The Quantitative Finance Library

Valuing Convertible Bonds with Stock Price, Volatility, Interest Rate, and Default Risk , P. Kovalov, V. Linetsky (2008)
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Quasi-monte carlo methods and pseudo-random numbers , H. Niederreiter (1978)
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Advanced Monte Carlo methods for barrier and related exotic options In this work, we present advanced Monte Carlo techniques applied to the pricing of barrier options and other related exotic contracts. It covers in particular the Brown- ian bridge approaches, the barrier shifting techniques (BAST) and their extensions as well. We leverage the link between discrete and continuous monitoring to de- sign efficient schemes, which can be applied to the Black-Scholes model but also to stochastic volatility or Merton?s jump models. This is supported by theoretical results and numerical experiments. , E. Gobet (2008)
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High order discretization schemes for the CIR process: application to Affine Term Structure and Heston models , A.Alfonsi (2008)
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A Class of Levy Process Models with almost exact calibration of both barrier and vanilla FX options Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange. The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important of the FX Options market is that barrier options, especially Double-no-touch (DNT) options, are now so activley traded that they are o longer considered, in ay way, exotic options. Instead, traders would, in principle, like ot use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we indtoruce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The seocnd stage fits to vanilla options. The model allows for jumps (ether finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ``overshoot'. , P.Carr, J.Crosby (2009)
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Improved Long-Period Generators Based on Linear Recurrences Modulo 2 , F. Panneton, P. L Ecuyer, and M. Matsumoto (2006)
Nb Clicks : 9721

A Comparative Analysis of Basket Default Swaps Pricing Using the Stein Method , E.Benhamou, D.Bastide, M.Ciuca (2007)
Nb Clicks : 8960

Random Numbers for Simulation , P. L Ecuyer (1990)
Nb Clicks : 8687

Is Multi-Factor Really Necessary to Price European Options in Commodity? The main result of this article is the presentation of the Distribution Match Method. This method applies to a general multi-factor pricing model under assumption of normal law drift. The idea is to find an equivalent one-factor model for European options. The equivalent model admits a weak solution, which has the same one-dimensional marginal probability distribution. Moreover, the one-dimensional distribution can be explicitly calculated under certain condition. This result can consequently induct closed formula for the future price and European option price. We apply these results to two well known commodity models, the Gabillon and the Gibson Schwartz model, to provide the price for the future price and a closed formula for the European options. , E. Benhamou, Z. Wang, A.G. Galli (2009)
Nb Clicks : 8685

No-Dynamic-Arbitrage and Market Impact , J. Gatheral (2008)
Nb Clicks : 8493

An Objected-Oriented Random-Number Package with Many Long Streams and Substreams , P. L Ecuyer, R. Simard, E. J. Chen, and W. D. Kelton (2002)
Nb Clicks : 8458

Time Dependent Heston Model The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [Hes93] or piecewise constant [MN03]. Hence, using a small volatility of volatility expansion and Malliavin calculus techniques, we derive an accurate analytical formula for the price of vanilla options for any time dependent Heston model (the accuracy is less than a few bps for various strikes and maturities). In addition, we establish tight error estimates. The advantage of this approach over Fourier based methods is its rapidity (gain by a factor 100 or more), while maintaining a competitive accuracy. From the approximative formula, we also derive some corollaries related first to equivalentHestonmodels (extending some work of Piterbarg on stochastic volatility models [Pit05b]) and second, to the calibration procedure in terms of ill-posed problems. , E. Benhamou, E. Gobet, M. Miri (2009)
Nb Clicks : 8342

Hedging Complex Barrier Options We show how several complex barrier options can be hedged using a portfolio of standard European options. These hedging strategies only involve trading at a few times during the option?s life. Since rolling, ratchet, and lookback options can be decomposed into a portfolio of barrier options, our hedging results also apply , P. Carr, A. Chou (2002)
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Fine-tune your smile: Correction to Hagan et al , J.Obloj (2007)
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An Improved Implied Copula Model and its Application to the Valuation of Bespoke CDO Tranches , J.Hull, A.White (2008)
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Probability distribution of returns in the Heston model with stochastic volatility We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow Jones data for 1982-2001 follow the scaling function for seven orders of magnitude. , A. Dragulescu, V.M. Yakovenko (2002)
Nb Clicks : 8215

Dynamic Factor Copula Model an factor copula model is the market standard model for multi-name credit derivatives. Its main drawback is that factor copula models exhibit correlation smiles when calibrating against market tranche quotes. We introduce a multi-period factor copula model to overcome the calibration deficiency of factor copula models by allowing the factor loadings to be time-dependent. Usually, multi-period factor copula models require multi-dimensional integration, typically computed by Monte Carlo simulation, which makes calibration extremely time consuming. In our model, the portfolio loss of a completely homogeneous pool possesses the Markov property, thus we can compute the portfolio loss distribution analytically without multi-dimensional integration. Numerical results demonstrate the efficiency and flexibility of our model to match market quotes. , K.Jackson, A.Kreinin, W.Zhang (2009)
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Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems We introduce a class of stable and second-order accurate finite difference schemes that resolve a number of problems when approximating the solution of option pricing models in computational finance using the finite difference method (FDM). In particular, we show how to avoid having to apply truncation methods to the domain of integration as well as the resulting ad-hoc experimentation in choosing suitable numerical boundary conditions that we must apply on the boundary of the truncated domain. Instead, we transform the PDE option model (which is originally defiined on a semi-infinite interval) to a PDE that is defined on a bounded interval. We then apply the elegant Fichera theory to help us determine which boundary conditions to apply to the transformed PDE. We show that the new system is well-posed by proving energy inequalities in the space of square-integrable functions. Having done that, we adapt the Alternating Direction Explicit (ADE) (a method that dates from the first half of the last century) method to compute an approximation to the solution of the transformed PDE. We prove that the explicit scheme is unconditionally stable and second-order accurate. The scheme is very easy to model, to program and to parallelise and the generalization to n-factor problems is easily motivated. We examine equity problems and we compare the ADE approach (in terms of accuracy and speedup) with more traditional finite difference schemes and with the Monte Carlo method. Finally, we stress-test the scheme by varying critical parameters over a spectrum of values and the results are noted. The methods in this article - in particular the combination of pure and applied techniques - are relatively new in the computational finance literature in our opinion, in particular the realization that a PDE on a semi-infinite interval can be transformed to one on a bounded interval and that the new PDE can be approximated by schemes other than Crank-Nicolson and its workarounds. Finally, we shall report on n-factor models in future articles. , D.Duffy (2009)
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Not-so-complex logarithms in the Heston model , C.Kahl, P.Jackel (2006)
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Modern Logarithms for the Heston Model , I.Fahrner (2007)
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Collateral and Credit Issues in Derivatives Pricing Regulatory changes are increasing the importance of collateral agreements and credit issues in over-the-counter derivatives transactions. This paper considers the nature of derivatives collateral agreements and examines the impact of collateral agreements, two-sided credit risk, funding costs, liquidity, and bid-offer spreads on the valuation of derivatives portfolios. , John Hull and Alan White (2014)
Nb Clicks : 7807

Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model , A.V.Haastrecht, A.Pelsser (2008)
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Fast strong approximation Monte-Carlo schemes for stochastic volatility models , C.Kahl, P.Jaeckel (2006)
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Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Discounting and Forwarding Yield Curves In this paper we revisit the problem of pricing and hedging plain vanilla single-currency interest rate derivatives using different yield curves for market coherent estimation of discount factors and forward rates with different underlying rate tenors (e.g. Euribor 3 months, 6 months,.etc.). Within such double-curve-single-currency framework, adopted by the market after the liquidity crisis started in summer 2007, standard single-curve no arbitrage relations are no longer valid and can be formally recovered through the introduction of a basis adjustment. Numerical results show that the resulting basis adjustment curves may display an oscillating micro-term structure that may induce appreciable effects on the price of interest rate instruments. Recurring to the foreign-currency analogy we also derive no arbitrage double-curve market-like formulas for basic plain vanilla interest rate derivatives, FRAs, swaps, caps/floors and swaptions in particular. These expressions include a quanto adjustment typical of cross-currency derivatives, naturally originated by the change between the numeraires associated to the two yield curves, that carries on a volatility and correlation dependence. Numerical scenarios confirm that such correction can be non-negligible, thus making unadjusted double-curve prices, in principle, not arbitrage free. , M.Bianchetti (2009)
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Efficient Simulation of the Double Heston Model Stochastic volatility models have replaced Black-Scholes model since they are able to generate a volatility smile. However, standard models fail to capture the smile slope and level movements. The Double-Heston model provides a more flexible approach to model the stochastic variance. In this paper, we focus on numerical implementation of this model. First, following the works of Lord and Kahl, we correct the analytical call option price formula given by Christoffersen et al. Then, we compare numerically the discretization schemes of Andersen, Zhu and Alfonsi to the Euler scheme. , P. Gauthier, D. Possamai (2009)
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Complex Logarithms in Heston-Like Models , R. Lord, C. Kahl (2008)
Nb Clicks : 7670

Hedging under the Heston Model with Jump-to-Default , P.Carr, W.Schoutens (2007)
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Models with time-dependent parameters using transform methods: application to Heston s model , A.Elices (2007)
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Dynamics of implied volatility surfaces The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to ?Vega? risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen?Lo`eve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach models and improves the well known ?sticky moneyness? rule used by option traders for updating implied volatilities. Our approach gives a justification for use of ?Vega?s for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors. , R. Cont, J. da Fonseca (2009)
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Finite Difference Based Calibration and Simulation In the context of a stochastic local volatility model, we present a numerical solution scheme that achieves full (discrete) consistency between calibration, finite difference solution and Monte-Carlo simulation. The method is based on an ADI finite difference discretisation of the model. , J.Andreasen, B.N.Huge (2010)
Nb Clicks : 7437

Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending We present a dialogue on Counterparty Credit Risk touching on Credit Value at Risk (Credit VaR), Potential Future Exposure (PFE), Expected Exposure (EE), Expected Positive Exposure (EPE), Credit Valuation Adjustment (CVA), Debit Valuation Adjustment (DVA), DVA Hedging, Closeout conventions, Netting clauses, Collateral modeling, Gap Risk, Re-hypothecation, Wrong Way Risk, Basel III, inclusion of Funding costs, First to Default risk, Contingent Credit Default Swaps (CCDS) and CVA restructuring possibilities through margin lending. The dialogue is in the form of a Q&A between a CVA expert and a newly hired colleague , Damiano Brigo (2011)
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Efficient Simulation of the Heston Stochastic Volatility Model , L.Andersen (2006)
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The Little Heston Trap , H.Albrecher, P.Mayer, W.Schoutens, J.Tistaert (2006)
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Pricing Options in an Extended Black Scholes Economy with Illiquidity : Theory and Empirical Evidence , U. Cetin, R. Jarrow, P. Protter, M. Warachka (2004)
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Calibrating Option Pricing Models with Heuristics Calibrating option pricing models to market prices often leads to optimisation problems to which standard methods (like such based on gradients) cannot be applied. We investigate two models: Heston?s stochastic volatility model, and Bates?s model which also includes jumps. We discuss how to price options under these models, and how to calibrate the parameters of the models with heuristic techniques. Sample Matlab code is provided. , M.Gilli, E.Schumann (2010)
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Closed Form Convexity and Cross-Convexity Adjustments for Heston Prices 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. , G.G.Drimus (2009)
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A Consistent Pricing Model for Index Options and Volatility Derivatives We propose and study a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing options on forward variance swaps and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns. An affine specification using L?vy processes as building blocks leads to analytically tractable pricing formulas for options on variance swaps as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying. , R.Cont, T.Kokholm (2009)
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Option Valuation Using Fast Fourier Transforms , P.Carr, D.Madan (1999)
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Gamma Expansion of the Heston Stochastic Volatility Model , P.Glasserman, K-K.Kim (2008)
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Stocks paying discrete dividends: modelling and option pricing In the Black-Scholes model, any dividends on stocks are paid continu- ously, but in reality dividends are always paid discretely, often after some announcement of the amount of the dividend. It is not entirely clear how such discrete dividends are to be handled; simple perturbations of the Black-Scholes model often fall into contradictions. Our approach here is to recognise the stock price as the net present value of all future dividends, and to model the (discrete) dividend process directly. The stock price pro- cess is then deduced, and various option-pricing formulae derived. The Black-Scholes model with continuous dividend payments results as a limit as the time between dividend payments goes to zero. 1 I , R. Korn, L.C.G. Rogers (2004)
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On the Heston Model with Stochastic Interest Rates In this article we discuss the Heston [17] model with stochastic interest rates driven by Hull-White [18] (HW) or Cox-Ingersoll-Ross [8] (CIR) processes. We define a so-called volatility compensator which guarantees that the Heston hybrid model with a non-zero correlation between the equity and interest rate processes is properly defined. Moreover, we propose an approximation for the characteristic function, so that pricing of basic derivative products can be efficiently done using Fourier techniques [12; 7]. We also discuss the effect of the approximations on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities. , L. A. Grzelak, K. Oosterlee (2009)
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On Models of Stochastic Recovery for Base Correlation This paper discusses various ways to add correlated stochastic recovery to the Gaussian Copula base correlation framework for pricing CDOs. Several recent models are extended to more general framework. It is shown that, conditional on the Gaussian systematic factor, negative forward recovery rate may appear in these models. This suggests that current static copula models of correlated default and recovery processes are inherently inconsistent. , Li, Hui (2009)
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A Practical Guide to Implied and Local Volatility We consider a stochastic local volatility model with domestic and foreign stochastic interest rates such that the volatilitydecomposes into a deterministic local volatility plus some bias terms. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, we are going to describe in detail the implementation of that model, focusing on the computation of a proper deterministic local volatility. To do so, we choose to generate an implied volatility surface without arbitrage in space and in time by parametrising a mixture of shifted lognormal densities under constraints and we use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. We will therefore need to devise an evolutionary algorithm that handle constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm combined with simple and robust constraints handling mechanisms we will propose a modified algorithm for solving our optimisation problem under constraints which greatly improves its performances. , D.A.Bloch (2010)
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Convergence Heston to SVI By an appropriate change of variables, we prove here that the SVI implied volatility parameterisation proposed in [2] and the large-time asymptotic of the Heston implied volatility derived in [1] do agree algebraically, thus confirming a conjecture proposed by J. Gatheral in [2] as well as proposing a simpler expression for the asymptotic implied volatility under the Heston model. , J.Gatheral, A.Jacquier (2010)
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Markovian Projection Onto a Heston Model , V. Piterbarg, A.Antonov, T.Misirpashaev (2007)
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No-Dynamic-Arbitrage and Market Impact , J.Gatheral (2009)
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Optimal Fourier inversion in semi-analytical option pricing At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities. , R. Lord, C. Kahl (2006)
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Liquidity Risk and Option Pricing Theory , R. A. Jarrow, P. Protter (2005)
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Convexity of option prices in the Heston model , J.Wang (2007)
Nb Clicks : 7127

Stochastic Recovery Model Applicable to First-to-Default Basket Pricing We propose a stochastic recovery framework for single-factor copula models, an extension of the Krekel model. It allows for efficient pricing of collateralized debt obligations and first-to-default baskets consistently with single-name pricing. We analyze such properties like recovery markdown, recovery rate correlation and implied base correlations. The application to first-to-default pricing is described in detail, including the analysis of obtained par spreads and sensitivity to correlation. In the deterministic recovery case, we recover the Hull and White model. , Roman Werpachowski (2009)
Nb Clicks : 7114