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.     

An Empirical Analysis of Growth Options of Japanese Electronics Firms

This paper provides some empirical support of real option models of corporate flexibility based on the analysis of the panel of Japanese electronics firms covering fiscal years 1999–2008. I apply random effect regression techniques and explain market value of firms’ growth options using several uncertainty proxies and firm-specific variables. Important contribution of the current study lies in empirical examining of the critical parameters of real option analysis of determinants of a firm’s investment behavior based on simple but effective structural model of multiple American-type options.     

A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson in J Financ Quant Anal 18(1):141–148 (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same efficiency as the quadratic approximation of Barone-Adesi and Whaley in J Financ 42:301–320 (1987), with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston’s stochastic volatility model, our method is shown to be extremely efficient and fairly accurate.     

Duration, factor sensitivities, and interest rate Greeks

In this paper, we introduce for interest rate sensitive assets the natural analogs of delta and gamma for equity options by considering the derivatives of asset prices with respect to the directions along which the forward rate curve may evolve. Macaulay duration and convexity, as well as stochastic duration considered in Cox et al. (J Business 52:51–61, 1979) and Munk (Rev Derivat Res 3:157–181, 1999), are easily obtained as special cases of these in which the derivatives are computed along parallel shifts and the direction of the forward rate volatilities, respectively. Moreover, we demonstrate using the example of the Ritchken and Sankarasubramanian (Math Financ 5:55–72, 1995) model that the hedging strategy based on these sensitivity measures provides a superior performance in comparison to the traditional duration based hedging approaches.      

On the cost of delayed currency fixing announcements

In Foreign Exchange Markets vanilla and barrier options are traded frequently. The market standard is a cutoff time of 10:00 a.m. in New York for the strike of vanillas and a knock-out event based on a continuously observed barrier in the inter bank market. However, many clients, particularly from Italy, prefer the cutoff and knock-out event to be based on the fixing published by the European Central Bank on the Reuters Page ECB37. These barrier options are called discretely monitored barrier options. While these options can be priced in several models by various techniques, the ECB source of the fixing causes two problems. First of all, it is not tradable, and secondly it is published with a delay of about 10–20 min. We examine here the effect of these problems on the hedge of those options and consequently suggest a cost based on the additional uncertainty encountered.      

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.     

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.      

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

Option pricing when correlations are stochastic: an analytical framework

In this paper we develop a novel market model where asset variances–covariances evolve stochastically. In addition shocks on asset return dynamics are assumed to be linearly correlated with shocks driving the variance–covariance matrix. Analytical tractability is preserved since the model is linear-affine and the conditional characteristic function can be determined explicitly. Quite remarkably, the model provides prices for vanilla options consistent with observed smile and skew effects, while making it possible to detect and quantify the correlation risk in multiple-asset derivatives like basket options. In particular, it can reproduce and quantify the asymmetric conditional correlations observed on historical data for equity markets. As an illustrative example, we provide explicit pricing formulas for rainbow “Best-of” options.     

CEO stock options and analysts’ forecast accuracy and bias

This paper investigates the relationship between CEO stock options and analysts’ earnings forecast accuracy and bias. A higher level of stock options may induce managers to undertake riskier projects, to change and/or reallocate their effort, and to possibly engage in gaming (such as opportunistic earnings and disclosure management). These managerial behaviors result in an increase in the complexity of forecasting and hence, less accurate analysts’ forecasts. Analysts’ optimistic forecast bias may also increase as the level of stock options pay increases. Because forecast complexity increases with stock options pay, analysts, needing greater access to management’s information to produce accurate forecasts, have incentives to increase the optimistic bias in their forecasts. Alternatively, a higher level of stock options pay may lead to improved disclosure because it better aligns managers’ and shareholders’ interests. The improved disclosure, in turn, may result in more accurate and less biased analysts’ forecasts. Our empirical evidence indicates that analysts’ earnings forecast accuracy decreases and forecast optimism increases as the level of CEO stock options increases. This evidence suggests that the incentive alignment effects of stock options are more than offset by the investment, effort allocation and gaming incentives induced by stock options grants to CEOs.     

Remarks on the Nonlinear Black-Scholes Equations with the Effect of Transaction Costs

We deal with the solvability and a weak formulation of nonlinear partial differential equations of Black-Scholes type for the pricing of options in the presence of transaction costs. Examples include the Hoggard–Whalley–Wilmott equation, which is introduced to model portfolios of European type options with transaction costs based on the idea of Leland. The cost structure gives rise to nonlinear terms. We show the existence and the uniqueness of solutions both in the classical and the weak sense, where the notion of weak solutions is introduced.     

A pure martingale dual for multiple stopping

In this paper, we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such, it is a natural generalization of the method in Rogers (Math. Finance 12:271–286, 2002) and Haugh and Kogan (Oper. Res. 52:258–270, 2004) for the standard stopping problem for American options. We term this representation a ‘pure martingale’ dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (Math. Finance 14:557–583, 2004). For the multiple dual representation, we propose Monte Carlo simulation methods which require only one degree of nesting.     

Momentum in Residential Real Estate

This paper examines whether there is return momentum in residential real estate in the U.S. Case and Shiller (American economic review 79(1):128–137, 1989) document evidence of positive return correlation in four U.S. cities. Similar to Jegadeesh and Titman’s (Journal of finance 56:699–720, 1993) stock market momentum paper, we construct long-short zero cost investment portfolios from more than 380 metropolitan areas based on their lagged returns. Our results show that momentum of returns in the U.S. residential housing is statistically significant and economically meaningful during our 1983 to 2008 sample period. On average, zero cost investment portfolios that buy past winning housing markets and short sell past losing markets earn up to 8.92% annually. Our results are robust to different sub-periods and more pronounced in the Northeast and West regions. While zero cost portfolios of residential real estate indices is not a tradable strategy, the implications of our results can be useful for builders, potential home owners, mortgage originators and traders of real estate options.     

Re-examining the investment-uncertainty relationship in a real options model

The main purpose of this paper is to re-examine the investment-uncertainty relationship in a real options model, and demonstrates that the Sarkar (J Econ Dyn Control 24:219–225, 2000) model is a special case of our model. This paper uses a general dynamic process, which incorporates mean reversion and jumps in a firm’s project earnings. We further derive a quasi-analytical form solution for the critical investment value and investment probability of a firm’s projects. From the simulation results, we find that an increase in uncertainty can always lead to an increase in the probability of investment, and thus has a positive impact on investment. These results, which differ from the findings of Sarkar (J Econ Dyn Control 24:219–225, 2000), could be explained by the mean-reversion and jump effects on a firm’s earnings.     

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.     

Banking fragility and liquidity creation: options as a substitute for deposits

Diamond and Rajan (J Finance 55:2431–2465, 2000; Am Econ Rev Papers Proc 91:422–425, 2001a; Carnegie–Rochester Conf Series Public Policy 54:37–71, 2001b; J Pol Econ 109:287–327, 2001c) have shown in a series of papers that it is precisely the fragility of their capital structure which allows banks to create liquidity. This is because the threat of runs by depositors forces bankers to extract full repayment on otherwise illiquid assets. This result has important implications for financial regulation, such as for capital requirements and deposit insurance. This note shows that put options held by bank owners dominate deposit financing in that they also discipline bankers but do not give rise to inefficient runs. Fragility is thus not necessary for liquidity creation in the Diamond–Rajan framework. This paper has substantially benefitted from the comments and suggestions of an anonymous referee.     

Risk-neutral compatibility with option prices

A common problem is to choose a “risk-neutral” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure” by completing the market with option prices. We do this by modeling the behavior of the stock price X, together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of “compatibility” between X and the prices of our options, and this poses some significant mathematical difficulties.     

Optimal exercise of executive stock options

Valuing executive stock options is a challenging problem, because the standard risk-neutral valuation of those options is not appropriate; the executive is not allowed to trade the stock of the firm, so is not operating in a complete market. As this paper shows, an executive holding many American-style call options on his firm’s stock will optimally exercise the options bit by bit, whereas a risk-neutral valuation of the options would assume that all are exercised at the same time. Comparative statics of the optimal exercise policy show many surprising features.      

Underwriter reputation and underpricing: evidence from the Australian IPO market

Dimovski and Brooks (J Intern Financ Mark Inst Money 14:267–280, 2004b) examined 358 Australian industrial and mining company initial public offerings (IPOs) from 1994 to 1999 to report that more money was left on the table by IPOs that engaged underwriters than those that did not engage underwriters. Loughran and Ritter (Autumn 5–37, 2004) suggested that the negative relation between underwriter reputation and underpricing has reversed in the 1990s with U.S. IPOs. The main purpose of this paper is to study the relationship between underwriter reputation and underpricing in terms of Australian IPO data. In this paper, we use 380 Australian industrial company IPOs from 1994 to 2004 to perform the empirical study. Our results suggest that more prestigious underwriters are associated with a higher level of underpricing. Other variables that are found to be significant in explaining the level of IPO underpricing are market sentiment, share options, total capital raised and underwriter options.     

Valuing reload options

Over the past quarter century, the use of stock options as pay for performance has grown enormously. Option grants now account for 32% of CEO pay—more than twice that of salaries. In addition options are now being granted to many more employees than before. During this same time period, there have been numerous innovations in the features on compensation options. One of these features is the reload—the grant of new options to replace shares tendered in the payment of the exercise. Within the past year, the long-delayed FASB requirement that options be expensed for financial reporting has finally become a fact. It is incumbent upon financial researchers to provide methods to achieve the goal of valuing options, not only to serve the accounting needs, but also to provide ways of determining their true costs and incentive effects. This paper analyzes the various forms of reload options and provides simple Black-Scholes like formulas for evaluating them. JEL Classification G13