Kristian R. Miltersen

Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates

We develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields. In the development of the model, we distinguish between forward and future convenience yields, a distinction that has not been recognized in the literature. Assuming normality of continuously compounded forward interest rates and convenience yields and log-normality of the spot price of the underlying commodity, we obtain closed-form solutions generalizing the Black-Scholes/Mertons formulas. We provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a European call option and the underlying futures contract have significant impact on the option prices.

Guaranteed Investment Contracts: Distributed and Undistributed Excess Return

Annual minimum rate of return guarantees are analyzed together with rules for distribution of positive excess return, i.e. investment returns in excess of the guaranteed minimum return. Together with the level of the annual minimum rate of return guarantee both the customers and the insurers fractions of the positive excess return are determined so that the market value of the insurers capital inflow (determined by the fraction of the positive excess return) equals the market value of the insurers capital outflow (determined by the minimum rate of return guarantee) at the inception of the contract. The analysis is undertaken both with and without a surplus distribution mechanism. The surplus distribution mechanism works through a bonus account that serves as a buffer in the following sense: in (‘bad’) years when the investment returns are lower than the minimum rate of return guarantee, funds are transferred from the bonus account to the customers account. In (‘good’) years when the investment returns are above the minimum rate of return guarantee, a part of the positive excess return is credited to the bonus account. In addition to characterizations of fair combinations of the level of the annual minimum rate of return guarantee and the sharing rules of the positive excess return, our analysis indicates that the presence of a surplus distribution mechanism allows the insurer to offer a much wider menu of contracts to the customer than without a surplus distribution mechanism.

Pricing rate of return guarantees in a Heath-Jarrow-Morton framework

Abstract Rate of return guarantees, included in many financial products, exist in two fundamentally different types. Maturity guarantees which are binding only at the expiration of the contract, and therefore, similar to financial options and multi-period guarantees which have the time to expiration divided into several subperiods with a binding guarantee for each subperiod. Relevant real-life examples are life insurance contracts and guaranteed investment contracts. We consider rate of return guarantees where the underlying rate of return is either (i) the rate of return on a stock investment or (ii) the short-term interest rates. Various types of these rate of return guarantees are priced in a general no-arbitrage Heath–Jarrow–Morton framework. We show that despite fundamental differences in the underlying rate of return processes ((i) or (ii)), the resulting pricing formulas for the guarantees are remarkably similar for maturity guarantees. For multi-period guarantees the presence of stochastic interest rates leads to intertemporal dependencies which complicates the valuation formulaes compared both to the case of maturity guarantees and the case of deterministic interest rates. Finally, we show how the term structure models of Vasicek (Vasicek, O., 1977. Journal of Financial Economics 5, 177–188) and Cox et al. (Cox, J.C., Ingersoll, Jr., J.E., Ross, S.A., 1985. Econometrica 53(2), 385–407) occur as special cases in our more general framework based on the model of Heath et al. (Heath, D., Jarrow, R.A., Morton, A.J., 1992. Econometrica 60 (1), 77–105).

Dynamic Spanning in the Consumption-Based Capital Asset Pricing Model

Under the assumptions of the Consumption-based Capital Asset Pricing Model (CCAPM), Pareto optimal consumption allocations are characterized by each agents consumption process being adapted to the filtration generated by the aggregate consumption process of the economy. The wealth processes of the agents, however, are adapted to the finer filtration generated by aggregate consumption and the conditional distribution of future aggregate consumption. Therefore, in order to achieve pareto optimal consumption allocations, a sufficiently varied set of assets must exist such that any wealth process adapted to this finer filtration can be implemented by dynamically trading in that set of assets. We provide sufficient conditions for the existence of such a set of assets based on dynamically trading contingent claims on aggregate consumption. In addition, we give sufficient conditions for the existence of equilibria in a dynamically effectively complete market in which agents are only able to trade in contingent claims on aggregate consumption, the market portfolio of firms, and a (numeraire) zero-coupon bond. We demonstrate the role of short- and long-term contingent claims on aggregate consumption for the implementation of Pareto optimal allocations inthe presence of short- and long-term risks. In addition, in the presence of personal risks, we demonstrate the role of insurance contracts. JEL Classification: G13.

Second Mortgages: Valuation and Implications for the Performance of Structured Financial Products

We provide an analytic valuation framework to value second mortgages and first lien mortgages when owners can take out a second lien. We then use the framework to value mortgage-backed securities (MBS) and, in particular, quantify the greater risk associated with MBS backed by first liens that have silent seconds"". Rating securities without accounting for the equity extraction option results in much higher ratings than warranted by expected loss. While the senior tranche’s rating should be A1 rather than Aaa in our benchmark calibration, the big losers from the equity extraction option are the mezzanine tranches who get wiped out.

Second Mortgages: Valuation and Implications for the Performance of Structured Financial Products: Second Mortgages

We provide an analytic valuation framework to value second mortgages and first lien mortgages when owners can take out a second lien. We then use the framework to value mortgage-backed securities (MBS) and, in particular, quantify the greater risk associated with MBS backed by first liens that have silent seconds"". Rating securities without accounting for the equity extraction option results in much higher ratings than warranted by expected loss. While the senior tranche’s rating should be A1 rather than Aaa in our benchmark calibration, the big losers from the equity extraction option are the mezzanine tranches who get wiped out.