The inherent feature of Bayesian empirical analysis is the dependence of posterior inference on prior parameters, which researchers typically specify. However, quantifying the magnitude of this dependence remains difficult. This article extends Infinitesimal Perturbation Analysis, widely used in classical simulation, to compute asymptotically unbiased and consistent sensitivities of posterior statistics with respect to prior parameters from Markov chain Monte Carlo inference via Gibbs sampling. The method demonstrates the possibility of efficiently computing the complete set of prior sensitivities for a wide range of posterior statistics, alongside the estimation algorithm using Automatic Differentiation. The method's application is exemplified in Bayesian Vector Autoregression analysis of fiscal policy in U.S. macroeconomic time series data. The analysis assesses the sensitivities of posterior estimates, including the Impulse response functions and Forecast error variance decompositions, to prior parameters under common Minnesota shrinkage priors. The findings illuminate the significant and intricate influence of prior specification on the posterior distribution. This effect is particularly notable in crucial posterior statistics, such as the substantial absolute eigenvalue of the companion matrix, ultimately shaping the structural analysis.

Measure-valued differentiation (MVD) is a relatively new method for computing Monte Carlo sensitivities, relying on a decomposition of the derivative of transition densities of the underlying process into a linear combination of probability measures. In computing the sensitivities, additional paths are generated for each constituent distribution and the payoffs from these paths are combined to produce sample estimates. The method generally produces sensitivity estimates with lower variance than the finite difference and likelihood ratio methods, and can be applied to discontinuous payoffs in contrast to the pathwise differentiation method. However, these benefits come at the expense of an additional computational burden. In this paper, we propose an alternative approach, called the absolute measure-valued differentiation (AMVD) method, which expresses the derivative of the transition density at each simulation step as a single density rather than a linear combination. It is computationally more efficient than the MVD method and can result in sensitivity estimates with lower variance. Analytic and numerical examples are provided to compare the variance in the sensitivity estimates of the AMVD method against alternative methods.

Notations and Conventions .......................................................................................................... 1 Chapter 1 ....................................................................................................................................... 2 Principal Value Distribution x − 1 (Problem 1.3) (9/25/2020) ............................................. 2 A Distribution u ∈ D(R) such that u = 1 x ⁄ on (0,∞) and u = 0 on (−∞, 0) (Problem 1.4) (9/25/2020) ........................................................................................................................... 4 An Interesting Example of a Non-Extendible Distribution (Problem 1.5) [2/26/2021]. ...... 5 Chapter 5 ....................................................................................................................................... 6 Convolution Equations on Forward Cones (Problem 5.5) (1/2/2021) .................................. 6 Chapter 6 ....................................................................................................................................... 8 Schwartz Kernel Theorem [Maintenance planned] (11/4/2020) ........................................... 8 Chapter 8 ..................................................................................................................................... 15 Structure Theorem for Tempered Distributions [Theorem 8.3.1] (5/11/2021) .................. 15

An inherent feature of Bayesian inference is posterior/prior (parameter) dependence. Locally, it canbe explained by the changes in posterior inference resulting from local changes in prior parameters.Yet, computing such sensitivities remains challenging given the analytical complexities and computational intensity of the widely used Markov Chain Monte Carlo. This paper extends Infinitesimal Perturbation Analysis, widely used in the classical simulation context to assess input sensitivities of stochastic dynamic systems, to compute prior parameter sensitivities of posterior statistics from Markov Chain Monte Carlo inference via Gibbs sampling. We show that, unlike the traditional finite differencing method for computing derivatives, our proposed method provides asymptotic unbiased and consistent derivative estimators of posterior statistics with respect to prior input parameters. Efficient computation of exact sensitivities is possible alongside the estimation algorithm via Automatic Differentiation tools, which avoid rerunning the chain. We illustrate the methods within Bayesian Vector Autoregression (BVAR) models, a benchmark tool in macroeconomic policy analysis and forecasting due to their ability to utilise informative shrinkage priors. We assess the prior robustness of both posterior parameter and impulse response inference in a standard smaller scale policy BVARfor US macroeconomic time series data under two versions of a common Minnesota shrinkage prior. The first formal sensitivity analysis shows that the exact prior specification can have substantial and complex influences on inference regarding the GDP responses under a government spending shock. Our results suggest that in the particular setting, the posterior impulse response function is less sensitive to the changes in the prior hyper-parameters if the Minnesota shrinkage pushes the model towards a random walk instead of a random walk with drifts..

Bayesian inference relies heavily on numerical Markov chain Monte carlo (MCMC) methods for the estimation of the typically intractable high-dimensional posterior distributions and requires specific inputs. In this paper we introduce a new general and efficient numerical approach to address important robustness concerns of MCMC analysis with respect to prior input assumptions, a major obstacle to wider acceptance of Bayesian inference, including MCMC algorithm performance (convergence) reflected in the dependence on the chain starting values. The approach builds on recent developments in sensitivity analysis of high-dimensional numerical integrals for classical simulation methods using automatic numerical differentiation methods to compute first order derivatives of algorithmic output with respect to all inputs. We introduce a range of new robustness measures based on Jacobian matrices of MCMC output w.r.t. to the two sets of input parameters, prior parameters and chain starting values, to enable researchers to routinely undertake a comprehensive sensitivity analysis of their MCMC results. The methods are implemented for a range of Gibbs samplers and illustrated using both simulated and real data examples.

The problem of estimating expectations of functions of conditional expectations using nested Monte Carlo simulation is studied. It is shown that typically the bias arising from non-linearity is in leading order inversely proportional to the number of sub-simulation paths when using a naive estimate. Various improved statistical estimators are introduced for the inner simulation. Applications to pricing of VIX derivatives, the computation of credit valuation adjustments and computation of Value-At-Risk are presented. It is shown that only small numbers of sub-paths are necessary for high accuracy.

In this paper, we propose a new regression-based sub-simulation-free upper bound. The method is based on approximating the martingale component of the lower bound process. The convergence and tightness of this upper bound are analyzed. In addition, the implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency of algorithms and the tightness of upper bounds. For a 20 rates Bermudan swaption, the distance between upper bounds with power enhancements and lower bounds is smaller than 10 basis points.

When pricing Bermudan derivatives by regression-based methods, foresight bias will appear in lower bounds when using a single simulation to estimate the exercise strategy and to compute lower bounds. In this paper, we propose a new method to remove this kind of bias without introducing an independent simulation. Numerical results indicate that the goodness of our method is comparable to that of using independent simulations. In addition, this method can be parallelized and enhanced by local regression. These improvements boost the accuracy and the time efficiency of lower bounds.

We introduce a new simulation algorithm for computing the Hessians of Bermudan swaptions and cancelable swaps. The resulting pathwise estimates are unbiased and accurate. Given the exercise strategy, the pathwise angularities are removed by a sequence of measure changes. The change of measure at each exercise time is chosen to be optimal in terms of minimizing the variance of the likelihood ratio terms. Numerical results for the Hessian of cancelable swaps are presented to demonstrate the speed and efficacy of the method.

A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. To this end, we present two methods for calibrating a local volatility model under correlated stochastic interest rates. This is achieved by first fitting a mixture model to market prices, and then determining the local volatility function that is consistent with this mixture model.

As observed by Carr and Lee (2009), variance swaps and other more complex volatility derivatives are increasingly being used by organizations to either trade volatility or hedge their portfolio’s v...

Prudential regulations require financial institutions to hold initial capital so that the possibility of ruin is very low. An important practical problem is to estimate the regulatory capital so the ruin probability is at the regulatory level, typically less than 0.1% over a finite-time horizon. Estimating probabilities of rare events is challenging, since naïve estimations via direct simulations of the surplus process is time consuming. In this paper, we present a stratification sampling algorithm for estimating finite-time ruin probabilities. We further introduce a sequence of measure changes to remove the pathwise discontinuities of the estimator, and compute unbiased first and second-order derivative estimates of the finite-time ruin probabilities with respect to both distributional and structural parameters. We then estimate the regulatory capital and its sensitivities. These estimates provide information to insurance companies for meeting prudential regulations as well as designing risk management strategies. Numerical examples are presented for the classical risk model, the Sparre Andersen risk model with interest and the periodic risk model with interest to demonstrate the speed and efficacy of our methodology.

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.

Credit value adjustment (CVA) and related charges have emerged as important risk factors following the Global Financial Crisis. These charges depend on uncertain future values of underlying products, and are usually computed by Monte Carlo simulation. For products that cannot be valued analytically at each simulation step, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their values. However, these functions do not necessarily provide accurate approximations to product values over all simulated paths and can result in biases that are difficult to control. Motivated by a novel characterization of the CVA as the value of an option with an early exercise opportunity at a stochastic time, we provide an approximation for CVA and other credit charges that rely only on the sign of the regression functions. The values are determined, instead, by pathwise deflated cash flows. A comparison of CVA for Bermudan swaptions and cancellable swaps shows that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

We analyse the primal-dual upper bound method for Bermudan options 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.

Power numeraires are defined. They are applied to the Black-Scholes model and the drift of the stock is derived. It is also shown how to use them to derive formulas for power options with barriers. A reinterpretation of contour-shifting in the context of characteristic function pricing is given. It is shown how to use power numeraires to improve the convergence of the COS method and numerical results are presented.

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.