A drift‐free simulation method for pricing commodity derivatives

Bermudan Swaptions in the LIBOR Market Model

A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries. Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities. Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day. This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it. While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads). In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections. Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.

A number of models have been successfully developed wherein the theoretical prices for caps, floors, and swaptions produced by the model are of the Black-76 form. In these models, discrete market rates are modelled like LIBOR rates in the LIBOR market models or forward swap rates in the swap market models; and under a suitable choice of numeraires, market rates can be modelled log normally. LIBOR caps and the market practice for pricing and quoting these instruments are discussed. It is shown that given a swap market model, the LIBOR rates will not be lognormal; thus, LIBOR market models and swap models are generally incompatible. Practice exercises are included.

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.KeywordsSABR volatility modelsSABR/LIBOR market modelsParallel simulated annealingGPUs

This paper focuses on designing and pricing commodity derivatives and insurance within a novel financial engineering framework that can be subsequently tested empirically using commodity price data. Optimal contract solutions are obtained and interpreted. We quantify explicitly how derivative prices and insurance premiums are affected by economic variables linked to commodity supply and demand. Our results generalize some existing commodity derivative pricing models and further show under which conditions there will be no trading of derivative instruments and insurance. We report GMM estimates of the model parameters for a large dataset of commodity futures. These results also contribute to a better understanding of the “financialization” of commodities.

In this paper an efficient calibration method for the multi-factor LIBOR Market Model (LMM) is proposed and is applied for the Japanese interest rate market. At first the joint calibration method in the cap and swaption market is demonstrated using a new parameterization for the correlation matrix in the LMM. Then we implement the proposed methodology for calibrating the Japanese cap and swaption markets, where the computational procedure is shown to be tractable and provides a practical estimation for the implied correlation matrix in the LMM. The empirical analysis also illustrates that Black's swaption volatilities through our calibration fit the market data almost exactly and that the estimated implied correlation matrix is smooth and stable.

Efficient Pricing and Greeks in the Cross-Currency LIBOR Market Model

LIBOR Market Model with Stochastic Volatility

The Libor market model has established itself as the benchmark model for interest rate derivatives. If the observed correlation and volatility surfaces cannot be reproduced by a model, we cannot hope to get meaningful prices; therefore the crucial task, before it comes to pricing and hedging, is to calibrate the model to given market data. An overview of the Libor market model is given and it is shown how to obtain a robust calibration. We show how to extend the model to include the market smile by making use of stochastic volatility. For these stochastic volatility Libor market models a new time homogeneous skew parametrization is introduced. Furthermore a new approximative terminal correlation formula based on the same parameter averaging technique used for pricing in a stochastic volatility Libor market model is presented. We look at a very robust initial calibration procedure; here we introduce some new approximations making it possible to use global optimizers. The optimal choice of local and global optimizers for each step of the calibration will be dealt with and an analysis of the robustness of the calibrated parameters is performed. Two extensions to the cross currency Libor market model introduced by Schlögl are developed. These make use of displacements and stochastic volatility to fit the observed skew and smile respectively. The calculation of the Greeks for exotic interest rate derivatives is crucial for hedging and therefore for trading these products. For a Libor market model this task can only be fulfilled by means of nontrivial Monte Carlo based methods. A new method for the calculation of the weights for the proxy simulation scheme method is developed making use of the conditional independence of the underlying from the stochastic volatility process, which can be used for any model for which this independence holds. The developed estimators are fast enough to be used in everyday practice.

A Libor Market Model with Default Risk

We consider a seasonal mean‐reverting model for energy commodity prices with jumps and Heston‐type stochastic volatility, and three nested models for comparison. By exploiting the affine form of the log‐spot models, we develop a general valuation framework for futures and discrete arithmetic Asian options. We investigate five major petroleum commodities from Europe (Brent crude oil, gasoil) and US (light sweet crude oil, gasoline, heating oil) and analyse the effects of the competing fitted spot models in futures pricing, Asian options pricing and hedging. We find evidence that price jumps and stochastic volatility are important features of the petroleum price dynamics.

Valuation of CMS Spread Options with Nonzero Strike Rates in the LIBOR Market Model

In this paper, we introduce a mean-field extension of the LIBOR market model (LMM) which preserves the basic features of the original model. Among others, these features are the martingale property, a directly implementable calibration and an economically reasonable parametrization of the classical LMM. At the same time, the mean-field LIBOR market model (MF-LMM) is designed to reduce the probability of exploding scenarios, arising in particular in the market-consistent valuation of long-term guarantees. To this end, we prove existence and uniqueness of the corresponding MF-LMM and investigate its practical aspects, including Black’s formula. Moreover, we present an extensive numerical analysis of the MF-LMM. The corresponding Monte Carlo method is based on a suitable interacting particle system which approximates the underlying mean-field equation.