Vulnerable Options Research Articles

CVA AND VULNERABLE OPTIONS IN STOCHASTIC VOLATILITY MODELS

This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

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.

Pricing vulnerable options under jump diffusion processes using double Mellin transform

In this paper, we price the vulnerable option under the assumption that both underlying and counterparty asset price follow correlated jump diffusion processes. By the two-dimensional Itô-Doeblin formula, we initially derive a partial integro-differential equation (PIDE) satisfied by the price of the vulnerable option with flexible jumps. Then, the double Mellin transform converts the PIDE into an ordinary differential equation (ODE), which possesses an explicit solution. Further, the inverse Mellin transform of the explicit solution to the ODE turns out to be a rigorous solution to the PIDE for vulnerable option. Finally, by carrying out some numerical experiments, we demonstrate that the PIDE’s solution produces price with a convincing degree of precision.

Pricing vulnerable options in a hybrid credit risk model driven by Heston–Nandi GARCH processes

This paper proposes a hybrid credit risk model, in closed form, to price vulnerable options with stochastic volatility. The distinctive features of the model are threefold. First, both the underlying and the option issuer’s assets follow the Heston–Nandi GARCH model with their conditional variance being readily estimated and implemented solely on the basis of the observable prices in the market. Second, the model incorporates both idiosyncratic and systematic risks into the asset dynamics of the underlying and the option issuer, as well as the intensity process. Finally, the explicit pricing formula of vulnerable options enables us to undertake the comparative statistics analysis.

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.

The valuation of vulnerable European options with risky collateral

This paper presents a model for valuing vulnerable European options with risky collateral under the assumption that the holder of vulnerable options could recover a proportion of the option value using the collateral account when default occurs. We describe the underlying asset and the risky collateral using correlated geometric Brownian motions and consider default risk in a reduced form model. An integral pricing formula of call options is derived when the default intensity follows an Ornstein–Uhlenbeck process. For practical purposes, we work under the default intensity captured by Cox–Ingersoll–Ross and Ornstein–Uhlenbeck processes respectively, and numerical results show that the differences in the values of vulnerable options under these two intensity processes are tiny. The impacts of risky collateral and default risk on option prices are illustrated. Specially, the effect of wrong (right) way risk can be reflected by the correlation between the underlying asset and default intensity.

Valuing vulnerable options with two underlying assets

ABSTRACT In this paper, we consider vulnerable options with two underlying assets, including options on the maximum of two underlying prices, options paying the best and cash, and options on the maximum of two geometric average prices. We derive the explicit pricing formulae of these options and perform numerical examples to illustrate prices of these options.

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.

CVA and vulnerable options pricing by correlation expansions

We consider the problem of computing the credit value adjustment (CVA) of a European option in presence of the wrong way risk in a default intensity setting. Namely we model the asset price evolution as solution to a linear equation that might depend on different stochastic factors and we provide an approximate evaluation of the option’s price, by exploiting a correlation expansion approach, introduced in Antonelli and Scarlatti (Finance Stoch 13:269–303, 2009). We also extend our theoretical analysis to include some further value adjustments, for instance due to collateralization and funding costs. Finally, in the CVA case, we compare the numerical performance of our method with the one recently proposed by Brigo and Vrins (Eur J Oper Res 269:1154–1164, 2018) and Brigo et al. (Innovations in insurance, risk and asset management, WSPC proceedings, 2018), in the case of a call option driven by a GBM correlated with a CIR default intensity. We additionally compare with the numerical evaluations obtained by other methods.

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.

Optimal risk-taking in corporate defined benefit plans under risk-shifting

PurposeThe purpose of this paper is to theoretically examine the risk-taking decision of corporate defined benefits (DB) plans. The equity holders’ investment problem that is represented by the position of a vulnerable option is solved.Design/methodology/approachThe simple traditional contingent claim approach is applied, which considers only the distributions of corporate cash flow, without the model expansions, such as market imperfections, needed to explain the firms’ behavior for DB plans in previous studies.FindingsThe authors find that the optimal solution to the equity holders’ DB investment problem is not an extreme corner solution such as 100 percent investment in equity funds as in the literature. Rather, the solution lies in the middle range, as is commonly observed in real-world economies.Originality/valueThe major value of this study is that it develops a clear mechanism for obtaining an internal solution for the equity holders’ DB investment problem and it provides the understanding that the base for corporate investment behavior for DB plans should incorporate the fact that in some cases the optimal solution is in the middle range. Therefore, the corporate risk-taking behavior of DB plans is harder to identify than the results of the empirical literature have predicted.

Pricing Vulnerable Option under Jump-Diffusion Model with Incomplete Information

In this paper, the closed-form pricing formula for the European vulnerable option with credit risk and jump risk under incomplete information was derived. Noise was introduced to the option writers assets while the underlying asset price and the value of corporation were assumed to follow the jump-diffusion processes. Finally the numerical experiment showed that jumps of underlying assets would increase the value of the option, but noise of corporation value was opposite.

Pricing Vulnerable Options with Correlated Credit Risk Under Jump-diffusion Processes When Corporate Liabilities Are Random

In this paper, we consider an improved model of pricing vulnerable options with credit risk. We assume that the vulnerable European options not only face default risk, but also face the rare shocks of the underlying assets and the counterparty assets. The dynamics of two correlated assets are modeled as a class of jump diffusion processes. Furthermore, we assume that the dynamic of the corporate liability is a geometric Brownian motion that is related to the underlying asset and the counterparty asset. Under this new framework, we give an explicit pricing formula of the vulnerable European options.

CVA and Vulnerable Options in Stochastic Volatility Models

In this work we want to provide a general principle to evaluate the CVA (Credit Value Adjustment) for a vulnerable option, that is an option subject to some default event, concerning the solvability of the issuer. CVA is needed to evaluate correctly the contract and it is particularly important in presence of WWR (Wrong Way Risk), when a credit deterioration determines an increase of the claim's price. In particular, we are interested in evaluating the CVA in stochastic volatility models for the underlying's price (which often fit quite well the market's prices) when admitting correlation with the default event. By cunningly using Ito's calculus, we provide a general representation formula applicable to some popular models such as SABR, Hull & White and Heston, which explicitly shows the correction in CVA due to the processes correlation. Later, we specialize this formula and construct its approximation for the three selected models. Lastly, we run a numerical study to test the formula's accuracy, comparing our results with Monte Carlo simulations.

Valuation of the Vulnerable Option Price Based on Mixed Fractional Brownian Motion

The pricing problem of a kind of European vulnerable option was studied. The mixed fractional Brownian motion and the jump process were used to characterize the evolution of stock prices. The closed-form solution to European option pricing was obtained by applying martingale measure transformation method. At the end of this paper, some numerical experiments were adopted to compare the new pricing formula introduced in this paper with the classical Black-Scholes pricing formula. The result showed that the new pricing formula conformed to the actual financial market. In fact, the option value is positively correlated with the underlying asset price and the company’s asset price and the jump process has significant influence on the value of option.

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 Vulnerable European Options under Lévy Process with Stochastic Volatility

This paper considers the pricing issue of vulnerable European option when the dynamics of the underlying asset value and counterparty’s asset value follow two correlated exponential Lévy processes with stochastic volatility, and the stochastic volatility is divided into the long-term and short-term volatility. A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility is governed by a mean-reverting process. Based on the proposed model, the joint moment generating function of underlying log-asset price and counterparty’s log-asset value is explicitly derived. We derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions. Finally, numerical simulations are provided to illustrate the effects of stochastic volatility, jump risk, and counterparty credit risk on the vulnerable option price.

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.

Pricing and hedging vulnerable option with funding costs and collateral

We explore the valuation and hedging strategies of a European vulnerable option with funding costs and collateralization for local volatility models. It is found that, in the absence of arbitrage opportunities, the option price must lie within a no-arbitrage band. The boundaries of no-arbitrage band are computed as solutions to backward stochastic differential equations (BSDEs in short) of replicating strategy and offsetting strategy. Under some conditions, we obtain the closed-form representations of the no-arbitrage band for local volatility models. In particular, the fully explicit expressions of the no-arbitrage band for Black–Scholes model and the constant elasticity of variance (CEV) model with time-dependent parameters are derived. Furthermore, we provide a strategy for the option holder by using the risky bond issued by the option writer to hedge the remaining potential losses. By virtue of numerical simulation, the impact of the default risk, funding costs and collateral can be observed visually.