Upper and lower solutions for regime-switching diffusions with applications in financial mathematics

This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique C2 solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute approximate solutions. An important feature of the proposed numerical method is that the unique solution is bracketed by the upper and lower approximate solutions, which provide an interval estimate of the unique solution function. We apply the general results to a regime-switching mean-reverting model and improve related results already reported in the literature. For the mean-reverting model, explicit upper and lower solutions are obtained and numerical integration methods are employed. In another case (Example 3 in section 5) a different regime-switching model is considered, where the general results apply, but only the upper solution is explicitly obtained. In that example, only the sequence of upper solutions is numerically constructed using finite difference methods. Numerical results are reported

Quantitative Techniques for Spread Trading in Commodity Markets

This thesis investigates quantitative techniques for trading strategies on two commodities, the difference of whose prices exhibits a long-term historical relationship known as mean-reversion. A portfolio of two commodity prices with very similar characteristics, the spread may be regarded as a distinct process from the underlying price processes so deserves to be modeled directly. To pave the way for modeling the spread processes, the fundamental concepts, notions, properties of commodity markets such as the forward prices, the futures prices, and convenience yields are described. Some popular commodity pricing models including both one and two factor models are reviewed. A new mean-reverting process to model the commodity spot prices is introduced. Some analytical results for this process are derived and its properties are analyzed. We compare the new one-factor model with a common existing one-factor model by applying these two models to price West Texas Intermediate (WTI) crude oil, and discuss its advantages and disadvantages. We investigate the recent behavioral change in the location spread process between WTI crude oil and Brent oil. The existing three major approaches to price a spread process namely cointegration, one-factor and two-factor models fail to fully capture these behavioral changes. We, therefore, extend the one-factor and two-factor spread models by including a compound Poisson process where jump sizes follow a double exponential distribution. We generalize the existing one-factor mean-reverting dynamics (Vasicek process) by replacing the constant diffusion term with a nonlinear term to price the spread process. Applying the new process to the empirical location spread between WTI and Brent crude oils dataset, it is shown how the generalized dynamics can rigorously capture the most important characteristics of the spread process namely high volatility, skewness and kurtosis. To consider the recent structural breaks in the location spread between WTI and Brent, we incorporate regime switching dynamics in the generalized model and Vasicek process by including two regimes. We also introduce a new mean-reverting random walk, derive its continuous time stochastic differential equation and obtain some analytical results about its solution. This new mean-reverting process is compared with the Vasicek process and its advantages discussed. We showed that this new model for spread dynamics is capable of capturing the possible skewness, kurtosis, and heavy tails in the transition density of the price spread process. Since the analytical transition density is unknown for this nonlinear stochastic process, the local linearization method is deployed to estimate the model parameters. We apply this method to empirical data for modeling the spread between WTI crude oil and West Texas Sour (WTS) crude oil. Finally, we apply the introduced trading strategies to empirical data for the location spread between WTI and Brent crude oils, analyze, and compare the profitability of the strategies. The optimal trading strategies for the spread dynamics in the cointegration approach and the one-factor mean-reverting process are discussed and applied to our considered empirical dataset. We suggest to use the stationary distribution to find optimal thresholds for log-term investment strategies when the spread dynamics is assumed to follow a Vasicek process. To incorporate essential features of a spread process such as skewness and kurtosis into the spread trading strategies, we extend the optimal trading strategies by considering optimal asymmetric thresholds

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail

Modelling FX smile : from stochastic volatility to skewness

Imperial Users onl

Interest rate models with Markov chains

Imperial Users onl

Understanding the Fine Structure of Electricity Prices.

This paper analyzes the special features of electricity spot prices derived from the physics of this commodity and from the economics of supply and demand in a market pool. Besides mean-reversion, a property they share with other commodities, power prices exhibit the unique feature of spikes in trajectories. We introduce a class of discontinuous processes exhibiting a jump-reversion component to properly represent these sharp upward moves shortly followed by drops of similar magnitude. Our approach allows to capture - for the first time to our knowledge - both the trajectorial and the statistical properties of electricity pool prices. The quality of the fitting is illustrated on a database of major US power markets.Energy price risk; Simulation; Calibration; Statistical estimations; Jump diffusions; Electricity prices;

Understanding the fine structure of electricity prices

This paper analyzes the special features of electricity spot prices derived from the physics of this commodity and from the economics of supply and demand in a market pool. Besides mean reversion, a property they share with other commodities, power prices exhibit the unique feature of spikes in trajectories. We introduce a class of discontinuous processes exhibiting a "jump-reversion" component to properly represent these sharp upward moves shortly followed by drops of similar magnitude. Our approach allows to capture—for the first time to our knowledge—both the trajectorial and the statistical properties of electricity pool prices. The quality of the fitting is illustrated on a database of major U.S. power markets

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail.volatility derivatives; operator methods; moment methods; conditional corridor variance swaps; variance knockout options

2008 (Winter)

Abstracts of the talks given at the 2008 Winter Colloquium