On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

Efficient,exact algorithms for asian options with multiresolution lattices

Asian options are a kind of path-dependent derivative. How to price such derivatives efficiently and accurately has been a long-standing research and practical problem. This paper proposes a novel multiresolution (MR) trinomial lattice for pricing European- and American-style arithmetic Asian options. Extensive experimental work suggests that this new approach is both efficient and more accurate than existing methods. It also computes the numerical delta accurately. The MR algorithm is exact as no errors are introduced during backward induction. In fact, it may be the first exact discrete-time algorithm to break the exponential-time barrier. The MR algorithm is guaranteed to converge to the continuous-time value. This revised version was published online in June 2006 with corrections to the Cover Date.     

A systematic and efficient simulation scheme for the Greeks of financial derivatives

Greeks are the price sensitivities of financial derivatives and are essential for pricing, speculation, risk management, and model calibration. Although the pathwise method has been popular for calculating them, its applicability is problematic when the integrand is discontinuous. To tackle this problem, this paper defines and derives the parameter derivative of a discontinuous integrand of certain functional forms with respect to the parameter of interest. The parameter derivative is such that its integration equals the differentiation of the integration of the aforesaid discontinuous integrand with respect to that parameter. As a result, unbiased Greek formulas for a very broad class of payoff functions and models can be systematically derived. This new method is applied to the Greeks of (1) Asian options under two popular Lévy processes, i.e. Merton's jump-diffusion model and the variance-gamma process, and (2) collateralized debt obligations under the Gaussian copula model. Our Greeks outperform the finite-difference and likelihood ratio methods in terms of accuracy, variance, and computation time.     

Efficient trinomial trees for local-volatility models in pricing double-barrier options

A local-volatility (LV) model captures the volatility smile while retaining the preference freedom of the Black–Scholes model. Past attempts to construct a smile-consistent tree for the LV surface do not guarantee validity. This paper presents an efficient and valid smile-consistent tree for the LV model. The only assumption is that the LV surface be upper- and lower-bounded. With this tree, double-barrier options can be priced with fast convergence even in the presence of volatility smile. This is confirmed numerically. An implied tree is also presented. It recovers the LV surface reasonably well.     

Unbiased and efficient Greeks of financial options

The price of a derivative security equals the discounted expected payoff of the security under a suitable measure, and Greeks are price sensitivities with respect to parameters of interest. When closed-form formulas do not exist, Monte Carlo simulation has proved very useful for computing the prices and Greeks of derivative securities. Although finite difference with resimulation is the standard method for estimating Greeks, it is in general biased and suffers from erratic behavior when the payoff function is discontinuous. Direct methods, such as the pathwise method and the likelihood ratio method, are proposed to differentiate the price formulas directly and hence produce unbiased Greeks (Broadie and Glasserman, Manag. Sci. 42:269–285, 1996). The pathwise method differentiates the payoff function, whereas the likelihood ratio method differentiates the densities. When both methods apply, the pathwise method generally enjoys lower variances, but it requires the payoff function to be Lipschitz-continuous. Similarly to the pathwise method, our method differentiates the payoff function but lifts the Lipschitz-continuity requirements on the payoff function. We build a new but simple mathematical formulation so that formulas of Greeks for a broad class of derivative securities can be derived systematically. We then present an importance sampling method to estimate the Greeks. These formulas are the first in the literature. Numerical experiments show that our method gives unbiased Greeks for several popular multi-asset options (also called rainbow options) and a path-dependent option.     

An improved combinatorial approach for pricing Parisian options

Parisian options are path-dependent options whose payoff depends on whether the underlying asset’s price remains continuously at or above a given barrier over a given time interval. Costabile’s (Decis Econ Finance 25(2):111–125, 2002b) algorithm for pricing Parisian options based on a combinatorial approach in binomial tree has a time complexity of O( n³){O\left( {n^{3}}\right)}. We improve that algorithm to yield one with a time complexity of only O(n²){O\left({n^{2}}\right)}.