Solving the dual Russian option problem by using change‐of‐measure arguments

Abstract

We apply the change‐of‐measure arguments of Shepp and Shiryaev (Theory of Probability and its Applications, 1994, 39, 103–119) to study the dual Russian option pricing problem proposed by Shepp and Shiryaev (Probability Theory and Mathematical Statistics: Lectures presented at the semester held in St. Peterburg, Russia, March 2 April 23, 1993, Amsterdam, the Netherlands: Gordon and Breach, 1996, pp. 209–218) as an optimal stopping problem for a one‐dimensional diffusion process with reflection. We recall the solution to the associated free‐boundary problem and give a solution to the resulting one‐dimensional optimal stopping problem by using the martingale approach of Beibel and Lerche (Statistica Sinica, 1997, 7, 93–108) and (Theory of Probability and its Applications, 2000, 45, 657–669).

Optimal stopping problems for running minima with positive discounting rates

“…For this purpose, we denote by U * (x, y) the value function of the optimal stopping problem which can be obtained from the one in (2.4) above or (5.1) below, by means of setting L = 0 there. It is shown in [38] (see also [13] for another derivation) that the function U * (x, y) ≡ V * (x, y; 0) ≡ W * (x, y; 0) with V * (x, y) ≡ V * (x, y; L) from (2.4) and W * (x, y) ≡ W * (x, y; L) from (5.1) admits the explicit expression in (5.16) below, under L = 0, and the optimal stopping time has the form below). (Note that a * here corresponds to θ in the notation of [38] and to 1/a * in the notations of [13].)…”

“…It is shown in [38] (see also [13] for another derivation) that the function U * (x, y) ≡ V * (x, y; 0) ≡ W * (x, y; 0) with V * (x, y) ≡ V * (x, y; L) from (2.4) and W * (x, y) ≡ W * (x, y; L) from (5.1) admits the explicit expression in (5.16) below, under L = 0, and the optimal stopping time has the form below). (Note that a * here corresponds to θ in the notation of [38] and to 1/a * in the notations of [13].) Suppose that h * (y) > a * y holds, for some y > 0.…”

“…holds. (Note that, due to the assumption that L > 0, the condition of (2.5) turns out to be stronger than the finiteness of the expected reward in (2.4) under L = 0 which is used in [38] and [13]. The verification of this condition is very complicated and is actually skipped below for the cases of different relations between the parameters of the model considered in the paper.)…”

“…In other words, the sellers can obtain the assets at the best prices Y τ they could have obtained up to the time of exercise τ with some additional fixed fee L > 0, but paid in the equivalent currency units at time 0, that is, e rτ (Y τ + L) (see [38] for the precise extensive formulation of the contract under L = 0). We recall that the problem of (2.4) for the case of L = 0 was actually solved is [38] (see also [13] for a solution by means of the change-of-measure arguments following [37]). Moreover, it follows from the result of [38; Section 2, Lemma] that, if 0 < r ≤ (µ − σ 2 /2) 2 /(2σ 2 ) and µ < σ 2 /2 holds, then V * (x, y) = 0, for all (x, y) ∈ E , under L = 0.…”

“…In the case of L = 0, the problems in (2.4) and (5.1) become the dual Russian option problems for selling short, which were formulated and explicitly solved by Shepp and Shiryaev [38], by means of reducing the initial problem to an optimal stopping problem for a two-dimensional (continuous) Markov process and solving the latter problem by using the smooth-fit and normal-reflection conditions. More recently, the problems in (2.4) and (5.1) in the case of L = 0 were solved in [13] by means of reducing the initial problems to optimal stopping problems for a one-dimensional diffusion process with reflection following the change-of-measure arguments from Shepp and Shiryaev [37]. Gerber et al [19] and Mordecki and Moreira [26] obtained closed form solutions to the perpetual Russian option problems for diffusions with negative exponential jumps.…”