Emilio Russo

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 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.

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.

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 Flexible Lattice Model for Pricing Contingent Claims under Multiple Risk Factors

The original binomial model is an easy-to-apply approximation procedure for valuing options under Black-Scholes assumptions. There is a single stochastic factor and the volatility, interest rate, and other parameters are known. However, weakening those assumptions typically produces a non-recombining lattice that blows up asymptotically when the number of time steps is increased for closer replication of the underlying continuous-time process. Various extensions of the basic lattice structure have been developed over the years. In this article, Russo and Staino provide a very general lattice model in the form of a “forward-shooting grid” that can handle three correlated risk factors: volatility, interest rate, and stock price. An innovation in the model is that volatility and the riskless interest rate are the primary state variables, while the asset price (whose returns process is a function of both volatility and the riskless rate) is treated as an auxiliary variable. The lattice determines the possible evolution of the volatility and interest rate, and the stock price is carried along as a set of possible values falling into discrete buckets at each node. The trivariate branching structure is represented in a lattice that allows eight branches from each node. This accommodates many of the standard continuous-time models, including non-zero correlation among the stochastic factors. A simulation exercise shows striking improvement in performance relative to earlier models in the literature.

A bivariate model for evaluating equity-linked policies with surrender option

This article proposes a bivariate lattice model for evaluating equity-linked policies embedding a surrender option when the underlying equity dynamics is described by a geometric Brownian motion with stochastic interest rate. The main advantage of the model stays in that the original processes for the reference fund and the interest rate are directly discretized by means of lattice approximations, without resorting to any additional transformation. Then, the arising lattices are combined in order to establish a bivariate tree where equity-linked policy premiums are computed by discounting the policy payoff over the lattice branches, and allowing early exercise at each premium payment date to model the surrender decision.

On Pricing Asian Options under Stochastic Volatility

American exercise presents difficulties for option valuation. For an in-the-money option, it becomes necessary at each point in time to consider whether to exercise or to hold on, based on expectations about the stock price at option expiration and also about the optionality value of potential exercise on every date when exercise will be possible in the future. An “Asian” option payoff presents its own valuation problems because the payoff is based on the arithmetic average of correlated lognormal prices, which is not lognormal. Adding stochastic volatility makes both of these problems much harder. But in this article, Russo and Staino are able to develop a lattice technique that deals with all three of these difficulties. The stochastic variable that the tree is built around is the variance, while both the current asset price and the running average of past prices to be included in calculating the payoff at expiration are carried along as auxiliary variables, in the form of sets of values at each node that span the range of possible values along all of the paths that lead to that node. Although there are several approximations in the procedure, accuracy is excellent, and execution is relatively fast.

A moment-matching method to generate arbitrage-free scenarios

We propose a new moment-matching method to build scenario trees that rule out arbitrage opportunities when describing the dynamics of financial assets. The proposed scenario generator is based on the monomial method, a technique to solve systems of algebraic equations. Extensive numerical experiments show the accuracy and efficiency of the proposed moment-matching method when solving financial problems in complete and incomplete markets.

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.