American options with stochastic dividends and volatility: A nonparametric investigation

TL;DR: In this paper, the authors provide a full discussion of the theoretical foundations of American option valuation and exercise boundaries and show how they depend on the various sources of uncertainty which drive dividend rates and volatility, and derive equilibrium asset prices, derivative prices and optimal exercise boundaries in a general equilibrium model.

About: This article is published in Journal of Econometrics.The article was published on 2000-01-01 and is currently open access. It has received 95 citations till now. The article focuses on the topics: Implied volatility & Volatility smile.

Summary

1 Introduction

• The early exercise feature of American option contracts considerably complicates their valuation.
• As noted before, the latter could apply to American as well as European contracts.
• By studying American options, their paper models both pricing and exercise strategies via nonparametric methods.
• In addition, their analysis features a combination of volatility ltering based on EGARCH models and nonparametric analysis hitherto not explored in the literature.

2 American option valuation with stochas-

• Tic dividends and volatility Much has been written on the valuation of American options.
• Two state variables are required to model a stochastic dividend yield which is imperfectly correlated with the volatility coe cients of the stock price process.
• In their representative agent economy, the value of this contract A useful property of the American option price is given next, Corollary 2.1 Consider the nancial market model with stochastic volatil- ity of Theorem 2.1.
• The determination of the excercise boundary, however, is a nontrivial step in this computation.

3 Nonparametric methods for American op-

• Tion pricing with stochastic volatility and dividends.
• The results in section 2 showed that the reduced forms for equilibrium American option prices and exercise decisions depend in a nontrivial way on two latent state processes Y and Z .
• Indeed, considering a fully speci ed parametric framework would require the computation of intricate expressions involving conditional expectations and identifying the exercise boundary which solves a recursive integral equation.
• It is the main reason why no attempts were made to compute prices and excercise decisions under such general conditions.
• The third subsection presents the estimation techniques and results while the nal one is devoted to testing the e ect of volatility and dividends on option valuation.

3.1 The generic reduced form speci cation

• By being nonparametric in both the formulation of the theoretical model and its econometric treatment, there are issues the authors cannot address.
• Nevertheless, the nonparametric approach does achieve the main goal of their econometric anaylsis, namely to determine whether the volatility and/or the dividend rate a ect the valuation of the contract and the exercise policy.
• 7For instance, suppose that in estimating nonparametrically the relations in (3.2) the authors nd that both and a ect B=K and C=K:.

3.2 Volatility measurement and estimation issues

• The rst step will consist of estimating the current state.
• Hence, the authors face the typical curse of dimensionality problem often encountered in nonparametric analysis.9.

3.2.1 Volatility measurement

• Practitioners regulary use the most recent past of the quadratic variation of S to extract volatility.
• Moreover, following Nelson (1992), even when misspeci ed, ARCH models still keep desirable properties regarding extracting the continuous time volatility.
• In the case of RiskMetricsTM for daily data, one sets = :94; a value which the authors retained for their computations.
• The computation of implied volatilities is discussed in Harvey and Whaley (1992a) and Fleming and Whaley (1994).
• They do take into account the dividend process.

3.2.2 Estimation issues

• It is beyond the scope and purpose of this paper to provide all the technical details.
• The parameter controls the level of neighboring in the following way.
• The characterization of the correlation in the data may be problematic in option price applications, however.
• This reference (chap. 6) also discusses the choice of the smoothing parameter in the context of nonparametric estimation from time series observations.
• More interestingly, Rilstone (1996) studies the generic problem of generated regressors, which is a regressor like ̂t, in a standard kernel-based regression model and shows how it a ects the convergence rates of the estimators while maintaining their properties of consistency and asymptotic normality.

3.3 Estimation results

• The authors focus their attention on the OEX contract which was also studied by Harvey and Whaley (1992a, b) and Fleming (1994).
• This is obviously not surprising as the option price is more sensitive to changes in volatility and to the volatility level itself over longer time horizons.
• These appear in Figure 2 and show that the results are robust with regard to the speci cation of volatility.
• Hence, based on this evidence the authors have to conclude that the emphasis on dividends alone in the pricing of OEX options, as articulated in Harvey and Whaley (1992a, b) and Fleming and Whaley (1994), is not enough to characterize option pricing in this market.
• The authors therefore report in Table 3 two statistics for each test, one based the entire sample and one based on the observation points with t >.

3.4 Nonparametric pricing of American call options

• In addition to the statistical issues involved in the speci cation of an option pricing functional the authors must also assess option pricing errors.
• To deal with volatility the authors compared two extremes, namely volatility days which reside in the rst and fourth quartile of the distribution.
• Moreover, the authors examined three maturities, namely 28, 56 and 84 days.
• In contrast, for high volatility the authors note that the nonparametric pricing schemes belong to the parametric range for medium maturities (56 days and 84 days) while the parametric models overprice for short maturities out- or at-the-money options.

4 Conclusion

• The authors considered American option contracts when the underlying asset or index has stochastic dividends and stochastic volatility.
• This situation is quite common in nancial markets and generalizes many cases studied in the literature so far.
• The theoretical models which were derived in section 2 yield fairly complex expressions which are di cult to compute.
• The authors approach also joins the recent e orts of applying nonparametric methods to option pricing.
• The method proposed in this paper has also substantial practical applications for users of OEX options.

Figures

Figure 1: Estimated call prices conditional on dividend and EGARCH volatility quartiles. | : rst quartile; { { : second quartile; : third quartile; : fourth quartile.

Figure 2: Estimated call prices conditional on dividend and implied volatility quartiles. | : rst quartile; { { : second quartile; : third quartile; : fourth quartile.

Table 2: Empirical quantiles of ltered conditional variances (̂2)

Figure 3: Estimated exercise boundary as a function of implied volatility.

Table 3: Goodness of t test statistics.

Table 4: American call option normalized prices (C=K).

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "American options with stochastic dividends and volatility: a nonparametric investigation" ?

In this paper, the authors study the effect of volatility on the performance of the OEX contract on the S & P100 stock index. 

Q2. What procedure was used to choose the bandwith parameter?

To choose the bandwith parameter the authors followed a procedure called generalized cross-validation, described in Craven and Wahba (1979) and used in the context of option pricing in Broadie et. al. (1995). 

Q3. What are the two critical assumptions used in the American option case?

1Two critical assumptions, namely (1) a constant dividend rate and(2) constant volatility, are often cited as restrictive and counter-factual. 

Q4. What is the main goal of the nonparametric approach?

the nonparametric approach does achieve the main goal of their econometric anaylsis, namely to determine whether the volatility and/or the dividend rate a ect the valuation of the contract and the exercise policy. 

Q5. What is the widely used kernel estimator of g in 3.1?

The most widely used kernel estimator of g in (3.11) is the NadarayaWatson estimator de ned byĝ (z) =Pn i=1K Zi zYiPni=1K Zi z ; (3.12) so thatĝ (Z1); : : : ; ĝ (Zn) 0 =WKn ( )Y; where Y = (Y1; : : : ; Yn) 0 and WKn is a n n matrix with its (i; j)-th element equal to K Zj Zi Pn k=1K Zk Zi : WKn is called the in uence matrix associated with the kernel K: 

Q6. What is the argument for the EGARCH volatilities?

The argument is that for a wide variety of misspeci ed ARCH models the di erence between the (EG)ARCH volatility estimates and the true underlying di usion volatilities converges to zero in probability as the length of the sampling time interval goes to zero at an appropriate rate. 

Q7. What papers were devoted to the subject?

Several papers were devoted to the subject, namely Nelson (1990, 1991, 1992, 1996a,b) and Nelson and Foster (1994, 1995), which brought together two approaches, ARCH and continuous time SV, for modelling time-varying volatility in nancial markets. 

Q8. What is the value of a contingent claim?

In this context, the value ofany contingent claim is simply given by its shadow price, i.e., the priceat which the representative agent is content to forgo holding the asset. 

Q9. What are the state variables required to model a stochastic dividend yield?

Two state variables are required tomodel a stochastic dividend yield which is imperfectly correlated with thevolatility coe cients of the stock price process. 

Q10. What is the simplest explanation for the volatility of options?

The results so far seem to suggest two things: (1) conditioning on t does not displace pricing of options and (2) the volatility e ect seems to be present only for large (fourth quartile) volatilities.