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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.

Monte-Carlo Sensitivities, Likelihood Ratio, Importance Sampling, Greeks, Proxy Simulation Scheme, Digital Option, Binary Option, Trigger Product, Target Redemption Note

Abstract:
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.

market model, calibration, Bermudan swaptions

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.

binomial trees, Richardson extrapolation, options, rate of convergence

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 thatsubstantial speed ups can be achieved. These techniques are shown to be particularly effective for computing deltas.

jump-diffusion, barrier option, Monte Carlo, importance sampling

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 CDO2 is straightforward and rapid.

portffolio credit derivatives, gamma process, CDO

Abstract:
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.

Monte Carlo Simulation, Pricing, Greeks, Variance Reduction, Auto-Callable, Trigger Product, Target Redemption Note

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.

LIBOR market model, drift approximation, Monte Carlo

Abstract:
Various drift approximations for the displaced-discussion 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.

LIBOR market model, predictor-corrector, discretization

Abstract:
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. 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.

option pricing, Dirichlet problem, maximum principle

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 is introduced. Two new classes of market models are developed and it is shown for them that similar results hold.

market model, complexity, Monte Carlo

Abstract:
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.

binomial tree, trinomial tree, American put option, speed

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.

binomial trees, Richardson extrapolation, options, rate of convergence

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.

Monte Carlo, callable, upper bounds

Abstract:
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.

early exercise, snowball, LIBOR market model, Monte Carlo simulation, American option

Abstract:
An algorithm for computing the drift in the LIBOR market model with additional idiosyncratic 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.

LIBOR market model, low factor, efficiency

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.

binomial trees, Richardson extrapolation, options, rate of convergence

Abstract:
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.

Bermudan option, LIBOR market model, early exercise, Monte Carlo

Abstract:
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.

Variance reduction, control variate, LIBOR market model, LMM, BGM, Markov-functional model, vega

Abstract:
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.

adjoint method, Delta, computational order, market model, Monte Carlo simulation

Abstract:
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.

LIBOR market model, BGM, range accrual, interpolation scheme, Monte Carlo, early exercise, Greeks, pathwise method, delta, vega

Abstract:
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.

adjoint method, Delta, Vega, computational order, market model, Monte Carlo simulation

Abstract:
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.

Asian options, GPU, CUDA

Abstract:
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.

LIBOR market model, LMM, BGM, Greeks, delta, vega, pathwise method, predictor-corrector

Abstract:
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.

Monte-Carlo Sensitivities, Greeks, Likelihood Ratio, Importance Sampling, Partial Proxy Simulation Scheme, Trigger Product, Discontinuous Pay-off, Digital Option, Auto-cap, Target Redemption Note

Abstract:
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.

Pathwise adjoint method, LIBOR market model, delta, vega

Abstract:
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.

first-hitting time, passage times, hitting-times, barrier, discretely-monitored, inverse Gaussian, stratified sampling, Monte-Carlo

Abstract:
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.

Price Sensitivities, Monte-Carlo Greeks, Partial Proxy Simulation Scheme, Minimal Partial Proxy Simulation Scheme, Pathwise Partial Proxy Method, Pathwise Minimal Partial Proxy Method, Discontinuous Pay-offs, Digital Options, Target Redemption Notes, LIBOR Market Model

Abstract:
In this paper, we show how to evolve a yield curve over time horizons of the order of years using a simple but effective semi-parametric method. The proposed technique preserves in the limit all the eigenvalues and eigenvectors of the observed changes in yields. It also recovers in a satisfactory way several important statistical features (unconditional variance, serial autocorrelation, distribution of curvatures, eigenvectors) of the real-world data. A simple financial explanation can be provided for the methodology. The possible financial applications are discussed.

Yield curve, semi-parametric, eigenvalues, eigenvectors