Shephard Model Research Articles

Barndorff‐Nielsen and Shephard model for hedging energy with quantity risk

AbstractIn this paper, the Barndorff‐Nielsen and Shephard (BN‐S) model is implemented to find an optimal hedging strategy in the presence of quantity risk for oil produced in the Bakken, a new region of oil extraction that is benefiting from fracking technology. Hedging and price risk management become much more involved with the inclusion of quantity risk. Explorers and drillers have uncertainty on the quantity of oil that would be extracted, and governments have uncertainty of the quantity of oil that will be extracted, sold, and available for imposing tax regimes. One of the main assumptions typically made in a portfolio model of hedging is that the quantity of inventory or demand is known. This is inappropriate in many hedging situations. Quantity risk compounds the difficulty of determining the optimal size of the position under both price and production risk. In this paper, we provide a novel way of handling the quantity risk in connection with the BN‐S model. The model is analyzed as a quadratic hedging problem and related analytical results are developed. The results indicate that oil can be optimally hedged with a combination of variance swaps and options. For various quantity risks, the model is implemented to analyze hedging decisions numerically for managing price risk in the Bakken oil commodities.

A Gamma Ornstein-Uhlenbeck Model Driven by a Hawkes Process

We propose an extension of the Gamma-OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the Gamma-OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the Gamma-OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the Gamma-OU Barndorff-Nielsen and Shephard model.

Moments of the asset price for the Barndorff-Nielsen and Shephard model

In this paper, we derive closed-form formulas for moments of the asset price in the Barndorff-Nielsen and Shephard (BN–S) stochastic volatility model. We also present similar results where the market is driven by a BN–S-type stochastic process. It is shown that in both cases the results depend on the cumulant transform of the background driving Levy process for the models. In this paper, we have also obtain various approximate expressions for the expected value of the square-root process for the shifted asset price with respect to the BN–S model.

Barndorff-Nielsen and Shephard model: oil hedging with variance swap and option

In this paper the Barndorff-Nielsen and Shephard (BN-S) model is implemented to find an optimal hedging strategy for the oil commodity from the Bakken, a new region of oil extraction that is benefiting from fracking technology. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The results indicate that oil can be optimally hedged with the use of a combination of options and variance swaps. Theoretical results related to the variance process are established and implemented for the analysis of the variance swap. In this paper we also determined the optimal amount of the underlying oil commodity that has to be held for minimizing the hedging error. The model and analysis are used to numerically analyze hedging decisions for managing price risk in Bakken oil commodities. From the numerical results, a number of important features of the usefulness of the Barndorff-Nielsen and Shephard model are illustrated.

Importance sampling applied to Greeks for jump–diffusion models with stochastic volatility

We develop a variance reduction technique, based on importance sampling in con- junction with the stochastic Robbins–Monro algorithm, for option prices of jump– diffusion models with stochastic volatility. This is done by combining the work developed by Arouna for pricing diffusion models, and extended by Kawai for Lévy processes without a Brownian component. We apply this technique to improve the numerical computation of derivative price sensitivities for general Lévy processes, allowing both Brownian and jump parts. Numerical examples are performed for both the Black–Scholes and Heston models with jumps and for the Barndorff–Nielsen–Shephard model to illustrate the efficiency of this numerical technique. The numerical results support that the proposed methodology improves the efficiency of the usual Monte Carlo procedures.

Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index

In this paper a couple of variance dependent instruments in the financial market are studied. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Veceř-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the “error term” are derived. Empirical data driven numerical examples are provided in support of the proposed price index.

Local risk-minimization for Barndorff-Nielsen and Shephard models

We obtain explicit representations of locally risk-minimizing strategies for call and put options in Barndorff-Nielsen and Shephard models, which are Ornstein–Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy processes, Arai and Suzuki (Int. J. Financ. Eng. 2:1550015, 2015) obtained a formula for locally risk-minimizing strategies for Levy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in (Arai and Suzuki in Int. J. Financ. Eng. 2:1550015, 2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, we introduce some numerical experiments for locally risk-minimizing strategies.

A GENERAL ORNSTEIN–UHLENBECK STOCHASTIC VOLATILITY MODEL WITH LÉVY JUMPS

We present a general class of stochastic volatility models with jumps where the stochastic variance process follows a Lévy-driven Ornstein–Uhlenbeck (OU) process and the jumps in the log-price process follow a Lévy process. This financial market model is a true extension of the Barndorff-Nielsen–Shephard (BNS) model class and can establish a weak link between log-price jumps and volatility jumps. Furthermore, we investigate the weak-link [Formula: see text]-OU-BNS model as a special case, where we calculate the characteristic function of the logarithmic price in closed form. The classical [Formula: see text]-OU-BNS model can be obtained as a limit of weak-link [Formula: see text]-OU-BNS models in the Skorokhod topology. We highlight that the weak-link property may be a useful model extension in the case of pricing barrier options.

Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets

The objective of this paper is to study the arbitrage free pricing of variance and volatility swaps for Barndorff-Nielsen and Shephard type Lévy process driven financial markets. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. In this paper, we obtain various approximate expressions for the pricing of volatility and variance swaps. We show that with the approximate formulas obtained from the Barndorff-Nielsen and Shephard model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. Pricing formulas proposed in this paper are simple to compute in real time and hence can be efficiently used in practical applications. Numerical results are provided in support of the accuracy of approximate formulas presented in this paper.

Option pricing and hedging for optimized Lévy driven stochastic volatility models

This paper pays attention to Ornstein-Uhlenbeck (OU) based stochastic volatility models with marginal law given by Classical Tempered Stable (CTS) distribution and Normal Inverse Gaussian (NIG) distribution, which are subclasses of infinite activity Lévy processes and are compared to finite activity Barndorff-Nielsen and Shephard (BNS) model. They are applied to option pricing and hedging in capturing leptokurtic features in asset returns and clustering effect in volatility that are consistently observed phenomena in underlying asset dynamics. The analytical formula of option pricing can be obtained through use of characteristic functions and Fast Fourier Transform (FFT) technique. Additionally, we introduce two hybrid optimization techniques such as hybrid Particle Swarm optimization (PSO) algorithm and hybrid Differential Evolution (DE) algorithm into parameters calibration schemes to improve the calibration quality for newly constructed models. Finally, we conduct experiments on Chinese emerging option markets to examine the performance of proposed models exploiting hybrid optimization techniques.

GENERALIZED BARNDORFF-NIELSEN AND SHEPHARD MODEL AND DISCRETELY MONITORED OPTION PRICING

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

Volatility, Variance and Covariance Swaps for BN-S Type Levy Process Driven Financial Markets

The objective of this paper is to study the arbitrage free pricing of the covariance swap for Barndorff-Nielsen and Shephard type L\'evy process driven financial markets. One of the major challenges in arbitrage free pricing of swap is to obtain an accurate pricing expression which can be used with good computational accuracy. In this paper we obtain analytic expressions for the pricing of the covariance swap. We show that with the analytic expressions obtained from the Barndorff-Nielsen and Shephard model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. The models and pricing formulas proposed in this paper are computable in real time and hence can be efficiently used in practical applications.

Numerical experiments on hedging cliquet options

In this paper, we conduct pricing and hedging experiments in order to check whether simple stochastic volatility models are capable of capturing forward volatility and forward skew risks correctly. As a reference, we use the Bergomi model, which treats these risks accurately. The results of our experiments show that the cost of poor volatility modeling in the Heston model, the Barndorff-Nielsen–Shephard model and a variance-gamma model with stochastic arrival is too high when pricing and hedging cliquet options.

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Pricing Asian options in a stochastic volatility model with jumps

We consider the problem of pricing arithmetic Asian options in the presence of stochastic volatility. By performing a change of numeraire introduced by Vĕcĕr, we derive a partial integro-differential equation (PIDE) for Asian options within Barndorff-Nielsen and Shephard (BNS) model framework. Then, a finite difference discretization is proposed for dealing with the terms containing the partial derivatives and a simple trapezoidal rule is used for the integral term due to jumps. Numerical experiments confirm that the developed methods are very efficient.

One Improved Option-Pricing Method Using the Barndorff-Nielsen and Shephard Model

The Black-Scholes Model (Black-Scholes-Merton Model) is well known among practitioners and researchers in the field of option pricing. The present study explains how to simulate the Levy process to price options and demonstrates that the Barndorff-Nielsen and Shephard Model is empirically superior to the Black-Scholes Model (Black-Scholes-Merton Model) in option pricing. This study also demonstrates how to simulate the Wiener process.

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models

Let denote the implied volatility at maturity t for a strike , where and is the current value of the underlying. We show that has a uniform (in x) limit as maturity t tends to infinity, given by the formula , for x in some compact neighbourhood of zero in the class of affine stochastic volatility models. Function is the convex dual of the limiting cumulant-generating function h of the scaled log-spot process. We express h in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen–Shephard model).

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

In the present paper we provide a semiexplicit valuation formula for Geometric Asian options, with fixed and floating strike under continuous monitoring, when the underlying stock price process exhibits both stochastic volatility and jumps. More precisely, we shall work in the Barndorff-Nielsen and Shephard (BNS) model framework. We shall provide some numerical illustrations of the results obtained.

THE STOCHASTIC VOLATILITY MODEL OF BARNDORFF-NIELSEN AND SHEPHARD IN COMMODITY MARKETS

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.