A parsimonious model for generating arbitrage-free scenario trees

Abstract

Simulation models of economic, financial and business risk factors are widely used to assess risks and support decision-making. Extensive literature on scenario generation methods aims at describing some underlying stochastic processes with the least number of scenarios to overcome the 'curse of dimensionality'. There is, however, an important requirement that is usually overlooked when one departs from the application domain of security pricing: the no-arbitrage condition. We formulate a moment matching model to generate multi-factor scenario trees for stochastic optimization satisfying no-arbitrage restrictions with a minimal number of scenarios and without any distributional assumptions. The resulting global optimization problem is quite general. However, it is non-convex and can grow significantly with the number of risk factors, and we develop convex lower bounding techniques for its solution exploiting the special structure of the problem. Applications to some standard problems from the literature show that this is a robust approach for tree generation. We use it to price a European basket option in complete and incomplete markets.

Pricing and Hedging GDP-linked Bonds in Incomplete Markets

“…We calibrate the arbitrage-free tree of Consiglio et al (2016a) for each country using asset returns from the Dimson-Marsh-Staunton Global Returns Data ( Dimson et al, 2002 ) and GDP data from Schularick and Taylor (2012) . For each country we use as traded assets its own T-bills, bonds and equity indices, plus the World bills, bonds and equity indices in local currency.…”

Pricing and hedging GDP-linked bonds in incomplete markets

“…We calibrate the arbitrage-free tree of Consiglio et al (2016a) for each country using asset returns from the DimsonMarsh-Staunton Global Returns Data ( Dimson et al, 2002 ) and GDP data from Schularick and Taylor (2012) . For each country we use as traded assets its own T-bills, bonds and equity indices, plus the World bills, bonds and equity indices in local currency.…”

“…Let us assume that we are supplied with a scenario tree endowed with both risk neutral Q and objective P probability measures, such as the one of Consiglio et al (2016a) , and let S 1 denote the GDP level of a given country. If the market is complete, then we can price any claim with payoffs contingent on the GDP through some function ( S 1 ), by discounting its expected payoff under Q .…”

Contingent Convertible Bonds for Sovereign Debt Risk Management

“…There have been significant advances in the calibration of scenario trees to match market observed moments for multiple risk factors for use in stochastic programming (Høyland and Wallace 2001;Klaassen 2002). Consiglio, Carollo, and Zenios (2016a) developed an arbitrage-free calibration procedure to match an arbitrary number of moments and obtain both risk neutral and objective probabilities, and is well suited for fitting trees to financial (risk free rates, CDS spreads), macroeconomic (GDP growth), and fiscal (primary balance) variables, as needed for our model.…”