Laplace Transforms and Installment Options

Abstract

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.

Installment options close to expiry

“…These two sets of expressions constitute the principal result of this paper. It is interesting to note that, provided L, which represents the continual input of cash in the form of installment payments, is positive, the location of the free boundary close to expiry is of the form x f (τ) ∼ τ(−lnτ), which is the same behavior as we found using an integral equation approach in [4], and also the same as that for the American put with D < r and the American call with D > r [2,5,13,16,17,21,25]. This differs from the x f (τ) ∼ x 1 √ τ behavior for the American put with D > r and the American call with D < r which was also the behavior encountered most often by Tao [26][27][28][29][30][31][32][33][34], who pioneered the method used here, in his studies of Stefan problems arising in melting and solidification.…”

“…Installment options are also mentioned in [7,14]. In [4], we used a partial Laplace transform to derive an integral equation for the location of the free boundary for an installment option, which we solved close to expiry.…”

A variational inequality arising from American installment call options pricing

“…There are some papers about install options, such as [3,6,10]. Particularly, there are a free boundary model in [10] and some numerical results about the model.…”

Efficient Numerical Valuation of Continuous Installment Options

“…Their applied technique suffers from a serious drawback, since the MEF method generates a discontinuity in the optimal stopping and early exercise boundaries. Alobaidi [2] analyzed the European case using the Laplace transformation to solve the free boundary problem. However, the method used is rather specific and not suitable for a numerical computation.…”