Connecting discrete and continuous path-dependent options
The paper deals with the relation of prices for discrete- and continuous-time version of path-dependent options including lookback, barrier and hindsight options. The authors present the correction terms that improve the approximation of discrete-time prices using continuous-time formulas. For barrier options the correction shifts the barrier to price a discrete option using the continuous formula. For lookbacks the correction shifts the expected maximum or minimum price. In particular, for hindsight options \[ V_{m}(S_{\pm},K)= V(S_{\pm}e^{\pm\beta_{1}\sigma\sqrt{T/m}}, Ke^{\pm\beta_{1}\sigma\sqrt{T/m}})e^{\mp\beta_{1}\sigma\sqrt{T/m}} +o(1/\sqrt{m}), \] where \(V\) denotes the price of a continuous-time version, \(V_{m}\) denotes the price of a discrete-time version of a hindsight option, in \(\pm\) and \(\mp\) the top case applies for calls and the bottom for puts, \(\beta_{1}=-\zeta(1/2)/\sqrt{2\pi}\), \(\zeta\) is the Riemann zeta function, \(S_{+}=\max_{0\leq u\leq t}S_{u}\), \(S_{-}=\min_{0\leq u\leq t} S_{u}\) in continuous-time cases and similar in discrete-time cases, the price \(S_{t}\) of single risky asset evolves according to \(dS_{t}=\nu S_{t}dt+\sigma S_{t}dw_{t}\), where \(w_{t}\) is a standard Wiener process, \(K\) is a strike price, \(m\) is the number of price-fixing dates. The discrete-time discrete-state lattice methods for determining accurate prices of discrete and continuous path-dependent options are developed.
scientific article; zbMATH DE number 1336675