# Working papers of Mark Joshi

Optimal Partial Proxy Method for Computing Gammas of Financial Products with Discontinuous and Angular Payoffs by Mark S. Joshi and Dan Zhu

We extend the limit optimal partial proxy method to compute second order sensitivities of financial products with discontinuous or angular payoffs by Monte Carlo simulation. The methodology is optimal in terms of minimizing the variance of likelihood ratios terms. Applications are presented for both equity options and interest rate products with discontinuous payoff structures. The first order optimal partial proxy method is also implemented to calculate the Hessians of insurance products with angular payoffs. Numerical results are presented which demonstrate the speed and efficacy of the method.

The Rate of Convergence of the Two-State Lattice Model for Pricing Vanilla Options by Mark S. Joshi and Chun Fung Kwok

Variations of the binomial tree model are reviewed and extensions to the two most efficient trees studied in a recent literature are proposed. Tian’s modified tree is extended to a more general class of tree, and the third order tree is extended to the seventh order tree. Analysis of the error of American put pricing in the binomial tree model is presented and new trees that have superior performance in pricing in-the-money American puts are developed. To further improve numerical results, a scheme that incorporates different trees is suggested.

The Multiplicative Dual for Multiple-Exercise Options by Mark S. Joshi and Nicholas Yap

We derive the first known multiplicative dual for Bermudan type options that can be exercised more than once. Our multiplicative dual possesses the almost sure property, thus making it a viable alternative to the additive one. A method to compute the multiplicative upper bound is presented and the primal-dual algorithm is naturally extended to the multiplicative multiple-exercise case.

We analyze the primal-dual upper bound method and prove that its bias is inversely proportional to the number of paths in sub-simulations for a large class of cases. We develop a methodology for estimating and reducing the bias. We present numerical results showing that the new technique is indeed effective.

We discuss the pricing of cancellable swaps using the displaced diffusion LIBOR market model using a multi-core graphics card. We demonstrate that over one hundred times speed up can be achieved in a realistic case.

Effective Sub-Simulation-Free Upper Bounds for the Monte Carlo Pricing of Callable Derivatives and Various Improvements to Existing Methodologies by Mark S. Joshi and Robert Tang, Journal of Economic Dynamics and Control, Feb 2014

We present a new non-nested approach to computing additive upper bounds for callable derivatives using Monte Carlo simulation. It relies on the regression of Greeks computed using adjoint methods. We also show that it is is possible to early terminate paths once points of optimal exercise have been reached. A natural control variate for the multiplicative upper bound is introduced which renders it competitive to the additive one. In addition, a new bi-iterative family of upper bounds is introduced which take a stopping time, an upper bound, and a martingale as inputs.

Optimal Limit Methods for Computing Sensitivities of Discontinuous Integrals Including Triggerable Derivative Securities by Jiun Hong Chan and Mark S. Joshi

We introduce a new approach to computing sensitivities of discontinuous integrals. The methodology is generic in that it only requires knowledge of the simulation scheme and the location of the integrand's singularities. The methodology is proven to be optimal in terms of minimizing the variance of the measure changes caused by the elimination of the discontinuities for finite bump sizes. An efficient adjoint implementation of the small bump-size limit is discussed, and the method is shown to be effective for a number of natural examples involving triggerable interest rate derivative securities.

Fourier Transforms, Option Pricing and Controls by Mark S. Joshi and Chao Yang

We incorporate a simple and effective control-variate into Fourier inversion formulas for vanilla option prices. The control-variate used in this paper is the Black-Scholes formula whose volatility parameter is determined in a generic non-arbitrary fashion. We analyze contour dependence both in terms of value and speed of convergence. We use Gaussian quadrature rules to invert Fourier integrals, and numerical results suggest that performing the contour integration along the real axis leads to the best pricing performance.

Accelerating Pathwise Greeks in the LIBOR Market Model by Mark S. Joshi and Alexander Wiguna , International Journal of Theoretical and Applied Finance,

In the framework of the displaced-diffusion LIBOR market model, we derive the pathwise adjoint method for the iterative predictor-corrector and Glasserman-Zhao drift approximations in the spot measure. This allows us to compute fast deltas and vegas under these schemes. We compare the discretisation bias obtained when computing Greeks with these methods to those obtained under the log-Euler and predictor-corrector approximations by performing tests with interest rate caplets and cancellable receiver swaps. The two predictor-corrector type methods were the most accurate by far. In particular, we found the iterative predictor-corrector method to be more accurate and slightly faster than the predictor-corrector method, the Glasserman-Zhao method to be relatively fast but highly inconsistent, and the log-Euler method to be reasonably accurate but only at low volatilities. Standard errors were not significantly different across all four discretisations.

Fast Gamma Computations for CDO Tranches by Mark S. Joshi and Chao Yang

We demonstrate how to compute first- and second-order sensitivities of portfolio credit derivatives such as synthetic collateralized debt obligation (CDO) tranches using algorithmic Hessian methods developed in Joshi and Yang (2010) in a single-factor Gaussian copula model. Our method is correct up to floating point error and extremely fast. Numerical result shows that, for an equity tranche of a synthetic CDO with 125 names, we are able to compute the whole Gamma matrix with computational times measured in seconds.

First and Second Order Greeks in the Heston Model by Jiun Hong Chan and Mark Joshi

In this paper, we present an efficient approach to compute the first and the second order price sensitivities in the Heston model using the algorithmic differentiation approach. Issues related to the applicability of the pathwise method are discussed in this paper as most existing numerical schemes are not Lipschitz in model inputs. Depending on the model inputs and the discretization step size, our numerical tests show that the sample means of price sensitivities obtained using the Lognormal scheme and the Quadratic-Exponential scheme can be highly skewed and have fat-tailed distribution while price sensitivities obtained using the Integrated Double Gamma scheme and the Double Gamma scheme remain stable.

Efficient Pricing and Greeks in the Cross-Currency LIBOR Market Model by Chris Beveridge, Mark Joshi, and Will Wright, Journal of Risk

We discuss the issues involved in an efficient computation of the price and sensitivities of Bermudan exotic interest rate derivatives in the cross-currency displaced diffusion LIBOR market model. Improvements recently developed for an efficient implementation of the displaced diffusion LIBOR market model are extended to the cross-currency setting, including the adjoint-improved pathwise method for computing sensitivities and techniques used to handle Bermudan optionality. To demonstrate the application of this work, we provide extensive numerical results on two commonly traded cross-currency exotic interest rate derivatives: cross-currency swaps (CCS) and power reverse dual currency (PRDC) swaps.

Algorithmic Hessians and the Fast Computation of Cross-Gamma Risk by Mark Joshi and Chao Yang, Transactions of the IIE

We introduce a new methodology for computing Hessians from algorithms for function evaluation, using backwards methods. We show that the complexity of the Hessian calculation is a linear function of the number of state variables times the complexity of the original algorithm. We apply our results to computing the Gamma matrix of multi-dimensional financial derivatives including Asian Baskets and cancellable swaps. In particular, our algorithm for computing Gammas of Bermudan cancellable swaps is order O(n^2) per step in the number of rates. We present numerical results demonstrating that the computing all n(n+1)/2 Gammas in the LMM takes roughly n/3 times as long as computing the price.

Fast Greeks for Markov-Functional Models Using Adjoint Pde Methods by Nick Denson and Mark Joshi

This paper demonstrates how the adjoint PDE method can be used to compute Greeks in Markov-functional models. This is an accurate and efficient way to compute Greeks, where most of the model sensitivities can be computed in approximately the same time as a single sensitivity using finite difference. We demonstrate the speed and accuracy of the method using a Markov-functional interest rate model, also demonstrating how the model Greeks can be converted into market Greeks.

Fast and Accurate Long Stepping Simulation of the Heston Stochastic Volatility Model by Jiun Hong Chan and Mark Joshi. Here is the C++ code including a Visual Studio 9.0 project for this scheme . Journal of Computational Finance

In this paper, we present three new discretization schemes for the Heston stochastic volatility model - two schemes for simulating the variance process and one scheme for simulating the integrated variance process conditional on the initial and the end-point of the variance process. Instead of using a short time-stepping approach to simulate the variance process and its integral, these new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new approaches.

Monte Carlo Bounds for Game Options Including Convertible Bonds by Chris Beveridge and Mark Joshi, Management Science

We introduce two new methods to calculate bounds for zero-sum game options using Monte Carlo simulation. These extend and generalise the duality results of Haugh--Kogan/Rogers and Jamshidian to the case where both parties of a contract have Bermudan optionality. It is shown that the Andersen--Broadie method can still be used as a generic way to obtain bounds in the extended framework, and we apply the new results to the pricing of convertible bonds by simulation.

Monte Carlo Market Greeks in the Displaced Diffusion LIBOR Market Model by Mark Joshi and Oh Kang Kwon,Journal of Risk

The problem of developing sensitivities of exotic interest rates derivatives to the observed implied volatilities of caps and swaptions is considered. It is shown how to compute these from sensitivities to model volatilities in the displaced diffusion LIBOR market model. The example of a cancellable inverse floater is considered.

Truncation and Acceleration of the Tian Tree for the Pricing of American Put Options by Ting Chen and Mark S. Joshi, Quantitative Finance

We present a new method for truncating binomial trees based on using a tolerance to control truncation errors and apply it to the Tian tree together with acceleration techniques of smoothing and Richardson extrapolation. For both the current (based on standard deviations) and the new (based on tolerance) truncation methods, we test different truncation criteria, levels and replacement values to obtain the best combination for each required level of accuracy. We also provide numerical results demonstrating that the new method can be 50% faster than previously presented methods when pricing American put options in the Black-Scholes model.

Fast Monte-Carlo Greeks for Financial Products With Discontinuous Pay-Offs by Jiun Hong Chan and Mark S. Joshi, Mathematical Finance

We introduce a new class of numerical schemes for discretizing processes driven by Brownian motions. These allow the rapid computation of sensitivities of discontinuous integrals using pathwise methods even when the underlying densities post-discretization are singular. The two new methods presented in this paper allow Greeks for financial products with trigger features to be computed in the LIBOR market model with similar speed to that obtained by using the adjoint method for continuous pay-offs. The methods are generic with the main constraint being that the discontinuities at each step must be determined by a one-dimensional function: the proxy constraint. They are also generic with the sole interaction between the integrand and the scheme being the specification of this constraint.

Fast and Accurate Pricing and Hedging of Long-Dated CMS Spread Options by Mark S. Joshi and Chao Yang, International Journal of Theoretical and Applied Finance

We present a fast method to price and hedge CMS spread options in the displaced-diffusion co-initial swap market model. Numerical tests demonstrate that we are able to obtain sufficiently accurate prices and Greeks with computational times measured in milliseconds. Further, we find that CMS spread options are weakly dependent on the at-the-money Black implied volatility skews.

Fast Sensitivity Computations for Monte Carlo Valuation of Pension Funds by Mark S. Joshi and David Pitt, ASTIN Bulletin

Sensitivity analysis, or so-called 'stress-testing', has long been part of the actuarial contribution to pricing, reserving and management of capital levels in both life and non-life assurance. Recent developments in the area of derivatives pricing have seen the application of adjoint methods to the calculation of option price sensitivities including the well-known 'Greeks' or partial derivatives of option prices with respect to model parameters. These methods have been the foundation for efficient and simple calculations of a vast number of sensitivities to model parameters in financial mathematics. This methodology has yet to be applied to actuarial problems in insurance or in pensions. In this paper we consider a model for a defined benefit pension scheme and use adjoint methods to illustrate the sensitivity of fund valuation results to key inputs such as mortality rates, interest rates and levels of salary rate inflation. The method of adjoints is illustrated in the paper and numerical results are presented. Efficient calculation of the sensitivity of key valuation results to model inputs is useful information for practising actuaries as it provides guidance as to the relative ultimate importance of various judgments made in the formation of a liability valuation basis.

Graphical Asian Options by Mark S. Joshi, Wilmott Journal

We study the problem of pricing an Asian option using CUDA on a graphics processing unit. We demonstrate that it is possible to get accuracy of 2E-4 in less than a fiftieth of a second.

Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals by Christopher Beveridge and Mark S. Joshi, SIAM Mathematical Finance Journal and International Journal of Theoretical and Applied Finance. (two pieces)

We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model. We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach to simulating the range accrual coupons. We consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic.

Fast and Accurate Greeks for the Libor Market Model by Nick Denson and Mark Joshi, Journal of Computational Finance

This paper derives the pathwise adjoint method for the predictor-corrector drift approximation in the displaced-diffusion LIBOR market model. We present a comparison of the Greeks between log-Euler and predictor-corrector, showing both methods have the same computational order but the latter to be much more accurate.

Pricing and Deltas of Discretely-Monitored Barrier Options Using Stratified Sampling on the Hitting-Times to the Barrier by Mark Joshi and Robert Tang, International Journal of Theoretical and Applied Finance

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.

Efficient Greek estimation in generic market models by Mark Joshi and Chao Yang, Algorithmic Finance

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.

Minimal Partial Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks by Jiun-Hong Chan and Mark Joshi, Journal of Computational Finance

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.

Flaming logs by Mark Joshi and Nick Denson, Wilmott Journal

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.

Fast Delta Computations in the Swap-Rate Market Model by Mark Joshi and Chao Yang , Journal of Economics Dynamics and Control

We develop an efficient algorithm to implement the adjoint method that computes sensitivities of an interest rate derivative (IRD) with respect to different underlying rates in the co-terminal swap-rate market model. The order of computation per step of the new method is shown to be proportional to the number of rates times the number of factors, which is the same as the order in the LIBOR market model.

Vega Control by Nick Denson and Mark Joshi, Risk Magazine

The calculation of prices and sensitivities of exotic interest rate derivatives in the LIBOR market model is often very time consuming. One approach that has been previously suggested is to use a Markov-functional model as a control variate for prices and deltas but not vegas. We present a new approach that is effective for prices, deltas and vegas. It achieves a standard error reduction by a factor of 10 for the price of a five-factor, twenty-year Bermudan swaption, and of 5 for its vega.

Practical Policy Iteration: Generic Methods for Obtaining Rapid and Tight Bounds for Bermudan Exotic Derivatives Using Monte Carlo Simulation by Christopher Beveridge, Mark S. Joshi and Robert Tang, Journal of Economics Dynamics and Control

We introduce a set of improvements which allow the calculation of very tight lower bounds for Bermudan derivatives using Monte Carlo simulation. These lower bounds can be computed quickly, and with minimal hand-crafting. Our focus is on accelerating policy iteration to the point where it can be used in similar computation times to the basic least-squares approach, but in doing so introduce a number of improvements which can be applied to both the least-squares approach and the calculation of upper bounds using the Andersen-Broadie method. The enhancements to the least-squares method improve both accuracy and efficiency. Results are provided for the displaced-diffusion LIBOR market model, demonstrating that our practical policy iteration algorithm can be used to obtain tight lower bounds for cancellable CMS steepener, snowball and vanilla swaps in similar times to the basic least-squares method.

Trinomial or Binomial: Accelerating American Put Option Price on Trees  by  Jiun Hong Chan, Mark S. Joshi, Robert Tang and Chao Yang, Journal of Futures Markets, Vol 29, Number 9, September 2009, 826--839

We investigate the pricing performance of eight trinomial trees and one binomial tree, which was found to be most effective in an earlier paper, under twenty different implementation methodologies for pricing American put options. We conclude that the binomial tree, the Tian third order moment matching tree with truncation, Richardson extrapolation and smoothing performs better than the trinomial trees.

Juggling Snowballs by Christopher Beveridge and Mark S. Joshi, Risk Magazine, December 2008

The pricing of snowball notes in the full-factor LIBOR market model is considered. The primary aspect of the problem considered is the early exercise feature, and it is shown how to characterize a class of sub-optimal points of exercise. By combining this characterization with least-squares regression on a suitable set of basis functions and using an extra trigger enhancement, it is shown that very tight lower bounds can be obtained in cases where previous methods required the use of sub-Monte Carlo simulations.

Conditional Analytic Monte Carlo Pricing Scheme for Auto-Callable Products by Christian Fries and Mark S. Joshi

In this paper we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout).  The Monte-Carlo pricing of the products with discontinuous payout is known to come with a high Monte-Carlo error. The numerical calculation of sensitivities (i.e., partial derivatives) of such prices by finite differences gives very noisy results, since the Monte-Carlo approximation (being a finite sum of discontinuous functions) is not smooth. Additionally, the Monte-Carlo error of the finite-difference approximation explodes as the shift size tends to zero.  Our method combines a product specific modification of the underlying numerical scheme, which is to some extent similar to an importance sampling and/or partial proxy simulation scheme and a reformulation of the payoff function into an equivalent smooth payout.  From the financial product we merely require that hitting of the stochastic trigger will result in an conditionally analytic value. Many complex derivatives can be written in this form. A class of products where this property is usually encountered are the so called auto-callables, where a trigger hit results in cancellation of all future payments except for one redemption payment, which can be valued analytically, conditionally on the trigger hit. From the model we require that its numerical implementation allows for a calculation of the transition probability of survival (i.e., non-trigger hit). Many models allows this, e.g., Euler schemes of Itô processes, where the trigger is a model primitive. The method presented is effective across a large range of cases where other methods fail, e.g. small finite difference shift sizes or short time to trigger reset (approaching maturity); this means that a practitioner can use this method and be confident that it will work consistently.

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, Journal of Risk, Vol 11, Number 4, Summer 2009, 87--108

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, Mathematical Finance, January 2010

Abstract: A new family of binomial trees as approximations to the Black--Scholes model is introduced. 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, Journal of Computational Finance, March 2008

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, Quantitative Finance, March 2009

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.

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, Quantitative Finance 2008

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.

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.