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.     

Barrier option pricing: a hybrid method approach

This paper adapts the hybrid method, a combination of the Laplace transformation and the finite-difference approach, to the pricing of barrier-style options. The hybrid method eliminates the time steps and provides a highly accurate and precise numerical solution that can be rapidly obtained. This method is superior to lattice methods when trying to solve barrier-style options. Previous studies have tried to solve barrier-style options; however, there have continually been several disadvantages. Very small time steps and stock node spaces are needed to avoid undesirable numerically induced oscillations in the solution of barrier option. In addition, all the intermediate option prices must be computed at each time step, even though one may be only interested in the terminal price of barrier-style complex options. The hybrid method may also solve more complex problems concerning barrier-style options with various boundary constraints such as options with a time-varying rebate. In order to demonstrate the accuracy and efficiency of the proposed scheme, we compare our algorithm with several well-known pricing formulas of barrier-type options. The numerical results show that the hybrid method is robust, and provides a highly accurate solution and fast convergence, regardless of whether or not the initial asset prices are close to the barrier.     

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.     

Analytical pricing of American options

By using the homotopy analysis method, we derive a new explicit approximate formula for the optimal exercise boundary of American options on an underlying asset with dividend yields. Compared with highly accurate numerical values, the new formula is shown to be valid for up to 2?years of time to maturity, which is ten times longer than existing explicit approximate formulas. The option price errors computed with our formula are within a few cents for American options that have moneyness (the ratio between stock and strike prices) from 0.8 to 1.2, strike prices of 100 dollars and 2?years to maturity.     

The Valuation of Volatility Options

This paper examines the valuation of European- and American-style volatilityoptions based on a general equilibrium stochastic volatility framework.Properties of the optimal exercise region and of the option price areprovided when volatility follows a general diffusion process. Explicitvaluation formulas are derived in four particular cases. Emphasis is placedon the MRLP (mean-reverting in the log) volatility model which has receivedconsiderable empirical support. In this context we examine the propertiesand hedging behavior of volatility options. Unlike American options,European call options on volatility are found to display concavity at highlevels of volatility.     

THE PRICING AND HEDGING OF LIMITED EXERCISE CAPS AND SPREADS

Capped options are barrier option spreads that automatically create simultaneous long and short positions. Exchange-traded capped options were introduced in 1991, though with limited volume. Such options, however, have traded on the over-the-counter markets for several years. Most of these options have the unusual feature that they automatically exercise when the underlying asset closes beyond a critical strike, making them a hybrid of European and American options. In this paper I present their boundary conditions and examine the prices, deltas, gammas, and thetas of caps as well as spreads constructed with European and American options. I also examine the effect of permitting exercise based only on the closing price as opposed to exercise at any time the critical strike is reached. I show that assuming that exercise can occur at any time can lead to serious pricing errors. The results have implications for the pricing of barrier options in general, which nearly always exercise early based only on the closing price.     

Parisian options with jumps: a maturity–excursion randomization approach

This paper introduces an analytically tractable method for the pricing of European and American Parisian options in a flexible jump–diffusion model. Our contribution is threefold. First, using a double Laplace–Carson transform with respect to the option maturity and the Parisian (excursion) time, we obtain closed-form solutions for different types of Parisian contracts. Our approach allows us also to analytically disentangle contributions of the jump and diffusion components for Parisian options in the excursion region. Second, we provide numerical examples and quantify the impact of jumps on the option price and the Greeks. Finally, we study the non-monotonic effects of volatility and jump intensity close to the excursion barrier, which are important for shareholders’ investment policy decisions in a levered firm.     

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.     

Analytical pricing of single barrier options under local volatility models

This paper considers a single barrier option under a local volatility model and shows that any down-and-in option can be priced by a combination of three standard European options whose volatility functions are connected through symmetrization. The symmetrized volatility function is approximated by a sequence of smooth functions that converges to the original one. An approximation formula is developed to price the standard European options with the approximated volatility functions. Finally, we apply the Aitken convergence accelerator to obtain an approximate price of the down-and-in option. Other single barrier options are priced in a similar fashion.     

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.     

Pricing exotic options in a regime switching economy: a Fourier transform method

This article considers the valuation of digital, barrier, and lookback options in a Markovian, regime-switching, Black–Scholes model. In Fourier space, integral representations for the option prices are derived via the theory on matrix Wiener–Hopf factorizations. Our main focus is on the 2-state case where the matrix Wiener–Hopf factorization is available analytically. A comparison to several numerical alternatives (analytical approximations, the Brownian bridge algorithm and a finite element scheme) demonstrates that the pricing formulas are easy to implement and lead to accurate price estimates.     

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.      

High-order accurate implicit finite difference method for evaluating American options

A numerical method is presented for valuing vanilla American options on a single asset that is up to fourth-order accurate in the log of the asset price, and second-order accurate in time. The method overcomes the standard difficulty encountered in developing high-order accurate finite difference schemes for valuing American options; that is, the lack of smoothness in the option price at the critical boundary. This is done by making special corrections to the right-hand side of the differnce equations near the boundary, so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr and Eaguet. Results of numerical experiments are presented comparing the method with more standard finite difference methods.     

A fast Fourier transform technique for pricing American options under stochastic volatility

This paper develops a non-finite-difference-based method of American option pricing under stochastic volatility by extending the Geske-Johnson compound option scheme. The characteristic function of the underlying state vector is inverted to obtain the vector’s density using a kernel-smoothed fast Fourier transform technique. The method produces option values that are closely in line with the values obtained by finite-difference schemes. It also performs well in an empirical application with traded S&P 100 index options. The method is especially well suited to price a set of options with different strikes on the same underlying asset, which is a task often encountered by practitioners.     

Model uncertainty and the pricing of American options

The virtue of an American option is that it can be exercised at any time. This right is particularly valuable when there is model uncertainty. Yet almost all the extensive literature on American options assumes away model uncertainty. This paper quantifies the potential value of this flexibility by identifying the supremum on the price of an American option when we do not impose a model, but rather consider the class of all models which are consistent with a family of European call prices. The bound is enforced by a hedging strategy involving these call options which is robust to model error.     

An Explicit Finite Difference Approach to the Pricing Problems of Perpetual Bermudan Options

The pricing problem of options with an early exercise feature, such as American options, is one of the important topics in mathematical finance. Pricing formulas for options with the early exercise feature, however, are not easy to obtain and the numerical methods are thus frequently required to derive the price of these options. The value function of perpetual Bermudan options is characterized with the partial differential equation and this is solved by the finite difference method in this article.     

Fourier transformation and the pricing of average-rate derivatives

In this article we propose a method to compute the density of the arithmetic average of a Markov process. This approach is then applied to the pricing of average rate options (Asian options). It is demonstrated that as long as a closed form formula is available for the discount bond price when the underlying process is treated as the riskless interest rate, analytical formulas for the density function of the arithmetic average and the Asian option price can be derived. This includes the affine class of term structure models. The Cox et al. (1985) square root interest rate process is used as an example. When the underlying process follows a geometric Brownian motion, a very efficient numerical method is proposed for computing the density function of the average. Extensions of the techniques to the cases of multiple state variables are also discussed.      

Pricing options on discrete realized variance with partially exact and bounded approximations

We derive efficient and accurate analytic approximation formulas for pricing options on discrete realized variance (DRV) under affine stochastic volatility models with jumps using the partially exact and bounded (PEB) approximations. The PEB method is an enhanced extension of the conditioning variable approach commonly used in deriving analytic approximation formulas for pricing discrete Asian style options. By adopting either the conditional normal or gamma distribution approximation based on some asymptotic behaviour of the DRV of the underlying asset price process, we manage to obtain PEB approximation formulas that achieve a high level of numerical accuracy in option values even for short-maturity options on DRV.     

Pricing American options under stochastic volatility and stochastic interest rates

We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters, this method can accurately price options with time-to-maturity up to several years. The main advantage of our approach over existing methods lies in its straightforward extension to models with stochastic volatility and stochastic interest rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility, and correlations on the American put price.