Valuation of American options under the CGMY model

In the present work, we concentrate on the analytical study of American options under the CGMY process. The decomposition formula of the American option and the integral equation for the optimal-exercise boundary are established in explicit forms. Moreover, an analytical approximation formula is obtained for the American value. This approximation is valid when time to maturity is either very short or very long. Numerical simulations are provided for European options, optimal-exercise prices and approximate values for American options.     

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.     

Commodity Spread Option with Cointegration

We derive the valuation formula of a European call option on the spread of two cointegrated commodity futures prices, based on the Gibson–Schwartz with cointegration (GSC) model. We also analyze the American commodity spread option including the early exercise premium representation and an analytical approximation valuation formulae with cointegration. In the numerical analysis, we compare the spread option values calculated by the GSC model and the Gibson–Schwartz (GS) model that ignores cointegration. Consistent with the intuition that the cointegration prevents the prices from diverging, the GSC model prices the commodity spread option lower than the GS model which have longer maturity of more than 6 years. In other words, the GS model may overprice the commodity spread options for those with longer maturity without taking account of cointegration. Thus, incorporating cointegration is important for valuation and hedging of long-term commodity spread options such as large scale oil refining plant developments.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.     

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.     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

Analytic approximation formulae for European crack spread options

In this paper, we investigate and compare the pricing of European crack spread call options under different underlying models. New proposed univariate and explicit constant elasticity of variance (CEV) models are assumed and new analytic approximation formulae in the form of asymptotic expansions are derived. As well we derive an analytic approximation formula based on an explicit version of two correlated Schwartz models. In order to compare the performance of our new formulae with the performance of current popular formulae, we calibrate market prices of short tenor heating oil crack spread call options (traded on the New York Mercantile Exchange) and empirically test their performances. Results from the analysis show that our univariate-based CEV formulae outperforms known univariate formulae in capturing market prices. Overall, however we found the explicit approach to be superior to the univariate approach and in particular our new explicit-based formulae performed best in capturing market prices for options with short tenor.     

The early exercise premia of America put options on stocks

Using the put-call parity, this paper finds that early exercise premia of short-lived American put options on stocks account for a significant portion of put prices. This finding holds even for out-of-the-money put options. The magnitude of the early exercise premia of American put options with no dividend is positvely related to the degree of moneyness, time to maturity of the put option, and the volatility. The magnitude of the early exercise premia of American put options with dividend is positvely related to the degree of moneyness and the risk-free interest rates.     

A general closed form option pricing formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram–Charlier series expansion, known as the Gauss–Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

     

American options under stochastic volatility: control variates,maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.     

Optimal strike prices of stock options for effort-averse executives

This paper evaluates the common practice of setting the strike prices of executive option plans at-the-money. Hall and Murphy [Hall, Brian, Murphy, Kevin J., 2000. Optimal exercise prices for executive stock options. American Economic Review 90 (2), 209–214] claim this practice to be optimal since it maximizes the sensitivity of compensation to firm performance. However, they do not incorporate effort and the possibility that managers are effort-averse into their model. We revisit this question while explicitly introducing these factors and allowing the reward package to include fixed wages, options, and stock grants. We simulate the manager’s effort choice and compensation as well as the value of shareholders’ equity under alternative compensation schemes, and identify schemes that are optimal. Our simulations indicate that, when abstracting from tax considerations, it is optimal to award managers with options that will most likely be highly valuable (i.e., substantially in-the-money) on their expiration date. Prior to 2006, the tax code and financial reporting standards provided incentives to award options that are closer to the money when issued than the options that were optimal in the absence of these considerations. Recent tax and reporting changes voided these incentives and thus we predict that these changes will induce firms to issue options with lower strike prices than those that were issued prior to 2006.     

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.     

Generalised Geske--Johnson Interpolation of Option Prices

Abstract:  This paper describes four separate option types as special cases of Bermudans with general inter–exercise and time to final maturity. This produces a surface with European, finite American, infinite Bermudan and infinite American options as special cases. This allows Geske–Johnson (1984) two–point pricing to be extended to consider time–to–maturity as well as time–between–exercise opportunities. Due to their position on this 'map', infinite Bermudans are christened Arctic options and their pricing solution is presented. Numerical comparisons to benchmark methods are made for call prices under GBM although the results here hold for other processes and for both puts and calls when symmetry arguments are invoked.     

Price Discovery in Near‐ and Away‐from‐the‐Money Option Markets

This paper examines the relative price discovery roles of near‐ and away‐from‐the‐money option markets. The evidence shows that, when considering multiple options with different strike prices jointly, option markets have an average information share of 17.6%. However, no individual option market dominates in the price discovery process, higher and lower trading activity options (i.e., near‐ and away‐from‐the‐money options, respectively) each contribute approximately equally to this process. The main implications of these results are that (1) collectively, option markets process a substantial amount of new stock price‐related information, and (2) looking across strike prices, option markets appear to be informationally nonredundant.     

Modelling implied volatility with OLS and panel data models

The paper performs an empirical estimation of time-varying volatility using OLS regression. Error Components, and Dummy Variable models, by regressing the implied volatility on time to maturity, the strike price and a dummy. Both the daily OLS equations and the panel data model provide more accurate estimates of Black and Scholes option prices than the bench-mark standard deviation of log returns. FT-SE 100 Index European options are used for empirical analysis.     

Partial differential equations for Asian option prices

In this article, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this paper, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and as they perform numerically well or better than the others, must be considered the preferred choice.     

Implementing Option Pricing Models When Asset Returns Are Predictable

The predictability of an asset's returns will affect the prices of options on that asset, even though predictability is typically induced by the drift, which does not enter the option pricing formula. For discretely-sampled data, predictability is linked to the parameters that do enter the option pricing formula. We construct an adjustment for predictability to the Black-Scholes formula and show that this adjustment can be important even for small levels of predictability, especially for longer maturity options. We propose several continuous-time linear diffusion processes that can capture broader forms of predictability, and provide numerical examples that illustrate their importance for pricing options.     

A simple efficient approximation to price basket stock options with volatility smile

This paper develops a new approach to obtain the price and risk sensitivities of basket options which have a volatility smile. Using this approach, the Black–Scholes model and the Stochastic Volatility Inspired model have been used to obtain an approximate analytical pricing formula for basket options with a volatility smile. It is found that our approximate formula is quite accurate by comparing it with Monte Carlo simulations. It is also proved the option value of our approach is consistent with the option value generated by Levy’s and Gentle’s approaches for typical ranges of volatility. Further, we give a theoretical proof that the option values from Levy’s and Gentle’s works are the upper bound and the lower bound, respectively, for our option value. The calibration procedure and a practical example are provided. The main advantage of our approach is that it provides accurate and easily implemented basket option prices with volatility smile and hedge parameters and avoids the need to use time-consuming numerical procedures such as Monte Carlo simulation.