Analytical solutions of optimal portfolio rebalancing

We study optimal portfolio rebalancing in a mean-variance type framework and present new analytical results for the general case of multiple risky assets. We first derive the equation of the no-trade region, and then provide analytical solutions and conditions for the optimal portfolio under several simplifying yet important models of asset covariance matrix: uncorrelated returns, same non-zero pairwise correlation, and a one-factor model. In some cases, the analytical conditions involve one or two parameters whose values are determined by combinatorial, rather than numerical, algorithms. Our results provide useful and interesting insights on portfolio rebalancing, and sharpen our understanding of the optimal portfolio.     

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.     

A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro-differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman–Kac representation of variational solutions to partial integro-differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow a wide range of underlying stochastic processes, comprising processes with Brownian part as well as a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and \(\alpha\)-semistable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial integro-differential equation, our results provide a rigorous basis for numerous applications in financial mathematics and in probability theory. We implement a Galerkin scheme to solve the corresponding pricing equation numerically and illustrate the effect of a killing rate.     

An Algorithmic Approach to Optimal Asset Liquidation Problems

This paper examines discrete-time optimal control problems arising in the context of optimal asset liquidation using recently published algorithms and code. We address these questions within a realistic framework, assuming that the order placement decisions must be adapted dynamically. Furthermore, we show how a duality-based technique can be used to assess the quality of our numerical solution.     

Explosion in the quasi-Gaussian HJM model

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low-dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.     

Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment

Abstract

This article considers the compound Poisson insurance risk model perturbed by diffusion with investment. We assume that the insurance company can invest its surplus in both a risky asset and the risk-free asset according to a fixed proportion. If the surplus is negative, a constant debit interest rate is applied. The absolute ruin probability function satisfies a certain integro-differential equation. In various special cases, closed-form solutions are obtained, and numerical illustrations are provided.     

On the valuation of fader and discrete barrier options in Heston's stochastic volatility model

We focus on closed-form option pricing in Heston's stochastic volatility model, where closed-form formulas exist only for a few option types. Most of these closed-form solutions are constructed from characteristic functions. We follow this closed-form approach and derive multivariate characteristic functions depending on at least two spot values for different points in time. The derived characteristic functions are used as building blocks to set up (semi-) analytical pricing formulas for exotic options with payoffs depending on finitely many spot values such as fader options and discretely monitored barrier options. We compare our result with different numerical methods and examine the computational accuracy.     

Robust and consistent estimation of generators in credit risk

Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation), one needs to compute the TPM and probabilities of default over a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by Bladt and Sorensen. [J. R. Stat. Soc. Ser. B (Stat. Method.), 2005, 67, 395–410] for a CTMC with an absorbing state for such estimation. This work’s contribution is threefold. Firstly, we provide directly computable closed form expressions for quantities appearing in the EM algorithm and associated information matrix, allowing to easy approximation of confidence intervals. Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under very weak conditions (for the TPM problem). Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), outperforms other known algorithms in several metrics, in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms.     

Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper, we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.     

Pricing Interest Rate Derivatives: A General Approach

The relationship between affine stochastic processes and bondpricing equations in exponential term structure models has beenwell established. We connect this result to the pricing of interestrate derivatives. If the term structure model is exponentialaffine, then there is a linkage between the bond pricing solutionand the prices of many widely traded interest rate derivativesecurities. Our results apply to m-factor processes with n diffusionsand l jump processes. The pricing solutions require at mosta single numerical integral, making the model easy to implement.We discuss many options that yield solutions using the methodsof the article.     

Optimal life insurance purchase and consumption/investment under uncertain lifetime

In this paper, we consider optimal insurance and consumption rules for a wage earner whose lifetime is random. The wage earner is endowed with an initial wealth, and he also receives an income continuously, but this may be terminated by the wage earner’s premature death. We use dynamic programming to analyze this problem and derive the optimal insurance and consumption rules. Explicit solutions are found for the family of CRRA utilities, and the demand for life insurance is studied by examining our solutions and doing numerical experiments.     

On the numerical stability of simulation methods for SDEs under multiplicative noise in finance

When simulating discrete-time approximations of solutions of stochastic differential equations (SDEs), in particular martingales, numerical stability is clearly more important than some higher order of convergence. Discrete-time approximations of solutions of SDEs with multiplicative noise, similar to the Black–Scholes model, are widely used in simulation in finance. The stability criterion presented in this paper is designed to handle both scenario simulation and Monte Carlo simulation, i.e. both strong and weak approximations. Methods are identified that have the potential to overcome some of the numerical instabilities experienced when using the explicit Euler scheme. This is of particular importance in finance, where martingale dynamics arise frequently and the diffusion coefficients are often multiplicative. Stability regions for a range of schemes are visualized and analysed to provide a methodology for a better understanding of the numerical stability issues that arise from time to time in practice. The result being that schemes that have implicitness in the approximations of both the drift and the diffusion terms exhibit the largest stability regions. Most importantly, it is shown that by refining the time step size one can leave a stability region and may face numerical instabilities, which is not what one is used to experiencing in deterministic numerical analysis.     

A general closed-form spread option pricing formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black–Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models.     

The Default Risk of Swaps

We characterize the exchange of financial claims from risky swaps. These transfers are among three groups: shareholders, debtholders, and the swap counterparty. From this analysis we derive equilibrium swap rates and relate them to debt market spreads. We then show that equilibrium swaps in perfect markets transfer wealth from shareholders to debtholders. In a simplified case, we obtain closed-form solutions for the value of the default risk in the swap. For interest-rate swaps, we obtain numerical solutions for the equilibrium swap rate, including default risk. We compare these with equilibrium debt market default risk spreads.     

Approximation methods for piecewise deterministic Markov processes and their costs

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.     

Agent‐based stochastic production lines design

An artificial agent‐based approach has been developed to improve the design and control of stochastic production lines. Genetic algorithms have been used as the premier agents learning mechanism. We benchmark our agent‐based approach with other well‐studied approaches such as infinitesimal perturbation analysis and mean‐value analysis methods. The performances of our agents are comparable with other approaches; in some cases, the agent‐based approach discovers even better solutions than the so‐called ‘optimal’ solutions by other approaches. The paper is one of the series of our work on multi‐agent intelligent enterprise modeling, and it seves as one of the most fundamental building blocks for other related works. © 2000 John Wiley & Sons, Ltd.     

Particle swarm optimization approach to portfolio construction

Particle swarm optimization (PSO) is an artificial intelligence technique that can be used to find approximate solutions to extremely difficult or impossible numeric optimization problems. Recently, PSO algorithms have been widely used in solving complex financial optimization problems. This paper presents a PSO approach to solve a portfolio construction problem, since this methodology is a population-based heuristic algorithm that is suitable for solving high-dimensional constrained optimization problems. The computational results show that PSO algorithms have advantages in optimizing the Sortino ratio, especially in speed, when the size of the portfolio is large.     

Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

In this paper we analyze a nonlinear Black–Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price V is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black–Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.