Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

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Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

This paper introduces the class of volatility modulated Levy-driven Volterra ($\mathcal{VMLV}$) processes and their important subclass of Levy semistationary ($\mathcal{LSS}$) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of $\mathcal{VMLV}$ processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.

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Modelling energy spot prices by volatility modulated Levy-driven Volterra processes

This paper introduces the class of volatility modulated Levy-driven Volterra (VMLV) processes and their important subclass of Levy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of VMLV processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.

/pdf/modelling-energy-spot-prices-by-volatility-modulated-levy-3vyvk4svxf.pdf

Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes

We consider the non-Gaussian stochastic volatility model of Barndorff-Nielsen and Shephard for the exponential mean-reversion model of Schwartz proposed for commodity spot prices We analyze the properties of the stochastic dynamics, and show in particular that the log-spot prices possess a stationary distribution defined as a normal variance-mixture model Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean-reversion of the stochastic volatility Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff-Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices An empirical example on UK spot data is included

The stochastic volatility model of barndorff-nielsen and shephard in commodity markets

Ambit processes are general stochastic processes based on stochastic integrals with respect to Levy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Levy noise analysis.

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Ambit Processes and Stochastic Partial Differential Equations

In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine two-factor mean-reverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward prices in terms of elementary functions. Through measure changes induced by the forward price process, we further obtain closed form pricing equations for spread options on the forward prices. For completeness, we address both an Actuarial and a risk-neutral approach to the valuation problem. Finally, we provide a calibration procedure and calibrate our model to the NYMEX Light Sweet Crude Oil spot and futures data, allowing us to extract the implied market prices of risk for this commodity.

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Energy spot price models and spread options pricing

It is well known that stochastic volatility is an essential feature of commodity spot prices. By using methods of singular perturbation theory, we obtain approximate but explicit closed‐form pricing equations for forward contracts and options on single‐ and two‐name forward prices. The expansion methodology is based on a fast mean‐reverting stochastic volatility driving factor and leads to pricing results in terms of constant volatility prices, their Deltas and their Delta‐Gammas. Both the standard single‐factor mean‐reverting spot model and a two‐factor generalization, in which the long‐run mean is itself mean‐reverting, are extended to include stochastic volatility and each is analysed in detail. The stochastic volatility corrections can be used to efficiently calibrate option prices and compute sensitivities.

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Samuel Hikspoors

Papers

Energy spot price models and spread options pricing

Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models