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In this study, a conditional expectation transform (CET) method was developed for solving high‐dimensional systems arising in vulnerable options pricing. The CET was formulated to recast the pricing of vulnerable options into that of exchange‐traded options under special terminal condition. Then, traditional option pricing methods are deployed to price the exchange‐traded options, and an analytical formula for vulnerable options is obtained. The vulnerable European option, vulnerable Barrier option, and vulnerable floating Lookback option were analyzed as examples to demonstrate the effectiveness and accuracy of the CET method. Numerical examples were provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

In this paper we provide an efficient methodology to compute the credit value adjustment of a European contingent claim subject to some default event concerning the issuer solvability, when the underlying and the default event are correlated. In particular, in a Black and Scholes market/CIR intensity-default model, we consider a second order expansion around the origin of a vulnerable call option with respect to a correlation parameter ρ , which may be used to describe the wrong way risk of the contract, measuring the dependence between the underlying asset price and the option's issuer default intensity. Numerical implementations of this approach are compared with the benchmark Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of European vulnerable option pricing under the mixed fractional Brownian motion with stochastic corporate liabilities and jumps. Assume that the dynamics of all assets including the underlying asset, the counterparty's assets and corporate liabilities are given by three interrelated jump-diffusion processes. Jumps are divided into a systematic component which affects the prices of all assets and an idiosyncratic component affecting each asset price. Under this framework, the pricing formulae for vulnerable options are derived using an actuarial approach. Further, we discuss some properties of the pricing formulae, which reveal that our model contains not only the traditional pure diffusion processes such as Black–Scholes model (J Polit Econ 81:637–659, 1973) and Klein model (J Bank Finance 20:1211–1229, 1996), but also the jump-diffusion processes such as Merton jump model (J Financial Econ 3:125–144, 1976), Tian et al. model (J Futures Mark 34:957–979, 2014) and Zhou et al. model (Acta Math Appl Sin-E 35:305–318, 2019). The numerical results show that the randomness of the corporate liabilities may reduce the expected payoff of the option holder at maturity time and also increase the likelihood of default, thus it reduces the vulnerable option price. [ABSTRACT FROM AUTHOR]

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.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate.

Parisian options, serving as substitutes for barrier options, are frequently embedded in complex financial derivatives. Despite the considerable mathematical challenges in pricing, their significant applications in finance and insurance have generated extensive studies in the literature. However, the impact of counterparty risk has not been taken into account so far. To address this gap, we develop a closed-form pricing framework for vulnerable Parisian options. Utilizing the Laplace transform and measure-change technique, we derive closed-form pricing formulas of Parisian options incorporating counterparty credit risk. Finally, we conduct numerical analyses to verify our pricing formulas' accuracy and efficiency. • To our best knowledge, this is the first study on vulnerable Parisian option pricing. • Closed-form pricing formulas are derived. • Our formulas outperform MC simulations and tree model in efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This study develops a closed-form solution for pricing vulnerable lookback options, which combines the Black–Scholes model for the underlying asset and a correlated jump–diffusion model for the issuer's asset to account for default risks. Our approach relies on applying Laplace transforms to establish a closed-form solution and compute them numerically using Laplace inversion algorithms. To determine the price, we derive the joint distribution of the first passage time of a drifted Brownian motion and the value of the correlated jump–diffusion. Through numerical examples, we demonstrate the accuracy of the numerical solutions obtained through our method and the stability of the algorithm. Subsequently, we conduct a numerical analysis to understand how counterparty risk influences option prices based on our formula. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

In this paper, we consider basket warrants with default risk in a stochastic volatility model, where the correlation between the underlying assets is stochastic. In addition, we take into account default risk of warrant issuers, and use a reduced form model to capture default risk. In the proposed framework, a closed-form approximation is provided and the effect of default risk is investigated numerically. [ABSTRACT FROM AUTHOR]

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.

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. [ABSTRACT FROM AUTHOR]

The pricing of options and other derivatives which are subject to the default risk of the writer usually requires the calibration of a sophisticated model and substantial effort in determining the input parameters. We propose a very simple method to incorporate correlated credit risk into the pricing of vulnerable derivatives. The approach is based upon some approximations of more complex models and requires a minimum of input parameters. It is therefore easily applicable and maintains the accuracy of sophisticated models to a large extent, as shown in numerical studies for call options, put options, and discount certificates. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present a new pricing model for vulnerable options, with time-varying variances for each asset described by Generalized Autoregressive Conditional Heteroscedasticity processes and correlated with the return of the asset. By connecting the underlying asset and the counterparty's assets through the market factor channel, the proposed model also captures stochastic correlation between the underlying asset return and the return of the counterparty's assets. The correlation depends on the levels of the variances of both assets and the market index as well. In the proposed framework, the closed-form solution for vulnerable options is derived and numerical results are presented to investigate the impact of counterparty default risk. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pricing issue of vulnerable options by assuming that the dynamics of all assets are governed by jump-diffusion processes with common factors in both continuous process and jump process components. Moreover, assume credit default event occurs when the value of the counterparty’s assets falls below the default barrier, which is stochastically affected by common factors as well. In the proposed framework, we derive a closed-form formula for vulnerable options and illustrate the impacts of stochastic barriers on option prices. Additionally, the U-shape curve appears when we investigate option prices against the volatility of default barriers. [ABSTRACT FROM AUTHOR]

We price vulnerable derivatives – i.e. derivatives where the counterparty may default. These are basically the derivatives traded on the over-the-counter (OTC) markets. Default is modelled in a structural framework. The technique employed for pricing is good deal bounds (GDBs). The method imposes a new restriction in the arbitrage free model by setting upper bounds on the Sharpe ratios (SRs) of the assets. The potential prices that are eliminated represent unreasonably good deals. The constraint on the SR translates into a constraint on the stochastic discount factor. Thus, tight pricing bounds can be obtained. We provide a link between the objective probability measure and the range of potential risk-neutral measures, which has an intuitive economic meaning. We also provide tight pricing bounds for European calls and show how to extend the call formula to pricing other financial products in a consistent way. Finally, we numerically analyse the behaviour of the good deal pricing bounds. [ABSTRACT FROM PUBLISHER]

Abstract: Following the framework of , this paper presents an improved method of pricing vulnerable options under jump diffusion assumptions about the underlying stock prices and firm values which are appropriate in many business situations. In contrast to model, jumps can be used to model sudden changes in stock prices and firm values. Further, with the jump risk, a firm can default instantaneously because of an unexpected drop in its value. Therefore, our model is able to provide sufficient conceptual insights about the economic mechanism of vulnerable option pricing. The numerical results show that a jump occurrence in firm values can increase the likelihood of default and reduce the vulnerable option prices. [Copyright &y& Elsevier]

• We provide analytical pricing formulae of vulnerable barrier options and vulnerable double barrier options. • We give the joint distribution of a special range of the two-dimensional correlated Brownian motions. • We numerically analyze the impacts of counterparty risk on option prices. In this paper, we provide analytical pricing formulae of vulnerable barrier options and vulnerable double barrier options. To obtain the price of vulnerable double barrier options, we give the joint distribution of a special range of the two-dimensional correlated Brownian motions. This result is of independent interest. Based on the derived formula, we numerically analyze the impacts of counterparty risk on option prices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate European and Asian options with default risk in an intensity-based model. By breaking down the risk into idiosyncratic and systematic components, we describe the underlying asset price using a two-factor stochastic volatility model and incorporate the correlation between the underlying asset and default risk. In the proposed framework, we obtain explicit pricing formulae of European and Asian options with default risk and illustrate the effects of default risk and systematic risk on option prices. Specially, prices of the options with default risk increase with systematic risk. By contrast, prices of the options with default risk drop as systematic risk increases when we keep the total initial volatility of the underlying asset unchanged. • We investigate vulnerable European and Asian options in an intensity-based model. • The underlying asset price is described by a two-factor stochastic volatility model. • The correlation between the underlying asset and default risk is incorporated. • We obtain explicit pricing formulae of vulnerable European and Asian options. • The effects of default risk and systematic risk on option prices are illustrated. [ABSTRACT FROM AUTHOR]