Integral and differential equations for the moments of multistate models in health insurance

The moments of the random future liabilities of health insurance policies are key quantities for studying distributional properties of the future liabilities. Assuming that the randomness of the future health status of individual policyholders can be described by a semi-Markovian multistate model, integral and differential equations are derived for moments of any order and for the moment generating function. Different representations are derived and discussed with a view to numerical solution methods.     

Exchange option pricing under stochastic volatility: a correlation expansion

Efficient valuation of exchange options with random volatilities while challenging at analytical level, has strong practical implications: in this paper we present a new approach to the problem which allows for extensions of previous known results. We undertake a route based on a multi-asset generalization of a methodology developed in Antonelli and Scarlatti (Finan Stoch 13:269–303, 2009) to handle simple European one-asset derivatives with volatility paths described by Ito’s diffusive equations. Our method seems to adapt rather smoothly to the evaluation of Exchange options involving correlations among all the financial quantities that specify the model and it is based on expanding and approximating the theoretical evaluation formula with respect to correlation parameters. It applies to a whole range of models and does not require any particular distributional property. In order to test the quality of our approximation numerical simulations are provided in the last part of the paper.     

Pricing options with Green's functions when volatility,interest rate and barriers depend on time

We derive the Green's function for the Black–Scholes partial differential equation with time-varying coefficients and time-dependent boundary conditions. We provide a thorough discussion of its implementation within a pricing algorithm that also accommodates American style options. Greeks can be computed as derivatives of the Green's function. Generic handling of arbitrary time-dependent boundary conditions suggests our approach to be used with the pricing of (American) barrier options, although options without barriers can be priced equally well. Numerical results indicate that knowledge of the structure of the Green's function together with the well-developed tools of numerical integration make our approach fast and numerically stable.     

Numerical Approach to Asset Pricing Models with Stochastic Differential Utility

In this paper employing two heuristic numerical schemes, we study the asset pricing models with stochastic differential utility (SDU), which is formulated by either of backward stochastic differential equations (BSDEs) or forward-backward stochastic differential equations (FBSDEs).The first scheme is based upon a traditional lattice algorithm of option pricing theories, involving the discretization scheme of coupled FBSDEs, which is combined with a technique of solving numerically a certain type of nonlinear equations with respect to the backward state variables. The second one is based upon the four step scheme of Ma et al. (1994) which solves quasi-linear partial differential equations associated with the FBSDEs. We demonstrate that our practical implementation algorithms can successfully solve the asset pricing models with generalized SDU and the large investor problem with market impact which are typical examples such that the usual four step scheme is difficult to implement. As other numerical applications we study the optimal consumption and investment policies of a representative agent with SDU, and the recoverability of preferences and beliefs from observed consumption data.     

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.     

On an automatic and optimal importance sampling approach with applications in finance

Calculating high-dimensional integrals efficiently is essential and challenging in many scientific disciplines, such as pricing financial derivatives. This paper proposes an exponentially tilted importance sampling based on the criterion of minimizing the variance of the importance sampling estimators, and its contribution is threefold: (1) A theoretical foundation to guarantee the existence, uniqueness, and characterization of the optimal tilting parameter is built. (2) The optimal tilting parameter can be searched via an automatic Newton’s method. (3) Simplified yet competitive tilting formulas are further proposed to reduce heavy computational cost and numerical instability in high-dimensional cases. Numerical examples in pricing path-dependent derivatives and basket default swaps are provided.     

American-type basket option pricing: a simple two-dimensional partial differential equation

We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent in basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the European-type comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte-Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.     

Partial differential equations for Asian option prices

In this article, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this paper, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and as they perform numerically well or better than the others, must be considered the preferred choice.     

A reduced PDE method for European option pricing under multi-scale,multi-factor stochastic volatility

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.     

Double knock-out Asian barrier options which widen or contract as they approach maturity

Barrier options are considered for Asian options using a differential equation method. Solutions are obtained in the form of Fourier series for barriers which expand or contract as they approach maturity. Rigorous bounds are obtained. It is shown that by differentiating with respect to a parameter, solutions for more general payoffs can be obtained.     

A systematic and efficient simulation scheme for the Greeks of financial derivatives

Greeks are the price sensitivities of financial derivatives and are essential for pricing, speculation, risk management, and model calibration. Although the pathwise method has been popular for calculating them, its applicability is problematic when the integrand is discontinuous. To tackle this problem, this paper defines and derives the parameter derivative of a discontinuous integrand of certain functional forms with respect to the parameter of interest. The parameter derivative is such that its integration equals the differentiation of the integration of the aforesaid discontinuous integrand with respect to that parameter. As a result, unbiased Greek formulas for a very broad class of payoff functions and models can be systematically derived. This new method is applied to the Greeks of (1) Asian options under two popular Lévy processes, i.e. Merton's jump-diffusion model and the variance-gamma process, and (2) collateralized debt obligations under the Gaussian copula model. Our Greeks outperform the finite-difference and likelihood ratio methods in terms of accuracy, variance, and computation time.     

The valuation of warrants: Implementing a new approach

The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option. When the underlying stock pays no dividends – and in some very restrictive cases when it does – a closed form solution to the differential equation subject to the appropriate boundary conditions, has been obtained. But, in some relevant cases such as the one in which the stock pays discrete dividends, no closed form solution has been found. This paper shows how to solve these equations by numerical methods. In addition, the optimal strategy for exercising American options is derived. A numerical illustration of the procedure is also presented.     

An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions

We propose a new method to calibrate the Vasicek and Cox--Ingersoll--Ross interest rate models from bond prices. We define an appropriate generating function and derive recursive relations between the derivatives of the generating function and the bond prices. The parameters of the Vasicek and CIR models are then obtained by solving a system of linearly independent equations arising from the recursive relations. We include numerical results that show the method’s accuracy when bond prices generated from the exact formulas are used.     

Functional Itô calculus†

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.     

Dynamic Investment Strategies to Reaction–Diffusion Systems Based upon Stochastic Differential Utilities

We study the dynamic investment strategies in continuous-time settings based upon stochastic differential utilities of Duffie and Epstein (Econometrica 60:353–394, 1992). We assume that the asset prices follow interacting Itô-Poisson processes, which are known to be the so-called reaction–diffusion systems. Stochastic maximum principle for stochastic control problems described by some backward-stochastic differential equations that are driven by Poisson jump processes allows us to derive the optimal investment strategies as well as optimal consumption. We shall furthermore propose a numerical procedure for solving the associated nested quasi-linear partial differential equations.     

Is value creation consistent with currency hedging?

This paper analyzes value creation through currency hedging in the Spanish market. The results show that the hedging with derivatives generated an average premium of 1.53% and that foreign currency debt generated 7.52%, with respect to company value approximated by Tobin's Q, while operational hedging does not affect company value. Moreover, in half of the observations corresponding to companies that hedged with derivatives, the value premium was between 0.08% and 0.99%. In the case of foreign currency debt, the range was between 1.79% and 10.37%. It demonstrates that the contribution of currency hedging to company value fluctuates considerable according to the volume of financial hedging. Thus, an empirical study of this aspect which only analyses the decision to hedge through dummy variables to define financial hedging, as empirical previous studies, can lead to biased results in terms of estimated premium amounts, because it assumes a homogenous treatment of companies regardless of hedging volumes.     

Pseudospectral methods for pricing options

Models with two or more risk sources have been widely applied in option pricing in order to capture volatility smiles and skews. However, the computational cost of implementing these models can be large—especially for American-style options. This paper illustrates how numerical techniques called ‘pseudospectral’ methods can be used to solve the partial differential and partial integro-differential equations that apply to these multifactor models. The method offers significant advantages over finite-difference and Monte Carlo simulation schemes in terms of accuracy and computational cost.     

A Simple Induction Approach and an Efficient Trinomial Lattice for Multi-State Variable Interest Rate Derivatives Models

This paper presents an alternative approach for interest rate lattice construction in the Ritchken and Sankarasubramanian (1995) framework. The proposed method applies a parsimonious induction technique to represent the distribution of auxiliary state variables and value interest rate derivatives. In contrast to other approaches, this technique requires no numerical interpolations, approximations and iterative procedures for pricing interest rate options using a simple backward induction and, therefore, provides significant computational advantages and flexibility with respect to existing implementations. Also, the proposed trinomial interest rate lattice specification provides for a further reduction in computational costs with additional flexibility. The results of this work can be extended to a class of derivatives pricing models with path dependent state variables and generalized to multi-factor models.     

Analyzing determinants of bond yield spreads with Bayesian Model Averaging

This paper analyzes determinants of country default risk in emerging markets, reflected by sovereign yield spreads. The results reported so far in the literature are heterogeneous with respect to significant explanatory variables. This could indicate a high degree of uncertainty about the “true” regression model. We use Bayesian Model Averaging as the model selection method in order to find the variables which are most likely to determine credit risk. We document that total debt, history of recent default, currency depreciation, and growth rate of foreign currency reserves as well as market sentiments are the key drivers of yield spreads.