The foreign exchange market comprises the largest global volume, so the pricing of foreign exchange options has always been a hot issue in the foreign exchange market. This paper treats the exchange ...rate as an uncertain process that is described by an uncertain fractional differential equation, and establishes a new uncertain fractional currency model. The uncertain process is driven by Liu process, and, with the application of the Mittag-Leffler function, the solution of the fractional differential equation in a Caputo sense is presented. Then, according to the uncertain fractional currency model, the pricing formulas of European and American currency options are given. Lastly, the two numerical examples of European and American currency options are given; the price of the currency option increased when p changed from 1.0 to 1.1, and prices with different p were all decreasing functions of exercise price K.
There exist some non-stochastic factors in the financial market, so the dynamics of the exchange rate highly depends on human uncertainty. This paper investigates the pricing problems of foreign ...currency options under the uncertain environment. First, we propose an currency model under the assumption that exchange rate, volatility, domestic interest rate and foreign interest rate are all driven by uncertain differential equations; especially, the exchange rate exhibits mean reversion. Since the analytical solutions of nested uncertain differential equations cannot always be obtained, we design a new numerical method, Runge–Kutta-99 hybrid method, for solving nested uncertain differential equations. The accuracy of the designed numerical method is investigated by comparison with the analytical solution. Subsequently, the quasi-closed-form solutions are derived for the prices of both European and American foreign currency options. Finally, in order to illustrate the rationality and the practicability of the proposed currency model, we design several numerical algorithms to calculate the option prices and analyze the price behaviors of foreign currency options across strike price and maturity.
Garman – Kohlhagen framework, which is an extension of the most popular Black-Scholes-Merton model, is often used by financial institutions in order to price options with a currency as underlying. ...These pricing techniques have in common the definition of a partial differential equation used for the definition of the future value of the derivative, called Fundamental PDE. The financial instruments characterization depends on the derivative pay-off and it is realized through the specification of the initial conditions (IC) and the Dirichlet’s boundary conditions (BC). For standard contracts, called plain-vanilla derivatives, and for a few class of non-standard instruments, called exotic derivatives, this problem can be solved analytically reaching a theoretical fair value through a closed formula (CF) valuation, otherwise a numerical method must be used. Classical numerical integration schemes, which have been implemented for this purpose, are Finite Difference Method (FDM) and Finite Elements Method (FEM). In the last ten years, financial engineering has focused on an innovative methodology for option pricing which has its foundations on Radial Basis Functions (RBF). This paper aims to examine how this technique works in the financial field and how this method can be applied to the fair-value determination of vanilla and exotic Forex options.
This article examines currency option pricing within a credible target zone arrangement where interventions at the boundaries push the exchange rate back into its fluctuation band. Valuation of such ...options is complicated by the requirement that the reflection mechanism should prevent the arbitrage opportunities that would arise if the exchange rate were to spend finite time on the boundaries. To prevent the latter, we superimpose instantaneously reflecting boundaries upon the familiar geometric Brownian motion (GBM) framework. We derive closed-form expressions for European call and put option prices and show that prices for the GBM model of Garman and Kohlhagen (
1983
) arise as the limit case for infinitely wide bands. We also illustrate that taking account of boundaries is of considerable economic value as erroneously using the unbounded-domain model of Garman and Kohlhagen (
1983
) easily overprices options by more than 100%.
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Barrier options are a class of highly path-dependent exotic options which present particular challenges to practitioners in all areas of the financial industry. They are traded heavily as ...stand-alone contracts in the Foreign Exchange (FX) options market, their trading volume being second only to that of vanilla options. The FX options industry has correspondingly shown great innovation in this class of products and in the models that are used to value and risk-manage them. FX structured products commonly include barrier features, and in order to analyse the effects that these features have on the overall structured product, it is essential first to understand how individual barrier options work and behave. FX Barrier Options takes a quantitative approach to barrier options in FX environments. Its primary perspectives are those of quantitative analysts, both in the front office and in control functions. It presents and explains concepts in a highly intuitive manner throughout, to allow quantitatively minded traders, structurers, marketers, salespeople and software engineers to acquire a more rigorous analytical understanding of these products. The book derives, demonstrates and analyses a wide range of models, modelling techniques and numerical algorithms that can be used for constructing valuation models and risk-management methods. Discussions focus on the practical realities of the market and demonstrate the behaviour of models based on real and recent market data across a range of currency pairs. It furthermore offers a clear description of the history and evolution of the different types of barrier options, and elucidates a great deal of industry nomenclature and jargon.
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There are no other books which focus on the topic of barrier options, despite it being a topic of great interest to many market practitioners, especially in FX The FX derivatives market is huge – the largest derivatives market in the world, and there is always interest in new materials on pricing of financial instruments in this space. This book provides a thorough treatment of a very under published aspect of FX – barrier options Through his work as a practitioner and trainer, the author is well placed to describe the latest best practice as well as historic approaches clearly with a balance of words, diagrams, graphs, mathematics and programming code. - The first book to analyse FX barrier options, an important and commonly traded class of exotic option, frequently traded (but not always understood) on the largest derivatives market in the world – FX
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Preface
Foreword
Glossary of Mathematical Notation
Contract Types
1 Meet the Products
1.1 Spot
1.1.1 Dollars per euro or euros per dollar?
1.1.2 Big figures and small figures
1.1.3 The value of Foreign
1.1.4 Converting between Domestic and Foreign
1.2 Forwards
1.2.1 The FX forward market
1.2.2 A formula for the forward rate
1.2.3 Payoff a forward contract
1.2.4 Valuation of a forward contract
1.3 Vanilla options 1.3.1 Put-Call Parity
1.4 Barrier-contingent vanilla options
1.5 Barrier-contingent payments
1.6 Rebates
1.7 Knock-in-knock-out (KIKO) options
1.8 Types of barriers
1.9 Structured products
1.10 Specifying the contract
1.11 Quantitative truisms
1.11.1 Foreign exchange symmetry and inversion
1.11.2 Knock-out plus knock-in equals no-barrier contract
1.11.3 Put-call parity
1.12 Jargon-buster
2 Living in a Black-Scholes World
2.1 The Black-Scholes model equation for spot price
2.2 The process for ln S
2.3 The Black-Scholes equation for option pricing
2.3.1 The lagless approach
2.3.2 Derivation of the Black-Scholes PDE
2.3.3 Black-Scholes model | hedging assumptions
2.3.4 Interpretation of the Black-Scholes PDE
2.4 Solving the Black-Scholes PDE
2.5 Payments
2.6 Forwards
2.7 Vanilla options
2.7.1 Transformation of the Black-Scholes PDE
2.7.2 Solution of the diffusion equation for vanilla options
2.7.3 The vanilla option pricing formulae
2.7.4 Price quotation styles
2.7.5 Valuation behaviour
2.8 Black-Scholes pricing of barrier-contingent vanilla options
2.8.1 Knock-outs
2.8.2 Knock-ins
2.8.3 Quotation methods
2.8.4 Valuation behaviour
2.9 Black-Scholes pricing of barrier-contingent payments
2.9.1 Payment in Domestic
2.9.2 Payment in Foreign
2.9.3 Quotation methods
2.9.4 Valuation behaviour
2.10 Discrete barrier options
2.11 Window barrier options
2.12 Black-Scholes numerical valuation methods
3 Black-Scholes Risk Management
3.1 Spot risk
3.1.1 Local spot risk analysis
3.1.2 Delta
3.1.3 Gamma
3.1.4 Results for spot Greeks
3.1.5 Non-local spot risk analysis
3.2 Volatility risk
3.2.1 Local volatility risk analysis
3.2.2 Non-local volatility risk
3.3 Interest rate risk
3.4 Theta
3.5 Barrier over-hedging
3.6 Co-Greeks
4 Smile Pricing
4.1 The shortcomings of the Black-Scholes model
4.2 Black-Scholes with term structure (BSTS)
4.3 The implied volatility surface
4.4 The FX vanilla option market
4.4.1 At-the-money volatility
4.4.2 Risk reversal
4.4.3 Buttery
4.4.4 The role of the Black-Scholes model in the FX vanilla options market
4.5 Theoretical Value (TV)
4.5.1 Conventions for extracting market data for TV calculations
4.5.2 Example broker quote request
4.6 Modelling market implied volatilities
4.7 The probability density function
4.8 Three things we want from a model
4.9 The local volatility (LV) model
4.9.1 It's the smile dynamics, stupid
4.10 Five things we want from a model
4.11 Stochastic volatility (SV) models
4.11.1 SABR model
4.11.2 Heston model
4.12 Mixed local/stochastic volatility (lsv) models
4.12.1 Term structure of volatility of volatility
4.13 Other models and methods
4.13.1 Uncertain Volatility (UV) models
4.13.2 Jump-diffusion models
4.13.3 Vanna-volga methods
5 Smile Risk Management
5.1 Black-Scholes with term structure
5.2 Local volatility model
5.3 Spot risk under smile models
5.4 Theta risk under smile models
5.5 Mixed local/stochastic volatility models
5.6 Static hedging
5.7 Managing risk across businesses
6 Numerical Methods
6.1 Finite-difference (FD) methods
6.1.1 Grid geometry
6.1.2 Finite-difference schemes
6.2 Monte Carlo (MC) methods
6.2.1 Monte Carlo schedules
6.2.2 Monte Carlo algorithms
6.2.3 Variance reduction
6.2.4 The Brownian Bridge
6.2.5 Early termination
6.3 Calculating Greeks
6.3.1 Bumped Greeks
6.3.2 Greeks from finite-difference calculations
6.3.3 Greeks from Monte Carlo
7 Further Topics
7.1 Managed currencies
7.2 Stochastic interest rates (SIR)
7.3 Real-world pricing
7.3.1 Bid-offer spreads
7.3.2 Rules-based pricing methods
7.4 Regulation and market abuse
A Derivation of the Black-Scholes Pricing Equations for Vanilla Options
B Normal and lognormal probability distributions
B.1 Normal distribution
B.2 Lognormal distribution
C Derivation of the local volatility function
C.1 Derivation in terms of call prices
C.2 Local volatility from implied volatility
C.3 Working in moneyness space
C.4 Working in log space
C.5 Specialization to BSTS
D Calibration of mixed local/stochastic volatility (LSV) models
E Derivation of Fokker-Planck equation for the local volatility model
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'FX Barrier Options are the subject of more in-depth study by practitioners than almost any other class of exotic options and yet they have been given relatively short shrift in the literature until now. Zareer Dadachanji's book brilliantly fills this gap. Readers are led gently but thoroughly from the basics to the state of the art with ample discussion throughout (and full mathematical details supplied in appendices). Highly recommended for beginners and experts alike.'
— Ben Nasatyr, Head of FX Quantitative Analysis, Citigroup 'Zareer Dadachanji's book on FX barrier options is clear, precise, and a pleasure to read. The derivations are as simple as possible while remaining correct, and the book displays a judicious blend of theory, modelling and practice. Students and practitioners will learn a lot (and not just about FX barrier options), and will do so with pleasure.'
— Riccardo Rebonato, Global Head of Rates and FX Analytics, PIMCO, and Visiting Lecturer, Mathematical Finance, Oxford University 'The market in FX barrier options has grown from a niche to the most liquid exotics market in the world, requiring models that are both very sophisticated and computationally efficient. The first of its kind, Dr Dadachanji's treatise is exclusively dedicated to the subject. The book requires few prerequisites but quickly builds to the state of the art in a clear and comprehensive manner. Undoubtedly it will be an indispensable companion to anyone involved in the subject or interested in learning it.'
— Vladimir Piterbarg, Head of Quantitative Analytics at Rokos Family Office 'This is the book I wish I'd had when I started my career as an FX quant – an insider's view of FX barrier option modelling from both a theoretical and practical perspective. It builds from the basic market set-up through to the latest techniques in an FX quant's toolkit.'
— Mark Jex, FX Quant in investment banks for 20 years, and pioneer of the mixed local/ stochastic volatility model 'As one who understands financial engineering from the trader's perspective as well as from the quant's, Zareer Dadachanji has written a very valuable book on FX barrier options. Requiring very little pre-requisite financial knowledge, it guides readers through the plethora of quantitative concepts, techniques and practical issues associated with these products. And it is an enjoyable read to boot.'
— Simon Hards, Global Head of FX Trading at Credit Suisse
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The first book to analyse FX barrier options, frequently traded (but not always understood) on the largest derivatives market in the world – FX
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Zareer Dadachanji is a quantitative analysis consultant with nearly two decades of corporate experience, mostly in financial quantitative modelling across a range of asset classes. He has spent 14 years working as a front-office quant at banks and hedge funds, including NatWest/RBS, Credit Suisse and latterly Standard C
Based on an uncertain currency model, put currency option pricing problem is discussed. Furthermore, we derive the pricing formulas of European and American put currency option. Meanwhile, this paper ...analyzes the relationship between pricing formulas and the relevant parameters. At last, two numerical examples presented in this paper illustrate the pricing formulas.
Technological innovation has changed the financial market significantly with the increasing application of high-frequency data in research and practice. This study examines the performance of ...intraday implied volatility (IV) in estimating currency options prices. Options quotations at a different trading time, such as the opening period, midday period and closing period of a trading day with one-month, two months’ and three months’ maturity, are employed to compute intraday IV for pricing currency options. We use the Mincer–Zarnowitz regression test to analyse the volatility forecast power of IV for three different forecast horizons (within a week, one week and one month). Intraday IV’s capability in estimating currency options price is measured by the mean squared error, mean absolute error and mean absolute percentage error measure. The empirical findings show that intraday IV is the key to accurately forecasting volatility and estimating currency options prices precisely. Moreover, IV at the closing period of the beginning of the week contains crucial information for options price estimation. Furthermore, the shorter maturity intraday IV is suitable for pricing options for a shorter horizon. In comparison, the intraday IV based on the longer maturity options subsumes appropriate information to price options with higher accuracy for the longer horizon. Our paper proposes a new approach to accurately pricing currency options using high-frequency data.
This paper presents an efficient currency option pricing model based on support vector regression (SVR). This model focuses on selection of input variables of SVR. We apply stochastic volatility ...model with jumps to SVR in order to account for sudden big changes in exchange rate volatility. We use forward exchange rate as the input variable of SVR, since forward exchange rate takes interest rates of a basket of currencies into account. Therefore, the inputs of SVR will include moneyness (spot rate/strike price), forward exchange rate, volatility of the spot rate, domestic risk-free simple interest rate, and the time to maturity. Extensive experimental studies demonstrate the ability of new model to improve forecast accuracy.
This paper introduces the intra-daily implied volatility (IDIV), a new volatility measure to price currency option accurately. The IDIV is developed based on the implied volatility estimated on ...equally spaced intra-daily intervals. This model captures the intra-daily level aggregate information related to foreign exchange (FX) behavior, which changes every five minutes. The implied volatility (IV) and realized volatility (RV) are widely accepted as good estimates of daily and intra-daily price volatility, respectively. Therefore, using the options pricing framework, we assess the capability of IDIV against IV and RV in pricing foreign currency options. A comparison of out-of-sample forecasts under both the F-test and Diebold-Mariano test reveals that the IDIV outperforms both the IV and the RV in estimating one-day-ahead option prices. In other words, the IDIV estimation framework provides a more accurate and efficient volatility estimate for pricing currency options. The findings of this study indicate that the forward looking intra-daily information of IDIV is appropriate to price options correctly rather than forward looking daily and historical intra-daily information is obtained by the IV and RV, respectively.
The implied volatility (IV) estimation process suffers from an obvious chicken-egg dilemma: obtaining an unbiased IV requires the options to be priced correctly and calculating an accurate option ...price (OP) requires an unbiased IV. We address this critical issue in two steps. First, the Granger causality test is employed, which confirms the chicken-and-egg problem in the IV computing process. Secondly, the concept of "moneyness volatility (MV)" is introduced as an alternative to IV. MV is modelled based on an option's moneyness (OM) during the life of the option's contract. The F-test, Granger-Newbold test and Diebold-Mariano test results consistently show that MV outperforms IV in estimating the exchange rate volatility for pricing options. Further, these series of tests across six major currency options substantiate the validity as well as the reliability of the results. We posit that MV offers a unique solution for pricing currency options accurately. PUBLICATION ABSTRACT
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