Antonino Zanette

Pricing and hedging American options by Monte Carlo methods using a Malliavin calculus approach

Following the pioneering papers of Fournié, Lasry, Lebouchoux, Lions and Touzi, an important work concerning the applications of the Malliavin calculus in numerical methods for mathematical finance has come after. One is concerned with two problems: computation of a large number of conditional expectations on one hand and computation of Greeks (sensitivities) on the other hand. A significant test of the power of this approach is given by its application to pricing and hedging American options. The paper gives a global and simplified presentation of this topic including the reduction of variance techniques based on localization and control variables. A special interest is given to practical implementation, number of numerical tests are presented and their performances are carefully discussed.

New insights on testing the efficiency of methods of pricing and hedging American options

Abstract With reference to the evaluation of the speed–precision efficiency of pricing and hedging of American Put options, we present and discuss numerical results obtained on the basis of four different large enough random samples according to the relevance of the American quality (relative importance of the early exercise opportunity) of the options. Here we provide a comparison of the best methods (lattice based numerical methods and an approximation of the American Premium analytical procedure) known in literature along with some key methodological remarks.

Monte Carlo methods for pricing and hedging American options in high dimension

We numerically compare some recent Monte Carlo algorithms devoted to the pricing and hedging American options in high dimension. In particular, the comparison concerns the quantization method of Barraquand-Martineau and an algorithm based on Malliavin calculus. The (pure) Malliavin calculus algorithm improves the precision of the computation of the delta but, merely for pricing purposes, is uncompetitive with respect to other Monte Carlo methods in terms of computing time. Here, we propose to suitably combine the Malliavin calculus approach with the Barraquand-Martineau algorithm, using a variance reduction technique based on control variables. Numerical tests for pricing and hedging American options in high dimension are given in order to compare the different methodologies.

Pricing Ratchet Equity-Indexed Annuities with Early Surrender Risk in a CIR++ Model

In this article we propose a lattice algorithm for pricing simple Ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR++ stochastic interest rates. In addition we present an asymptotic expansion technique that permits us to obtain a first-order approximation formula for the price of simple Ratchet EIAs without early surrender risk and without a global minimum contract value. Numerical comparisons show the reliability of the proposed methods.

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 (2014) the Black and Scholes framework seems to be inappropriate for such a 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 cliquet options by tree methods

This paper focuses on the problem of pricing the cliquet options which provide a guaranteed minimum annual return. The tree method which we propose simplifies the standard binomial Cox–Ross–Rubinstein approach which, in this context, is problematic from a computational point of view. Our technique provides very efficient and reliable evaluations in a Black-Scholes framework with piecewise constant interest rates and volatilities.

A hybrid tree/finite-difference approach for Heston-Hull-White type models

We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods

Efficient pricing of swing options in Lévy-driven models

We consider the problem of pricing swing options with multiple exercise rights in Lévy-driven models. We propose an efficient Wiener–Hopf factorization method that solves multiple parabolic partial integro-differential equations associated with the pricing problem. We compare the proposed method with a finite difference algorithm. Both proposed deterministic methods are related to the dynamic programming principle and lead to the solution of a multiple optimal stopping problem. Numerical examples illustrate the efficiency and the precision of the proposed methods.

Pricing and hedging GMWB in the Heston and in the Black–Scholes with stochastic interest rate models

In this paper, we approach the problem of valuing a particular type of variable annuity called GMWB when advanced stochastic models are considered. As remarked by Yang and Dai (Insur Math Econ 52(2):231–242, 2013), and Dai et al. (Insur Math Econ 64:364–379, 2015), the Black–Scholes framework seems to be inappropriate for such a long maturity products. Also Chen et al. (Insur Math Econ 43(1):165–173, 2008) show that the price of GMWB variable annuities is very sensitive to the interest rate and the volatility parameters. We propose here to use a stochastic volatility model (the Heston model) and a Black–Scholes model with stochastic interest rate (the Black–Scholes Hull–White model). For this purpose, we consider four numerical methods: a hybrid tree-finite difference method, a hybrid tree-Monte Carlo method, an ADI finite difference scheme and a Standard Monte Carlo method. These approaches are employed to determine the no-arbitrage fee for a popular version of the GMWB contract and to calculate the Greeks used in hedging. Both constant withdrawal and dynamic withdrawal strategies are considered. Numerical results are presented, which demonstrate the sensitivity of the no-arbitrage fee to economic and contractual assumptions as well as the different features of the proposed numerical methods.

Fast binomial procedures for pricing Parisian/ParAsian options

The discrete procedures for pricing Parisian/ParAsian options depend, in general, on three dimensions: time, space, time spent over the barrier. Here we present some combinatorial and lattice procedures which reduce the computational complexity to second order. In the European case the reduction was already given by Lyuu and Wu (Decisions Econ Finance 33(1):49–61, 2010) and Li and Zhao (J Deriv 16(4):72–81, 2009), in this paper we present a more efficient procedure in the Parisian case and a different approach (again of order 2) in the ParAsian case. In the American case we present new procedures which decrease the complexity of the pricing problem for the Parisian/ParAsian knock-in options. The reduction of complexity for Parisian/ParAsian knock-out options is still an open problem.