Conditional sampling for barrier option pricing under the LT method (Q2873131)

Barrier options are financial instruments whose payoffs depend on an underlying asset hitting a specified level or not during some period. In the present paper a conditional sampling method for quasi-Monte Carlo sampling under the linear transformation (LT) algorithm for barrier options is considered.NEWLINENEWLINEA conditional sampling scheme for pricing knock-out barrier options under the linear transformation algorithm from \textit{J. Imai} and \textit{K. S. Tan} [``A general dimension reduction technique for derivative pricing'', J. Comput. Finance 10, 129--155 (2006)] is developed. The proposed method is compared with the existing conditional Monte Carlo scheme of Glasserman and Staum and it is shown that a substantial variance reduction is achieved. The extended method allows for pricing knock-in barrier options. To obtain a further variance reduction, a root-finding method is introduced.NEWLINENEWLINEIn the introduction a brief review of barrier options is given.NEWLINENEWLINEIn Sections 2 and 3 an overview of the LT algorithm of Imai and Tan and the conditional sampling scheme of \textit{P. Glasserman} and \textit{J. Staum} [Oper. Res. 49, No. 6, 923--937 (2001; Zbl 1163.91398)] is given. It is assumed that the risk-neutral dynamics of the assets are given in the Black-Scholes world by the equality NEWLINE\[NEWLINE d S_i(t) = r S_i(t) + \sigma_i S_i(t) dW_i(t), \quad i = 1,2, \ldots, n, NEWLINE\]NEWLINE where \(S_i(t)\) denotes the price of asset \(i\) at time \(t,\) \(r\) is the risk-free interest rate, and \(\sigma_i\) is the volatility of asset \(i.\) Here, \(W = (W_1(t), \ldots, W_n(t))\) is an \(n\)-dimensional Brownian motion. The details of the Monte Carlo techniques for pricing options for this model are presented and discussed.NEWLINENEWLINEIn Section 4 the main results of the paper are presented. An alternative conditional sampling scheme compatible with the LT algorithm is derived. First, the Glasserman-Staum algorithm for an up-and-in option is reminded. The concept of the new approach to the LT algorithm is developed. In Theorem 4.1 it is shown that the estimator based on conditional sampling proposed by the authors is unbiased. In Theorem 4.2 it is shown that using a regular Monte Carlo method, the proposed estimator for an up-and-out option has reduced variance. In Theorem 4.3 it is shown that using a randomly shifted QMC, the proposed estimator for an up-and-out option has a reduced variance.NEWLINENEWLINEIn Section 5 several numerical examples to illustrate the introduced method are given. The Sobol' sequences and a digital shifting of them are used. Examples, like a single barrier Asian basket, a double barrier binary Asian, a single barrier Asian and a binary barrier are considered.NEWLINENEWLINEIn Section 6 the algorithm presented by the authors is extended to knock-in options. In Theorem 6.1 it is shown that the estimator for knock-in options based on conditional sampling is unbiased. In Theorem 6.2 it is shown that when the new method uses a randomly shifted QMC method, the defined estimator has a reduced variance. Through two examples -- a continuous down-and-in put with knock-in condition and a basket Asian with knock-in -- the results are verified.NEWLINENEWLINEIn Section 7 the concept of root-finding is presented and through two examples -- a single barrier Asian basket and a down-and-in put option is illustrated.NEWLINENEWLINEIn Section 8 the conditional sampling methods for QMC sampling under the LT algorithm for barrier options are summarized.