Forward curve

The forward curve is a function graph in finance that defines the prices at which a contract for future delivery or payment can be concluded today. For example, a futures contract forward curve is prices being plotted as a function of the amount of time between now and the expiry date of the futures contract (with the spot price being the price at time zero). The forward curve represents a term structure of prices. Forward interest rate A forward interest rate is a type of interest rate that is specified for a loan that will occur at a specified future date. As with current interest rates, forward interest rates include a term structure which shows the different forward rates offered to loans of different maturities. According to the unbiased expectations hypothesis, forward interest rates predict spot interest rates at the time the loan is actually made, but many analysts dispute whether this is true, as it ignores durational risk. This figure is part of the lending & cr

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Anders Trolle

Commodity derivatives are becoming an increasingly important part of the global derivatives market. Here we develop a tractable stochastic volatility model for pricing commodity derivatives. The model features unspanned stochastic volatility, quasi-analytical prices of options on futures contracts, and dynamics of the futures curve in terms of a low-dimensional affine state vector. We estimate the model on NYMEX crude oil derivatives using an extensive panel data set of 45,517 futures prices and 233,104 option prices, spanning 4082 business days.We find strong evidence for two predominantly unspanned volatility factors.

Oxford University Press2009

Efficient Pricing of Energy Derivatives

Anders Trolle

I present a tractable framework, first developed in Trolle and Schwartz (2009), for pricing energy derivatives in the presence of unspanned stochastic volatility. Among the model features are i) a perfect fit to the initial futures term structure, ii) a fast and accurate Fourier-based pricing formula for European-style options on futures contracts, enabling efficient calibration to liquid plain-vanilla exchange-traded derivatives, and iii) the evolution of the futures curve being described in terms of a low-dimensional affine state vector, making the model ideally suited for pricing complex energy derivatives and real options by simulation. I also consider an extension of the framework that takes jumps in spot prices into account.

Palgrave Macmillan US2014

Exponential-Polynomial Families and the Term Structure of Interest Rates

Damir Filipovic

Exponential-polynomial families like the Nelson-Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with inter-temporal modelling. We characterize the consistent Ito processes which have the property to provide an arbitrage free interest rate model when representing the parameters of some bounded exponential-polynomial type function. This includes in particular diffusion processes. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families should rather not be used for modelling the term structure of interest rates.

Bernoulli Society for Mathematical Statistics and Probability2000

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In finance, a futures contract (sometimes called futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between pa

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