As renewable energy increasingly penetrates into power grid systems, new challenges arise for system operators to keep the systems reliable under uncertain circumstances, while ensuring high utilization of renewable energy. This paper presents unit commitment (UC) which takes into account the volatility of wind power generation. The UC problem is solved with the forecasted intermittent wind pow...

Terrorism, political system instability and currency rate fluctuations are the three most evident issues of 21st century. In this study, comparative analysis is performed to check the impact of all these issues on PSX Volatility. EGARCH (1,1) approach is used on four different kinds of data collected from 1st January 2000 to 31st December 2015. Terrorist events, FX return fluctuations with rest...

Concerning price processes, the fact that the volatility is not constant has been observed for a long time. So we deal with models as dXt = μtdt + σtdWt where σ is a stochastic process. Recent works on volatility modeling suggest that we should incorporate jumps in the volatility process. Empirical observations suggest that simultaneous jumps on the price and the volatility [8, 9] exist. The hy...

Abstract We consider the pricing of a range of volatility derivatives, including volatility and variance swaps and swaptions. Under risk-neutral valuation we provide closed-form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility. We describe a general partial differential equation framework for derivatives that have an extra de...

A better understanding of the empirical dynamics of Black-Scholes implied volatility surface has long been of considerable interest to both practitioners and academics. Basing on some findings about the ad hoc Black-Scholes valuation approach suggested in Dumas, Flemming and Whaley (1998), this essay studies the empirical performance of various volatility function forms that characterize the re...

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneou...

They should reduce their equity position. We study the portfolio problem of a long-horizon investor that allocates between a risk-less and a risky asset in an environment where both volatility and expected returns are time-varying. We find that investors, regardless of their horizon, should substantially decrease risk exposure after an increase in volatility. Ignoring variation in volatility le...

Financial theory suggests residual volatility should not be related to expected returns due to diversification. A robust portfolio construction process reduces volatility by investing in a large number of assets that span different geographies and industries. Firm-specific volatility largely washes out. What is left is systematic volatility related to common sources of return variation. Wes Cri...

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the...