Option Pricing with Discrete Rebalancing

Abstract

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative changes in the stock price. The stock price follows a marked point process and the market is incomplete. We first characterize the equivalent martingale measures. An explicit pricing formula based on the minimal martingale measure is then provided together with the hedging strategy underlying portfolio adjustments. Two examples illustrate our pricing framework: a jump process driven by a latent geometric Brownian motion and a marked Poisson process. We establish the convergence to the Black-Scholes model when the triggering price increment shrinks to zero. For the empirical application we use IBM, France Telecom and CAC 40 intraday transaction data, and compare option prices given by the marked Poisson model, the Black-Scholes model and observed option prices.

Keywords: weak convergence, incomplete market, option pricing, minimal martin-gale measure, discrete rebalancing, marked point process

JEL Classification: D52, G13