Analysis of model implied volatility for jump diffusion models: Empirical evidence from the Nordpool market

In this paper we examine the importance of mean reversion and spikes in the stochastic behaviour of the underlying asset when pricing options on power. We propose a model that is flexible in its formulation and captures the stylized features of power prices in a parsimonious way. The main feature of the model is that it incorporates two different speeds of mean reversion to capture the differences in price behaviour between normal and spiky periods. We derive semi-closed form solutions for European option prices using transform analysis and then examine the properties of the implied volatilities that the model generates. We find that the presence of jumps generates prominent volatility skews which depend on the sign of the mean jump size. We also show that mean reversion reduces the volatility smile as time to maturity increases. In addition, mean reversion induces volatility skews particularly for ITM options, even in the absence of jumps. Finally, jump size volatility and jump intensity mainly affect the kurtosis and thus the curvature of the smile with the former having a more important role in making the volatility smile more pronounced and thus increasing the kurtosis of the underlying price distribution

Modelling short and long-term risks in power markets: Empirical evidence from Nord Pool

In this paper we propose a three-factor spike model that accounts for different speeds of mean reversion between normal and spiky shocks in the Scandinavian power market. In this model both short and long-run factors are unobservable and are hence estimated as latent variables using the Kalman filter. The proposed model has several advantages. First, it seems to capture in a parsimonious way the most important risks that practitioners face in the market, such as spike risk, short-term risk and long-term risk. Second, it explains the seasonal risk premium observed in the market and improves the fit between theoretical and observed forward prices, particularly for long-dated forward contracts. Finally, closed-form solutions for forward contracts, derived from the model, are consistent with the fact that the correlation between contracts of different maturities is imperfect. The resulting model is very promising, providing a very useful policy analysis and financial engineering tool to market participants for risk management and derivative pricing particularly for long-dated contracts.Electricity derivatives Kalman filter Affine jump diffusion models

Analysis of model implied volatility for jump diffusion models: Empirical evidence from the Nordpool market

In this paper we examine the importance of mean reversion and spikes in the stochastic behaviour of the underlying asset when pricing options on power. We propose a model that is flexible in its formulation and captures the stylized features of power prices in a parsimonious way. The main feature of the model is that it incorporates two different speeds of mean reversion to capture the differences in price behaviour between normal and spiky periods. We derive semi-closed form solutions for European option prices using transform analysis and then examine the properties of the implied volatilities that the model generates. We find that the presence of jumps generates prominent volatility skews which depend on the sign of the mean jump size. We also show that mean reversion reduces the volatility smile as time to maturity increases. In addition, mean reversion induces volatility skews particularly for ITM options, even in the absence of jumps. Finally, jump size volatility and jump intensity mainly affect the kurtosis and thus the curvature of the smile with the former having a more important role in making the volatility smile more pronounced and thus increasing the kurtosis of the underlying price distribution.Affine jump diffusion models Implied volatility Volatility skew Electricity derivatives Risk management