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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)
    Nb Clicks : 8184
  • Valuing Convertible Bonds with Stock Price, Volatility, Interest Rate, and Default Risk , P. Kovalov, V. Linetsky (2008)
    Nb Clicks : 8106
  • Quasi-monte carlo methods and pseudo-random numbers , H. Niederreiter (1978)
    Nb Clicks : 7985
  • 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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  • High order discretization schemes for the CIR process: application to Affine Term Structure and Heston models , A.Alfonsi (2008)
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  • No-Dynamic-Arbitrage and Market Impact , J. Gatheral (2008)
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  • A Comparative Analysis of Basket Default Swaps Pricing Using the Stein Method , E.Benhamou, D.Bastide, M.Ciuca (2007)
    Nb Clicks : 5451
  • 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)
    Nb Clicks : 5343
  • 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 : 5328
  • 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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  • Random Numbers for Simulation , P. L Ecuyer (1990)
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  • 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 : 5148
  • An Improved Implied Copula Model and its Application to the Valuation of Bespoke CDO Tranches , J.Hull, A.White (2008)
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  • 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)
    Nb Clicks : 4993
  • Improved Long-Period Generators Based on Linear Recurrences Modulo 2 , F. Panneton, P. L Ecuyer, and M. Matsumoto (2006)
    Nb Clicks : 4969
  • 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)
    Nb Clicks : 4959
  • 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 : 4941
  • 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)
    Nb Clicks : 4814
  • 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)
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  • Target Redemption Note , Y.Kuen Kwok, C.Chiu Chu (2006)
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  • Not-so-complex logarithms in the Heston model , C.Kahl, P.Jackel (2006)
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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)
    Nb Clicks : 4610
  • 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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  • Local Volatility Enhanced by a Jump to Default A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of -1.5. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and FORD over the period October 2004 to September 2007. , P.Carr, D.P.Madan (2010)
    Nb Clicks : 4531
  • 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)
    Nb Clicks : 4482
  • Robust Replication of Volatility Derivatives In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive delta-hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation. , P.Carr, R.Lee (2003)
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  • Fast strong approximation Monte-Carlo schemes for stochastic volatility models , C.Kahl, P.Jaeckel (2006)
    Nb Clicks : 4419
  • Models with time-dependent parameters using transform methods: application to Heston s model , A.Elices (2007)
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  • Efficient Simulation of the Heston Stochastic Volatility Model , L.Andersen (2006)
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  • Two-Dimensional Markovian Model for Dynamics of Aggregate Credit Loss , A.V.Lopatin, T.Misirpashaev (2007)
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  • Complex Logarithms in Heston-Like Models , R. Lord, C. Kahl (2008)
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  • The Valuation of Convertible Bonds With Credit Risk , E. Ayache, P.A. Forsyth, K.R. Vetzal (2003)
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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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  • Fine-tune your smile: Correction to Hagan et al , J.Obloj (2007)
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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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  • Introduction to Weather Derivative Pricing , S.Jewson (2004)
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  • No-Dynamic-Arbitrage and Market Impact , J.Gatheral (2009)
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  • Option Valuation Using Fast Fourier Transforms , P.Carr, D.Madan (1999)
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  • Modern Logarithms for the Heston Model , I.Fahrner (2007)
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  • CDS with Counterparty Risk in a Markov Chain Copula Model with Joint Defaults In this paper we study the counterparty risk on a payer CDS in a Markov chain model of two reference credits, the firm underlying the CDS and the protection seller in the CDS. We first state few preliminary results about pricing and CVA of a CDS with counterparty risk in a general set-up. We then introduce a Markov chain copula model in which wrong way risk is represented by the possibility of joint defaults between the counterpart and the firm underlying the CDS. In the set-up thus specified we have semi-explicit formulas for most quantities of interest with regard to CDS counterparty risk like price, CVA, EPE or hedging strategies. Model calibration is made simple by the copula property of the model. Numerical results show adequation of the behavior of EPE and CVA in the model with stylized features. , S.Crepey, M.Jeanblanc, B.Zargari (2009)
    Nb Clicks : 4131
  • Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator , M. Matsumoto and T. Nishimura (1998)
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  • Volatility Interpolation We present an effcient algorithm for interpolation and extrapolation of a discrete set of European option prices into a an arbitrage consistent full double continuum in expiry and strike of option prices. The method is based on an application of the fully implicit finite difference method and related to the local variance gamma model of Carr (2008). In a numerical example we show how the model can fitted to all quoted prices in the SX5E option market (12 expiries, each with roughtly 10 strikes) in 0.05 seconds of CPU time. , J.Andreasen, B.N.Huge (2010)
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  • The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing , A.Kalemanova, B.Schmid, R.Werner (2007)
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  • Liquidity Risk and Option Pricing Theory , R. A. Jarrow, P. Protter (2005)
    Nb Clicks : 4111
  • 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 : 4109
  • 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)
    Nb Clicks : 4108
  • The Little Heston Trap , H.Albrecher, P.Mayer, W.Schoutens, J.Tistaert (2006)
    Nb Clicks : 4089
  • Markovian Projection Onto a Heston Model , V. Piterbarg, A.Antonov, T.Misirpashaev (2007)
    Nb Clicks : 4087
  • Fx Barriers With Smile Dynamics , G.Baker, R.Beneder, A.zliber (2004)
    Nb Clicks : 4072
  • Convexity of option prices in the Heston model , J.Wang (2007)
    Nb Clicks : 4063

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