Understanding Index Option Returns

It is acknowledged by the leading researchers that the fundamental force behind the emergence of advanced option pricing models was the result of abundant empirical research analysis, leading to the fact that the asset return distribution is non-log-normal. It obviously resulted in strong moneyness and maturity pricing biases of Black–Scholes (BS). The concept defined the assertive distributional assumptions for Nifty (benchmark index of India), because the stated view supported the idea and its implications majorly focused on establishing the well-protected alternative model. The core concept of almost all the option pricing models was substantially indicating towards the flexible distributional structure. Such behaviour not only correlated underlying stock returns and its volatility but also subdued the level of skewness and kurtosis. Therefore, this article endeavours to investigate empirically the out-of-sample moneyness–maturity forecasting performance of deterministic volatility option pricing models during the recent waves of economic imbalance. It forecasts S&P CNX Nifty 50 index options of India against significant BS option pricing model and also attempts to find out forecasting performances of a family of deterministic volatility option pricing models, namely, Practitioner Black-Scholes (PBS), Gram–Charlier (GC) and Constant Elasticity of Variance (CEV). The accuracy of models’ prices testify to the relative market price of Nifty index options using error metrics. For the purpose of this research, the prices are compared analytically through continuously updating the parameters of the models using cross-sectional option data, almost on a daily basis. The underlying focus of the article emphasizes on how to investigate the parameters of various models, pertaining specifically to a period of turbulence in the Indian economy. Eventually, while giving verdict to the results deduced, we measure out that the PBS model has smaller out-of-sample valuation errors in pricing Nifty Index options than the CEV and GC. However, it remains challenging to root out price bias completely out of all models we have framed out and applied.

Comovement and Volatility Risk Premia

Explaining the Level of Credit Spreads: Option-Implied Jump Risk Premia in a Firm Value Model

Employee stock options are widely used. SFAS 123(R), Share-Based Payment, issued in December 2004 requires that all entities use a fair-value-based option pricing model, adjusted for the unique characteristics of the options, to recognize expenses for employee stock options at the grant date. The Cox-Ross-Rubinstein (CRR) model – a binomial lattice model that converges to the Black-Scholes (BS) model when the number of periods to expiration approaches infinity - meets the requirements of SFAS 123(R). The CRR model factors in the often complex characteristics of employee stock options and always gives a lower option value than the BS model. This paper demonstrates the implementation of the CRR model using Microsoft Excel and shows that constant dividend yield, early exercise, and cliff vesting can be easily factored into the CRR model. Keywords SFAS 123(R), Employee Stock Options, Fair-Value-Based Option Pricing Model, CRR Model

Global Currency Misalignments, Crash Sensitivity, and Downside Insurance Costs

Finance models as metaphors

Compared with the developed countries, Chinese stock index option market developed late and accumulated less historical data. At the same time, it has the characteristics that many developed countries' stock option markets do not have. Black-Scholes Model is an important model in the field of option pricing. It is not enough for a financial model to be reasonable in theory, but it needs to be tested by the actual market. Therefore, based on the main transaction data of the CSI 300 stock index option contracts from 2010 to 2020, this paper analyzes the compliance of the CSI 300 stock index option market to the basic assumptions of the B-S option pricing model, the main parameters of the CSI 300 stock index options in the B-S option pricing model, and the fitting degree of the B-S model's prediction and actual prices. The result shows that there is a certain deviation between the pricing conclusions of the B-S model and the actual option prices. This paper analyzes the reasons for the deviation from many aspects, and puts forward the corresponding improvement strategies that the users of the model should adopt, so as to improve the applicability of the B-S option pricing model in Chinese stock index option market.

While there is a general consensus that forward rates have little if any power to forecast changes in spot rates, there is less consensus on the existence of time-varying premiums in forward rates. Progress has been made toward understanding foreign exchange risk premiums as empirical research continues, but the existence of forward foreign exchange premiums remains controversial. Unlike previous research, we have chosen to construct empirical asset pricing models that directly link the risk premiums to the state variables or risk factors that are thought to be the fundamental economic determinants of forward exchange pricing. We first link the foreign exchange risk premiums to real interest rate differentials, excess national equity market return differentials, and term structure slope differentials based on the various international parity conditions. We find that variations in risk premiums are closely linked to variations in risky asset return differentials. Accordingly, based on the modern asset pricing theory, we propose a time-varying risk premium model which links the risk premiums to macroeconomic and financial uncertainty. We find that volatilities from macroeconomic uncertainty and financial markets of the exchange countries can account for the variations in risk premiums and most of the state variables are priced. In particular, one common factor has been identified, namely the US real output shocks, which affect the pricing for all foreign currencies. The implication is that a shock in the US market might produce an uncertainty, transmitting into the national currency market. Finally, to distinguish the possibly short-lived risk premium and/or market inefficiency, we use high frequency data that relates the risk premiums to relative volatility perceived in the national equity markets. In general, the result is consistent with the risk premium hypothesis where a positive speculative profit is a reward for bearing a relatively higher risk. This dissertation contributes to existing literature by offering further evidence regarding the pricing of forward foreign exchange rates and time-varying risk premiums. Issues such as peso problems, learning, irrational speculative bubbles, as well as the effect of transaction costs will definitely play some roles in explaining forward rate determination and are left to future research.

In this paper a new method is designed to account price of military software, which is based on option pricing models. Firstly, it is introduced about the basic knowledge of option pricing model. And then the military software pricing model is provided based on Black-Scholes option model and binomial tree model. Several variables in reference to military software price are redefined in the related models. Then, in order to calculate the software price, we build up system of equations, of which one is Black-Scholes equation and the other one is binomial tree equation. Lastly, a case study is given to demonstrate the feasibility of the model. The study results and sensitivity analysis show the factors that influence the real price of software and the consistency of two classic option pricing models in practical application. In addition, one of the factors—volatility plays an important role in military software pricing based on Black-Scholes model, and there is a further research to make it clear and objective.

We use a novel pricing model to filter times series of diffusive volatility and jump intensity from S&P 500 index options. These two measures capture the ex-ante risk assessed by investors. We find that both components of risk vary substantially over time, are quite persistent, and correlate with each other and with the stock index. Using a simple general equilibrium model with a representative investor, we translate the filtered measures of ex-ante risk into an ex-ante risk premium. We find that the average premium that compensates the investor for the risks implicit in option prices, 10.1 percent, is about twice the premium required to compensate the same investor for the realized volatility, 5.8 percent. Moreover, the ex-ante equity premium that we uncover is highly volatile, with values between 2 and 32 percent. The component of the premium that corresponds to the jump risk varies between 0 and 12 percent.

Free (Almost) Variance Insurance

In recent years, quantitative researchers used a wide range of models to price options, from the Black-Scholes model to more complex models such as the Heston model. This paper aims to analyze the effectiveness of the Black-Scholes model and the Binomial Tree model by using them to price Berkshire Hathaway’s equity options and European-style S&P 100 index options. The method used in this paper is gathering the market data of the options first. Second, using the data gathered to price the options by applying the Black-Scholes and Binomial Tree models. Third, comparing the derived theoretical price with the market price by getting the Sum of Square Errors. Lastly, determining the best model for each type of option. Through this research, the author found that comparing the two models, the Binomial Tree model derives a smaller Sum of Square Errors when pricing European-style index options, and the Black-Scholes model derives a smaller Sum of Square Errors when pricing American equity options. Thus, the Binomial Tree model is a better model to price European-style index options, and the Black-Scholes model is more effective when pricing American equity options.

Volatility implied by an option pricing model is seen as the market participants' assessment of volatility. Past studies documented that implied volatility based on an option pricing model is found to outperform the historical volatility in forecasting future realised volatility. Thus, this study examines the implied volatility smiles and term structures in the Australian S&P/ASX 200 index options from the year 2001 to 2010, which covers the global financial crisis in the mid-2007 until the end of 2008. The results show that the implied volatility rises significantly during the crisis period, which is three time the rate before crisis.

We construct a no-arbitrage term structure model with jumps in the entire state vector at deterministic times but of random magnitudes. Jump risk premia are allowed for. We show that the model implies a closed-form representation of yields as a time-inhomogeneous affine function of the state vector, and derive other theoretical implications. We apply the model to the term structure of US Treasury rates, estimated at the daily frequency, allowing for jumps on days of employment report announcements. Our model can match the empirical fact that the term structure of interest rate volatility has a hump-shaped pattern on employment report days (but not on other days). The model also produces patterns in bond risk premia that are consistent with the empirical finding that much of the time-variation in excess bond returns accrues at times of important macroeconomic data releases.

High-Frequency Betas and Cross Section of Expected Stock Returns in Southeast Asia