We develop a conditional sampling scheme for pricing knock-out barrier options under the Linear Transformations (LT) algorithm from Imai and Tan (2006). We compare our new method to an existing conditional Monte Carlo scheme from Glasserman and Staum (2001), and show that a substantial variance reduction is achieved. We extend the method to allow pricing knock-in barrier options and introduce a root-finding method to obtain a further variance reduction. The effectiveness of the new method is supported by numerical results.
Barrier options are financial instruments whose payoff depend on an underlying asset hitting a specified level or not during some period. The simplest example would be a regular knock-out call option, where the payoff is set to zero if the option hits a certain barrier level during the lifetime of the option, and results in the payoff of a regular call option otherwise. These options are popular because they cost at most the amount of their vanilla counterparts as there is a higher probability of ending up with nothing. For more background information on barrier options and their applications, see for instance [4], [5], [28] and [29]. For sufficiently simple products there exist analytical formulas for the value, see, e.g., all the previous references, or [13]. When the product is too complex, one must resort to numerical methods to obtain the value. Furthermore, if the barrier is not monitored continuously but discretely, no accurate closed form formulas are available. Several papers exist dealing with discretely monitored barrier options, e.g., [16] and [30].
A straightforward simulation method for pricing discretely monitored options is to sample the assets in each monitoring date. However, when a barrier condition is introduced in the payoff, only a minority of the paths may lead to non-zero values if the probability of satisfying the barrier condition is low. This leads to significant increases in the variance of the estimator. Glasserman and Staum [11] remedy this problem by introducing a sampling scheme where the asset with the barrier condition is sampled conditional on survival in each monitoring date. In [16] a different scheme is used based on the hitting time of the asset as the sampling variable instead of the asset increments. Both of these papers use Monte Carlo to sample the necessary variates.
In this paper we use low-discrepancy points to generate the uniform variates for sampling the asset paths. The merit of using low-discrepancy points for valuing financial products is discussed in [24] and [25]. Since then quasi-Monte Carlo methods have often been applied to applications in finance, e.g., [19,2,9,17,18,21]. Care must be taken however in the construction of the asset paths, see for instance [22] or [26], where it is shown how the path construction method can have a large influence on the convergence rate, and also [27], which discusses the influence on the nature of the discontinuity. Imai and Tan [14] provide an alternative to the Brownian Bridge or PCA construction, called the Linear Transformation method (or LT for short). They provide an optimal construction, in some sense, by minimizing the effective dimension of the underlying payoff function. We reconcile this LT method with the idea of conditional sampling, to gain a “best of two worlds” approach for valuing barrier options. Even though our method can be applied using other covariance matrix decompositions, such as PCA and Cholesky, we will use the LT method because of its proven performance, see [14].
The outline of our paper is as follows. In Sections 2 and 3 we present an overview of the LT algorithm from [14] and the conditional sampling scheme from [11]. Section 4 contains the main results of our paper: a modified conditional sampling scheme compatible with the LT algorithm. In Section 5 we give several numerical examples to illustrate our method. Then in Sections 6 and 7 we present two extensions to our sampling scheme. The first extension deals with knock-in options, and the second one is a modification to combine the quasi-Monte Carlo method with an analytical integration using root-finding. We summarize our results in Section 8.
We assume a Black-Scholes world in which the risk-neutral dynamics of the assets are given by dS i (t) = rS i (t)dt + σ i S i (t)dW i (t), i = 1, . . . , n,
where S i (t) denotes the price of asset i at time t, r is the risk-free interest rate and σ i the volatility of asset i. Also, W = (W 1 (t), . . . , W n (t)) is an n-dimensional Brownian motion, with dW i dW j = ρ ij dt. For more information on this type of market model, see for instance [7], [10], [13] or [28]. When resorting to Monte Carlo techniques for pricing options under this model, asset paths need to be sampled at each time point. In what follows, we assume an equally spaced time discretization ∆t = T /m for simplicity, but we note that all results can be applied without using this assumption. Under this assumption, we have t j = j∆t. Under Black-Scholes dynamics we simulate the trajectories according to
, where we use the prime to denote the transpose of a vector. Then W is multivariate normally distributed with covariance matrix
, where
One approach to simulate W is to decompose the matrix Σ as
where C is the Cholesky factor (see, e.g., [10]) and calculate
where z is an mn-dimensional vector of i.i.d. standard normally distributed variables. Assuming that we have a European option payoff represented as max f ( W ), 0 th
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