A Simple Formula for the Expected Rate of Return of an Option over a Finite Holding Period

Under conditions consistent with the Black-Scholes formula, a simple formula is developed for the expected rate of return of an option over a finite holding period possibly less than the time to expiration of the option. Under these conditions, surprisingly, the expected future value of a European option, even prior to expiration, is shown equal to the current Black-Scholes value of the option, except that the expected future value of the stock at the end of the holding period replaces the current stock price in the Black-Scholes formula and the future value of a riskless invesment of the striking price replaces the striking price. An extension of this result is used to approximate moments of the distribution of returns from an option portfolio.     

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.     

Optimal Insurance Design Under a Value-at-Risk Framework

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. JEL Classification No: G22     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

Volatility spreads and earnings announcement returns

Prior research documents that volatility spreads predict stock returns. If the trading activity of informed investors is an important driver of volatility spreads, then the predictability of stock returns should be more pronounced during major information events. This paper investigates whether the predictability of equity returns by volatility spreads is stronger during earnings announcements. Volatility spreads are measured by the implied volatility differences between pairs of strike price and expiration date matched put and call options and capture price pressures in the option market. During a two-day earnings announcement window, the abnormal returns to the quintile that includes stocks with relatively expensive call options is more than 1.5% greater than the abnormal returns to the quintile that includes stocks with relatively expensive put options. This result is robust after measuring volatility spreads in alternative ways and controlling for firm characteristics and lagged equity returns. The degree of announcement return predictability is stronger when volatility spreads are measured using more liquid options, the information environment is more asymmetric, and stock liquidity is low.     

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.     

The Price Effect of Option Introduction

This paper examines the price effect of option introduction from 1974 to 1980. The introduction of individual options causes a permanent price increase in the underlying security, beginning approximately three days before introduction. The price effect appears to be associated with introduction, and not announcement, throughout the sample period. Excess returns volatility declines with option introduction. Systematic risk is unchanged. There is a positive relation between the price increase and a measure of activity in the options market.     

Option pricing under a stressed-beta model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model which naturally induces stochastic volatility in the asset price. Under this Stressed-Beta model, the mechanism is relatively simple: the slope coefficient—which measures asset returns relative to market returns—switches between two values, depending on the market being above or below a given level. After specifying the model, we use it to price European options on the asset. Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in option markets. This opens the possibility of forward-looking calibration of the slope coefficients, using option data, as illustrated in the paper.     

Additional evidence on transparency and bank financial performance

This paper examines the relationship between volatility and the probability of occurrence of expected extreme returns in the Canadian market. Four measures of volatility are examined: implied volatility from firm option prices, conditional volatility calculated using an EGARCH model, idiosyncratic volatility, and expected shortfall. A significantly positive relationship is observed between a firm's idiosyncratic volatility and the probability of occurrence of an extreme return in the subsequent month for firms. A 10% increase in idiosyncratic volatility in a given month is associated with the probability of an extreme shock in the subsequent month (top or bottom 1.5% of the returns distribution) of 26.4%. Other firm characteristics, including firm age, price, volume and book‐to‐market ratio, are also shown to be significantly related to subsequent firm extreme returns. The effects of conditional and implied volatility are mixed. The E‐GARCH and expected shortfall measures of conditional volatility are consistent with mean reversion: high short term realizations of conditional volatility foreshadow a lower probability of extreme returns.     

The dark side of financial innovation: A case study of the pricing of a retail financial product

The offering prices of 64 issues of a popular retail structured equity product were, on average, almost 8% greater than estimates of the products' fair market values obtained using option pricing methods. Under reasonable assumptions about the underlying stocks' expected returns, the mean expected return estimate on the structured products is slightly below zero. The products do not provide tax, liquidity, or other benefits, and it is difficult to rationalize their purchase by informed rational investors. Our findings are, however, consistent with the recent hypothesis that issuing firms might shroud some aspects of innovative securities or introduce complexity to exploit uninformed investors.     

On the qualitative effect of volatility and duration on prices of Asian options

We show that under the Black–Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility. This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the Black–Scholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the Black–Scholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options.     

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.     

Implied Risk Aversion Parameter from Option Prices

This paper estimates the representative investor's coefficient of relative risk aversion using option price data. Estimation is carried out using the method of simulated moments. Employing the following assumptions: a) agents have constant proportional risk averse preferences, b) complete markets exist, and c) asset returns are distributed lognormally, an objective function is constructed within the equivalent martingale measure framework. Unlike the case of equity markets, the implied risk aversion parameter from option prices is quite low and stays between zero and one.     

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.     

Nominal rigidities,asset returns,and monetary policy

Asset-return implications of nominal price and wage rigidities are analyzed in general equilibrium. Nominal rigidities, combined with permanent productivity shocks, increase expected excess returns on production claims. This is mainly explained by consumption dynamics driven by rigidity-induced changes in employment and markups. An interest-rate monetary policy rule affects asset returns. Stronger (weaker) rule responses to inflation (output) increase expected excess returns. Policy shocks substantially increase asset-return volatility. Price rigidity heterogeneity produces cross-sectoral differences in expected returns. The model matches important macroeconomic moments and the Sharpe ratio of stock returns, but only captures a small fraction of the observed equity premium.     

The Price of Correlation Risk: Evidence from Equity Options

We study whether exposure to marketwide correlation shocks affects expected option returns, using data on S&P100 index options, options on all components, and stock returns. We find evidence of priced correlation risk based on prices of index and individual variance risk. A trading strategy exploiting priced correlation risk generates a high alpha and is attractive for CRRA investors without frictions. Correlation risk exposure explains the cross-section of index and individual option returns well. The correlation risk premium cannot be exploited with realistic trading frictions, providing a limits-to-arbitrage interpretation of our finding of a high price of correlation risk.     

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.     

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.     

On the Upper Bound of a Call Option

Using only a weak set of assumptions, Merton (1973) shows that the upper bound of a European or American call option on a non-dividend paying stock is the underlying stock price: a result which is often extended to options on dividend paying stocks. In this short technical piece we show that the underlying stock price is in fact not the least upper bound of either a European or an American call option on a stock that pays one or more known dividends prior to maturity. Based on Merton's (1973) original framework, new upper bounds are established which depend on the size(s) of the dividend(s) compared to the size of the strike. JEL Classification: G12, G13     

Optimal Risk Management Using Options

This article provides an analytical solution to the problem of an institution optimally managing the market risk of a given exposure by minimizing its Value-at-Risk using options. The optimal hedge consists of a position in a single option whose strike price is independent of the level of expense the institution is willing to incur for its hedging program. This optimal strike price depends on the distribution of the asset exposure, the horizon of the hedge, and the level of protection desired by the institution. Moreover, the costs associated with a suboptimal choice of exercise price are economically significant.