Massimo Costabile

Option pricing under regime-switching jump-diffusion models

We present an explicit formula and a multinomial approach for pricing contingent claims under a regime-switching jump-diffusion model. The explicit formula, obtained as an expectation of Merton-type formulae for jump-diffusion processes, allows to compute the price of European options in the case of a two-regime economy with lognormal jumps, while the multinomial approach allows to accommodate an arbitrary number of regimes and a generic jump size distribution, and is suitable for pricing American-style options. The latter algorithm discretizes log-returns in each regime independently, starting from the highest volatility regime where a recombining multinomial lattice is established. In the remaining regimes, lattice nodes are the same but branching probabilities are adjusted. Derivative prices are computed by a backward induction scheme.

A Simplified Approach to Approximate Diffusion Processes Widely Used in Finance

We propose a simplified approach to approximate a variety of heteroskedastic diffusions widely used in finance to describe the evolution of state variables such as equity prices, short interest rates, and others. In contrast to the common approach based on approximating a new homoskedastic process obtained by transforming the original heteroskedastic one, we build up binomial and trinomial trees that directly discretize the initial process. Despite this, the proposed approximation models are based on recombining lattices that converge weakly to the corresponding limiting diffusion. Numerical results show that the proposed algorithms are efficient and that they compute accurate prices.

A Forward Shooting Grid Method for Option Pricing with Stochastic Volatility

One of the most common sources of path dependency in derivatives arises when the volatility is stochastic. This is apparent in the basic binomial model, where time-varying volatility causes the lattice to splinter rather than recombine, leading to n 2 different nodes at the nth time step instead of n + 1 nodes in a tree that recombines. Various methods have been developed to deal with that problem within a lattice framework, by constructing three-dimensional lattices with both the price and the volatility as state variables. An alternative technique is the forward shooting grid that builds a lattice for the stock price and carries along a set of possible values for the volatility at each price node as auxiliary variables. But both of those approaches can run into problems with negative transition probabilities and difficulty in achieving the right correlation between returns and volatility changes. In this article, Costabile, Massabó, and Russo develop a different forward shooting grid approach, in which the squared volatility is the primary path-dependent variable and stock prices are carried along as the auxiliary variables. Negative transition probabilities are avoided, and the procedure produces highly accurate results very efficiently in a compact tree.

A lattice-based model to evaluate variable annuities with guaranteed minimum withdrawal benefits under a regime-switching model

We consider the problem of evaluating variable annuities with a guaranteed minimum withdrawal benefit under a regime-switching model. We propose a trinomial lattice model to approximate the evolution of the investment fund value and the policy value at inception is computed through a backward induction scheme. Finally, the insurance fee is computed as the solution of the equation that makes the contract actuarially fair. Numerical results are reported to illustrate the consistency of the proposed model.

Computing finite-time survival probabilities using multinomial approximations of risk models

We consider the problem of computing finite-time survival probabilities for various risk models. We develop an approximating discrete-time multinomial lattice that mimics the evolution of the corresponding continuous risk process. A simple recursive algorithm to compute survival probabilities is described. Numerical results show that the proposed scheme yields accurate values in all the considered cases.

La valutazione di opzioni implicite nei mutui bancari

We analyze the problem of pricing implicit options embedded in mortgages and provide a general framework which can be extended to other types of implicit contingent claims. In particular, we deal with the problem of pricingprepaymentoptions andcap for floating rate mortgages.

Evaluating Variable Annuities with GMWB When Exogenous Factors Influence the Policy-Holder Withdrawals

We propose a model for evaluating variable annuities with guaranteed minimum withdrawal benefits in which a rational policy-holder, who would withdraw the optimal amounts maximizing the current policy value only with respect to the endogenous variables of the evaluation problem, acts in a more realistic context where her/his choices may be influenced by exogenous variables that may lead to withdraw sub-optimal amounts. The model is based on a trinomial approximation of the personal sub-account dynamics that, despite the presence of a downward jump due to the payed withdrawal at each anniversary of the contract, guarantees the reconnecting property. A backward induction scheme is used to compute the insurance fair fee paid for the guarantee.

A Path-Independent Humped Volatility Model for Option Pricing

Abstract This article presents a path-independent model for evaluating interest-sensitive claims in a Heath–Jarrow–Morton (1992, Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 60, pp. 77–105) framework, when the volatility structure of forward rates shows the deterministic and stationary humped shape analysed by Ritchken and Chuang (2000, Interest rate option pricing with volatility humps, Review of Derivatives Research, 3(3), pp. 237–262). In our analysis, the evolution of the term structure is captured by a one-factor short rate process with drift depending on a three-dimensional state variable Markov process. We develop a lattice to discretize the dynamics of each variable appearing in the short rate process, and establish a three-variate reconnecting tree to compute interest-sensitive claim prices. The proposed approach makes the evaluation problem path-independent, thus overcoming the computational difficulties in managing path-dependent variables as it happens in the Ritchken–Chuang (2000, Interest rate option pricing with volatility humps, Review of Derivatives Research, 3(3), pp. 237–262) model.