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blog maintained by Matt Davey, who is director of Lab49. lots of interesting posts to find for implementing UI and ideas on architecture for trading applications eg., check out the cool statistical arbitrage UI in http://mdavey.wordpress.com/2011/01/11/statarb-user-experience-ux-part-8/ Discuss this paper
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Following are links to some nice software which will help understand quantitative trading or high frequency finance: 1. Tradelink at http://code.google.com/p/tradelink/ is an excellent open source code available. Good thing about it is it uses C# which has nice debugging support, so that you will be able to understand program flow better. You can view the startup guide on the tradelink site, or use the following link for a quick start: http://www.quantcode.com/modules/smartfaq/faq.php?faqid=93
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We present a new and general technique for obtaining closed form expansions for prices of options in the Heston model, in terms of Black-Scholes prices and Black-Scholes greeks up to arbitrary orders. We then apply the technique to solve, in detail, the cases for the second order and third order expansions. In particular, such expansions show how the convexity in volatility, measured by the Black-Scholes volga, and the sensitivity of delta with respect to volatility, measured by the Black-Scholes vanna, impact option prices in the Heston model. The general method for obtaining the expansion rests on the construction of a set of new probability measures, equivalent to the original pricing measure, and which retain the affine structure of the Heston volatility diffusion. Finally, we extend our method to the pricing of forward-starting options in the Heston model. Discuss this paper
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by GRAEME WEST ABSTRACT Recently the SABR model has been developed to manage the option smile which is observed in derivatives markets. Typically, calibration of such models is straightforward as there is adequate data available for robust extraction of the parameters required asinputs to the model. The paper considers calibration of the model in situations where input data is very sparse. Although this will require some creative decision making, the algorithms developed here are remarkably robust and can be used confidently for mark to market and hedging of option portfolios. KEY WORDS: SABR model, equity derivatives, volatility skew calibration, illiquid markets Discuss this paper
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slides and some example code by Paolo Foschi Table of Contents I 1. Binomial Model Mono-period binomial model Parameter Calibration Early exercise Computational and numerical issues 2 Finite Differences Discretization Explicit Method Implicit Method The Black and Scholes PDE Bi-dimensional equations 3 Monte Carlo Numerical Integration Monte Carlo Method Generating Random Variables Variance Reduction Techniques SDE Integration 4 Fourier Transform methods Exponential Damping Time Value Approach Discuss this paper
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by Damiano Brigo and Massimo Morini In this work we analyze market payoffs of Credit Default Swaps (CDS) and we derive rigorous standard market formulas for pricing options on CDS. Formulas are based on modelling CDS spreads which are consistent with simple market payoffs, and we introduce a subfiltration structure allowing all measures to be equivalent to the risk neutral measure. Then we investigate market CDS spreads through change of measure and consider possible choices of rates for modelling a complete term structure of CDS spreads. We also consider approximations and apply them to pricing of specific market contracts. Results are derived in a probabilistic framework similar to that of Jamshidian (2004).
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by Abel Elizalde Some investors in the Collateralized Debt Obligations (CDOs) market have been publicly accused of not fully understanding the risks and dynamics of these products. They won Discuss this paper
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Yinqiu Lu, Salih Neftci Abstract. We investigate the theoretical and empirical difference between the standard convexity adjustment and Forward Libor Model in a particular case of two-period Constant Maturity Swaps. Using daily data from 1991 to 1997, we simulate the di erence (spread) between the two-period CMS swap rates calculated by convexity adjustment and Forward Libor Model. The spread reaches 8.49 basis points in some cases, and correlation coecients between spread and one-year, two-year cap volatilities are 0.8750 and 0.7939, respectively. Moreover, convexity adjustment yields CMS swap rates higher than Forward Libor Model does. Since the pricing using Forward Libor Model would be exact, we conclude that the convexity adjustment overestimates CMS swap rates. In this paper, we simulate two-period CMS swap, and it is reasonable to believe that the spread will be much bigger for longer period CMS swap or other convex instrument.
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Christian L. Dunis and Richard Ho Traditional quantitative porfolio construction relies on the analysis of correlations between assets. Over the last 10 years, following the generalised use of JP Morgan (1994) RiskMetrics approach, quantitative portfolio managers have made a growing use of conditional correlations. If correlations are indeed time-varying, unfortunately their many changes make them in practise a difficult tool to use when managing quantitative portfolios, as the frequent rebalancing they imply may be very costly. In this paper, we use the concept of cointegration which relies on the long-term relationship between time series, and thus assets, to devise quantitative European equities portfolios in the context of two applications: a classic index tracking strategy and a long-short equity market neutral strategy. We use data from the Dow Jones EUROStoxx50 index and its constituent stocks from 4 January 1999 to 30 June 2003. Our results show that the designed portfolios are strongly cointegrated with the benchmark and indeed demonstrate good tracking performance. In the same vein, the long-short market neutral strategy generates steady returns under adverse market circumstances but, contrary to expectations, does not minimise volatility. Keywords: cointegration, index tracking, market neutral strategy, portfolio optimisation, vector autoregression models. Discuss this paper
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William C. Johnson I develop a new model to predict firm stock returns from firm bond yield information. The model uses a no-arbitrage argument to predict the bond value in default (recovery rate) and the expected stock return for the firm. The model has explanatory power comparable to the traditional CAPM, firm size, and firm book-to-market factors. The model implies that bond prices are cointegrated with stock prices for a firm and we provide empirical evidence that this is the case. We show that price discovery generally occurs in the bonds of a firm, not in the stock, consistent with bond traders being better informed than stock traders.
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PATRICK S. HAGAN Here we focus on a single class of deals, the constant maturity swaps, caps, and floors. We develop a framework that leads to the standard methodology for pricing these deals, and then use this framework to systematically improve the pricing. Discuss this paper
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Timo Salminen This thesis examines a forward rate market model which is used for simulating the development of multiple successive LIBOR forward rates. The main goal is to find the best available method for calibrating the model to current market expectations. Also the model plausibility for pricing swaption based products is studied. In the thesis different calibration methods Discuss this paper
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D. M. Pooley, K. R. Vetzaly, P. A. Forsythz Discontinuities in the payo function (or its derivatives) can cause inaccuracies for numerical schemes when pricing nancial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: averaging the initial data, shifting the grid, and a projection method. By themselves, these techniques are not sucient to restore expected behaviour. However, when combined with a special timestepping method, high accuracy is achieved. Examples are provided for one and two factor option pricing problems.
With a rich spectrum of maturities and tenors to contend with, the toughest aspect of pricing interest rate options is calibrating models of forward rates to market data. Here, Damiano Brigo and Fabio Mercurio present a scheme for simultaneously calibrating swaption volatilities to covariance parameters in the forward Libor model, reducing the need for Monte Carlo simulation Discuss this paper
These tutorials explain the C++ language from its basics up to the newest features of ANSI-C++, including basic concepts such as arrays or classes and advanced concepts such as polymorphism or templates. The tutorial is oriented in a practical way, with working example programs in all sections to start practicing each lesson right away. Discuss this paper
Abstract This paper develops closed-form solutions for options on credit spreads with GARCH models. We extend the mean-reverting model proposed in Longstaff and Schwartz (1995) and we use the Heston and Nandi's (1999) GARCH specification rather than the traditional lognormal. Our model, being more flexible, captures better the empirical properties of observed credit spreads and contains Longstaff and Schwartz (1995) model as a special case. GARCH coefficients are estimated using spread levels for corporate bonds.
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Lecturer. Harry Zheng, In this course we conduct a general survey of state-of-the-art research results and implementation techniques on credit risk modelling, valuation, and management. We discuss both structural models and intensity models. We also cover copula models and credit migration models. We illustrate how to use these models to price credit-related instruments (defaultable bonds, vulnerable options, credit default swaps). We also explain some industry-standard programs (CreditGrades, CreditMetrics).
Abstract This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.
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This article is taken from a supplement that originally appeared with the February issue of Risk magazine Alex Puaca is chairman of Dart, a derivatives software house and part of the Intercapital Group Alex@icap.com Discuss this paper
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Andreas J. Grau Peter A. Forsyth Kenneth R. Vetzal School of Computer Science University of Waterloo, Canada
Abstract In practice, convertible bonds can often be called only if notice is given to the olders. Most methods for valuing convertible bonds assume that the bond is continuously callable. In this paper, we develop an accurate PDE method for valuing convertible bonds with a finite notice period. Example computations are presented which illustrate the effect of varying notice periods. The results are compared with a recently published approximation method. Discuss this paper
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Oliver Chen A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Abstract The thesis contains the construction of a new class of pricing models for credit derivatives. The underlying stochastic process models the credit quality of individual obligors within the wider context of all firms that issue debt. The challenge is to reconcile a broad set of statistical and pricing information within a model where the main observables can be computed as integrals or sums of well-known functions, and the other observables are obtainable through efficient numerical schemes. In the first part, we use continuous processes that are based on square-root diffusion processes. Parameterized local volatility is introduced by a measure change and a coordinate transformation in such a way that the stochastic process retains integrability. Jumps in the process are introduced by subordinating on a random time-change. With this framework we are able to match empirical data on credit processes. Adding a drift enables us to introduce a measure change to the pricing measure to achieve consistency with market prices. In the second part, we discretise the stochastic process in such a way that node-tonode transition probabilities can be computed as sums of orthogonal polynomials. In this way, computing efficiency is increased while the empirical properties of the model remain intact. Lastly, we give applications of this credit model. Risk-neutral transition probabilities are calculated and compared with those calculated using previous models. Credit default ii swaps are priced. A mapping to equity in which the equity process is a martingale is constructed. This mapping to equity allows us to price, in particular, equity default swaps. Empirical results from the mapping indicate that the constant elasticity of variance (CEV) process would be an appropriate pure diffusion approximation to our process to price equity default swaps, and results using the CEV process are presented.
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by C. Albanese Imperial College, University of London
presented at the Second International Conference on Credit Risk, HEC Montral, Canada April 16th, 2004
Summary I discuss credit barrier models which can be fitted to both the credit transition matrix and credit spread curves under the real-world and risk-neutral measures, respectively. The underlying stochastic process is in a newly discovered class which is solvable in analytically closed form and is characterized by state dependent volatility and jumps and can accomodate stochastic volatility. This class extends at once most such models in the literature. I discuss the estimation procedure of the real-world measure based on the credit transition matrix and the calibration procedure for the risk neutral measure based on spread curve data. I present new results on implied credit migration rates and estimate liquidity convenience yields for forward yield spreads.
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Wolfram Boenkost Lucht Probst Associates GmbH, 60311 Frankfurt Wolfgang M. Schmidt Abstract Cross currency swaps are powerful instruments to transfer assets or liabilities from one currency into another. The market charges for this a liquidity premium, the cross currency basis spread, which should be taken into account by the valuation methodology. We describe and compare two valuation methods for cross currency swaps which are based upon using two different discounting curves. The first method is very popular in practice but inconsistent with single currency swap valuation methods. The second method is consistent for all swap valuations but leads to mark-to-market values for single currency off market swaps, which can be quite different to standard valuation results.
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Jian Wang Abstract The Heston model is a stochastic volatility model. We show that the option price in the Heston model is convex in the underlying asset for convex contract functions. We verify this using the explicit formula for European call options and extend to the general case using an approximation argument. Some other properties of the Heston model are also discussed. Finally, we illustrate the results using numerical methods.
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This article is submitted by Professor Yue Kuen KWOK, Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China for the Encyclopedia of Financial Engineering and Risk Management.
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