Introduction to Financial Mathematics Concepts and Computational Methods
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Term Structure Models of Commodity Prices
TERM STRUCTURE MODELS OF COMMODITY PRICES Cahier de recherche du Cereg n°2003–9 Delphine LAUTIER ABSTRACT. This review article describes the main contributions in the literature on term structure models of commodity prices. A first section is devoted to the theoretical analysis of the term structure. It confines itself primarily to the traditional theories of commodity prices and to their explanation of the relationship between spot and futures prices. The normal backwardation and storage theories are however a bit limited when the whole term structure is taken into account. As a result, there is a need for an extension of the analysis for long-term horizon, which constitutes the second point of the section. Finally, a dynamic analysis of the term structure is presented. Section two is centred on term structure models of commodity prices. The presentation shows that these models differ on the nature and the number of factors used to describe uncertainty. Four different factors are generally used: the spot price, the convenience yield, the interest rate, and the long-term price. Section three reviews the main empirical results obtained with term structure models. First of all, simulations highlight the influence of the assumptions concerning the stochastic process retained for the state variables and the number of state variables. Then, the method usually employed for the estimation of the parameters is explained. Lastly, the models’ performances, namely their ability to reproduce the term structure of commodity prices, are presented. Section four exposes the two main applications of term structure models: hedging and valuation. Section five resumes the broad trends in the literature on commodity pricing during the 1990s and early 2000s, and proposes futures directions for research. -
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. -
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). -
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. -
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. -
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. -
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. -
EC3070 FINANCIAL DERIVATIVES GLOSSARY Ask Price the Bid Price
EC3070 FINANCIAL DERIVATIVES GLOSSARY Ask price The bid price. Arbitrage An arbitrage is a financial strategy yielding a riskless profit and requiring no investment. It commonly amounts to the successive purchase and sale, or vice versa, of an asset at differing prices in different markets. i.e. it involves buying cheap and selling dear or selling dear and buying cheap. Bid A bid is a proposal to buy. A typical convention for vocalising a bid is “p for n”: p being the proposed unit price and n being the number of units or contracts demanded. Backwardation Backwardation describes a situation where the amount of money required for the future delivery of an item is lower than the amount required for immediate delivery. Backwardation is a signal that the item in question is in short supply. The opposite market condition to backwardation is known as contango, which is when the spot price is lower than the futures price. In fact, there is some ambiguity in the usage of the term. According to the definition above, backwardation is when Fτ|0 <S0, where Fτ|0 is the current price for a delivery at time τ and S0 is the current spot price. In an alternative definition, backwardation exists when Fτ|0 <E(Fτ|t) with 0 <t<τ, which is when the expected future price at a later date exceeds the futures price settled at time t = 0. In modern usage, this is called normal backwardation. Buyer A buyer is a long position holder who has agreed to accept the delivery of a commodity at some future date. -
Copyrighted Material
k Trim Size: 6in x 9in Sinclair583516 bindex.tex V1 - 05/04/2020 9:51 P.M. Page 219 INDEX Bid-ask spreads (total cost component), 200 A Bitcoin, 172–173, 172f Acceleration, 60 Black-Scholes-Merton (BSM) Adjustment repair, 86 assumptions/equation/model, 2–4, 6, Alpha decay, 15–16 Anchoring, 19 108, 113, 115, 183, 192 Arbitrage counterparty risk, 178–179 Black-Scholes-Merton (BSM) PDE, 5, 116 Arbitrage pricing theory (APT), 63 Bonds, 49–50 Bonferroni’s correction, usage, 23 k ASPX option profits, taxation, 187 219 k Asymmetric implied volatility skew, 191 Broken wing butterflies/condors, 99–100 ATM, 104, 106t, 123, 131t, 137 Butterflies, 95–100, 96f, 98t At-the-money covered call, delta level, 135 BXMD/BXM/BXY, 133, 134t Autocorrelation, 73f, 91 Availability heuristic, 19 C Average return, calculation, 24 Calendar spread, 100–102 Calls, 55f, 129–136, 129f, 132f, 142 B call spread, 130f, 131t, 142f, 143t Back-tested rules, performer sample, 23 implied volatility, level (example), 140t Backwardation, 61, 62 put-call parity/relationship, 95, 115, 116 Badly priced butterfly, returns (summary strike call, 118f, 120f statistics), 98t Capital asset pricing model (CAPM), Badly priced condor, returns (summary development, 63 statistics), 98t Cash flows, taxation rate, 6 Bankruptcy, risk (increase),COPYRIGHTED66 Catastrophe MATERIAL theory, 20 Bayesian model, usage, 30 Chicago Board Options Exchange (CBOE), 16, Behavioral biases, inefficiency, 67–68 36, 45–46, 133, 134f Behavioral finance, 16–21 CNDR index, 45, 45f Best, defining (possibility), 121 Commissions (total cost component), 200 Beta, 67–68, 68t, 82, 196 Commodities, 47–49, 48t, 49t BFLY index, 45, 46f Condors, 95–100, 96f, 97f, 98t, 104t Biases, 18–20, 23, 30, 113 Confidence, 61–62, 71–75, 80 k k Trim Size: 6in x 9in Sinclair583516 bindex.tex V1 - 05/04/2020 9:51 P.M. -
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. -
Essays in Risk Management for Crude Oil Markets
Essays in Risk Management for Crude Oil Markets by Abdullah Al Mansour A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Applied Economics Waterloo, Ontario, Canada, 2012 c Abdullah Al Mansour 2012 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract This thesis consists of three essays on risk management in crude oil markets. In the first essay, the valuation of an oil sands project is studied using real options approach. Oil sands production consumes substantial amount of natural gas during extracting and upgrading. Natural gas prices are known to be stochastic and highly volatile which introduces a risk factor that needs to be taken into account. The essay studies the impact of this risk factor on the value of an oil sands project and its optimal operation. The essay takes into account the co-movement between crude oil and natural gas markets and, accordingly, proposes two models: one incorporates a long-run link between the two markets while the other has no such link. The valuation problem is solved using the Least Square Monte Carlo (LSMC) method proposed by Longstaff and Schwartz (2001) for valuing American options. The valuation results show that incorporating a long-run relationship between the two markets is a very crucial decision in the value of the project and in its optimal operation. -
Momentum Strategies in Commodity Futures Markets
EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE 393-400 promenade des Anglais 06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 32 53 E-mail: [email protected] Web: www.edhec-risk.com Momentum Strategies in Commodity Futures Markets Joëlle Miffre Associate Professor of Finance, EDHEC Business School Georgios Rallis Cass Business School, Ph.D. Student ABSTRACT The article tests for the presence of short-term continuation and long-term reversal in commodity futures prices. While contrarian strategies do not work, the article identifies 13 profitable momentum strategies that generate 9.38% average return a year. A closer analysis of the constituents of the long-short portfolios reveals that the momentum strategies buy backwardated contracts and sell contangoed contracts. The correlation between the momentum returns and the returns of traditional asset classes is also found to be low, making the commodity-based relative-strength portfolios excellent candidates for inclusion in well-diversified portfolios. Keywords: Commodity futures, Momentum, Backwardation, Contango, Diversification JEL classification codes: G13, G14 Author for correspondence: Joëlle Miffre, Associate Professor of Finance, EDHEC Business School, 393 Promenade des Anglais, 06202, Nice, France, Tel: +33 (0)4 93 18 32 55, e-mail: [email protected] A version of this paper is forthcoming in the Journal of Banking and Finance. The authors would like to thank Chris Brooks and two anonymous referees for helpful comments. EDHEC is one of the top five business schools in France owing to the high quality of its academic staff (104 permanent lecturers from France and abroad) and its privileged relationship with professionals that the school has been developing since its establishment in 1906.