Pricing Vulnerable European Options under Lévy Process with Stochastic Volatility

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

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.

An exhaustive survey of research on pricing foreign exchange options with stochastic volatility and stochastic interest rates is presented in this article. The article also includes contributions to pricing of currency options with fixed volatility and deterministic interest rates, domestic and foreign, and in a systematic way extend to results obtained for cases which include stochastic volatility and stochastic interest rates. Since the literature in pricing foreign exchange currency option is quite extensive, the article considers some of the more important contributions.

This paper is concerned with analytical approximation for option pricing while the underlying asset price follows a geometric mean-reverting diffusion process, which is often used in real and commodity options. Two methods are developed, one employing the idea of Edgeworth series expansion and another based on Taylor series expansion. Numerical tests show that the second method significantly outperforms the first one, particularly for long-maturity options. In addition, we compare the analytical approximation prices with that obtained using an alternative log-normal model for commodities and find out for many cases the discrepancy in option prices is remarkable.

American options with stochastic dividends and volatility: A nonparametric investigation

<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.

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

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

Static Hedging of Standard Options

We price options so as to take into account the existence of memory (short or long) characterizing the stochastic processes that generate prices, volatility and interest rates. In particular, we propose a model for Bull Spread options in a Mixed Modified Fractional Hull-White-Vasicek stochastic volatility and stochastic interest rate model. We propose a specific Bull Spread Vulnerable option pricing based on MMFHWV model.

A multiscale extension of the Margrabe formula under stochastic volatility