Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

Abstract

We price a contingent claim liability using the utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε > 0 in two cases with and without a contingent claim liability. Using the computations from the heuristic argument in Whalley & Wilmott [37] we provide a rigorous derivation of the asymptotic expansion of the value function in powers of ε 1/3 in both cases with and without a contingent claim liability. Additionally, using utility indifference method we derive the price of the contingent claim liability up to order ε. In both cases, we also obtain a "nearly optimal" strategy, whose expected utility asymptotically matches the leading terms of the value function.

Trading With Small Price Impact

“…For proportional transaction costs, a number of corresponding results have been obtained, formally (Whalley and Wilmott ; Kallsen and Muhle‐Karbe ) and rigorously (Bichuch ; Bouchard et al. ; Possamai and Royer ).…”

“…Weakening these regularity assumptions to European call and put options, for example, is an open problem even in simpler models with proportional transaction costs (Bichuch ; Possamai and Royer ).…”

Option Pricing and Hedging With Small Transaction Costs

“…In particular, the Landau symbols O(·) and o(·) refer to pointwise estimates, with the implicit assumption of enough regularity in time and states to eventually turn these into an estimate of the expected utility generated by the approximately optimal policy. Rigorous proofs have been worked out in the present setting for the Black-Scholes model (Bichuch 2011;Guasoni and Muhle-Karbe 2015), and by Soner and Touzi (2012) for an infinite-horizon consumption problem in a Markovian setup.…”

“…Motivated by previous asymptotic results (Whalley and Wilmott 1997;Bichuch 2011;Guasoni and Muhle-Karbe 2015;Soner and Touzi 2012), we assume that the deviations of the optimal strategy with transaction costs from the frictionless optimizer are asymptotically proportional to the cubic root of the spread:…”

Consistency of option prices under bid–ask spreads

“…In the literature on option pricing under transaction costs, it is usually assumed that the bid and ask of the underlying are constant multiples of a mid‐price (often assumed to be geometric Brownian motion). This mid‐price is then used as trigger to decide whether an option should be exercised, followed by physical delivery (Bichuch, ; Davis, Panas, & Zariphopoulou, ; Whalley & Wilmott, ). The assumption that such a constant‐proportion mid‐price triggers exercise seems to be rather ad hoc, though.…”