DOCSLIB.ORG

Valuation of American Basket Options Using Quasi-Monte Carlo Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Valuation of American Basket Options Using Quasi-Monte Carlo Methods Valuation of American Basket Options using Quasi-Monte Carlo Methods d-fine GmbH Christ Church College University of Oxford A thesis submitted in partial fulfillment for the MSc in Mathematical Finance September 26, 2009 Abstract Title: Valuation of American Basket Options using Quasi-Monte Carlo Methods Author: d-fine GmbH Submitted for: MSc in Mathematical Finance Trinity Term 2009 The valuation of American basket options is normally done by using the Monte Carlo approach. This approach can easily deal with multiple ran- dom factors which are necessary due to the high number of state variables to describe the paths of the underlyings of basket options (e.g. the Ger- man Dax consists of 30 single stocks). In low-dimensional problems the convergence of the Monte Carlo valuation can be speed up by using low-discrepancy sequences instead of pseudo- random numbers. In high-dimensional problems, which is definitely the case for American basket options, this benefit is expected to diminish. This expectation was rebutted for different financial pricing problems in recent studies. In this thesis we investigate the effect of using different quasi random sequences (Sobol, Niederreiter, Halton) for path generation and compare the results to the path generation based on pseudo-random numbers, which is used as benchmark. American basket options incorporate two sources of high dimensional- ity, the underlying stocks and time to maturity. Consequently, different techniques can be used to reduce the effective dimension of the valuation problem. For the underlying stock dimension the principal component analysis (PCA) can be applied to reduce the effective dimension whereas for the time dimension the Brownian Bridge method can be used. We an- alyze the effect of using these techniques for effective dimension reduction on convergence behavior. To handle the early exercise feature of American (basket) options within the Monte Carlo framework we consider two common approaches: The Threshold approach proposed by Andersen (1999) and the Least-Squares Monte Carlo (LSM) approach suggested by Longstaff and Schwartz (2001). We investigate both pricing methods for the valuation of American (bas- ket) options in the equity market. Contents 1 Introduction 1 2 Basket Options 4 2.1 Introduction . .4 2.2 Different Types of Basket Options . .4 2.2.1 Plain Vanilla Basket Options . .4 2.2.2 Exchange Options . .5 2.2.3 Asian Basket Options . .6 2.3 American-style Basket Options . .7 3 Monte Carlo Simulation 8 3.1 Introduction . .8 3.2 Mathematical Foundations . .9 3.2.1 Strong Law of Large Numbers . .9 3.2.2 Error Estimation . 10 3.2.3 Central Limit Theorem . 10 3.3 Pseudo-Random Numbers . 11 3.4 Quasi-Random Numbers . 13 3.4.1 Low Discrepancy and Integration Error . 14 3.4.2 Halton Sequences . 16 3.4.3 Sobol Sequences . 18 3.4.4 Niederreiter Sequences . 19 3.5 The curse of dimensionality . 22 4 Effective Dimension Reduction 23 4.1 Introduction . 23 4.2 Principal Component Analysis . 23 4.3 Brownian Bridge . 25 i 5 Valuation of American Basket Options 28 5.1 Introduction . 28 5.2 Model Framework . 29 5.3 Monte Carlo Path Construction . 30 5.4 Valuation of Early Exercise Feature . 33 5.4.1 Least-Squares Approach . 33 5.4.2 Threshold Approach . 35 6 Comparison of Numerical Methods 37 6.1 Introduction . 37 6.2 Data . 38 6.3 Software Implementation . 38 6.4 Results . 40 6.4.1 Classical Put Options . 40 6.4.2 Exchange Options . 43 6.4.3 Basket Options . 45 7 Conclusions 49 A Data 51 Bibliography 51 ii List of Tables 6.1 Statistics for different paths generation methods for American Dax call valuation . 46 A.1 Description Dax constituents . 51 A.2 Correlation matrix for Dax constituents for April 30, 2009 . 52 iii List of Figures 3.1 Two-dimensional projections of pseudo-random sequences . 13 3.2 Two-dimensional projections of Halton sequences . 17 3.3 Two-dimensional projections of Sobol sequences . 20 3.4 Two-dimensional projections of Niederreiter sequences . 21 4.1 Brownian bridge construction . 26 6.1 Convergence properties plain-vanilla American put . 42 6.2 Convergence properties American exchange options using Underlying- Transformation ............................... 44 6.3 Convergence properties American Dax call for different random num- ber generators using Least-Squares approach . 48 A.1 Convergence properties of American put for different random number generators using Least-Squares approach . 53 A.2 Convergence properties of American put for different random number generators using Threshold approach . 54 A.3 Convergence properties of American exchange options for different ran- dom number generators . 55 A.4 Convergence properties American Dax call for pseudo-random num- bers and Sobol sequences . 56 A.5 Convergence properties American Dax call for Niederreiter and Halton sequences . 57 iv Chapter 1 Introduction The pricing and optimal exercise of options with early exercise features is one of the most challenging problems in mathematical finance. The problem becomes even more complex, if the model for the economy has more than one state variable, i.e. more than one factor affecting the value of the option as well as the optimal exercise mo- ment. American basket options are a prominent example for derivatives with early exercise features and multiple state variables. Due to their construction principle bas- ket options generally represent an advantageous alternative to hedge risky positions of several assets, since trading only one basket option instead of several single-asset options decreases transaction costs. Usually lattice methods or finite differences will be used to price American or Bermudan style options. However, these techniques are inefficient for high dimen- sional problems (i.e. problems with more than three state variables) and they are very difficult to apply on path-dependent options. For higher dimensions it would be relatively cheap to use simulation techniques since its computational cost does not increase exponentially as compared to other methods and they are relatively simple to apply. However, for a long time, simulation techniques seemed unapplicable to options with early exercise features. This is due to their forward-construction prin- ciple and their path-by-path generation. Since contingent claims with early exercise features are traded in all important derivative markets, many suggestions have been made during the last years to price such options by simulation, and in particular to de- termine the optimal early-exercise strategy. Examples can be found in Tilley (1993), Barraquand and Martineau (1995), Carriere (1996), Broadie and Glasserman (1997), Andersen (1999), Longstaff and Schwartz (2001), or Ib´anezand Zapatero (2004). In this case the starting point will be the application of standard Monte Carlo simulation with pseudo-random numbers, eventually combined with some variance 1 reduction techniques such as antithetic variables. However, using pseudo-random numbers can be quite slow, due to the relatively poor convergence rate of O(M −1=2) for M sample paths. Here, the application of Quasi-Monte Carlo simulation (QMC), e.g. using Sobol, Niederreiter or Halton sequences, can improve the performance of Monte Carlo simulations dramatically, since in optimal cases the convergence rate of QMC is O(log(M)dM −1), where d represents the dimension of the integration problem. Therefore, for small values of d the convergence is due to the construc- tion principle of low-discrepancy sequences significantly higher as for standard Monte Carlo. However, for higher dimensional problems this advantage will diminish, since the convergence strongly depends on the dimension of the integration problem for which reason the application of QMC to price American basket option seems not ad- vantageous. Especially for American options it cannot be expected that QMC works, since due to regression or optimization over the sample paths the sample points have interactions. However, Chaudhary (2005) discovered a significant speedup in the nu- merical results for the Least-squares approach proposed by [25]. Therefore we also expect to find this improvement for the convergence of the valuation of American basket options when applying QMC. Moreover, earlier empirical studies for finance problems, see among others [30, 29, 12], have shown in fact significant improvements using QMC even in high dimensional financial pricing problems up to dimension 360. The main reason is that many practical problems in mathematical finance have a low effective dimension { a notion introduced by [12] { i.e. either the integrand depends only of several variables, or the integrand can be approximated by a sum of functions with each depending only on a small number of variables at the same time. Sev- eral techniques to reduce the effective dimension have been successfully applied for derivative pricing problems. The most prominent are the Brownian bridge method for path construction (BB) first applied by Moskowitz and Caflisch (1996) and Caflisch et al. (1997) and the principal component analysis (PCA) first used by Acworth et al. (1997) to identify the underlyings which contribute the largest part of the total variance. Besides this empirical studies Wang and Sloan (2007) provides a theoretical analysis of the possible convergence order of QMC used in conjunction with BB or PCA. However, all these studies only analyze the convergence properties of QMC with BB or with PCA, but, to the best of our knowledge, not the combined application of QMC with both { BB and PCA. In our analysis we consider the valuation of American basket options as a promi- nent example for a high-dimensional derivative pricing problem, whereby we reduce 2 the effective dimension by applying the Brownian bridge method (BB) and the Princi- pal Component Analysis (PCA) to reduce the effective time and underlying stock di- mension. We focus our analysis on the convergence properties and numerical stability by applying the different techniques as well as to develop a path construction method which takes both effective dimension reduction techniques into account.
Load more

Recommended publications
  • Regression-Based Monte Carlo for Pricing High-Dimensional American-Style Options
    Regression-Based Monte Carlo For Pricing High-Dimensional American-Style Options Niklas Andersson [email protected] Ume˚aUniversity Department of Physics April 7, 2016 Master's Thesis in Engineering Physics, 30 hp. Supervisor: Oskar Janson ([email protected]) Examiner: Markus Adahl˚ ([email protected]) Abstract Pricing different financial derivatives is an essential part of the financial industry. For some derivatives there exists a closed form solution, however the pricing of high-dimensional American-style derivatives is still today a challenging problem. This project focuses on the derivative called option and especially pricing of American-style basket options, i.e. options with both an early exercise feature and multiple underlying assets. In high-dimensional prob- lems, which is definitely the case for American-style options, Monte Carlo methods is advan- tageous. Therefore, in this thesis, regression-based Monte Carlo has been used to determine early exercise strategies for the option. The well known Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) has been implemented and compared to Robust Regression Monte Carlo (RRM) by C.Jonen (2011). The difference between these methods is that robust regression is used instead of least square regression to calculate continuation values of American style options. Since robust regression is more stable against outliers the result using this approach is claimed by C.Jonen to give better estimations of the option price. It was hard to compare the techniques without the duality approach of Andersen and Broadie (2004) therefore this method was added. The numerical tests then indicate that the exercise strategy determined using RRM produces a higher lower bound and a tighter upper bound compared to LSM.
    [Show full text]
  • FX BASKET OPTIONS - Approximation and Smile Prices - ‌
    M.Sc. Thesis FX BASKET OPTIONS - Approximation and Smile Prices - ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ Author Patrik Karlsson, [email protected] Supervisors Biger Nilsson (Department of Economics, Lund University) Abstract Pricing a Basket option for Foreign Exchange (FX) both with Monte Carlo (MC) techniques and built on different approximation techniques matching the moments of the Basket option. The thesis is built on the assumption that each underlying FX spot can be represented by a geometric Brownian motion (GBM) and thus have log normally distributed FX returns. The values derived from MC and approximation are thereafter priced in a such a way that the FX smile effect is taken into account and thus creating consistent prices. The smile effect is incorporated in MC by assuming that the risk neutral probability and the Local Volatility can be derived from market data, according to Dupire (1994). The approximations are corrected by creating a replicated portfolio in such a way that this replicated portfolio captures the FX smile effect. ‌ ‌ Sammanfattning (Swedish) Prissättning av en Korgoption för valutamarknaden (FX) med hjälp av både Monte Carlo-teknik (MC) och approximationer genom att ta hänsyn till Korgoptionens moment. Vi antar att varje underliggande FX-tillgång kan realiseras med hjälp av en geometrisk Brownsk rörelse (GBM) och därmed har lognormalfördelade FX-avkastningar. Värden beräknade mha. MC och approximationerna är därefter korrigerade på ett sådant sätt att volatilitetsleendet för FX-marknaden beaktas och därmed skapar konsistenta optionspriser. Effekten av volatilitetsleende överförs till MC-simuleringarna genom antagandet om att den risk neutral sannolikheten och den lokala volatilitet kan härledas ur aktuell marknadsdata, enligt Dupire (1994).
    [Show full text]
  • Pricing of Arithmetic Basket Options by Conditioning 3 Sometimes to Poor Results
    PRICING OF ARITHMETIC BASKET AND ASIAN BASKET OPTIONS BY CONDITIONING G. DEELSTRAy,∗ Universit´eLibre de Bruxelles J. LIINEV and M. VANMAELE,∗∗ Ghent University Abstract Determining the price of a basket option is not a trivial task, because there is no explicit analytical expression available for the distribution of the weighted sum of the assets in the basket. However, by conditioning the price processes of the underlying assets, this price can be decomposed in two parts, one of which can be computed exactly. For the remaining part we first derive a lower and an upper bound based on comonotonic risks, and another upper bound equal to that lower bound plus an error term. Secondly, we derive an approximation by applying some moment matching method. Keywords: basket option; comonotonicity; analytical bounds; moment matching; Asian basket option; Black & Scholes model AMS 2000 Subject Classification: Primary 91B28 Secondary 60E15;60J65 JEL Classification: G13 This research was carried out while the author was employed at the Ghent University y ∗ Postal address: Department of Mathematics, ISRO and ECARES, Universit´eLibre de Bruxelles, CP 210, 1050 Brussels, Belgium ∗ Email address: [email protected], Tel: +32 2 650 50 46, Fax: +32 2 650 40 12 ∗∗ Postal address: Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281, building S9, 9000 Gent, Belgium ∗∗ Email address: [email protected], [email protected], Tel: +32 9 264 4895, Fax: +32 9 264 4995 1 2 G. DEELSTRA, J. LIINEV AND M. VANMAELE 1. Introduction One of the more extensively sold exotic options is the basket option, an option whose payoff depends on the value of a portfolio or basket of assets.
    [Show full text]
  • A Stochastic and Local Volatility Pricing Framework
    Multi-asset derivatives: A Stochastic and Local Volatility Pricing Framework Luke Charleton Department of Mathematics Imperial College London A thesis submitted for the degree of Master of Philosophy January 2014 Declaration I hereby declare that the work presented in this thesis is my own. In instances where material from other authors has been used, these sources have been appropri- ately acknowledged. This thesis has not previously been presented for other MPhil. examinations. 1 Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Abstract In this thesis, we explore the links between the various volatility modelling concepts of stochastic, implied and local volatility that are found in mathematical finance. We follow two distinct routes to compute new terms for the representation of stochastic volatility in terms of an equivalent local volatility. In addition to this, we discuss a framework for pricing multi-asset options under stochastic volatility models, making use of the local volatility representations derived earlier in the thesis. Previous ap- proaches utilised by the quantitative finance community to price multi-asset options have relied heavily on numerical methods, however we focus on obtaining a semi- analytical solution by making use of approximation techniques in our calculations, with the aim of reducing the time taken to price such financial instruments.
    [Show full text]
  • Stock Basket Option Tutorial | Finpricing
    Equity Basket Option Pricing Guide FinPricing Equity Basket Summary ▪ Equity Basket Option Introduction ▪ The Use of Equity Basket Options ▪ Equity Basket Option Payoffs ▪ Valuation ▪ Practical Guide ▪ A Real World Example Equity Basket Equity Basket Option Introduction ▪ A basket option is a financial contract whose underlying is a weighted sum or average of different assets that have been grouped together in a basket. ▪ A basket option can be used to hedge the risk exposure to or speculate the market move on the underlying stock basket. ▪ Because it involves just one transaction, a basket option often costs less than multiple single options. ▪ The most important feature of a basket option is its ability to efficiently hedge risk on multiple assets at the same time. ▪ Rather than hedging each individual asset, the investor can manage risk for the basket, or portfolio, in one transaction. ▪ The benefits of a single transaction can be great, especially when avoiding the costs associated with hedging each and every individual component. Equity Basket The Use of Basket Options ▪ A basket offers a combination of two contracdictory benefits: focus on an investment style or sector, and diversification across the spectrum of stocks in the sector. ▪ Buying a basket of shares is an obvious way to participate in the anticipated rapid appreciation of a sector, without active management. ▪ An investor bullish on a sectr but wanting downside protection may favor a call option on a basket of shares from that sector. ▪ A trader who think the market overestimates a basket’s volatility may sell a butterfly spread on the basket.
    [Show full text]
  • Local Volatility FX Basket Option on CPU and GPU
    www.nag.co.uk Local Volatility FX Basket Option on CPU and GPU Jacques du Toit1 and Isabel Ehrlich2 Abstract We study a basket option written on 10 FX rates driven by a 10 factor local volatility model. We price the option using Monte Carlo simulation and develop high performance implementations of the algorithm on a top end Intel CPU and an NVIDIA GPU. We obtain the following performance figures: Price Runtime(ms) Speedup CPU 0.5892381192 7,557.72 1.0x GPU 0.5892391192 698.28 10.83x The width of the 98% confidence interval is 0.05% of the option price. We confirm that the algorithm runs stably and accurately in single precision, and this gives a 2x performance improvement on both platforms. Lastly, we experiment with GPU texture memory and reduce the GPU runtime to 153ms. This gives a price with an error of 2.60e-7 relative to the double precision price. 1 Introduction Financial markets have seen an increasing number of new derivatives and options available to trade. As these options are ever more complex, closed form solutions for pricing are often not readily available and we must rely on approximations to obtain a fair price. One way in which we can accurately price options without a closed form solution is by Monte Carlo simulation, however this is often not the preferred method of pricing due to the fact that it is computationally demanding (and hence time-consuming). The advent of massively parallel computer hardware (multicore CPUs with AVX, GPUs, Intel Xeon Phi also known as Intel Many Integrated Core3 or MIC) has created renewed interest in Monte Carlo methods, since these methods are ideally suited to the hardware.
    [Show full text]
  • Basket Options and Implied Correlations: a Closed Form Approach
    Basket options and implied correlations: a closed form approach Svetlana Borovkova Free University of Amsterdam CFC conference, London, January 17-18, 2007 1 • Basket option: option whose underlying is a basket (i.e. a portfolio) of assets. + • Payoff of a European call basket option: ( B ( T ) − X ) B(T) is the basket value at the time of maturity T , X is the strike price. 2 Commodity baskets • Crack spreads: Qu * Unleaded gasoline + Qh * Heating oil - Crude • Soybean crush spread: Qm * Soybean meal + Qo * Soybean oil - Soybean • Energy company portfolios: Q1 * E1 + Q2 * E2 + … + Qn En, where Qi’s can be positive as well as negative. 3 Motivation: • Commodity baskets consist of two or more assets with negative portfolio weights (crack or crush spreads), Asian-style.\ • The valuation and hedging of basket (and Asian) options is challenging because the sum of lognormal r.v.’s is not lognormal. • Such baskets can have negative values, so lognormal distribution cannot be used, even in approximation. • Most existing approaches can only deal with baskets with positive weights or spreads between two assets. • Numerical and Monte Carlo methods are slow, do not provide closed formulae. 4 Our approach: • Essentially a moment-matching method. • Basket distribution is approximated using a generalized family of lognormal distributions : regular, shifted, negative regular or negative shifted. The main attractions: • applicable to baskets with several assets and negative weights, easily extended to Asian-style options • allows to apply Black-Scholes formula • provides closed form formulae for the option price and greeks 5 Regular lognormal, shifted lognormal and negative regular lognormal 6 Assumptions: • Basket of futures on different (but related) commodities.
    [Show full text]
  • Basket Option Pricing and Implied Correlation in a Lévy Copula Model
    Basket option pricing and implied correlation in a Levy´ copula model Daniel¨ Linders∗ Wim Schoutensy Version: January 10, 2015 Abstract In this paper we employ the Levy´ copula model to determine basket option prices. More precisely, basket option prices are determined by replacing the distribution of the real basket with an appropriate approximation. For the approximate basket we determine the underlying characteristic function and hence we can derive the related basket option prices by using the Carr-Madan formula. Two approaches are considered. In the first approach, we replace the arithmetic sum by an appropriate geometric average, whereas the second approach can be considered as a three-moments-matching method. Numerical examples illustrate the accuracy of our approximations; several Levy´ models are calibrated to market data and basket option prices are determined. In a last part we show how our newly designed basket option pricing formula can be used to define implied Levy´ correlation by matching model and market prices for basket op- tions. Our main finding is that the implied Levy´ correlation smile is flatter than its Gaussian counterpart. Furthermore, if (near) at-the-money option prices are used, the corresponding implied Gaussian correlation estimate is a good proxy for the implied Levy´ correlation. Keywords: basket options, characteristic function, implied correlation, Levy´ market, Variance-Gamma. 1 Introduction Nowadays, an increased volume of multi-asset derivatives is traded. An example of such a derivative is a basket option. The basic version of such a multivariate product has the same characteristics as a vanilla option, but now the underlying is a basket of stocks instead of a single stock.
    [Show full text]
  • Spread and Basket Option Pricing: an Application to Interconnected Power Markets
    University of Milano-Bicocca Department of Quantitative Methods for Economics and Business Doctoral Thesis in Mathematics for Financial Markets Analysis Spread and Basket Option Pricing: an Application to Interconnected Power Markets Author: Supervisor: Ruggero Caldana Gianluca Fusai Milan, October 2012 Dedicated to my family. Abstract. An interconnector is an asset that gives the owner the right, but not the obligation, to transmit electricity between two locations each hour of the day over a prefixed time period. The financial value of the interconnector is given by a series of options that are written on the price differential between two electricity markets, that is, a strip of European options on an hourly spread. Since the hourly forward price is not directly observable on the market, Chapter 1 proposes a practical procedure to build an hourly forward price curve, fitting both base load and peak load forward quotations. One needs a stochastic model, a valuation formula, and a calibra- tion method to evaluate interconnection capacity contracts. To capture the main features of the electricity price series, we model the energy price log-returns for each hour with a non-Gaussian mean-reverting stochastic process. Unfortunately no explicit solution to the spread option val- uation problem is available. Chapter 2 develops a method for pricing the generic spread option in the non-Gaussian framework by extending the Bjerksund and Stensland (2011) approximation to a Fourier transform framework. We also obtain an upper bound on the estimation error. The method is applicable to models in which the joint characteristic function of the underlying assets is known analytically.
    [Show full text]
  • The Pricing of Basket Options: a Weak Convergence Approach
    The Pricing of Basket Options: A Weak Convergence Approach Lijun Bo ∗ Yongjin Wang y Abstract We consider a limit price of basket options in a large portfolio where the dynamics of basket assets is described as a CEV jump diffusion system. The explicit representation of the limiting price is established using weak convergence of empirical measure valued processes generated by the system. As an application, the closed-form formula of the limit price is derived when the price dynamics of basket assets follows a mixed-double exponential jump-diffusion system. AMS 2000 subject classifications: 3E20, 60J20. Keywords and phrases: Basket options, Jump diffusion, Weak convergence. 1 Introduction A basket option is an option whose payoff is a weighted sum or average of prices of the two or more risky assets that have been grouped together in a basket at maturity. In general, it is difficult to price basket options explicitly since the joint distribution of the underlying basket asset price process are unknown due to multi-dimensionality, in particular for the multi-dimensional jump-diffusion basket asset price dynamics. Instead, some works have focused on the development of fast and accurate approximation techniques and establishment of sharp lower and upper bounds for basket option prices, see [12] therein they review and compare six different methods for valuing basket options in a systematic way, and discuss the influence of model parameters on the performance of the different approximations. Recently [14] apply the asymptotic expansion method to find the approximating value of the lower bound of European style basket call option prices for a local volatility jump-diffusion model.
    [Show full text]
  • Hedging Basket Options by Using a Subset of Underlying Assets
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Su, Xia Working Paper Hedging Basket Options by Using a Subset of Underlying Assets Bonn Econ Discussion Papers, No. 14/2006 Provided in Cooperation with: Bonn Graduate School of Economics (BGSE), University of Bonn Suggested Citation: Su, Xia (2006) : Hedging Basket Options by Using a Subset of Underlying Assets, Bonn Econ Discussion Papers, No. 14/2006, University of Bonn, Bonn Graduate School of Economics (BGSE), Bonn This Version is available at: http://hdl.handle.net/10419/22959 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons
    [Show full text]
  • Energy Markets I: First Models
    Energy Markets I: First Models Rene´ Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Banff, May 2007 Carmona Energy Markets, Banff Plan of the Course Commodity Markets Production, Transportation, Storage, Delivery Spot / Forward Markets Spread Option Valuation Why Spread Options First Asset Valuation Gas and Power Markets Physical / Financial Contracts Price Formation Load and Temperature Weather Markets Weather Exposure Temperature Options More Asset Valuation Plant Optionality Valuation Financial Valuation Valuing Storage Facilities Emission Markets Carmona Energy Markets, Banff Deregulated Electiricty Markets No More Utilities monopolies Vertical Integration of production, transportation, distribution of electricity Unbundling Open competitive markets for production and retail (Typically, grid remains under control) New Price Formation Constant supply - demand balance (Market forces) Commodities form a separate asset class! LOCAL STACK – MERIT ORDER (plant on the margin) Carmona Energy Markets, Banff Role of Financial Mathematics & Financial Engineering Support portfolio management (producer, retailer, utility, investment banks, ...) Different data analysis (spot, day-ahead, on-peak, off-peak, firm, non-firm, forward,··· , negative prices) New instrument valuation (swing / recall / take-or-pay options, weather and credit derivatives, gas storage, cross commodity derivatives, .....) New forms of hedging using physical assets Perfected by GS & MS (power plants, pipelines,
    [Show full text]