Vulnerable Options Research Articles

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

In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numeraire technique, proposed by Geman et al. [ 14 ], is employed to obtain a semi-analytical pricing formula for an vulnerable European option in the presence of regime switching effect. The method is numerically implemented using the multinomial approach in Costabile et al. [ 6 ]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.

VALUATION OF VULNERABLE OPTIONS UNDER THE DOUBLE EXPONENTIAL JUMP MODEL WITH STOCHASTIC VOLATILITY

In this paper, we extend the framework of Klein [15] [Journal of Banking & Finance 20: 1211–1229] to a general model under the double exponential jump model with stochastic volatility on the underlying asset and the assets of the counterparty. Firstly, we derive the closed-form characteristic functions for this dynamic. Using the Fourier-cosine expansion technique, we get numerical solutions for vulnerable European put options based on the characteristic functions. The inverse fast Fourier transform method provides a fast numerical algorithm for the twice-exercisable vulnerable Bermuda put options. By virtue of the modified Geske and Johnson method, we obtain an approximate pricing formula of vulnerable American put options. Numerical simulations are made for investigating the impact of stochastic volatility on vulnerable 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 Market Prices of Common Jump Risks under Regime-Switching Models

This paper investigates the valuation of vulnerable European options considering the market prices of common systematic jump risks under regime-switching jump-diffusion models. The way of regime-switching Esscher transform is adopted to identify an equivalent martingale measure for pricing vulnerable European options. Explicit analytical pricing formulae for vulnerable European options are derived by risk-neutral pricing theory. For comparison, the other two cases are also considered separately. The first case considers all jump risks as unsystematic risks while the second one assumes all jumps risks to be systematic risks. Numerical examples for the valuation of vulnerable European options are provided to illustrate our results and indicate the influence of the market prices of jump risks on the valuation of vulnerable European options.

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.

Pricing Vulnerable Options with Jump Clustering

AbstractThis paper presents a valuation of vulnerable European options using a model with self‐exciting Hawkes processes that allow for clustered jumps rather than independent jumps. Many existing valuation models can be regarded as special cases of the model proposed here. Using numerical analyses, this study also performs sensitivity analyses and compares the results to those of existing models for European call options. The results show that jump clustering has a significant impact on the option value. © 2017 Wiley Periodicals, Inc. Jrl Fut Mark

PRICING VULNERABLE EUROPEAN OPTIONS WITH STOCHASTIC CORRELATION

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.

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.

Pricing vulnerable path-dependent options using integral transforms

In the over-the-counter (OTC) markets, the holders of many contracts are vulnerable to counterparty credit risk. Because of this issue, vulnerable options must be considered. In addition, in a financial environment, the pricing of path-dependent options yields many interesting mathematical challenges. In this paper, we study the pricing of vulnerable path-dependent options using double Mellin transforms to investigate an explicit (closed) form pricing formula or semi-analytic formula in each path-dependent option.

ANALYTICAL VALUATION OF VULNERABLE OPTIONS IN A DISCRETE-TIME FRAMEWORK

In this paper, we present a pricing model for vulnerable options in discrete time. A Generalized Autoregressive Conditional Heteroscedasticity process is used to describe the variance of the underlying asset, which is correlated with the returns of the asset. As for counterparty default risk, we study it in a reduced form model and the proposed model allows for the correlation between the intensity of default and the variance of the underlying asset. In this framework, we derive a closed-form solution for vulnerable options and investigate quantitative impacts of counterparty default risk on option prices.

Pricing vulnerable options with stochastic default barriers

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.

A Data Driven Approach to Calculate Optimum Collateral Amount for Vulnerable Option

Background/Objectives: The main objective of this paper is to present a method that determines optimum collateral amount at which the risk of the venerable option is same as the exchange traded risk. Methods/Statistical Analysis: Mathematical models have been presented in this paper that is related to binomial tree building, Venerable option pricing. An algorithm has also been presented to calculate minimum collateral amount. Experimental models demonstrate the Convergence of Collateral amount and Sensitivity of Vulnerable Option Price to Model Parameters, and Correctness of the optimum collateral amount. Findings: A methodology has been presented in this paper that can be used to compute maximum collateral amount that must be supported by the writer of the option at which the venerable option becomes as risky as the exchange traded risk. This methodology can also be sued when the option writers chooses a fixed collateral amount when the underlying price is greater than the fixed price. A navel binomial decision has been developed and presented considering no assumption of the underlying distribution. It has been found that the price of an option with credit risk converges to exchange traded option as the collateral amount reaches a certain optimal value. The option writer in this can case can use the excess collateral amount for some other purpose. It has also been found that rules that are related to plain vanilla option need not be followed for calculating venerable option.

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.

Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy

In this paper, we use a Markov-modulated regime switching approach to model various states of the economy, and study the pricing of vulnerable European options when the dynamics of the underlying asset value and the asset value of the counterparty follow two correlated jump-diffusion processes under regime switching. The correlation is modelled by both the diffusion parts and the pure jump parts which describe the uncertainty of the value of the risky assets. We develop a method to determine an equivalent martingale measure and a parsimonious representation of the risk-neutral density is provided. Based on this, we derive an analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, and implement the formula through numerical Laplace inversion.

The European Vulnerable Option Pricing with Jumps Based on a Mixed Model

In this paper, we combine the reduced-form model with the structural model to discuss the European vulnerable option pricing. We define that the default occurs when the default process jumps or the corporate goes bankrupt. Assuming that the underlying asset follows the jump-diffusion process and the default follows the Vasicek model, we can have the expression of European vulnerable option. Then we use the measure transformation and martingale method to derive the explicit solution of it.

Policy neutrality and uncertainty: an info-gap perspective

Reducing uncertainty is a central goal of intelligence analysis. 'Reducing uncertainty' can mean (1) reduce ignorance or ambiguity or potential for surprise in describing situations or intentions, or (2) reduce adverse impacts of ignorance, ambiguity or surprise on decision outcomes. We make two claims. First, the second meaning needs greater attention in intelligence analysis. Uncertainty itself isn't pernicious, but adverse impact of surprise is. Some policy options are less vulnerable to uncertainty than others. These less vulnerable (i.e. more robust) options can tolerate more uncertainty. Analysts should identify policy options that are robust to uncertainty. Second, reducing the impact of uncertainty requires awareness of policymakers' goals. This needn't conflict with analysts' policy neutrality. Tension between neutrality and involvement arises in economics, engineering, and medicine. The method of info-gap robust-satisficing supports decision making under uncertainty in these and other disciplines. Implications for intelligence analysis are explored in this paper. We discuss the assessment of Iraqi WMD capability in 2002.

Pricing vulnerable European options under a two-sided jump model via Laplace transforms

In this paper, we propose a model for pricing the vulnerable European options when the value of the firm's asset of the counterparty follows a double exponential jump-diffusion process and correlates with the underlying asset value of the option. The two-sided jump process can model sudden down and up changes in the firm's asset value, which provides more economic insights on vulnerable options pricing. Incorporating the early default and payoff conditions, and based on the explicit formula of the joint Laplace transforms of the first passage times for the two-sided jump process and a correlated Bownian motion derived by us, we present a simple analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, one for the space and one for the time, and the formula can be implemented by numerical Laplace inversion. The numerical results show that our pricing formula is correct and efficient.

The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment

Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. With the changes of measures, the traditional pricing method is simplified and the general pricing formula is obtained for the European vulnerable option with stochastic interest rate. At the same time, the explicit expression for it comes into being.

Pricing black-scholes options with correlated credit risk and jump risk

This article follows the framework of Klein (1996) to present 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 Klein’s (1996) model, jumps allow not only for sudden changes in stock prices and firm values, but also for a firm to 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. In particular, an analytical pricing formula for vulnerable European options under jump diffusion model is derived. The numerical results show that a jump occurrence in firm values can increase the likelihood of default and reduce the vulnerable option prices.