Pricing Vulnerable Options

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

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 article, we generalize the classical Edgeworth series expansion used in the option pricing literature. We obtain a closed-form pricing formula for European options by employing a generalized Hermite expansion for the risk neutral density. The main advantage of the generalized expansion is that it can be applied to heavy-tailed return distributions, a case for which the standard Edgeworth expansions are not suitable. We also show how the expansion coefficients can be inferred directly from market option prices.

<p style='text-indent:20px;'>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 numéraire technique, proposed by Geman et al. [<xref ref-type="bibr" rid="Geman">14</xref>], 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. [<xref ref-type="bibr" rid="MC">6</xref>]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.

We consider a general option pricing framework incorporating the double Heston stochastic volatility, stochastic interest rate, jumps and Markov regime switching. Under the proposed framework, we derive the analytical pricing formulae for European options using Fourier transform technique. Numerical examples illustrate that the option prices and the implied volatility curves under different regimes vary clearly, and the effects of regime-switching and jumps on the option price differ.

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&amp;P 500 index.

In 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.

This expository article highlights the significance of Monte Carlo simulation in pricing of options. We discuss the various types of financial derivatives, particularly options and their classifications. The discrete and continuous time models for the underlying assets are dwelled upon. We consider a geometric Brownian motion (GBM) based model for stock price process and discuss the payoffs of plain vanilla as well as path-dependent European-type options, namely, barrier, lookback, and Asian. We mention the option pricing formula for plain vanilla European option and describe the Monte Carlo approach to option pricing with illustrative algorithms and results for some of these options.

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.

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.

Government guarantees are often priced by appealing to an analogy with option pricing techniques pioneered by Black and Scholes (1973) and Merton (1973). Of course, in reality, the government does not purchase options in the open market. Instead, barring a Ponzi game opportunity, the government must adjust tax revenue or spending in response to shocks in the value of the asset being insured for the government to maintain its intertemporal budget constraint. Nonetheless, this article derives an exact mathematical relationship between fiscal policy and option pricing. Explicitly deriving the option pricing equivalence from first principles proves that the analogy is, in fact, correct and suitable for use by public finance economists and government officials to estimate the cost of government guarantees. Starting from first principles also has the benefit of showing that an additional mathematical term, which multiplies the option price, is necessary to capture the pay-as-you-go nature of most perpetual governme nt guarantees-- a term that is missing in the previous work. The author develops the analysis in the context of a highly relevant example: investing the Social Security trust fund in equities. Advocates of this policy have argued that the higher expected returns would help prefund future benefits and, therefore, would require less of a tax increase on future workers. Several government agencies have scored hypothetical legislation and have concluded likewise. The results herein show just the opposite. By focusing on expected returns, trust fund investment creates an instant windfall for current workers and is mathematically equivalent to a tax increase on all future workers, relative to a baseline policy of maintaining the current payroll tax rate. National saving is reduced. The author computes President Clinton's proposal to invest in equities to be equivalent to increasing the future payroll tax by 0.8 percent in perpetuity. A policy recommendation to invest about 40 percent of the trust fund in stocks analyzed recently by the Social Security Administration is equ ivalent to a 21 percent increase in the future payroll tax. INTRODUCTION As Robert Merton noted in his Nobel address, the use of option pricing technology has extended well beyond financial operations and has been used in numerous valuation problems, including government guarantees. The list is extensive and includes deposit insurance, student loan guarantees, loans to business, the government right to change quotas, and many other guarantees. [1] Of course, in reality, the government does not purchase options for the purpose of hedging. Instead, barring a Ponzi game opportunity, the government must adjust tax revenue or spending in response to shocks in the value of the asset being insured for the government to maintain its intertemporal budget constraint. Government policy then becomes state contingent and the government's intertemporal budget constraint must be taken explicitly into account. Although most analyses of guarantees rationalize using options pricing based on an analogy, this article starts with fiscal policy instruments and proves that the analogy is, in fact, true--with a modification. The author derives an exact mathematical equivalence between state-contingent fiscal policies and option pricing. Deriving implicit option prices from the government's intertemporal budget constraint not only brings an even greater sense of reality to the case for using arbitrage pricing technology to value government guarantees, but it is also important because many government guarantees are not prefunded. The option pricing formulae must be augmented with a multiplicative term that recognizes the pay-as-you-go nature of most perpetual guarantees. As a relevant example, this article considers the proposal to alter the Social Security trust fund's portfolio from exclusively holding government debt to holding equity as well. …

In this paper, we study the pricing of vulnerable options, which are exposed to the option issuer’s default risk. We develop a pricing framework that integrates a reduced-form model for default risk with the 4/2 stochastic volatility model for the underlying asset. A feature of our model is the correlation between the issuer’s default intensity and the systematic component of the stochastic volatility. Using the characteristic function method and properties of the Grasselli transform, we derive an analytical pricing formula for a European vulnerable call option. Finally, we conduct numerical experiments to illustrate the impact of significant parameters, such as the recovery rate, default intensity, and the specific parameters of the 4/2 model. The results show that the 4/2 model component, which distinguishes it from the standard Heston model, has a significant effect on option prices.

Are There Critical Levels of Stochastic Volatility for Early Option Exercise?

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

As the subprime credit crisis has attracted attention to financial derivative instruments, more frequently arises questions about fairvalue calculations. Over the time, different models had been introduced. All of those models take into account factors affectingprices. Mostly, factors used in calculations on the same type of financial instruments are approximately the same. Therefore questionarises, which factor affects price more and which less, with no matter which model would be used for fair value calculations. One offinancial derivative instrument types is options. Options are agreements, which give to option buyer rights to buy or sell underlyingasset. While the seller or writer of option has obligation to buy or sell underlying asset. This research aims to explore the impact offactors on option fair value calculations and evaluate most important ones from those, which could be chosen by option buyer orseller. To reach the aim of research following tasks are developed: 1) review of fair value calculation models; 2) compare results ofusage of different models and changes in affecting factors; 3) highlight differences between option price affecting factors, modelsused in calculations and results provided. Research includes literature review and analysis of option pricing results. Option pricecalculations are based on historical option prices, using black-Scholes and Binomial option pricing models. KEYWORDS: options, fair value models, strike price, exercise date. DOI: http://dx.doi.org/ 10.15181/rfds.v11i3.617