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Greeks in Monte Carlo
  • Flaming Logs This paper extends the pathwise adjoint method for Greeks to the displaced-diffusion LIBOR market model and also presents a simple way to improve the speed of the method. The speed improvements of approximately 20% are achieved without using any additional approximations to those of Giles and Glasserman. , N.Denson, M.S.Joshi (2009)
  • Minimal Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks In this paper, we present a generic framework known as the minimal partial proxy simulation scheme. This framework allows stable computation of the Monte-Carlo Greeks for financial products with trigger features via finite difference approximation. The minimal partial proxy simulation scheme can be considered as a special case of the partial proxy simulation scheme (Fries and Joshi, 2008b) as a measure change (weighted Monte Carlo) is performed to prevent path-wise discontinuities. However, our approach differs in term of how the measure change is performed. Specifically, we select the measure change optimally such that it minimises the variance of the Monte-Carlo weight. Our method can be applied to popular classes of trigger products including digital caplets, autocaps and target redemption notes. While the Monte-Carlo Greeks obtained using the partial proxy simulation scheme can blow up in certain cases, these Monte-Carlo Greeks remain stable under the minimal partial proxy simulation scheme. Standard errors for Vega are also significantly lower under the minimal partial proxy simulation scheme. , J.Hong Chan, M.S.Joshi (2009)
  • Efficient Greek Estimation in Generic Market Models We first develop an efficient algorithm to compute Deltas of interest rate derivatives for a number of standard market models. The computational complexity of the algorithms is shown to be proportional to the number of rates times the number of factors per step. We then show how to extend the method to efficiently compute Vegas in those market models. , M.S.Joshi, C.Yang (2009)
  • Stable Monte-Carlo Sensitivities of Bermudan Callable Products n this paper we discuss the valuation and sensitivities of financial products with early exercise rights (e.g., Bermudan options) using a Monte-Carlo simulation. The usual way to value early exercise rights is the backward algorithm. As we will point out, the Monte-Carlo version of the backward algorithm is given by an unconditional expectation of a random variable whose paths are discontinuous functions of the initial data. This results in noisy sensitivities, when sensitivities are calculated from finite differences of valuations. We present a simple localized smoothing of the Monte-Carlo backward algorithm which results in stable, variance reduced sensitivities. In contrast to other payoff smoothing methods, the smoothed backward algorithm will converge to the true Bermudan value in the Monte-Carlo limit. However, it looses the property of being a strict lower bound. The method is easy to implement since it is a simple modification to the pricing algorithm and it is independent of the underlying model. , C.P.Fries (2009)
  • Sensitivity estimates for portfolio credit derivatives using Monte Carlo , P.Glasserman, Z.Chen (2008)
  • A Note on Monte Carlo Greeks using the Characteristic Function , J. Kienitz (2008)
  • A Note on Monte Carlo Greeks for Jump Diffusions and other Levy models , Jorg Kienitz (2008)
  • Localized Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks , C.P.Fries (2007)
  • Malliavin Greeks without Malliavin Calculus , P.Glasserman, N.Chen (2006)
  • Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks , C.P.Fries, M.S.Joshi (2006)
  • Proxy Simulation Schemes for generic robust Monte-Carlo sensitivities, process oriented importance sampling and high accuracy drift approximation (with applications to the LIBOR Market Model) We consider a generic framework for generating likelihood ratio weighted Monte Carlo simu- lation paths, where we use one simulation scheme K? (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K* such that EQ (f (K* ) | Ft ) = EQ (f (K? ) ? w | Ft ). (1) This is done numerically in every time step, on every path. This makes the approach independent of the product (the function f in (1)) and even of the model, it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method [5] and Malliavin?s Calculus [11, 18] reconsidered on the level of the discrete numerical simulation scheme. Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin?s Calculus). The framework is completely generic and may be used for - high accuracy drift approximations, - process oriented importance sampling and the - robust calculation of partial derivatives of expectations w.r.t. model parameters (i.e. sen- sitivities, aka. Greeks) by applying finite differences by reevaluating the expectation with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking. , C.P.Fries, J.Kampen (2005)
  • Smoking Adjoints : fast evaluation of Greeks in Monte Carlo calculations , M.B. Giles, P. Glasserman (2005)
  • Rapid And Accurate Development of Prices and Greeks for Nth To Default Credit Swaps in the Li Model , M.Joshi, D.Kainth (2004)
  • Pricing and Deltas of Discretely-Monitored Barrier Options Using Stratified Sampling on the Hitting-Times to the Barrier We develop new Monte Carlo techniques based on stratifying the stock s hitting-times to the barrier for the pricing and Delta calculations of discretely-monitored barrier options using the Black-Scholes model. We include a new algorithm for sampling an Inverse Gaussian random variable such that the sampling is restricted to a subset of the sample space. We compare our new methods to existing Monte Carlo methods and find that they can substantially improve convergence speeds. , M.S.Joshi, R.Tang (2003)
  • Monte Carlo evaluation of Greeks for multidimensional barrier and lookback options , E.Gobet, G.Bernis, A.Kohastu-Higa (2003)
  • Malliavin Calculus for Monte Carlo Methods in Finances , E. Benhamou (2002)
  • Malliavin Calculus applied to Finance , M.Montero, A.Kohatsu-Higa (2002)
  • An Introduction to Malliavin Calculus , P.K.Friz (2002)
  • A Generalisation of Malliavin Weighted Scheme for Fast Computation of the Greeks , E.Benhamou (2001)
  • Applications of Malliavin Calculus to Monte Carlo Methods in Finance , E.Fournie, JM.Lasry, J.Lebuchoux, PL.Lions, N.Touzi (1999)

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