Essays on Computational Problems in Insurance

This dissertation consists of two chapters. The first chapter establishes an algorithm for calculating capital requirements. The calculation of capital requirements for financial institutions usually entails a reevaluation of the company\u27s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. The current chapter proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. Our approach is motivated by a well-known method for pricing non-European derivatives. We study convergence of the algorithm and analyze the resulting estimate for practically important risk measures. Moreover, we address the problem of how to choose the regressors, and show that an optimal choice is given by the left singular functions of the corresponding valuation operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. The second chapter discusses another application of regression-based methods, in the context of pricing variable annuities. Advanced life insurance products with exercise-dependent financial guarantees present challenging problems in view of pricing and risk management. In particular, due to the complexity of the guarantees and since practical valuation frameworks include a variety of stochastic risk factors, conventional methods that are based on the discretization of the underlying (Markov) state space may not be feasible. As a practical alternative, this chapter explores the applicability of Least-Squares Monte Carlo (LSM) methods familiar from American option pricing in this context. Unlike previous literature we consider optionality beyond surrendering the contract, where we focus on popular withdrawal benefits - so-called GMWBs - within Variable Annuities. We introduce different LSM variants, particularly the regression-now and regression-later approaches, and explore their viability and potential pitfalls. We commence our numerical analysis in a basic Black-Scholes framework, where we compare the LSM results to those from a discretization approach. We then extend the model to include various relevant risk factors and compare the results to those from the basic framework

The least squares method for option pricing revisited

It is shown that the the popular least squares method of option pricing converges even under very general assumptions. This substantially increases the freedom of creating different implementations of the method, with varying levels of computational complexity and flexible approach to regression. It is also argued that in many practical applications even modest non-linear extensions of standard regression may produce satisfactory results. This claim is illustrated with examples

Sequential Monte Carlo pricing of American-style options under stochastic volatility models

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Comparing optimal convergence rate of stochastic mesh and least squares method for Bermudan option pricing

We analyze the stochastic mesh method (SMM) as well as the least squares method (LSM) commonly used for pricing Bermudan options using the standard two phase methodology. For both the methods, we determine the decay rate of mean square error of the estimator as a function of the computational budget allocated to the two phases and ascertain the order of the optimal allocation in these phases. We conclude that with increasing computational budget, while SMM estimator converges at a slower rate compared to LSM estimator, it converges to the true option value whereas LSM estimator, with fixed number of basis functions, usually converges to a biased value

Pricing American Options using Monte Carlo Method

This thesis reviewed a number of Monte Carlo based methods for pricing American options. The least-squares regression based Longstaff-Schwartz method (LSM) for approximating lower bounds of option values and the Duality approach through martingales for estimating the upper bounds of option values were implemented with simple examples of American put options. The effectiveness of these techniques and the dependencies on various simulation parameters were tested and discussed. A computing saving technique was suggested to reduce the computational complexity by constructing regression basis functions which are orthogonal to each other with respect to the natural distribution of the underlying asset price. The orthogonality was achieved by using Hermite polynomials. The technique was tested for both the LSM approach and the Duality approach. At the last, the Multilevel Mote Carlo (MLMC) technique was employed with pricing American options and the effects on variance reduction were discussed. A smoothing technique using artificial probability weighted payoff functions jointly with Brownian Bridge interpolations was proposed to improve the Multilevel Monte Carlo performances for pricing American options