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In this chapter we introduce different types of options and their characteristics. Then, we develop put-call parity theorems for European, American, and futures options. Finally, we discuss option strategies and their investment applications.
In this chapter, we introduce the basic concepts of call and put options. Second, we discuss the Black-Scholes option pricing model and its application. Third, we discuss how to apply the option pricing theory in capital structure issue. Finally, the warrant, one type of equity options, is discussed in detail.
In this chapter, we first introduce the basic concepts of call and put options. Then we show how the simple one period binominal call option pricing model can be derived. Finally, we show how a generalized binominal option pricing model can be derived.
In this chapter, we extend the binomial option pricing model to a multinomial option pricing model. Then we derive the multinomial option pricing model and apply it to the limiting case of Black and Scholes model. Finally, we introduce a lattice framework for option pricing model and its application to option valuation.
In this chapter, we review two famous models on binomial option pricing, Rendleman and Barter (RB 1979) and Cox et al. (CRR 1979). We show that the limiting results of the two models both lead to the celebrated Black-Scholes formula. From our detailed derivations, CRR is easy to follow if one has the advanced level knowledge in probability theory but the assumptions on the model parameters make its...
In this chapter, we first introduce normal distribution, lognormal distribution, and their relationship. Then we discuss multivariate normal and lognormal distributions. Finally, we apply both normal and lognormal distributions to derive Black-Scholes formula under the assumption that the rate of stock price follows a lognormal distribution.
The main purpose of this chapter is to present the American option pricing model on stock with dividend payment and without dividend payment. A Microsoft Excel program for evaluating this American option pricing model is also presented.
This paper compares the American option prices with one known dividend under two alternative specifications of the underlying stock price: displaced log normal and log normal processes. Many option pricing models follow the standard assumption of the Black–Scholes model (Journal of Political Economy 81:637–659, 1973) in which the stock price, follows a log normal process. However, in order to reach...
The purpose of this paper is to develop certain relatively recent mathematical discoveries known generally as stochastic calculus, or more specifically as Itô’s Calculus and to also illustrate their application in the pricing of options. The mathematical methods of stochastic calculus are illustrated in alternative derivations of the celebrated Black–Scholes–Merton model. The topic is motivated by...
In this paper we review the renowned Constant Elasticity of Variance (CEV) option pricing model and give the detailed derivations.There are two purposes of this article. First, we show the details of the formulae needed in deriving the option pricing and bridge the gaps in deriving the necessary formulae for the model. Second, we use a result by Feller to obtain the transition probability density...
In this chapter, we assume that the volatility of option price model is stochastic instead of deterministic. We apply such assumption to the nonclosed-form solution developed by Scott (Journal of Finance and Quantitative Analysis 22(4):419–438, 1987) and the closed-form solution of Heston (The Review of Financial Studies 6(2):327–343, 1993). In both cases, we consider a model in which the variance...
In this chapter, we introduce the definitions of Greek letters. We also provide the derivations of Greek letters for call and put options on both dividends-paying stock and non-dividends stock. Then we discuss some applications of Greek letters. Finally, we show the relationship between Greek letters, with one of the examples from the Black– Scholes partial differential equation.
This paper extends the generalized Cox-Ross- Rubinstein (hereafter GCRR) model of Chung and Shih (2007). We provide a further analysis of the convergence rates and patterns based on various GCRR models. The numerical results indicate that the GCRR-XPC model and the GCRR-JR $$(p = 1/2)$$ model (defined in Table 34.1) outperform the other GCRR models for pricing European calls and American puts...
This paper examines a variety of methods for extracting implied probability distributions from option prices and the underlying. The paper first explores nonparametric procedures for reconstructing densities directly from options market data. I then consider local volatility functions, both through implied volatility trees and volatility interpolation. I then turn to alternative specifications of...
It has been well documented that using the Black-Scholes model to price options with different strikes generates the so-called volatility smile. Many previous papers have attributed the smile to the normality assumption in the Black-Scholes model. Hence, they generalize the Black-Scholes model to incorporate a richer distribution. In contrast to previous studies, our model allows for not only a richer...
Recent studies have extended the Black–Scholes model to incorporate either stochastic interest rates or stochastic volatility. But, there is not yet any comprehensive empirical study demonstrating whether and by how much each generalized feature will improve option pricing and hedging performance. This paper fills this gap by first developing an implementable option model in closed-form that admits...
In this chapter we introduce the application of the characteristic function in financial research. We consider the technique of the characteristic function useful for many option pricing models. Two option pricing models are derived in details based on the characteristic functions.
The payoffs on the expiration dates of Asian options depend on the underlying asset’s average price over some prespecified period rather than on its price at expiration. In this chapter we outline the possible applications of these options and describe the different methodologies and techniques that exist for their evaluation as well as their advantages and disadvantages.
We have developed a modified Edgeworth binomial model with higher moment consideration for pricing European or American Asian options. If the number of the time steps increases, our numerical algorithm is as precise as that of Chalasani et al. (1999), with lognormal underlying distribution for benchmark comparison. If the underlying distribution displays negative skewness and leptokurtosis, as often...
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