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/Merton's 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.
Hansen M, Miltersen KR. Minimum rate of return guarantees: the Danish case. Scand. Actuarial J. 2002; 4: 280 -318. We analyze minimum rate of return guarantees for life-insurance (investment) contracts and pension plans with a smooth surplus distribution mechanism. We speci cally model the smoothing mechanism used by most Danish life-insurance companies and pension funds. The annual distribution of bonus will be based on this smoothing mechanism after taking the minimum rate of return guarantee into account. In addition, based on the contribution method the customer will receive a nal (non-negative) undistributed surplus when the contract matures.We consider two different methods that the company can use to collect payment for issuing these minimum rate of return guarantee contracts: the direct method where the company gets a xed (percentage) fee of the customer's savings each year, e.g. 0.5% in Denmark, and the indirect method where the company gets a share of the distributed surplus. In both cases we analyze how to set the terms of the contract in order to have a fair contract between an individual customer and the company.Having analyzed the one-customer case, we turn to analyzing the case with two customers. We consider the consequences of pooling the undistributed surplus over two inhomogeneous customers. This implies setting up different mechanisms for distributing nal bonus (undistributed surplus) between the customers.
We derive a unified model that gives closed form solutions for caps and floors written on interest rates as well as puts and calls written on zero-coupon bonds. The crucial assumption is that simple interest rates over a fixed finite period that matches the contract, which we want to price, are log-normally distributed. Moreover, this assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility. CLOSED FORM SOLUTIONS FOR interest rate derivatives, in particular caps, floors,and bond options, have been obtained by a number of authors for Markovian term structure models with normally distributed interest rates or alternatively log-normally distributed bond prices (see, for example, Jamshidian (1989, 1991a); Heath, Jarrow, and Morton (1992); Brace and Musiela (1994); Geman, El Karoui, and Rochet (1995)). These models support Black-Scholes type formulas most frequently used by practitioners for pricing bond options and swaptions. Unfortunately, these models imply negative interest rates with positive probabilities, and hence they are not arbitrage free in an economy with opportunities for riskless and costless storage of money. Briys, Crouhy, and Schobel (1991) apply the Gaussian framework to derive closed form solutions for caps, floors, and European zero-coupon bond options. To exclude the influence of negative forward rates on the pricing of zero-coupon bond options, they introduce an additional boundary condition. As shown by Rady and Sandmann (1994) these pricing formulas are only supported by a term structure model with an absorbing boundary for the forward rate at zero, where the absorbing probability is not negligible, which for a term structure model is a quite problematic assumption. The Journal of FinanceAlternatively, modeling log-normally distributed interest rates avoids the problems of negative interest rates. However, as shown by Morton (1988) and Hogan and Weintraub (1993), these rates explode with positive probability, implying zero prices for bonds and hence also arbitrage opportunities. Furthermore, so far, no closed form solutions are known for these models.As observed by Sandmann and Sondermann (1994), the problems of exploding interest rates result from an unfortunate choice of compounding period of the interest rates modeled, namely the continuously compounding rate. Assuming that the continuously compounded interest rate is log-normally distributed results in "double exponential" expressions, i.e., the exponential function is itself an argument of an exponential function, thus giving rise to infinite expectations of the accumulation factor and of inverse bond prices under the martingale measure. The problem disappears as shown in Sandmann and Sondermann (1994) if, instead of assuming that the continuously compounded interest rates are log-normally distributed, one assumes that simple interest rates over a fixed finite period are log-normally distributed. In practice, interest rates, both spot and forward, are quoted as simple rates per annum (yea...
Abstract. 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 customer's and the insurer's fractions of the positive excess return are determined so that the market value of the insurer's capital inflow (determined by the fraction of the positive excess return) equals the market value of the insurer's 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 customer's 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.
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