Pricing equity and debt tranches of collateralized funds of hedge fund obligations: An approach based on stochastic time change and Esscher-transformed martingale measure

Abstract

Abstract. Collateralized Funds of Hedge Fund Obligations (CFOs) are relatively recent structured finance products linked to the performance of underlying funds of hedge funds. The capital structure of a CFO is similar to traditional Collateralized Debt Obligations (CDOs), meaning that investors are offered different rated notes and equity interest. CFOs are structured as arbitrage market value CDOs. The fund of funds manager actively manages the fund to maximize total return while limiting price volatility within the guidelines of the structure. The aim of this paper is to provide a useful framework to evaluate Collateralized Funds of Hedge Fund Obligations, that is pricing the equity and the debt tranches of a CFO. The basic idea is to evaluate each CFO liability as an option written on the underlying pool of hedge funds. The value of every tranche depends on the evolution of the collateral portfolio during the life of the contract. Care is taken to distinguish between the fund of hedge funds and its underlying hedge funds, each of which is itself a portfolio of various securities, debt instruments and financial contracts. The proposed model incorporates skewness, excess-kurtosis and is able to capture more complex dependence structures among hedge fund log-returns than the mere correlation matrix. With this model it is possible to describe the impact of an equivalent change of probability measure not only on the marginal processes but also on the underlying dependence structure among hedge funds.

Valuation of Collateralized Funds of Hedge Fund Obligations: A Basket Option Pricing Approach

“…Due to jumps size correlation, this process is more flexible in modelling dependence among hedge funds compared to the one presented in [9]. In particular,…”

“…To compute the Characteristic function (Cf) of the MVG process we follow the same procedure of [9]. The Laplace Exponent of the Gamma subordinator is…”

“…Modelling dependence in this way allows to introduce two sources of co-movement among the NAV of different hedge funds. First, the use of a common stochastic clock introduces a new business time, in which all the market operates, it means all prices jump simultaneously [2,3,6,9]. Secondly, jump sizes are correlated [2,5,8].…”

“…To evaluate dynamically the collateral portfolio NAV it is necessary to model the joint risk neutral evolution of the underlying hedge funds and at the same time any CFO's structural features like coupon payments, over collateralization test, liquidity profile, equity distribution rules, management fees and so on have to be taken into account. In [9] the physical dependence among hedge fund log-returns is introduced through a Gamma stochastic time change of a Multivariate Brownian motion with drift, with independent components [2,6]. From a methodological stand point, the main limitation of that model lies in a low degree of flexibility: in particular, it is not able to replicate the correlations observed in the market.…”

“…In a recent contribution Tassinari and Corradi [9] developed a model to price equity and debt tranches of a Collateralized fund of hedge fund Obligations (CFO). The basic idea developed therein is to compute the fair price of each tranche as its risk neutral expected payoff, discounted at the risk free rate.…”