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.     

Maximum entropy distributions inferred from option portfolios on an asset

We obtain the maximum entropy distribution for an asset from call and digital option prices. A rigorous mathematical proof of its existence and exponential form is given, which can also be applied to legitimise a formal derivation by Buchen and Kelly (J. Financ. Quant. Anal. 31:143–159, 1996). We give a simple and robust algorithm for our method and compare our results to theirs. We present numerical results which show that our approach implies very realistic volatility surfaces even when calibrating only to at-the-money options. Finally, we apply our approach to options on the S&P 500 index.     

Futures Options and the Volatility of Futures Prices

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at-the-money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

Valuation of vulnerable American options with correlated credit risk

This article evaluates vulnerable American options based on the two-point Geske and Johnson method. In accordance with the Martingale approach, we provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures. Employing Richardson’s extrapolation gets the values of vulnerable American options. To demonstrate the accuracy of our proposed method, we use numerical examples to compare the values of vulnerable American options from our proposed method with the benchmark values from the least-square Monte Carlo simulation method. We also perform sensitivity analyses for vulnerable American options and show how the prices of vulnerable American options vary with the correlation between the underlying assets and the option writer’s assets.      

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.     

The Expected Return and Exercise Time of Merton-style Real Options

We analyse the rate of return and expected exercise time of Merton-style options (1973) employed in many real option situations where the possibility of exercise is both perpetual and American in nature. Using risk-neutral and risk-adjusted pricing techniques, Merton-style options are shown to have an expected return that is a constant percentage of the option value and independent of the proximity to the critical exercise boundary. Merton options thus remain at the same point on the Security Market Line, unlike European options whose position and rate of return change dynamically. We also present formulae for the expected time and discounted times to exercise and analyse the dependency of these variables on volatility.     

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.     

Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.     

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.     

Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme

This paper will demonstrate how European and American option prices can be computed under the jump-diffusion model using the radial basis function (RBF) interpolation scheme. The RBF interpolation scheme is demonstrated by solving an option pricing formula, a one-dimensional partial integro-differential equation (PIDE). We select the cubic spline radial basis function and adopt a simple numerical algorithm (Briani et al. in Calcolo 44:33–57, 2007) to establish a finite computational range for the improper integral of the PIDE. This algorithm reduces the truncation error of approximating the improper integral. As a result, we are able to achieve a higher approximation accuracy of the integral with the application of any quadrature. Moreover, we a numerical technique termed cubic spline factorisation (Bos and Salkauskas in J Approx Theory 51:81–88, 1987) to solve the inversion of an ill-conditioned RBF interpolant, which is a well-known research problem in the RBF field. Finally, our numerical experiments show that in the European case, our RBF-interpolation solution is second-order accurate for spatial variables, while in the American case, it is second-order accurate for spatial variables and first-order accurate for time variables.     

Sequential American Exchange Property Options

Property development activities often occur in stages, which are appropriately modeled as sequential American exchange property options, where there are interim expenditures required in order to keep the property development options “alive”. Normally American exchange options require a numerical solution, but herein there is a new closed-form approximate solution, which is computationally efficient and accurate. This method combines repeats of Margrabe European exchange and Geske compound option solutions with tight upper boundaries of either American perpetuities or European exchange options with a high volatility. Illustrations are provided of the sensitivity of the real sequential options and optimal timing to changes in several parameters, which provide a framework for property policy (tax, subsidy and regulatory) guidelines and for property development strategy evaluation. There are several plausible applications of these real option models in commercial and residential property development, within commercial property leases, with regard to switching tenants, and agricultural alternatives.     

A hyperbolic diffusion model for stock prices

In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.     

Do option traders on value and growth stocks react differently to new information?

This study compares the changes in implied volatilities of options on Nasdaq 100 and Russell 2000 value and growth portfolios, for the time period of 2004 and 2005. Following the methodologies in Stein (J Finance 44:1011–1024, 1989) and Heynen et al. (J Financ Quant Anal 29:31–56, 1994), we attempt to infer whether there are systematic differences in the degree of overreactions between value and growth options. The empirical evidence indicates that the reactions to information by investors in growth options, as proxied by options on Nasdaq 100 and Russell 2000 growth, are stronger than those of Russell 2000 value. Whether these reactions can be considered as overreacting, however, is not entirely conclusive. Nevertheless, the results imply that difference in investors’ behavior and styles is one potential explanation for the value stock effect.     

Path-dependent game options: a lookback case

The game option, which is also known as Israel option, is an American option with callable features. The option holder can exercise the option at any time up to maturity. This article studies the pricing behaviors of the path-dependent game option where the payoff of the option depends on the maximum or minimum asset price over the life of the option (i.e., the game option with the lookback feature). We obtain the explicit pricing formula for the perpetual case and provide the integral expression of pricing formula under the finite horizon case. In addition, we derive optimal exercise strategies and continuation regions of options in both floating and fixed strike cases.     

Pricing equity options everywhere

Finite difference methods are a popular technique for pricing American options. Since their introduction to finance by Brennan and Schwartz their use has spread from vanilla calls and puts on one stock to path-dependent and exotic options on multiple assets. Despite the breadth of the problems they have been applied to, and the increased sophistication of some of the newer techniques, most approaches to pricing equity options have not adequately addressed the issues of unbounded computational domains and divergent diffusion coefficients. In this article it is shown that these two problems are related and can be overcome using multiple grids. This new technique allows options to be priced for all values of the underlying, and is illustrated using standard put options and the call on the maximum of two stocks. For the latter contract, I also derive a characterization of the asymptotic continuation region in terms of a one-dimensional option pricing problem, and give analytic formulae for the perpetual case.     

Empirical Study of Nikkei 225 Options with the Markov Switching GARCH Model

This paper investigates the pricing of Nikkei 225 Options using the Markov Switching GARCH (MSGARCH) model, and examines its practical usefulness in option markets. We assume that investors are risk-neutral and then compute option prices by using Monte Carlo simulation. The results reveal that, for call options, the MSGARCH model with Student’s t-distribution gives more accurate pricing results than GARCH models and the Black–Scholes model. However, this model does not have good performance for put options.     

Fast accurate binomial pricing

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.     

An adjusted binomial model for pricing Asian options

We propose a model for pricing both European and American Asian options based on the arithmetic average of the underlying asset prices. Our approach relies on a binomial tree describing the underlying asset evolution. At each node of the tree we associate a set of representative averages chosen among all the effective averages realized at that node. Then, we use backward recursion and linear interpolation to compute the option price.     

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.