Three essays on the valuation of embedded derivatives in financial contracts.

摘要

The financial markets have undergone paramount deregulations over the past decades: allowing the exchange rates to float, freeing the interest rates and decontrol of commodity markets, which led to increased volatility throughout all these markets. As a result, derivatives have become increasingly important as a means of managing and mitigating the financial risks created by these volatilities. The latest estimates from Bank of International Settlements (2004) suggest that the notional amount of the global derivatives market account for {dollar}270 trillion.; Derivative contracts are either traded on an exchange or off exchanges (over-the-counter OTC). The exchange-traded derivatives are usually standardized contracts, by doing so, the exchanges aggregate a large number of participants in order to build liquidity in these contracts. On the other hand, the participants on OTC markets seek more customized-instruments whose risk-return characteristics closely meet their individual needs. Along with all these stand-alone derivative contracts, many derivatives are bundled with more standard financial contracts (a host or base contract), from which they cannot be separated.; The embedded derivative arise from deliberate financial engineering and so intentionally shifting of certain risks between parties. Indeed, by adding these special features in the base contracts, we create a ""hybrid instrument"", whose terms (such as the price, maturity, size or even the underlyings) are different from the host contract. As a matter of fact, the whole cashflow structure and the risk profiles of the two vehicles are different.; An example of host contracts that are very often structured with embedded options are the fixed income securities. Indeed, these debt vehicles are usually leveraged by including caps (call provisions), floors (put provisions) and equity conversion features, hence creating an equity-interest rate indexed financial instrument. The global market of convertible bonds accounted for {dollar}560 billion (Morgan Stanley as of January 2002), and many arbitrage hedge funds are among the active players on this market which contributes to its liquidity. The convertible bonds represent an alternative financing channel essentially for growth companies. Indeed, for those companies that have not reached yet a stable cash and revenue generation, are faced with high financing costs due to their associated risk. On the other hand, the high growth potential they offer is reflected in all their equity-related securities. Thus, the convertible bonds allow them to access the bond market at significantly lower financing cost by giving the bondholders the flexibility to switch into share-holders and benefit from future growth potentials. The valuation of convertible bonds along with other embedded options is challenging since it requires at least three underlying factors to be taken into account: the interest rate dynamics, the default probability and the equity value. The first and second essays present different valuation models for convertible bonds with call and put provisions. Under each framework, we propose different numerical methods for solving efficiently these pricing problems.; In the first essay, we propose a dynamic programming formulation for pricing Bermudan embedded options in a bond. Three provisions are considered: the conversion, the call and the put options. We investigate three numerical approaches for solving the corresponding stochastic dynamic programming problem. In the first approach, we approximate the value function piecewise and solve the corresponding approximated Bellman equation in closed form. The second approach is a combination of piecewise interpolation and Quasi-Monte Carlo integration and the third approach is based on a polynomial fitting of the value function. Based on our numerical investigations, the two first techniques prove to be numerically efficient and stable in pricing the embedded options. The po