Connecting discrete and continuous path-dependent options

Abstract

Abstract. This paper develops methods for relating the prices of discrete-and continuous-time versions of path-dependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closed-form solutions for continuous option prices to approximate their discrete counterparts. We also develop discrete-time discrete-state lattice methods for determining accurate prices of discrete and continuous path-dependent options. In several cases, the lattice methods use correction terms based on the connection between discrete-and continuous-time prices which dramatically improve convergence to the accurate price.

Chapter 8 Discrete Barrier and Lookback Options

“…Of course, as we mentioned, the straightforward result from the central limit theorem, which has error o(1), does not give a good approximation. An approximation based on the results from sequential analysis (see, e.g., Siegmund, 1985 with the error order o(1/ √ m ) is given in Broadie et al (1999), whose proof is simplified in Kou (2003) and Hörfelt (2003). We will review these results in Section 4.…”

“…It is well known that the straightforward binomial tree is not efficient in pricing discrete and lookback barrier options, due to the inefficiencies in computing discrete extreme values of the sample paths involved in the payoffs. Broadie et al (1999) proposed an enhanced trinomial tree method which explicitly uses the continuity correction in Broadie et al (1997) and a shift node. A Dirichlet lattice method based on the conditional distribution via Brownian bridge is given in Kuan and Webber (2003).…”

“…(2) Direct Monte Carlo simulation or standard binomial trees may be difficult, and can take hours to produce accurate results; see Broadie et al (1999). (3) Although the central limit theorem asserts that as m → ∞ the difference between the discretely and continuously monitored barrier options should be small, it is well known that the numerical differences can be surprisingly large, even for large m; see, e.g., the table in Section 4.…”

“…Kou (2003) covered all eight cases with a simpler proof (see also Hörfelt, 2003). The continuity corrections for discrete lookback options are given in Broadie et al (1999).…”

Pricing financial claims contingent upon an underlying asset monitored at discrete times

“…Consequently, for even relatively few dates, numerical evaluation becomes very inefficient. To overcome the mis-pricing, a wide variety of numerical techniques have been proposed in the literature, including recent noteworthy additions [5][6][7][8][9][10][11][12][13][14][15]. This work is a revision of an article by two of the authors [16] in which an exact analytic expression for the down-out option price was obtained as the solution to the Black-Scholes PDE.…”