Put option

Abstract: This paper provides simple, analytic approximations for pricing exchange-traded American call and put options written on commodities and commodity futures contracts. These approximations are accurate and considerably more computationally efficient than finite-difference, binomial, or compound-option pricing methods. OPTIONS WRITTEN ON A wide variety of commodities and commodity futures contracts' now trade in the U.S. and Canada. Nearly all these options are American style2 and thus have early exercise premiums implicitly embedded in their prices. Unlike the European-style option-pricing problems, however, analytic solutions for the American option-pricing problems have not been found, and the pricing of American options has usually resorted to finite-difference, binomial, or, more recently, compound-option approximation methods. While these approximation methods yield accurate American option values, they are cumbersome and expensive to use. The purpose of this paper is to provide an accurate, inexpensive method for pricing American call and put options written on commodities and commodity futures contracts. The development of the "quadratic" approximation method is contained in Section I. Commodity option and commodity futures option contracts are defined, the underpinnings of commodity option valuation are discussed, and the solutions to the European call and put option-pricing problems are presented. Unlike the non-dividend-paying stock option case, it is shown that the American call option written on a commodity, as well as the American put option, may optimally be exercised prior to expiration. The approximation methods for the American call and put option values are then derived in the

Abstract: This paper presents a methodology for arriving at empirical estimates of deposit insurance premiums from market data by using isomorphic relationships betweeen equity and a call option, and insurance and a put option. The data utilizes the market value of equity to solve for the asset value and its volatility. Market perceptions of FDIC bailout policies are explicitly modeled so as to eliminate the bias in inverted values of assets and their volatility. Sensitivity analyses are performed to show that rank orderings based on premiums are robust to changes in specification, thus facilitating allocation of aggregate premium across banks. WHILE ECONOMISTS HAVE LONG argued in favor of risk-adjusted deposit insurance as both more equitable and more efficient than the current system of flatrate premiums, various recent developments have further contributed to an increasing dissatisfaction with the current system. First, both the banking industry and the government seem to be tending to the view that deregulation of the banking industry would be necessary in order to meet more sophisticated future demands on the industry as well as desirable as a policy means of stimulating greater competition among banks. Moreover, a sudden rise in the incidence of bank failures,1 and the vulnerability of the U.S. banks to the socalled international debt crisis have served to bring to the fore concern about the health of the banking industry. In the absence of deposit insurance, riskier banks will be able to attract deposits only at higher rates, and these higher costs of funding serve as built-in market-regulated incentives to limit excessive risk-taking by banks. As introduction of deposit insurance makes deposits equally risk-free across banks, these incentives disappear, and regulation and close supervision of the banking industry must necessarily replace them as deterrents to excessive risk-taking. Thus, when insurance is offered at a flat premium, regulation is designed to ensure that the risk posed to the insurer-both asset and financial risk-is appropriately uniform

Abstract: As applied to the behavior of homeowners with mortgages, option theory predicts that mortgage prepayment or default will be exercised if the call or put option in in the money by some specific amount. Our analysis: tests the extent to which the option approach can explain default and prepayment behavior; evaluates the practical importance of modeling both options simultaneously; and models the unobserved herterogeneity of borrowers in the home mortgage market. The paper presents a unified model of the competing risks of mortgage termination by prepayment and default, considering the two hazards as dependent competing risks which are estimated jointly. It also accounts for the unobserved heterogeneity among borrowers, and estimates the unobserved heterogeneity simultaneously with the parameters and baseline hazards associated with prepayment and default functions. Our results show that the option model, in its most straightforward version, does a good job of explaining default and prepayment; but it is not enough by itself. The simultaneity of the options is very important empirically in explaining behavior. The results also show that there exists significant heterogeneity among mortgage borrowers. Ignoring this heterogeneity results in serious errors in estimating the prepayment behavior of homeowners.

Abstract: An analytic solution to the American put problem is derived herein. The hedge ratio and other derivatives of the solution are presented. The formula derived implies an exact duplicating portfolio for the American put consisting of discount bonds and stock sold short. The formula is extended to consider put options on stocks paying cash dividends. A polynomial expression is developed for evaluating these formulae. Values and hedge ratios for puts on both dividend and nondividend paying stocks are calculated, tabulated, and compared with values derived by numerical integration and binomial approximation. As with European options, evaluating an analytic formula is more efficient than approximating the stock price process or the partial differential equation by binomial or finite difference methods. Finally, applications of this American put solution are discussed.