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

In this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

This paper obtains an analytic pricing formula for vulnerable European options. This formula allows the option writer to default at any time during the life of the option, and the recovery value of the option in default to be an amount equal to the writerfs asset-to-debt ratio multiplied by the economic equivalent of the option right, rather than its default-free value. Numerical results indicate that the economic equivalent of the option right, assumed as its default-free value, is inappropriate, and a default, restricted to occur only at the exercise date, seriously understates the impact of credit risk on option prices. Key words: vulnerable option, option pricing, credit risk

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.

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.

This article proposes a closed pricing formula for European options when the return of the underlying asset follows extended normal distribution, that is, any different degrees of skewness and kurtosis relative to the normal distribution induced by the Black-Scholes model. The moment restriction is suggested, so that the pricing model under any arbitrary distribution for an underlying asset must satisfy the arbitrage-free condition. Numerical experiments and comparison of empirical performance of the proposed model with the Black-Scholes, ad hoc Black-Scholes, and Gram-Charlier distribution models are carried out. In particular, an estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 index options is made, and in-sample and out-of-sample tests are performed. These results not only support the previous finding that the actual density of the underlying asset shows skewness to the left and high peaks, but also demonstrate that the present model has good explanatory power for option prices. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:845–871, 2005

In this paper we discuss an option pricing problem in a hidden Markovian regime-switching model with a stochastic interest rate and volatility. Regime switches are attributed to structural changes in an hidden economic environment and are described by a continuous-time, finite-state, unobservable Markov chain. The model is then applied to the valuation of a standard European option. By means of the standard separation principle, filtering and option valuation problems are separated. Robust filters for the hidden states of the economy and their robust filtered estimates of unknown parameters from the expectation maximization algorithm are presented based on standard techniques in filtering theory. Then an explicit expression of a conditional characteristic function relevant to option pricing is presented and the valuation of the option is discussed using the inverse Fourier transformation approach. Using the limiting behavior of the conditional characteristic function, an efficient implementation of the transform inversion integral is considered. Numerical experiments are given to illustrate the flexibility of filtering algorithms and the significance of regime-switching in option pricing.

Multiscale stochastic elasticity of variance for options and equity linked annuity; A Mellin transform approach

ABSTRACTThis article mainly investigates risk-minimizing European currency option pricing and hedging strategy when the spot foreign exchange rate is driven by a Markov-modulated jump-diffusion model. We suppose the domestic and foreign money market floating interest rates, the drift, and the volatility of the exchange rate dynamics all depend on the state of the economy, which is modeled by a continuous-time hidden Markov chain. The model considered in this article will provide market practitioners with flexibility in characterizing the dynamics of the spot foreign exchange rate. Using the minimal martingale measure, we obtain a system of coupled partial-differential-integral equations satisfied by the currency option price and find the corresponding hedging strategies and the residual risk. According to simulation of currency option prices in the special case of double exponential jump-diffusion regime-switching model, we further discuss and show the effects of the parameters on the prices.

Compound Poisson and compound Bernoulli perturbations are introduced in addition to normal errors in the AR(1) model for discrete time version of log-price process. The formula for expected present value of European Call Option is extended to these cases. Other than European option, the valuation of Asian option is also considered and formulas for Geometric Asian option for AR(1) log-price process incorporating compound Poisson and compound Bernoulli jumps also obtained.

In the current literature, the focus of credit-risk analysis has been either on the valuation of risky corporate bond and credit spread or on the valuation of vulnerable options, but never both in the same context. There are two main concerns with existing studies. First, corporate bonds and credit spreads are generally analyzed in a context where corporate debt is the only liability of the firm and a firm’s value follows a continuous stochastic process. This setup implies a zero short-term spread, which is strongly rejected by empirical observations. The failure of generating non-zero short-term credit spreads may be attributed to the simplified assumption on corporate liabilities. Because a corporation generally has more than one type of liability, modeling multiple liabilities may help to incorporate discontinuity in a firm’s value and thereby lead to realistic credit term structures. Second, vulnerable options are generally valued under the assumption that a firm can fully pay off the option if the firm’s value is above the default barrier at the option’s maturity. Such an assumption is not realistic because a corporation can find itself in a solvent position at option’s maturity but with assets insufficient to pay off the option. The main contribution of this study is to address these concerns. The proposed framework extends the existing equity-bond capital structure to an equity-bond-derivative setting and encompasses many existing models as special cases. The firm under study has two types of liabilities: a corporate bond and a short position in a call option. The risky corporate bond, credit spreads, and vulnerable options are analyzed and compared with their counterparts from previous models. Numerical results show that adding a derivative type of liability can lead to positive short-term credit spreads and various shapes of credit-spread term structures that were not possible in previous models. In addition, we found that vulnerable options need not always be worth less than their default-free counterparts. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:301–327, 2001

Options are crucial to global financial markets. However, there is rarely existing literature combining a specific option and a certain company, with simultaneous analysis and conclusions of the option. Based on the ordinary least square (OLS) regression and Black-Scholes (BS) model, we design two simulated options for Morgan Stanley, which are European options and power options. Then the option's value is identified in this article. Besides, through the sensitivity analysis of the two options and their related parameters, we get some results and conclusions about the characteristics of the two options. In general, the value of options is affected by independent variables and parameters. Compared with a European option, a power option has a higher risk and higher future expected return. This paper provides better advice for investors, on the premise of fully considering various possible situations, compare the advantages and disadvantages of the two options, and make their own investment choices.

The impact of market frictional costs on pricing insurance and financial products in a regime-switching environment has not been well-explored. This paper introduces a general pricing model for European options which incorporates market frictional costs, regime switches and model uncertainty. Regime switches are due to changes in an economic environment. Model uncertainty is attributed to misspecification of transition intensities for economic regimes. The selling and buying prices of a European option are determined through stochastic optimal control and nonlinear partial differential equations. A fair value is determined by a closed-form solution to a minimization problem based on a relative entropy. The fair value is consistent with the one obtained using the Esscher transform, which is an important tool in actuarial science. Numerical methods and results for implementing the pricing model are presented. The results indicate that after controlling for the model uncertainty, market frictional costs are more significant than regime switches in accounting for the fair, selling and buying prices.

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.

The main purpose of this paper is to study the stock price jump process is more common than Poisson process -a kind of special renewal process-the incident time interval is independent random variable sequence and identical subordinate to Gamma distribution. The object of study is European bi-direction option, It also known as the double two-way options. Also, European bi-direction option is an exotic option. Then, build up stochastic differential equation under the circumstance of the market no arbitrage based on stochastic analysis and martingale theory. Based on the above discussion, we obtain the European bi-direction option pricing formulas under jump diffusion model by simple mathematical induce by means of martingale method.

This paper develops a numerical method for pricing European options with regime-switching volatility and interest rate governed by the Black-Scholes equation. The method is based on a finite volume method with a spatial discretization and implicit time stepping technique. We show that the system matrix of the discretized system is an M-matrix and present an algorithm for solving the problem. Numerical experiment are implemented for some European option problems to illustrate the usefulness of this method.