Fred Benth - Academia.edu (original) (raw)

Papers by Fred Benth

Stochastics An International Journal of Probability and Stochastic Processes, 1998

We study convergence rates for generalized random variables, ihese results will be applied to stu... more We study convergence rates for generalized random variables, ihese results will be applied to study a stochastic finite element approach to stochastic partial differential equations

Mathematical Finance, 2001

SSRN Electronic Journal, 2000

ABSTRACT In energy markets, the use of quanto options have increased significantly in the recent ... more ABSTRACT In energy markets, the use of quanto options have increased significantly in the recent years. The payoff from such options are typically triggered by an energy price and a measure of temperature and are thus suited for managing both price and volume risk in energy markets. Using an HJM approach we derive a closed form option pricing formula for energy quanto options, under the assumption that the underlying assets are log-normally distributed. Our approach encompasses several interesting cases, such as geometric Brownian motions and multifactor spot models. We also derive delta and cross-gamma hedging parameters. Furthermore, we illustrate the use of our model by an empirical pricing exercise using NYMEX traded natural gas futures and CME traded HDD temperature futures for New York and Chicago.

The markets for electricity, gas and temperature have distinctive features, which provide the foc... more The markets for electricity, gas and temperature have distinctive features, which provide the focus for countless studies. For instance, electricity and gas prices may soar several magnitudes above their normal levels within a short time due to imbalances in supply and demand, yielding what is known as spikes in the spot prices. The markets are also largely influenced by seasons, since power demand for heating and cooling varies over the year. The incompleteness of the markets, due to nonstorability of electricity and temperature as well as limited storage capacity of gas, makes spot-forward hedging impossible. Moreover, futures contracts are typically settled over a time period rather than at a fixed date. All these aspects of the markets create new challenges when analyzing price dynamics of spot, futures and other derivatives. This book provides a concise and rigorous treatment on the stochastic modeling of energy markets. Ornstein-Uhlenbeck processes are described as the basic m...

We generalize a result by Karatzas and Shreve, (15) to the multi- dimensional case. A viscosity s... more We generalize a result by Karatzas and Shreve, (15) to the multi- dimensional case. A viscosity solution approach is taken to show that the value function of the multi-dimensional monotone follower problem coincides with the integral of the value function of associated stopping problems. The connection holds under a strong factorization property of the running cost function.

We propose a mean-reverting model for the spot price dynamics of elec- tricity which includes sea... more We propose a mean-reverting model for the spot price dynamics of elec- tricity which includes seasonality of the prices and spikes. The dynamics is a sum of non-Gaussian Ornstein-Uhlenbeck processes with jump processes giving the normal vari- ations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot

. We investigate an innite horizon investment-consumption model in which a singleagent consumes a... more . We investigate an innite horizon investment-consumption model in which a singleagent consumes and distributes her wealth between a risk-free asset (bank account) and severalrisky assets (stocks) whose prices are governed by Levy (jump-diusion) processes. We supposethat transactions between the assets incur a transaction cost proportional to the size of thetransaction. The problem is to maximize the total utility of

Quantum Probability and Infinite Dimensional Analysis - Proceedings of the 29th Conference, 2010

ABSTRACT The authors of this paper consider a Lévy process X=(X t ) t≥0 and are interested in com... more ABSTRACT The authors of this paper consider a Lévy process X=(X t ) t≥0 and are interested in computing the following quantity Δ(X):=∂ ∂xE[f(x+X T )]·(1) In the context of exponential Lévy models in mathematical finance, this corresponds to the Delta of an option with payoff function f (in terms of the log-price). They discuss different methods for the computation of Δ(X), i.e., (i) the density method, where ∂ ∂xE[f(x+X T )]=E[f(x+X T )π], with π an appropriate weight; (ii) the conditional density method, where they use the Lévy-Khintchine representation of X into drift plus Brownian motion plus jumps, and then apply the density method only to the Brownian part by first conditioning on it. Then, they consider the approximation of the Lévy process X by X ε , where jumps smaller than ε have been truncated and replaced by a suitably scaled Brownian motion. They show that convergence of X ε to X as ε↓0 implies convergence of Δ(X ε ) to Δ(X). Finally, they study the properties of the different methods for computing (1), in particular, their respective variance, and discuss the consequences for numerical implementations.

Statistical Tools for Finance and Insurance, 2011

Applied Mathematics and Optimization - APPL MATH OPT, 2003

We show that the value function of a singular stochastic control problem is equal to the integral... more We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.

... Laser Cooling and Stochastics 53 and Bx(p) will denote the ball in RD with radius ? and cente... more ... Laser Cooling and Stochastics 53 and Bx(p) will denote the ball in RD with radius ? and center x. We shall refer to Bo(r), for some small r, as the 'trap', this corresponding ... For y outside the ball Bq(1) various forms of X(y) are considered. We shall discuss three model types: ...

International Journal of Theoretical and Applied Finance, 2009

We consider the problem of utility indifference pricing of a put option written on a non-tradeabl... more We consider the problem of utility indifference pricing of a put option written on a non-tradeable asset, where we can hedge in a correlated asset. The dynamics are assumed to be a two-dimensional geometric Brownian motion, and we suppose that the issuer of the option have exponential risk preferences. We prove that the indifference price dynamics is a martingale with respect to an equivalent martingale measure (EMM) Q after discounting, implying that it is arbitrage-free. Moreover, we provide a representation of the residual risk remaining after using the optimal utility-based trading strategy as the hedge.Our motivation for this study comes from pricing interest-rate guarantees, which are products usually offered by companies managing pension funds. In certain market situations the life company cannot hedge perfectly the guarantee, and needs to resort to sub-optimal replication strategies. We argue that utility indifference pricing is a suitable method for analysing such cases.We provide some numerical examples giving insight into how the prices depend on the correlation between the tradeable and non-tradeable asset, and we demonstrate that negative correlation is advantageous, in the sense that the hedging costs become less than with positive correlation, and that the residual risk has lower volatility. Thus, if the insurance company can hedge in assets negatively correlated with the pension fund, they may offer cheaper prices with lower Value-at-Risk measures on the residual risk.

Stochastics An International Journal of Probability and Stochastic Processes, 2005

Stochastics An International Journal of Probability and Stochastic Processes, 2013

ABSTRACT We study the robustness of option prices to model variation after a change of measure wh... more ABSTRACT We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.

Stochastics An International Journal of Probability and Stochastic Processes, 1996

Generalized stochastic processes in the space are studied, where is the dual of a space of smooth... more Generalized stochastic processes in the space are studied, where is the dual of a space of smooth random variables (over the white noise probability space) constructed via L -norms weighted with exponentials of the Ornstein-Uhlenbeck operator. This class of processes allows for the construction of generalized stochastic integrals following Itoô's recipe. Moreover, the notion of the (sub-, super-, semi-) martingale

Stochastics An International Journal of Probability and Stochastic Processes, 2002

Stochastic Analysis and Applications, 2005

Stochastic Analysis and Applications, 2002

Scandinavian Journal of Statistics, 2007

Stochastics An International Journal of Probability and Stochastic Processes, 1998

We study convergence rates for generalized random variables, ihese results will be applied to stu... more We study convergence rates for generalized random variables, ihese results will be applied to study a stochastic finite element approach to stochastic partial differential equations

Mathematical Finance, 2001

SSRN Electronic Journal, 2000

ABSTRACT In energy markets, the use of quanto options have increased significantly in the recent ... more ABSTRACT In energy markets, the use of quanto options have increased significantly in the recent years. The payoff from such options are typically triggered by an energy price and a measure of temperature and are thus suited for managing both price and volume risk in energy markets. Using an HJM approach we derive a closed form option pricing formula for energy quanto options, under the assumption that the underlying assets are log-normally distributed. Our approach encompasses several interesting cases, such as geometric Brownian motions and multifactor spot models. We also derive delta and cross-gamma hedging parameters. Furthermore, we illustrate the use of our model by an empirical pricing exercise using NYMEX traded natural gas futures and CME traded HDD temperature futures for New York and Chicago.

The markets for electricity, gas and temperature have distinctive features, which provide the foc... more The markets for electricity, gas and temperature have distinctive features, which provide the focus for countless studies. For instance, electricity and gas prices may soar several magnitudes above their normal levels within a short time due to imbalances in supply and demand, yielding what is known as spikes in the spot prices. The markets are also largely influenced by seasons, since power demand for heating and cooling varies over the year. The incompleteness of the markets, due to nonstorability of electricity and temperature as well as limited storage capacity of gas, makes spot-forward hedging impossible. Moreover, futures contracts are typically settled over a time period rather than at a fixed date. All these aspects of the markets create new challenges when analyzing price dynamics of spot, futures and other derivatives. This book provides a concise and rigorous treatment on the stochastic modeling of energy markets. Ornstein-Uhlenbeck processes are described as the basic m...

We generalize a result by Karatzas and Shreve, (15) to the multi- dimensional case. A viscosity s... more We generalize a result by Karatzas and Shreve, (15) to the multi- dimensional case. A viscosity solution approach is taken to show that the value function of the multi-dimensional monotone follower problem coincides with the integral of the value function of associated stopping problems. The connection holds under a strong factorization property of the running cost function.

We propose a mean-reverting model for the spot price dynamics of elec- tricity which includes sea... more We propose a mean-reverting model for the spot price dynamics of elec- tricity which includes seasonality of the prices and spikes. The dynamics is a sum of non-Gaussian Ornstein-Uhlenbeck processes with jump processes giving the normal vari- ations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot

. We investigate an innite horizon investment-consumption model in which a singleagent consumes a... more . We investigate an innite horizon investment-consumption model in which a singleagent consumes and distributes her wealth between a risk-free asset (bank account) and severalrisky assets (stocks) whose prices are governed by Levy (jump-diusion) processes. We supposethat transactions between the assets incur a transaction cost proportional to the size of thetransaction. The problem is to maximize the total utility of

Quantum Probability and Infinite Dimensional Analysis - Proceedings of the 29th Conference, 2010

ABSTRACT The authors of this paper consider a Lévy process X=(X t ) t≥0 and are interested in com... more ABSTRACT The authors of this paper consider a Lévy process X=(X t ) t≥0 and are interested in computing the following quantity Δ(X):=∂ ∂xE[f(x+X T )]·(1) In the context of exponential Lévy models in mathematical finance, this corresponds to the Delta of an option with payoff function f (in terms of the log-price). They discuss different methods for the computation of Δ(X), i.e., (i) the density method, where ∂ ∂xE[f(x+X T )]=E[f(x+X T )π], with π an appropriate weight; (ii) the conditional density method, where they use the Lévy-Khintchine representation of X into drift plus Brownian motion plus jumps, and then apply the density method only to the Brownian part by first conditioning on it. Then, they consider the approximation of the Lévy process X by X ε , where jumps smaller than ε have been truncated and replaced by a suitably scaled Brownian motion. They show that convergence of X ε to X as ε↓0 implies convergence of Δ(X ε ) to Δ(X). Finally, they study the properties of the different methods for computing (1), in particular, their respective variance, and discuss the consequences for numerical implementations.

Statistical Tools for Finance and Insurance, 2011

Applied Mathematics and Optimization - APPL MATH OPT, 2003

We show that the value function of a singular stochastic control problem is equal to the integral... more We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.

... Laser Cooling and Stochastics 53 and Bx(p) will denote the ball in RD with radius ? and cente... more ... Laser Cooling and Stochastics 53 and Bx(p) will denote the ball in RD with radius ? and center x. We shall refer to Bo(r), for some small r, as the 'trap', this corresponding ... For y outside the ball Bq(1) various forms of X(y) are considered. We shall discuss three model types: ...

International Journal of Theoretical and Applied Finance, 2009

We consider the problem of utility indifference pricing of a put option written on a non-tradeabl... more We consider the problem of utility indifference pricing of a put option written on a non-tradeable asset, where we can hedge in a correlated asset. The dynamics are assumed to be a two-dimensional geometric Brownian motion, and we suppose that the issuer of the option have exponential risk preferences. We prove that the indifference price dynamics is a martingale with respect to an equivalent martingale measure (EMM) Q after discounting, implying that it is arbitrage-free. Moreover, we provide a representation of the residual risk remaining after using the optimal utility-based trading strategy as the hedge.Our motivation for this study comes from pricing interest-rate guarantees, which are products usually offered by companies managing pension funds. In certain market situations the life company cannot hedge perfectly the guarantee, and needs to resort to sub-optimal replication strategies. We argue that utility indifference pricing is a suitable method for analysing such cases.We provide some numerical examples giving insight into how the prices depend on the correlation between the tradeable and non-tradeable asset, and we demonstrate that negative correlation is advantageous, in the sense that the hedging costs become less than with positive correlation, and that the residual risk has lower volatility. Thus, if the insurance company can hedge in assets negatively correlated with the pension fund, they may offer cheaper prices with lower Value-at-Risk measures on the residual risk.

Stochastics An International Journal of Probability and Stochastic Processes, 2005

Stochastics An International Journal of Probability and Stochastic Processes, 2013

ABSTRACT We study the robustness of option prices to model variation after a change of measure wh... more ABSTRACT We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.

Stochastics An International Journal of Probability and Stochastic Processes, 1996

Generalized stochastic processes in the space are studied, where is the dual of a space of smooth... more Generalized stochastic processes in the space are studied, where is the dual of a space of smooth random variables (over the white noise probability space) constructed via L -norms weighted with exponentials of the Ornstein-Uhlenbeck operator. This class of processes allows for the construction of generalized stochastic integrals following Itoô's recipe. Moreover, the notion of the (sub-, super-, semi-) martingale

Stochastics An International Journal of Probability and Stochastic Processes, 2002

Stochastic Analysis and Applications, 2005

Stochastic Analysis and Applications, 2002

Scandinavian Journal of Statistics, 2007