In this paper, we study financial markets with stochastic volatilities driven by fractional Brownian motion with Hurst index H1/2. Our models include fractional versions of Ornstein-Uhlenbeck, Vasicek, geometric Brownian motion and continuous-time GARCH models. We price variance and volatility...

The valuation of the variance swaps for local Levy based stochastic volatility with delay (LLBSVD) is discussed in this paper. We provide some analytical closed forms for the expectation of the realized variance for the LLBSVD. As applications of our analytical solutions, we fit our model to 10...

In this paper, we model financial markets with semi-Markov volatilities and price covarinace and correlation swaps for this markets. Numerical evaluations of varinace, volatility, covarinace and correlations swaps with semi-Markov volatility are presented as well. The novelty of the paper lies...

Some commodity prices, like oil and gas, exhibit the mean reversion, unlike stock price. It means that they tend over time to return to some long-term mean.In this paper we consider a risky asset S_t following the mean-reverting stochastic process. The aim of this paper is to obtain an explicit...

The valuation of the variance swaps for local stochastic volatility with delay and jumps is discussed in this paper. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. Besides, we also present a lower bound for...

The jumps in stock market volatility are found to be so active that this discredits many recently proposed stochastic volatility models without jumps (Bollerslev et al (2008)). The most convincing evidence comes from recent nonparametric work using high-frequency data as in Barndorff-Nielsen and...

In this paper, we introduce an extension to the LIBOR Market model that is suitable to incorporate both sudden market shocks as well as changes in the overall economic climate into the interest rate dynamics. This is achieved by substituting the simple diffusion process of the original LIBOR...

In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique and Laplace transform method to evaluate...

In these lectures' notes I would like to introduce forwards, futures and options, and to review some results on Black-Scholes-73 and Black-76 models for positive prices, and also on alternatives models for negative prices for option valuation of futures contracts.I will focus on the first model...

In this paper, we show numerically how to calculate the price of bond options, swaps, caps and floors for Levy one-factor stochastic interest rate models via partial integro-differential equations (PIDE). These models include, in particular, Ornshtein-Uhlenbeck (1930), Vasicek (1977),...