Victor Vaugirard

Pricing catastrophe bonds by an arbitrage approach

Abstract This paper develops a simple arbitrage approach to valuing insurance-linked securities, which accounts for catastrophic events and interest rate randomness, notwithstanding a framework of non-traded underlyings. It shows that holders of catastrophe bonds are in a short position on one-touch binary options based upon risk-tracking indexes that obey jump-diffusion processes. Using first-passage time distributions, this contribution provides a closed-form valuation expression in the context of pure crashes, while it resorts to numerical simulations in the case of mid-range catastrophes. Comparative statics results point out that the term structure of yield spreads of catastrophe bonds is hump-shaped as for corporate bonds.

Valuing catastrophe bonds by Monte Carlo simulations

This paper reports fairly accurate simulations of insurance-linked securities within an arbitrage-free framework, while accounting for catastrophic events and allowing for stochastic interest rates. Assessing these contingent claims exhibits features of instability rooted in the discontinuity of the payoffs of binary options around their threshold, which is magnified by possible jumps in their underlying dynamics. The error made while simulating path-dependent digital options whose underlyings obey geometric Brownian motion is used to control the estimation of digital options whose underlyings follow jump-diffusion processes. Comparative statics results highlight the hump shape of the term structure of catbond yield spreads.

The valuation of nature-linked bonds with exchange rate risk

This paper develops an arbitrage approach to pricing insurance bonds that bear currency risk. Bondholders are shown to have a short position on path-dependent digital options written on risk-tracking indices. It implements the technique of forward-neutral change of numeraire and comes down to computing first-passage-time distributions of drifted Brownian motions. We derive closed-form formulas or perform simulations depending on whether interest rates are deterministic or stochastic. Along the way, we evaluate outside-barrier currency call options. Then this research studies the effects of both nature risk and exchange rate uncertainty and shows that the former is more significant than the latter.

Monte Carlo applied to exotic digital options

This paper tailors Monte Carlo simulations to the scope of binary options whose underlying dynamics obey jump-diffusion or jump-mean-reverting processes and may not be traded. In the process, the existence of well-defined arbitrage prices is justified notwithstanding a framework of incomplete markets. The all-or-nothing feature of digital options makes simulations unstable in the vicinity of their threshold, which entails the implementation of variance reduction techniques. An extension to stochastic interest rates highlights the fact that probabilistic techniques and simulations can be married to further improve the accuracy of the estimations.

FIRST PASSAGE TIMES FOR RISK-TRACKING PROXIES

This paper determines first passage time distributions with a two-fold emphasis. The focus is first set on interest rate randomness. It derives a closed-form solution in the case of moving boundaries, indexed on risk-free bonds, and where interest rates obey mean-reverting processes and underlyings follow lognormal diffusion processes. It turns next to the underlyings, which may not be exchange-traded and whose dynamics obey jump-diffusion processes. It builds an equilibrium valuation framework and determines the rational-expectations equilibrium price of digital options. As those underlyings may be risk-tracking indices, the article can be applied to pricing insurance-linked securities, such as catastrophe bonds.

Hitting Time and Time Change

This paper determines first‐passage time distributions with a twofold emphasis on the dynamics of the state variables and interest rate uncertainty. Underlyings follow two‐dimensional geometric Brownian motions, Ornstein–Uhlenbeck processes or Poisson jump‐diffusion processes, and boundaries are either fixed or indexed on risk‐free bonds. Forward‐neutral changes of numeraire enable one to derive generic valuation expressions, while changing time allows one to determine closed‐form solutions for geometric Brownian motions and moving barriers. In turn, the latter formulas are used to reduce the variance of Monte Carlo simulations in the case of jump‐diffusion processes, by means of the control variate method.