Static hedging and pricing American options

This paper utilizes the static hedge portfolio (SHP) approach of Derman et al. [Derman, E., Ergener, D., Kani, I., 1995. Static options replication. Journal of Derivatives 2, 78–95] and Carr et al. [Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190] to price and hedge American options under the Black-Scholes (1973) model and the constant elasticity of variance (CEV) model of Cox [Cox, J., 1975. Notes on option pricing I: Constant elasticity of variance diffusion. Working Paper, Stanford University]. The static hedge portfolio of an American option is formulated by applying the value-matching and smooth-pasting conditions on the early exercise boundary. The results indicate that the numerical efficiency of our static hedge portfolio approach is comparable to some recent advanced numerical methods such as Broadie and Detemple [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211–1250] binomial Black-Scholes method with Richardson extrapolation (BBSR). The accuracy of the SHP method for the calculation of deltas and gammas is especially notable. Moreover, when the stock price changes, the recalculation of the prices and hedge ratios of the American options under the SHP method is quick because there is no need to solve the static hedge portfolio again. Finally, our static hedging approach also provides an intuitive derivation of the early exercise boundary near expiration.     

Analytical approximations for the critical stock prices of American options: a performance comparison

Many efficient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (J Finance 42:301–320, 1987) and Barone-Adesi and Elliott (Stoch Anal Appl 9(2):115–131, 1991), the lower bound approximation in Broadie and Detemple (Rev Financial Stud 9:1211–1250, 1996), the tangent approximation in Bunch and Johnson (J Finance 55(5):2333–2356, 2000), the Laplace inversion method in Zhu (Int J Theor Appl Finance 9(7):1141–1177, 2006b), and the interpolation method in Li (Working paper, 2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.     

Confined exponential approximations for the valuation of American options

We provide an alternative analytic approximation for the value of an American option using a confined exponential distribution with tight upper bounds. This is an extension of the Geske and Johnson compound option approach and the Ho et al. exponential extrapolation method. Use of a perpetual American put value, and then a European put with high input volatility is suggested in order to provide a tighter upper bound for an American put price than simply the exercise price. Numerical results show that the new method not only overcomes the deficiencies in existing two-point extrapolation methods for long-term options but also further improves pricing accuracy for short-term options, which may substitute adequately for numerical solutions. As an extension, an analytic approximation is presented for a two-factor American call option.     

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.     

Implied stopping rules for American basket options from Markovian projection

This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and Black–Scholes models. In high dimensions, nonlinear PDE methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. From a numerical analysis point of view, we split the given non-smooth high-dimensional problem into two subproblems, namely one dealing with a smooth high-dimensionality integration in the parameter space and the other dealing with a low-dimensional, non-smooth optimal stopping problem in the projected state space. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early exercise boundary of the option by solving an HJB PDE in the projected, low-dimensional space. The resulting near-optimal early exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow one to assess the accuracy of the proposed pricing method. Indeed, our approximate early exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to 50. In these examples, the resulting option price relative errors are only of the order of few percent.     

Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions

In this paper, we develop an efficient payoff function approximation approach to estimating lower and upper bounds for pricing American arithmetic average options with a large number of underlying assets. The crucial step in the approach is to find a geometric mean which is more tractable than and highly correlated with a given arithmetic mean. Then the optimal exercise strategy for the resultant American geometric average option is used to obtain a low-biased estimator for the corresponding American arithmetic average option. This method is particularly efficient for asset prices modeled by jump-diffusion processes with deterministic volatilities because the geometric mean is always a one-dimensional Markov process regardless of the number of underlying assets and thus is free from the curse of dimensionality. Another appealing feature of our method is that it provides an extremely efficient way to obtain tight upper bounds with no nested simulation involved as opposed to some existing duality approaches. Various numerical examples with up to 50 underlying stocks suggest that our algorithm is able to produce computationally efficient results.     

Valuation of American partial barrier options

This paper concerns barrier options of American type where the underlying asset price is monitored for barrier hits during a part of the option’s lifetime. Analytic valuation formulas of the American partial barrier options are provided as the finite sum of bivariate normal distribution functions. This approximation method is based on barrier options along with constant early exercise policies. In addition, numerical results are given to show the accuracy of the approximating price. Our explicit formulas provide a very tight lower bound for the option values, and moreover, this method is superior in speed and its simplicity.     

Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model

We analyze American put options in a hyper-exponential jump-diffusion model. Our contribution is threefold. Firstly, by following a maturity randomization approach, we solve the partial integro-differential equation and obtain a tight lower bound for the American option price. Secondly, our method allows to disentangle the contributions of jumps and diffusion for the early exercise premium. Finally, using American-style options on the S&P 100 index from January 2007 until December 2012, we estimate various hyper-exponential specifications and investigate the implications for option pricing and jump-diffusion disentanglement. We find that jump risk accounts for a large part of the early exercise premium.     

New analytical option pricing models with Weyl–Titchmarsh theory

Abstract

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.     

EARLY EXERCISE OF AMERICAN INDEX OPTIONS

We show that exercise of American call options on stock indexes frequently occurs before expiration and attribute this early exercise to the “wild card” option that results from the cash settlement exercise process. The wild card represents an “implied option” to sell the index option at the fixed settlement price; it is therefore a put option on the index call option. We derive a simple one-period valuation model using compound option pricing. Analysis of observed early exercise demonstrates that the wild card feature is a factor influencing early exercise of index options.     

A general closed-form spread option pricing formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black–Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models.     

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.     

The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.     

Extending quadrature methods to value multi-asset and complex path dependent options

The exposition of the quadrature (QUAD) method (Andricopoulos, Widdicks, Duck, and Newton, 2003. Universal option valuation using quadrature methods. Journal of Financial Economics 67, 447–471 (see also Corrigendum, Journal of Financial Economics 73, 603 (2004)) is significantly extended to cover notably more complex and difficult problems in option valuations involving one or more underlyings. Trials comparing several techniques in the literature, adapted from standard lattice, grid and Monte Carlo methods to tackle particular types of problem, show that QUAD offers far greater flexibility, superior convergence, and hence, increased accuracy and considerably reduced computational times. The speed advantage of QUAD means that, even under the curse of dimensionality, it is not necessary to resort to Monte Carlo methods (certainly for options involving up to five underlying assets). Given the universality and flexibility of the method, it should be the method of choice for pricing options involving multiple underlying assets, in the presence of many features, such as early exercise or path dependency.     

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.     

An alternative approach to the valuation of American options and applications

In this paper we examine the structure of American option valuation problems and derive the analytic valuation formulas under general underlying security price processes by an alternative but intuitive method. For alternative diffusion processes, we derive closed-form analytic valuation formulas and analyze the implications of asset price dynamics on the early exercise premiums of American options. In this regard, we introduce useful and interesting diffusion processes into American option-pricing literature, thus providing a wide range of choices of pricing models for various American-type derivative assets. This work offers a useful analytic framework for empirical testing and practical applications such as the valuation of corporate securities and examining the impact of options trading on market micro-structure.     

A closed-form solution to American options under general diffusion processes

This paper investigates American option pricing under general diffusion processes. Specifically, the underlying asset price is assumed to follow a diffusion process in which both the dividend yield and volatility are functions of time and the underlying asset price. Using the generalized homotopy analysis method, the determination of the early exercise boundary is separated from the valuation procedure of American options. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. In addition, the corresponding optimal early exercise boundary and the Greeks are obtained in closed-form solutions. A nonlinear sequence transformation, the Padé technique, is used to effectively accelerate the convergence of the partial sums of the infinite series. As the homotopy constructed in this paper is based on a generalized deformation with a shape parameter and kernel function, the error of the homotopic approximation could be reduced further for a fixed order. Numerical examples demonstrate the validity, effectiveness, and flexibility of the proposed approach.     

A Refined Binomial Lattice for Pricing American Asian Options

We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the Cox-Ross-Rubinstein (1979) binomial lattice of stock prices. Each node in the lattice is partitioned into nodelets, each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the Hull-White (1993) method. From the backward-recursive upper-bound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditional-expectation based idea from Rogers-Shi (1995) and Chalasani-Jha-Varikooty (1998). Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n^4/20 for n > 14 periods.