Theory and Calibration of HJM with Shape Factors

Roncoroni, Andréa; Guiotto, Paolo

doi:10.1007/978-3-662-12429-1_19

Cited by 5 publications

(7 citation statements)

References 6 publications

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“…The Bergomi model for future variance is introduced in [6] and [7], and a consistency condition for the drift in futures curves for variance is given in [8]. Term structure and the associated Heath-Jarrow-Morton (HJM) framework are discussed in [9,15,23,24]. In the past decade there has been a lot of research addressing the causal relationship between SPX and VIX options, which includes some re-evaluation of widely-used SVMs and a search for new models to fit both markets.…”

Section: Background Literaturementioning

confidence: 99%

Section: Background Literaturementioning

confidence: 99%

“…. , which satisfy a linear system of ordinary differential equations obtained by Ito's formula from the stochastic differential equations of the factor process (24),…”

Section: The Double Nelson Modelmentioning

confidence: 99%

“…(see background on HJM framwwork in[9,15,24]). Differentiating the HJM condition with respect to T and using equation (35) yields,α(t, T ) = −ν(t, T )σ * (X t )∇f T −t (X t ) .…”

mentioning

confidence: 99%

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Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

Papanicolaou

2018

SSRN Journal

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This paper shows how to recover stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function as the output of an inverse problem, with the inputs given by a VIX futures market model. Analysis will show that some conditions need to be met in order for there to not be any inter-model arbitrage or mis-priced derivatives. Given these conditions the inverse problem can be solved. Several models are analyzed and explored numerically to gain a better understanding of the theory and its limitations.

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Section: Background Literaturementioning

confidence: 99%

Section: Background Literaturementioning

confidence: 99%

“…. , which satisfy a linear system of ordinary differential equations obtained by Ito's formula from the stochastic differential equations of the factor process (24),…”

Section: The Double Nelson Modelmentioning

confidence: 99%

“…(see background on HJM framwwork in[9,15,24]). Differentiating the HJM condition with respect to T and using equation (35) yields,α(t, T ) = −ν(t, T )σ * (X t )∇f T −t (X t ) .…”

mentioning

confidence: 99%

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Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

Papanicolaou

2018

SSRN Journal

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“…The classical way goes by specifying a stochastic model for the spot price, and from this model derive the dynamics of forward prices based on no-arbitrage principles (see Lucia and Schwartz (2002), Cartea and Figueroa (2005), Roncoroni and Geman (2006), Benth, Kallsen, and Meyer-Brandis (2007), Garcia, Klüppelberg, and Müller (2011), Barndorff-Nielsen, Benth, and Veraart (2013), Weron and Zator (2014), and Benth, Klüppelberg, Müller, and Vos (2014)). The alternative is to follow the Heath-Jarrow-Morton approach and to specify the dynamics of the forward prices directly, as it has been done in Roncoroni and Guiotto (2001), , Weron and Borak (2008) and Kiesel, Schindlmayr, and Boerger (2009). All these studies model the forward prices using multifactor models driven by Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

A space-time random field model for electricity forward prices

Benth

Paraschiv

2018

Journal of Banking & Finance

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Stochastic models for forward electricity prices are of great relevance nowadays, given the major structural changes in the market due to the increase of renewable energy in the production mix. In this study, we derive a spatio-temporal dynamical model based on the Heath-Jarrow-Morton (HJM) approach under the Musiela parametrization, which ensures an arbitrage-free model for electricity forward prices. The model is fitted to a unique data set of historical price forward curves. As a particular feature of the model, we disentangle the temporal from spatial (maturity) effects on the dynamics of forward prices, and shed light on the statistical properties of risk premia, of the noise volatility term structure and of the spatio-temporal noise correlation structures. We find that the short-term risk premia oscillates around zero, but becomes negative in the long run. We identify the Samuelson effect in the volatility term structure and volatility bumps, explained by market fundamentals. Furthermore we find evidence for coloured noise and correlated residuals, which we model by a Hilbert space-valued normal inverse Gaussian Lévy process with a suitable covariance functional.

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Consistent time‐homogeneous modeling of SPX and VIX derivatives

Papanicolaou

2022

Mathematical Finance

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This paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Therefore, consistent modeling for both SPX and VIX should involve an SVM that can be obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function by solving an inverse problem where the input is the VIX function given by a market model. Analysis will show conditions necessary for there to be a unique solution to this inverse problem. The models are consistent if the recovered volatility function is non-negative. Examples are presented to illustrate the theory, to highlight the issue of negativity in solutions, and to show the potential for inconsistency in non-Markov settings.

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Theory and Calibration of HJM with Shape Factors

Cited by 5 publications

References 6 publications

Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

A space-time random field model for electricity forward prices

Consistent time‐homogeneous modeling of SPX and VIX derivatives

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