Smooth simultaneous calibration of the LMM to caplets and co-terminal swaptions by Ferdinando Ametrano and Mark S. Joshi.
We introduce a new calibration methodology that allows perfect fitting of the displaced diffusion LIBOR market model to caplets and co-terminal swaptions, whilst avoiding global optimizations. The approach works by regarding a forward rate as a difference of swap-rates and then bootstrapping through rates one by one.Comparing discretisations of the Libor market model in the spot measure by Chris Beveridge, Nick Denson and Mark S. Joshi
Various drift approximations for the displaced-diffusion LIBOR market model in the spot measure are compared. The advantages, disadvantages and implementation choices for each of predictor-corrector and the Glasserman--Zhao method are discussed. Numerical tests are carried out and we conclude that the predictor-corrector method is superior.The Convergence of Binomial Trees for Pricing the American Put by Mark S. Joshi
Abstract: We study 20 different implementation methodologies for each of 11 different choices of parameters of binomial trees and investigate the speed of convergence for pricing American put options numerically. We conclude that the most effective methods involve using truncation, Richardson extrapolation and sometimes smoothing. We do not recommend use of a European option as a control. The most effective trees are the Tian third order moment matching tree and a new tree designed to minimize oscillations.
Achieving Higher Order Convergence for The Prices of European Options In Binomial Trees by Mark S. Joshi
Abstract: A new family of binomial trees as approximations to the Black--Scholes model isintroduced. For this class of trees, the existence of complete asymptotic expansions for the prices of vanilla European options is demonstrated and the first three terms are explicitly computed. As special cases, a tree with third order convergence is constructed and the conjecture of Leisen and Reimer that their tree has second order convergence is proven.
Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks by Christian Fries and Mark S. Joshi, to appear in the Journal of Computational Finance
Abstract: We consider a generic framework which allows to calculate robust Monte-Carlo sensitivities seamlessly through simple finite difference approximation. The method proposed is a generalization and improvement of the proxy simulation scheme method (Fries and Kampen, 2005). As a benchmark we apply the method to the pricing of digital caplets and target redemption notes using LIBOR and CMS indices under a LIBOR Market Model. We calculate stable deltas, gammas and vegas by applying direct finite difference to the proxy simulation scheme pricing. The framework is generic in the sense that it is model and almost product independent. The only product dependent part is the specification of the proxy constraint. This allows for an elegant implementation, where new products may be included at small additional costs
Achieving smooth asymptotics for the prices of European options in binomial trees by Mark S. Joshi,
Abstract: A new binomial approximation to the Black–Scholes model is introduced. It is shown that for digital options and vanilla European call and put options that a complete asymptotic expansion of the error in powers of 1/n exists. This is the first binomial tree for which such an asymptotic expansion has been shown to exist.
Intensity Gamma: a new approach to pricing portfolio credit derivatives, by Mark S. Joshi and Alan M. Stacey, Risk Magazine July 2006
Abstract: We develop a completely new model for correlation of credit defaults based on a financially intuitive concept of business time similar to that in the Variance Gamma model for stock price evolution. Solving a simple equation calibrates each name to its credit spread curve and we show that the overall model can be calibrated to the market base correlation curve of a tranched CDO index. Once this calibration is performed, obtaining consistent arbitrage-free prices for non-standard tranches, products based on different underlying names and even more exotic products such as $\mathrm{CDO}^2$ is straightforward and rapid.
Using Monte Carlo
simulation and importance sampling to rapidly obtain jump-diffusion
prices of continuous barrier options by Mark S. Joshi and Terence
Leung. Journal of Computational Finance, July 2007
Abstract. The problem of pricing a
continuous barrier option in a jump-diffusion model is studied. It is
shown that via an effective combination of importance sampling and
analytic formulas that substantial speed ups can be achieved. These
techniques are shown to be particularly effective for computing deltas.
Rapid computation of drifts in a reduced factor LIBOR Market Model by Mark S. Joshi, Wilmott May 2003
Abstract: An algorithm of order number of factors times number of rates for the computing the drifts of all the rates in the LIBOR market model. This is better than the naive algorithm which is of order number of rates squared.
Monte Carlo bounds for callable products with non-analytic break costs by Mark S. Joshi
Abstract. The pricing of callable derivative products with complicated pay-offs is studied. A new method for finding upper bounds by Monte Carlo simulation is introduced, this relies on modelling the callable product directly. The method has a wide range of applicability and is shown to be effective for Asian tail products. Presentation
Achieving decorrelation and speed simultaneously in the LIBOR market model by Mark S. Joshi, Journal of Risk, 2006
Abstract. An algorithm for computing the drift in the LIBOR market model with additional idiosynchratic terms is introduced. This algorithm achieves a computational complexity of order equal to the number of common factors times the number of rates. It is demonstrated that this allows better matching of correlation matrices in reduced-factor models.
A simple derivation of and improvements to Jamshidian's and Rogers' upper bound methods for Bermudan options by Mark S. Joshi, Applied Mathematical Finance July 2007
Abstract. Rogers’ method for upper bounds for Bermudan options is rephrased in terms of buyers and sellers prices. It is shown how to deduce Jamshidian’s upper bound result in a simple fashion from Roger’s method, including the case of possibly zero final pay-off. Both methods are improved by ruling out exercise at suboptimal points. It is also shown that it is possible to use sub- Monte Carlo simulations to estimate the value of the hedging portfolio at intermediate points in the Jamshidian method without jeopardizing its status as upper bound.
Effective implementation of generic market models by Mark S. Joshi and Lorenzo Liesch, ASTIN Bulletin Dec 2007
Abstract. A number of standard market models are studied. For each one, algorithms of computational complexity equal to the number of rates times the number of factors to carry out the computations for each step are introduced. Two new classes of market models are developed and it is shown for them that similar results hold.
New and robust drift approximations for the LIBOR market model by Mark S. Joshi and Alan Stacey, to appear in Quantitative Finance
Abstract: We present four new methods for approximating the drift in the LIBOR market model. These are compared to a variety of existing methods including PPR, Glasserman-Zhao and predictor-corrector. We see that two of them which use correlation adjustments to better approximate the drift are more effective than existing methods.
Option Pricing and the Dirichlet problem by Mark S. Joshi, Wilmott Magazine, 2006
It is well-known that the Dirichlet problem for the Laplacian on a reasonably smooth compact domain in Rn can be solved using Brownian motion. Indeed the result was found by Kakutani in 1944, [3, 4]. In this note, I want to discuss how this result can be reinterpreted financially. Our objective is to increase our intuition about the problem rather than to attempt to prove new results.
See also www.quarchome.org
Here's my advice sheet for those wanting to work as a quantitative analyst in finance. It was originally aimed at pure maths PhDs, but lots of other people seem to like it.
I have now released xlw 2.1 formerly called xlwPlus. xlw is a package for creating xlls in C++ with minimal effort. An xll is a way of adding new functions to EXCEL xlwPlus has various added features, the most important is that the interfacing code is generated automatically by an extra routine instead of being coded manually. If you have any questions, please ask them on xlw-users mailing list on sourceforge.
The code for C++ design patterns and derivatives pricing is available here.
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Coming soon: More mathematical finance.