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

Papers by Fred Benth

Convergence rates for finite elementapproximations of stochastic partial differential equations

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

Pricing and Hedging Quanto Options in Energy Markets

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 discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differe... more We discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differential equations.

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

Portfolio Optimization in a L¨|vy Market with Intertemporal Substitution and Transaction Costs

. 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

Lévy Models Robustness and Sensitivity

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

Weather derivatives (WD) are different from most financial derivatives because the underlying wea... more Weather derivatives (WD) are different from most financial derivatives because the underlying weather cannot be traded and therefore cannot be replicated by other financial instruments. The market price of risk (MPR) is an important parameter of the associated equivalent martingale measures used to price and hedge weather futures/options in the market. The majority of papers so far have priced non-tradable assets assuming zero MPR, but this assumption underestimates WD prices. We study the MPR structure as a time dependent object with concentration on emerging markets in Asia. We find that Asian Temperatures (Tokyo, Osaka, Beijing, Teipei) are normal in the sense that the driving stochastics are close to a Wiener Process. The regression residuals of the temperature show a clear seasonal variation and the volatility term structure of CAT temperature futures presents a modified Samuelson effect. In order to achieve normality in standardized residuals, the seasonal variation is calibrated with a combination of a fourier truncated series with a GARCH model and with a local linear regression. By calibrating model prices, we implied the MPR from Cumulative total of 24hour average temperature futures (C24AT) for Japanese Cities, or by knowing the formal dependence of MPR on seasonal variation, we price derivatives for Kaohsiung, where weather derivative market does not exist. The findings support theoretical results of reverse relation between MPR and seasonal variation of temperature process. . The financial support from the Deutsche Forschungsgemeinschaft via SFB 649 "Ökonomisches Risiko", Humboldt-Universität zu Berlin is gratefully acknowledged. 1 derivatives has attracted the attention of many researchers. and Alaton, Djehiche and Stillberger fitted Ornstein-Uhlenbeck stochastic processes to temperature data and investigated future prices on temperature indices. analyse heteroscedasticity in temperature volatily and Benth , and Benth, Saltyte Benth and Koekebakker develop the non-arbitrage framework for pricing different temperature derivatives prices.

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

... 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 &amp;amp;amp;#x27;trap&amp;amp;amp;#x27;, this corresponding ... For y outside the ball Bq(1) various forms of X(y) are considered. We shall discuss three model types: ...

UTILITY INDIFFERENCE PRICING OF INTEREST-RATE GUARANTEES

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

Under general conditions stated in Rheinländer [32], we prove that in a stochastic volatility mar... more Under general conditions stated in Rheinländer [32], we prove that in a stochastic volatility market the Radon-Nikodym density of the minimal entropy martingale measure can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein-Stein and Heston stochastic volatility models.

A note on convergence of option prices and their Greeks for Lévy models

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.

On the martingale property for generalized stochastic processes

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

We investigate an infinite horizon investment-consumption model in which a single agent consumes ... more We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimization problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.

Stochastic Analysis and Applications, 2005

We analyze the classical Merton's portfolio optimization problem when the risky asset follows an ... more We analyze the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman equation is a two-dimensional nonlinear parabolic partial differential equation. We produce an explicit solution to this equation by reducing it to a simpler one-dimensional linear parabolic equation. This reduction is achieved through a Cole-Hopf type transformation, recently introduced in portfolio optimization theory by Zariphopoulou . A verification argument is then used to prove that this solution coincides with the value function of the control problem. The optimal investment strategy is also given explicitly. On the technical side, the main problem we are facing here is the necessity to identify conditions on the parameters of the control problem ensuring uniform integrability of a family of random variables that roughly speaking are the exponentials of squared Wiener integrals. Date: March 30, 2005.

Stochastic Analysis and Applications, 2002

Scandinavian Journal of Statistics, 2007

This paper analyzes the weather derivatives traded at the Chicago Mercantile Exchange (CME), with... more This paper analyzes the weather derivatives traded at the Chicago Mercantile Exchange (CME), with futures and options written on different temperature indices. We propose to model the temperature dynamics as a continuous-time autoregressive process with lag p and seasonal variation. The choice p = 3 turns out to be sufficient to explain the temperature dynamics observed in Stockholm, Sweden, where we fit the model to more than 40 years of daily observations. The main finding is a clear seasonal variation in the regression residuals, where temperature shows high variability in winter, low in autumn and spring, and increasing variability towards the early summer. Our model allows for derivations of explicit prices for several futures and options. Note that the volatility term structure of futures written on the cumulative average temperature has a modified Samuelson effect, where the volatility prior to the measurement period increases, except for the last part, where it may decrease.