Binomial basis for linear information dynamics: real options, dividends and the valuation of equity

Analytical research has confirmed that real options give rise to the kind of nonlinearities observed in practice between equity prices and the figures appearing on corporate financial statements. We develop these real option values in terms of a quasi 'supply-side' model of linear information dynamics based on simple discrete time binomial filtration processes. Our analysis shows that the linear models that pervade the empirical (and analytical) work of the area, will almost certainly suffer from an omitted variables problem. Parameter estimation will then be inconsistent and inefficient.     

Consumption-based equity valuation

Using a CCAPM-based risk-adjustment model, we perform yearly valuations of a large sample of stocks listed on NYSE, AMEX, and NASDAQ over a 30-year period. The model differs from standard valuation models in the sense that it adjusts forecasted residual income for risk in the numerator rather than through a risk-adjusted cost of equity in the denominator. The risk adjustments are derived based on assumptions about the time-series properties of residual income returns and aggregate consumption rather than on historical stock returns. We compare the performance of the model with several implementations of standard valuation models, both in terms of median absolute valuation errors (MAVE) and in terms of excess returns on simple investment strategies based on the differences between model and market prices. The CCAPM-based valuation model yields a significantly lower MAVE than the best performing standard valuation model. Both types of models can identify investment strategies with subsequent excess returns. The CCAPM-based valuation model yields time-series of realized hedge returns with more and higher positive returns and fewer and less negative returns compared with the time-series of realized hedge returns based on the best performing standard valuation model for holding periods from 1 to 5 years. In a statistical test of 1-year-ahead excess return predictability based on the models’ implied pricing errors, the CCAPM-based valuation model is selected as the better model. Using the standard series of aggregate consumption and the nominal price index, a reasonable level of relative risk aversion, and calibrated growth rates in the continuing value at each valuation date, the CCAPM-based valuation model produces small risk adjustments to forecasted residual income and low continuing values. Compared with standard valuation models, it relies less on estimated parameters and speculative elements when aggregating residual earnings forecasts into a valuation.     

Real (adaptation) options and the valuation of equity: some empirical evidence

Data from 1,374 firms across four broad industrial groupings are used to assess the contribution that real (adaptation) options make to overall equity values. The analysis indicates that real (adaptation) options make a significant contribution to the equity value of firms with a market to book ratio (of equity) of around unity or less. As the market to book ratio grows beyond this level, however, the contribution made by real (adaptation) options decays quickly away and equity values are mainly comprised of the present value of the dividends that firms are expected to pay. This means that for around one in every five of the firms in our sample real (adaptation) options make a significant contribution to overall equity value. Thus, while linear equity valuation models would seem to be appropriate for the substantial majority of firms on which our sample is based, there is a sizeable minority of firms where real (adaptation) options have a significant impact on equity values. For this latter group of firms there will be a non-linear relationship between equity value and its determining variables. This has important implications for the regression procedures that are applied in this area of accounting research.     

Familiarity bias and earnings-based equity valuation

Review of Quantitative Finance and Accounting - This study examines whether investors’ familiarity bias affects their earnings-based equity valuation. Building on theoretical and empirical...     

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.     

The valuation of compound options

This paper presents a theory for pricing options on options, or compound options. The method can be generalized to value many corporate liabilities. The compound call option formula derived herein considers a call option on stock which is itself an option on the assets of the firm. This perspective incorporates leverage effects into option pricing and consequently the variance of the rate of return on the stock is not constant as Black-Scholes assumed, but is instead a function of the level of the stock price. The Black-Scholes formula is shown to be a special case of the compound option formula. This new model for puts and calls corrects some important biases of the Black-Scholes model.     

Seasonality and the valuation of commodity options

Price movements in many commodity markets exhibit significant seasonal patterns. However, given an observed futures price, a deterministic seasonal component at the price level is not relevant for the pricing of commodity options. In contrast, this is not true for the seasonal pattern observed in the volatility of the commodity price. Analyzing an extensive sample of soybean, corn, heating oil and natural gas options, we find that seasonality in volatility is an important aspect to consider when valuing these contracts. The inclusion of an appropriate seasonality adjustment significantly reduces pricing errors in these markets and yields more improvement in valuation accuracy than increasing the number of stochastic factors.     

Rainbow trend options: valuation and applications

Asset selection and timing decisions are major investment concerns. To resolve these issues simultaneously, a new class of rainbow trend options is proposed. The diversification effect of rainbow options can reduce the importance of asset selection decisions and trend options can mitigate unfavorable effects on market entry and exit decisions. We consider a general framework to facilitate the derivation of analytic pricing formulas for simple, pure, and Asian rainbow trend options using the martingale pricing method. The properties of these options and their Greeks are analyzed. We also investigate the performance of the dynamic delta hedging strategy for issuers of rainbow trend options. Last, this paper explores the applications of rainbow trend options for hedging price risks, designing executive stock options, modifying countercyclical capital buffer proposed by Basel Committee, and acting as control variates of the Monte Carlo simulation.     

Liquidation triggers and the valuation of equity and debt

Many bankruptcy codes implicitly or explicitly contain net-worth covenants, which provide the firm’s bondholders with the right to force reorganization or liquidation if the value of the firm falls below a certain threshold. In practice, however, default does not necessarily lead to immediate change of control or to liquidation of the firm’s assets by its debtholders. To consider the impact of this on the valuation of corporate securities, we develop a model in which liquidation is driven by a state variable that accumulates with time and severity of distress. We model a dynamic grace period for the liquidation event. Recent or severe distress events may have greater impact on the liquidation trigger. Our model can be applied to a wide array of bankruptcy codes and jurisdictions.     

The analytic valuation of American options

No analytic solution exists for the valuation of American optionswritten on futures contracts and foreign currencies for whichearly exercise may be optimal. This article formulates the Americanoption valuation problem in economically and mathematicallymeaningful ways. This enables us to derive valuation formulasfor American options. The properties associated with the optimalexercise boundary are examined, and a numerical technique toimplement the valuation formulas is presented.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

Accounting data and asset valuation: Theory

Firms and divisions which are not traded on organized exchanges are often valued without the benefit of market data. Accounting data is used instead. One suggested approach is to use accounting beta as a proxy for market return beta. In the context of the Arbitrage Pricing Theory, we provide a theoretical justification for such a procedure. Our results provide a set of sufficient conditions so that return betas and accounting betas are equal. Our results also suggest a general methodology for evaluating projects and untraded firms using accounting data. The method underlying the derivation here is very general and can be applied in deriving testable restrictions between fundamentals, broader in context than that of accounting variables.     

Equity value,implied cost of equity and shareholders’ real options

This study investigates the effects of shareholders’ real options on (i) firm financial performance and (ii) estimations of the implied cost of equity. After measuring the equity value of steady‐state operations using the residual income model, and the abandonment and expansion options using the Black‐Scholes option pricing model, I find that firms with a large expansion (abandonment) option value experience better (worse) financial performance than those with a small such value. I also find that ignoring these options results in a downward bias in implied cost of equity estimates by an average of 1.23 percentage points.     

Asian equity markets: growth,opportunities, and challenges

Asian equity markets have grown significantly in size since the early 1990s, driven by strong international investor inflows, growing regional financial integration, capital account liberalization, and structural improvements to markets. The development of equity markets provides a more diversified set of channels for financial intermediation to support growth, thus bolstering medium-term financial stability. At the same time, as highlighted by the May–June 2006 market corrections, the increasing role of stock markets potentially changes the nature of macroeconomic and financial stability risks, as well as the policy requirements for dealing with these risks.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

Two-dimensional risk-neutral valuation relationships for the pricing of options

The Black–Scholes model is based on a one-parameter pricing kernel with constant elasticity. Theoretical and empirical results suggest declining elasticity and, hence, a pricing kernel with at least two parameters. We price European-style options on assets whose probability distributions have two unknown parameters. We assume a pricing kernel which also has two unknown parameters. When certain conditions are met, a two-dimensional risk-neutral valuation relationship exists for the pricing of these options: i.e. the relationship between the price of the option and the prices of the underlying asset and one other option on the asset is the same as it would be under risk neutrality. In this class of models, the price of the underlying asset and that of one other option take the place of the unknown parameters.      

Forward-neutral valuation relationships for options on zero coupon bonds

This paper extends the literature on Risk-Neutral Valuation Relationships (RNVRs) to derive valuation formulae for options on zero coupon bonds when interest rates are stochastic. We develop Forward-Neutral Valuation Relationships (FNVRs) for the transformed-bounded random walk class. Our transformed-bounded random walk family of forward bond price processes implies that (i) the prices of the zero coupon bonds are bounded below at zero and above at one, and (ii) negative continuously compounded interest rates are ruled out. FNVRs are frameworks for option pricing, where the forward prices of the options are martingales independent of the market prices of risk. We illustrate the generality and flexibility of our approach with models that yield several new closed-form solutions for call and put options on discount bonds.     

The effect of mergers on implied volatility of equity options

This paper uses a sample of 25 large mergers from 1996 to 2004 to study the effect of mergers on the implied volatilities of equity options. The results indicate a statistically significant increase in volatility beyond the amount predicted if the transaction were effectively nothing more than a portfolio combination of the target and acquirer. The disparity suggests that, at least for the first 18 months after the transaction becomes effective, market participants expect mergers to increase risk. Integration risk and uncertainty about the extent to which efficiency gains and greater market power are realized are possible explanations for the discrepancy.     

Pricing options in incomplete equity markets via the instantaneous Sharpe ratio

We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Saá-Requejo (J Polit Econ 108:79–119, 2000) and of Björk and Slinko (Rev Finance 10:221–260, 2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque et al. (Derivatives in financial markets with stochastic volatility. Cambridge University Press, 2000).     

Closed-form valuation of American call options on stocks paying multiple dividends

An American call option on a stock paying a single known dividend can be valued using the Roll–Geske–Whaley formula. This paper extends the Roll–Geske–Whaley model to the n dividends case by using the generalized n-fold compound option model. In this way this paper offers a closed-form solution for American options on stocks paying n known discrete dividends. Moreover, the model also offers the critical values of the early exercise boundaries at each ex-dividend date instant, making it easy to define an early exercise strategy. Numerical examples are included to illustrate this approach.