Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

Option prices and stock market momentum: evidence from China

Option prices tend to be correlated to past stock market returns due to market imperfections. We unprecedentedly examine this issue on the SSE 50 ETF option in the Chinese derivatives market. To measure the price pressure in the options market, we construct an implied volatility spread based on pairs of the SSE 50 ETF option with identical expiration dates and strike prices. By regressing the implied volatility spread on past stock returns, we find that past stock returns exert a strong influence on the pricing of index options. Specifically, we find that SSE 50 ETF calls are significantly overvalued relative to SSE 50 ETF puts after stock price increases and the reverse is also true after the stock price decreases. Moreover, we validate the momentum effects in the underlying stock market to be responsible for the price pressure. These findings are both economically and statistically significant and have important implications.     

SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.     

The information content of ETF options

We examine the information content of China's Shanghai Stock Exchange (SSE) 50 ETF options introduced in 2015. Trading volume and implied volatilities of calls versus puts differ markedly: trading volume is consistently higher for calls, and implied volatility is higher for puts. Put-call volume and implied volatility ratios are not good predictors of future SSE 50 returns. Implied volatility follows a right-skewed smirk across strike prices, indicating a tendency among option traders to turn bullish and expect the stock market to recover from the June 2015 market crash. The options market dominates the price discovery process, with an average information leadership share of 67%. Our price discovery results persist during the COVID outbreak.     

The information content of implied volatilities and model-free volatility expectations: Evidence from options written on individual stocks

We measure the volatility information content of stock options for individual firms using option prices for 149 US firms and the S&P 100 index. We use ARCH and regression models to compare volatility forecasts defined by historical stock returns, at-the-money implied volatilities and model-free volatility expectations for every firm. For 1-day-ahead estimation, a historical ARCH model outperforms both of the volatility estimates extracted from option prices for 36% of the firms, but the option forecasts are nearly always more informative for those firms that have the more actively traded options. When the prediction horizon extends until the expiry date of the options, the option forecasts are more informative than the historical volatility for 85% of the firms. However, at-the-money implied volatilities generally outperform the model-free volatility expectations.     

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.     

Overstatement of Implied Variance in the Dollar/Yen Currency Option Market

This paper investigates the predictive power of implied variancesextracted from the dollar/yen option prices. Implied variances areestimated from transaction prices of currency options traded on PHLXusing the option pricing model of Garman and Kohlhagen (1983). Incontrast to recent findings on stock and stock index options, theout-of-sample tests indicate that the implied variance is an upwardbiased estimator of future variance; and that the variance forecastsfrom GARCH and historical models do not contain significantincremental information in predicting future variance. Tradingstrategies are also developed to exploit the observed overstatementof variance in the dollar/yen option market. Traders that can executethe delta-neutral trading strategies at the observed markettransaction prices could lock in a significant profits during theperiod examined. However, for investors that facing highertransaction costs, the magnitude of the profits is generally notlarge enough to allow for abnormal risk-adjusted profits.     

An IFRS 2 and FASB 123 (R) Compatible Model for the Valuation of Employee Stock Options

In this paper, we show how employee stock options can be valued under the new reporting standards IFRS 2 and FASB 123 (revised) for share-based payments. Both standards require companies to expense employee stock options at fair value. We propose a new valuation model, referred to as Enhanced American model, that complies with the new standards and produces fair values often lower than those generated by traditional models such as the Black–Scholes model or the adjusted Black–Scholes model. We also provide a sensitivity analysis of model input parameters and analyze the impact of the parameters on the fair value of the option. The valuation of employee stock options requires an accurate estimation of the exercise behavior. We show how the exercise behavior can be modeled in a binomial tree and demonstrate the relevance of the input parameters in the calibration of the model to an estimated expected life of the option. JEL Classification G13, G30     

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.

     

THE PROPOSED INTRODUCTION OF FUTURES‐STYLE MARGINING IN THE UNITED STATES: AN AUSTRALIAN COMPARISON

We extend the quadratic approximation method to examine American‐style options traded using futures‐style margining and show that an early exercise premium can exist when the cost of carry is negative. Empirical results based on a reduced form of the model using futures‐style call options traded on the Australian All Ordinaries Share Price Index are consistent with previous research: call option early exercise premiums are economically zero. Full option prices are examined by comparing observed futures‐style with theoretical stock‐style values. We find futures‐style values exceed stock‐style values and argue that the increase results from improvements in liquidity. The findings are particularly relevant given the pending decision at the Commodity Futures Trading Commission to introduce a futures‐style system in the United States. JEL classification: G13, C13     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

Finance, Politics, and the Accounting for Stock Options

The politics of option accounting crosses party lines, reflecting both the interests of the affected constituencies and the desire for power over standard setting. House Bill HR-3574, which mandates an assumption of zero stock price volatility, runs counter to the recently passed Financial Accounting Standards Board (FASB) rule requiring fair-value expensing of stock options. For any option issued at or out of the money, where strike prices are normally set, expense recognition is zero under this bill's mandated assumption.
Besides excessive use of stock options, the lack of a "final peace" in the option accounting war appears to have encouraged another questionable corporate practice. This article examines a sample of "six-and-one restructurings," exchanges of options in which expensing of re-priced (deep out-of-the-money) options can be avoided if employees wait at least six months and one day before receiving new options. The authors found that market-adjusted stock prices tend to decrease during the six-month period before the strike price is reset. This result provides one more reason why companies should be required to use fair-value option pricing models to expense options.     

Informed Trading in Stock and Option Markets

We investigate the contribution of option markets to price discovery, using a modification of Hasbrouck's (1995) "information share" approach. Based on five years of stock and options data for 60 firms, we estimate the option market's contribution to price discovery to be about 17% on average. Option market price discovery is related to trading volume and spreads in both markets, and stock volatility. Price discovery across option strike prices is related to leverage, trading volume, and spreads. Our results are consistent with theoretical arguments that informed investors trade in both stock and option markets, suggesting an important informational role for options.     

Pricing options on mean reverting underliers

For mean reverting base probabilities, option pricing models are developed, using an explicit measure change induced by the selection of a terminal time and a terminal random variable. The models employed are the square root process and an OU equation driven by centred variance gamma shocks. VIX options are calibrated using the square root process. The OU equation driven by centred variance gamma shocks is applied in pricing options on the ratio of the stock price for J. P. Morgan Chase (JPM) to the Exchange Traded Fund for the financial sector with ticker XLF. For the purposes of calibrating the ratio option pricing model to market data, we indirectly infer the prices for stock options on JPM from the prices for options on the ratio, by hedging the conditional value of JPM options given XLF, using options on XLF. The implied volatilities for the options on the ratio are then indirectly observed to be fairly flat. This suggests that for JPM, the use XLF as a benchmark is a possibly good choice. It is shown to perform better than the use of the S&P 500 index. Furthermore, though the use of an unrelated stock price like Johnson and Johnson as a benchmark for JPM provides as a good fit as does the use of XLF, this comes at the cost of requiring a considerable smile for the implied volatilities on the ratio options and hence a more complex model for the implied distribution on the ratio.     

Financial inverse problem and reconstruction of?infinitely divisible distributions with Gaussian component

We consider an inverse problem of partial integro-differential equations of market prices of call options with many maturities and strike prices for geometric Lévy processes. We show the well-posedness (reconstruction, uniqueness, and stability) of the inverse problem among the class of infinitely divisible distributions with analyticity.     

Implied Binomial Trees

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.     

Transactions data tests of efficiency of the Chicago board options exchange

This paper performs lower boundary condition tests based on rational pricing of call options and an implied standard deviation test based on the bid/ask prices of options. These efficiency tests attempt to closely approximate conditions in the option markets to avoid the pitfalls indicated by Phillips and Smith (1980). The tests use transactions data and account for the effects of stock and option bid/ask prices, simultaneity of stock and option prices, depth of market, execution lag and transaction costs. The small and relatively infrequent profits due to market mispricing disappear in the lower boundary tests when transaction costs are taken into account. Frequent violations of the tighter boundary conditions in the implied standard deviation test are reported, but the estimated profits cannot be unambiguously attributed to option market inefficiency.     

Modeling stock pinning

This paper investigates the effect of hedging strategies on the so-called pinning effect, i.e. the tendency of stock's prices to close near the strike price of heavily traded options as the expiration date nears. In the paper we extend the analysis of Avellaneda and Lipkin, who propose an explanation of stock pinning in terms of delta hedging strategies for long option positions. We adopt a model introduced by Frey and Stremme and show that, under the original assumptions of the model, pinning is driven by two effects: a hedging-dependent drift term that pushes the stock price toward the strike price and a hedging-dependent volatility term that constrains the stock price near the strike as it approaches it. Finally, we show that pinning can be generated by simulating trading in a double auction market. Pinning in the microstructure model is consistent with the Frey and Stremme model when both discrete hedging and stochastic impact are taken into account.     

Revisiting the incentive effects of executive stock options

We develop a multiperiod framework to evaluate the incentive effects of executive stock options (ESOs). For a given increase in the grant-date firm stock price (and a concurrent increase in return volatility), the increment of total value at the vesting date acts as a proxy for the incentive effects of ESOs. If the option is attached to the existing contract without adjusting cash compensation, we suggest that a firm should not always fix the strike price to the grant-date stock price; instead, the strike price should vary with the length of the vesting period. We also show that, compared with at-the-money options, restricted stock generates greater incentives to increase stock prices in some scenarios, especially when equity-based awards are vested early. If the vesting period is long, the firm could grant options instead of restricted stock to maximize incentives.     

Maturity and exercise price of executive stock options

Using a simple three-period model in which a manager can gather information before making an investment decision, this paper studies optimal contracts with various stock options. In particular, we show how the exercise price of executive stock options is related to a base salary, the size of the option grant, leverage, and the riskiness of a desired investment policy. The optimal exercise price increases in the size of grant and the base salary and decreases in leverage and the riskiness of a desired investment policy. Other things equal, the optimal exercise price of European options with a longer maturity should increase more for an increase in the base salary and the size of grant and decrease more for an increase in leverage than the one with a shorter maturity. The optimal exercise price of American options is determined by the optimal exercise prices of European options with different maturities. Given the fixed exercise price, the size of the option grant does not decrease in the face value of debt.