Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal the Black and Scholes framework seems to be inappropriate for such long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions

Pricing and Hedging GLWB and GMWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

La valutazione di polizze assicurative Variable Annuities di tipo GLWB e GMWB ha attratto l'attenzione sia del mondo accademico, sia di quello finanziario. Diversi precedenti lavori di ricerca hanno segnalato che il modello di Black e Scholes risulta inadatto per prodotti con una maturit\ue0 cos\uec lunga. La Tesi propone di utilizzare un modello a volatilit\ue0 stocastica (modello di Heston) e un modello di Black-Scholes con tasso d'interesse stocastico (modello di Hull-White). A tal proposito la Tesi presenta quattro metodi numerici per la valutazione della Variable Annuities di tipo GLWB e GMWB: un metodo ibrido che coniuga metodi ad albero e metodi con equazioni alle derivate parziali, un metodo ibrido che utilizza metodi ad albero e metodi Monte Carlo, un metodo ADI con differenze finite, ed un metodo Monte Carlo Standard. Questi metodi sono utilizzati per determinare il costo in un mercato provo di arbitraggi per le versioni pi\uf9 popolari delle due polizze, e inoltre per calcolare le Greche utilizzate per la copertura. Le strategie del cliente considerate sono sia di tipo costante, sia di tipo dinamico (inclusa la recessione totale). Sono inoltre presentati risultati numerici che dimostrano la sensibilit\ue0 del valore delle polizze alle ipotesi di natura economica, contrattuale e demografica.Evaluation of Variable Annuity insurance policies GLWB and GMWB type has attracted the attention of both the academic world and real world financial markets. Several previous research studies have reported that the Black-Scholes model is unsuitable for products with such a long maturity. The Thesis proposes to use a stochastic volatility model (Heston model) and a Black-Scholes model with stochastic interest rate (Hull-White model). In this regard, the Thesis presents four numerical methods for the evaluation of the GLWB Variable Annuity and GMWB type: a hybrid method that combines tree methods and methods with partial differential equations, a hybrid method using tree methods and Monte Carlo methods, an ADI method with finite differences, and a standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB and GMWB contracts, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions

Fractals and the lead crack airframe lifing framework

Abstract The physically short crack regime is primary region of interest in the design and sustainment of highly optimised metallic aircraft. The authors have previously shown that by characterising a fracture surface using fractals concept produces a crack growth model similar to that first proposed by Frost and Dugdale in 1958. This provides a scientific basis to the crack growth model. Further investigations revealed that for short cracks these models predict that crack growth is exponentially related to the applied load history. This observation has led to a practical aircraft lifing approach applicable to the short crack regime known as the lead crack framework. This paper summarises the fractality of metallic fracture surfaces, presents examples of the crack growth behaviour in complex structures, and summarises some useful crack growth tools

Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

International audienceIn this paper we propose an efficient method to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski's approach [33], we implement here a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is obtained by means of Gaussian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but it is not accurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the proposed algorithm is fast and reliable, and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality

TAXATION OF A GMWB VARIABLE ANNUITY IN A STOCHASTIC INTEREST RATE MODEL

Modeling taxation of Variable Annuities has been frequently neglected, but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity. In particular, they consider the simple Black\u2013Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long-term products, and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policyholder behavior, as observed by Gouden\ue8ge et al. (2018). In this paper, we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement

Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

In this paper, we propose an efficient method for computing the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. We consider specif- ically options written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski’s approach, we im- plement here a backward dynamic programming algorithm, based on a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is in turn obtained by means of Gaus- sian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but remains inaccurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, allowing us to treat very large baskets while reducing the variance of price estimators. Numerical tests demonstrate that the proposed algorithm is fast and reli- able, and able to handle American options on very large baskets of assets as well, thus overcoming the problem of the curse of dimensionality