Narrow Results

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

We introduce a new approach for modeling the prepayments of a mortgage pool and show how it can be used to value mortgage pools and agency mortgage-backed securities. We describe the full spectrum of refinancing behavior using a notion of refinancing efficiency. Our approach has two distinguishing features: (1) our primary focus is on understanding the market value of a mortgage, in contrast with standard models that strive (often unsuccessfully) to predict future cash flows, and (2) we use two separate yield curves, one for modeling mortgage cash flows and the other for MBS cash flows.

We use order book data combined with tick data to analyze the supply curve models of liquidity issues in stock and option market trading. We show that supply curves really exist, and further that for highly liquid stocks they are linear. For slightly less liquid stocks the supply curve tends to be jump linear.

There is no margin settlement mechanism for existing covered warrants in Taiwan, thus the credit risk of the warrant issuer must be considered when investors evaluate the price of a covered warrant. This paper applies the vulnerable option valuation model to empirically study the difference in the theoretical value of a vulnerable warrant, Black–Scholes option price and the market price of warrant by using the Taiwan warrant data. Empirical results show that the theoretical value of a vulnerable warrant is lower than the Black–Scholes non-vulnerable option value and its market value.

Existing studies on the informational content of at-the-money implied volatility (ATMIV) and past realized volatility (PRV) and the relation between the two have mainly focused on a single short forecast horizon and conclude that ATMIV outperforms PRV. We examine the relation between implied and realized volatility over both short and longer forecasting horizons to provide a forecasting competition. We analytically demonstrate the option maturity effect on the sensitivity of the implied volatility (IV) estimation. As time to maturity increases, vega increases but at a decreasing rate, up to . At shorter (longer) maturities, small pricing changes should have greater (smaller) corrective impact on IVs. We find that IV outperforms PRV over a one month forecast horizon. However, as the forecast horizon increases, PRV outperforms IV and subsumes the information contained in it. These mixed results may be attributed to the reduced efficiency of longer dated sections of the S&P 500 options index market we analyze.

This study creates, models and calibrates the prices of real estate investment trusts (REITs) within an option pricing framework. Unlike Barraclough et al. [(2013). Using option prices overpayments and synergies in M&A transactions. The Review of Financial Studies, 26(3), 695-772] and Marcato and Sebehela [(2022). The paradoxical prices of options. Review of Pacific Basin Financial Markets & Policies, 25(2), 2250009], this study went further and incorporated new information in the form of news. The results show that for a target (acquiring) firm, the exponential risk premium is made up of the cost of carry plus (less) new news. The same illustration applies equally in a merged entity.

In financial markets, there are situations where investors have the future stock prices according to the experts’ evaluations rather than historical data. Thus, the estimations of the stock prices contain much subjective imprecision instead of randomness. This paper discusses a portfolio investment with options in such a kind of situation. Treating the stock index price as an uncertain variable, we build an uncertain mean-chance portfolio model based on uncertainty theory and provide the equivalent form of the model. Furthermore, we make a comparison of the optimal expected return between portfolio investment with options and without options. An important conclusion is reached: The portfolio investment with options produces a no less expected return than that without options. In addition, we make sensitivity analysis and get two vital corresponding results. As an illustration, a numerical example is presented as well. The numerical results reveal that the options should be considered in portfolio investment. And the call option with maximum exercise price is most valuable per premium cost with the same exercise date.

I examine the effects of return predictability on option prices for the market portfolio in the presence of stochastic volatility and/or stochastic interest rates. The analysis is implemented in an equilibrium framework where a consistent option pricing model is derived with the return predictability and stochastic volatility and the precise link between the actual and the risk neutral measures is endogenized. The equilibrium analysis indicates that the return predictability is induced by the mean-reverting and heteroskedastic features of aggregate dividends. It is shown that risk-neutral option pricing model with the stochastic volatility and/or stochastic interest rates can be consistent with return predictability. Numerical results suggest that (i) models with either perfect predictability or no predictability will significantly overprice long-term options across different strike prices when the return of the underlying exhibits modest predictability; (ii) the stochastic volatility does not affect option prices in a significant way when asset return predictability is properly reflected in the actual stock price process; (iii) when return predictability is correctly specified, the effects of stochastic interest rates are not uniform.

In this paper, we consider price weighted-volatility swap and price weighted-variance swap. The underlying asset considered in this paper is assumed to follow a general stochastic differential equation and exhibits stochastic volatility. We obtain analytical pricing formulas for the weighted-variance swap and approximate expression for the weighted-volatility swap. Nice bounds for the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to implement in real time and can be applied efficiently for practical applications. We consider the problem of hedging volatility swap with variance swap and obtain analytical formula for the hedge ratio. We also consider a problem of hedging an asset with variance swap and option. We determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted-variance swap. A numerical example is also provided.

US tax laws provide investors an incentive to time the sales of their bonds to minimize tax liability. This grants a tax timing option that affects bond value. In reality, corporate bond investors’ tax-timing strategy is complicated by risk of default. In this chapter, we assess the effects of taxes and stochastic interest rates on the timing option value and equilibrium price of corporate bonds by considering discount and premium amortization, multiple trading dates, transaction costs, and changes in the level and volatility of interest rates. We find that the value of tax-timing option account for a substantial proportion of corporate bond price and the option value increases with bond maturity and credit risk.

In this study, we document a novel lead–lag relation between historical and implied volatilities based on China’s CSI300 Index options. We show that historical volatilities have incremental information when we predict implied volatilities, and this pattern tends to be more stable for put options than for call options. Moreover, we reveal that this lead– lag relation is relevant to option terms and time horizons of historical volatilities, which means implied volatilities of long-term options are more likely to be properly predicted by long-term historical volatilities on average. Finally, we find that speculative trading might explain our results.

This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model.

In this paper, a novel algorithm for numerical solutions of American option models is presented in terms of the precision integration and wavelet. We discretize the space variable employing quasi–wavelet appropriately so that the resulting semi–discrete system of equations is a system of ordinary differential equation, and then we apply precise integration method to the semi-discrete system. Numerical results show that the proposed algorithm is practicable and efficient.

In this paper, an algorithm for numerical solutions of American option models is presented in terms of the precision integration. We discretize the space variable employing finite differences appropriately so that the resulting semi-discrete system of equations is a system of ordinary differential equation, and then we apply precise integration method to the semi-discrete system, and analyze errors of the method. Numerical results show that the proposed algorithm is practicable and efficient.

The following sections are included:

• INTRODUCTION

• DISCRETE AND CONTINUOUS RANDOM VARIABLES

• PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES
  • Probability Distribution
  • Probability Function and Cumulative Distribution Function

• EXPECTED VALUE AND VARIANCE FOR DISCRETE RANDOM VARIABLES
  • Expected Value for Earnings per Share
  • Expected Value of Ages of Students
  • Expected Value and Variance: Defective Tires
  • Expected Value and Variance: Commercial Lending Rate

• THE BERNOULLI PROCESS AND THE BINOMIAL PROBABILITY DISTRIBUTION
  • The Bernoulli Process
  • Binomial Distribution
  • Probability Function
  • Mean and Variance

• THE HYPERGEOMETRIC DISTRIBUTION
  • The Hypergeometric Formula
  • Mean and Variance

• THE POISSON DISTRIBUTION AND ITS APPROXIMATION TO THE BINOMIAL DISTRIBUTION
  • The Poisson Distribution
  • The Poisson Approximation to the Binomial Distribution

• JOINTLY DISTRIBUTED DISCRETE RANDOM VARIABLES
  • Joint Probability Function
  • Marginal Probability Function
  • Conditional Probability Function
  • Independence

• EXPECTED VALUE AND VARIANCE OF THE SUM OF RANDOM VARIABLES
  • Covariance and Coefficient of Correlation Between Two Random Variables
  • Expected Value and Variance of the Summation of Random Variables X and Y
  • Expected Value and Variance of Sums of Random Variables

• Summary

• Appendix 6A The Mean and Variance of the Binomial Distribution

• Appendix 6B Applications of the Binomial Distribution to Evaluate Call Options
  • What Is an Option?
  • The Simple Binomial Option Pricing Model
  • The Generalized Binomial Option Pricing Model¹⁰

• Questions and Problems