monte Carlo simulations to show the accuracy of the pricing formulas and provide graphs to illustrate the properties of options.

A Binomial Tree Approach to Pricing Vulnerable Option in a Vague World

The aim of this paper is pricing the vulnerable options in a vague world. Due to the vulnerability of financial markets and the economy environment in the real world, investors cannot always have precise information about firm value and default recovery rate in vulnerable option pricing. Therefore, following the framework of Klein in 1996, a fuzzy binomial tree pricing model is derived by modelling the firm value and default recovery rate as fuzzy numbers. The numerical results show that the precise information assumption about the firm value and recovery rate in Klein model may lead to underestimate the credit risk on the values of vulnerable options. This study aims to provide insights for future research on defaultable options pricing under imprecise market information.

Pricing vulnerable options with stochastic volatility

In this paper, we investigate the pricing issue of vulnerable options with stochastic volatility by decomposing stochastic volatility into the long-term and short-term volatility. We describe the short-term fluctuation of stochastic volatility using a mean-reverting process, and assume the long-term volatility to be a constant. Based on the proposed model, we derive a pricing formula of vulnerable options in a special case. Numerical results are presented to illustrate the impacts of two stochastic volatility components on vulnerable option prices.

The martingale approach for vulnerable binary option pricing under stochastic interest rate

We consider the vulnerable option pricing problem when the stochastic interest rate is driven by a Hull-White model. Based on the firm value model, we suppose that the stock prices, assets and liabilities of a company follow the relevant O-U processes. We adopt the martingale approach to determine the equivalent martingale measure for pricing the vulnerable binary option, the analytical pricing formula of the vulnerable binary options is derived.

Differences in the Prices of Vulnerable Options with Different Counterparties

AbstractIn this paper, a new pricing model is proposed to investigate the differences in the prices of vulnerable options with different counterparties. I start by specifying the dynamics of the market portfolio, and then break down the risk of the underlying asset and the assets of the counterparties into systematic and idiosyncratic risk, which allows me to distinguish the counterparties from these two kinds of risk. Finally, the derived pricing formulae are used to illustrate the differences between vulnerable option prices. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:148–163, 2017

A closed form solution for vulnerable options with Heston’s stochastic volatility

Over-the-counter stock markets in the world have been growing rapidly and vulnerability to default risks of option holders traded in the over-the-counter markets became an important issue, in particular, since the global finance crisis and Eurozone crisis. This paper studies the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics. In this paper, we obtain a closed form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula and find how the stochastic volatility effect on the Black–Scholes price as well as on the decreasing speed of the option price with credit risk depends on moneyness.