Roland Mallier

Laplace transforms and American options

Laplace transform methods are used to study the valuation of American call and put options with constant dividend yield, and to derive integral equations giving the location of the optimal exercise boundary. In each case studied, the main result of this paper is a nonlinear Fredholm-type integral equation for the location of the free boundary. The equations differ depending on whether the dividend yield is less than or exceeds the risk-free rate. These integral equations contain a transform variable, so the solution of the equations would involve finding the free boundary that satisfies the equations for all values of this transform variable. Expressions are also given for the transform of the value of the option in terms of this free boundary.

LAPLACE TRANSFORMS AND INSTALLMENT OPTIONS

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

On the optimal exercise boundary for an American put option

An American put option is a derivative financial instrument that gives its holder the right but not the obligation to sell an underlying security at a pre-determined price. American options may be exercised at any time prior to expiry at the discretion of the holder, and the decision as to whether or not to exercise leads to a free boundary problem. In this paper, we examine the behavior of the free boundary close to expiry. Working directly with the underlying PDE, by using asymptotic expansions, we are able to deduce this behavior of the boundary in this limit.

ASYMPTOTIC ANALYSIS OF AMERICAN CALL OPTIONS

American call options are financial derivatives that give the holder the right but not the obligation to buy an underlying security at a pre-determined price. They differ from European options in that they may be exercised at any time prior to their expiration, rather than only at expiration. Their value is described by the Black-Scholes PDE together with a constraint that arises from the possibility of early exercise. This leads to a free boundary problem for the optimal exercise boundary, which determines whether or not it is beneficial for the holder to exercise the option prior to expiration. However, an exact solution cannot be found, and therefore by using asymptotic techniques employed in the study of boundary layers in fluid mechanics, we find an asymptotic expression for the location of the optimal exercise boundary and the value of the option near to expiration.

Laplace transforms and the American straddle

We address the pricing of American straddle options. We use partial Laplace transform techniques due to Evans et al. (1950) to derive a pair of integral equations giving the locations of the optimal exercise boundaries for an American straddle option with a constant dividend yield.

A Green′s function for a convertible bond using the Vasicek model

We consider a convertible security where the underlying stock price obeys a lognormal random walk and the risk-free rate is given by the Vasicek model. Using a Laplace transform in time and a Mellin transform in the stock price, we derive a Green′s function solution for the value of the convertible bond.

Interest rate swaps under CIR

We consider fixed-for-floating interest rate swaps under the assumption that interest rates are given by the mean-reverting Cox-Ingersoll-Ross model. By using a Greens function approach, we derive analytical expressions for the values of both a vanilla swap and an in-arrears swap.

Evaluating approximations to the optimal exercise boundary for American options

We consider series solutions for the location of the optimal exercise boundary of an American option close to expiry. By using Monte Carlo methods, we compute the expected value of an option if the holder uses the approximate location given by such a series as his exercise strategy, and compare this value to the actual value of the option. This gives an alternative method to evaluate approximations. We find the series solution for the call performs excellently under this criterion, even for large times, while the asymptotic approximation for the put is very good near to expiry but not so good further from expiry.

Interactions between pairs of oblique waves in a Bickley jet

Abstract We consider the weakly nonlinear spatial evolution of a pair of varicose oblique waves and a pair of sinuous oblique waves superimposed on an inviscid Bickley jet, with each wave being slightly amplified on a linear basis. The two pairs are assumed to both be inclined at the same angle to the plane of the jet. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the modes. These equations are discussed and solved numerically, and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.

The American straddle close to expiry

We address the pricing of American straddle options. We use a technique due to Kim (1990) to derive an expression involving integrals for the price of such an option close to expiry. We then evaluate this expression on the dual optimal exercise boundaries to obtain a set of integral equations for the location of these exercise boundaries, and solve these equations close to expiry.