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Article Excerpt
1. Introduction
The adequacy of an option pricing model is typically evaluated in an out-of-sample pricing exercise. We naturally prefer the method that minimizes the price differences to the observed market prices. However, the choice of the particular loss function for the in-sample estimation and the out-of-sample evaluation influences the result of that model selection process. Christoffersen and Jacobs (2004) show that the evaluation loss can be minimized by taking the same loss function for in-sample estimation and out-of-sample evaluation. In contrast, empirical researchers are often inconsistent in their choice of the loss functions. They do not align the estimation and evaluation loss functions, so the results of these studies may be misguiding. In the statistics literature it has already been argued that the choice of the loss function is part of the specification of the statistical model under consideration (e.g., Engle 1993). Therefore, it may happen that a misspecified model outperforms a "correctly specified" model, if different loss functions in estimation and evaluation are used.
The majority of empirical option valuation studies use different loss functions at the estimation and evaluation stages; examples are Hutchinson et al. (1994), Bakshi et al. (1997), Chernov and Ghysels (2000), Heston and Nandi (2000), and Pan (2002). The results of these studies regarding model selection are therefore questionable. In contrast, Dumas et al. (1998) and Lehnert (2003) tested the out-of-sample performance of their model using identical loss functions in the estimation and evaluation stages.
Although Christoffersen and Jacobs (2004) show the importance of the loss function in option valuation, they do not recommend one particular loss function. However, the particular loss function used in their empirical analysis characterizes the model specification under consideration. Therefore, it is still possible that even if the loss functions are aligned, a mis-specified model could outperform a "correctly specified" model when the "inappropriate" loss function is used. They correctly suggest that the alignment is more a rule of thumb than a general theorem and that the usefulness has to be evaluated in empirical work. The general problem with loss functions is that the choice of a particular one is heavily subjective and determined by the user of the option valuation model. Depending on the particular purpose of the model, like hedging, speculating, or market making, one loss function is preferred. Using different loss functions, the user puts more or less weight on the correct pricing of options with different moneyness.
In option valuation, not only the pricing model plays an important role, but also the parameter values of these pricing models. Parameters are usually estimated based on historical data. When a particular phenomenon is not present in the historical data, the parameters of the distribution function that are intended to account for the phenomenon are estimated with considerable uncertainty. Uncertainty in the parameter estimates leads to uncertainty in the forecasted future price process and, hence, uncertainty in the out-of-sample root mean squared pricing error (RMSE). We will show that it is important to take estimation risk into account. Estimation risk refers to the fact that point estimates of parameters, resulting from an estimation procedure, do not necessarily correspond to the underlying true parameters. There is still uncertainty about these true values. The tradeoff between the RMSE associated with the parameter estimates and the uncertainty embedded in reported pricing errors plays an important role here.
The aim of this paper is to provide an empirical selection approach to arrive at the most suitable loss function for a given data set and given the purpose of the model. A related method was proposed by Bams et al. (2005) to evaluate value-at-risk models, and in this paper we apply similar logic to the problem of loss function selection in an option valuation context. In our view, such an approach should deal with uncertainty in the reported out-of-sample pricing errors that stems from parameter uncertainty.
In the next section, we set up the econometric framework. We explain our testing procedure using a standard option pricing model, the so-called ad hoc Black-Scholes model. In [section]3, we describe the data, [section]4 provides a description of our option pricing procedure, and the empirical results for the standard model are presented in [section]5. In [section]6, we demonstrate that the results are insensitive to the choice of the underlying option pricing model and replicate the analysis for a more sophisticated GARCH option pricing model. Finally, [section]7 provides conclusions and suggestions for future research.
2. Econometric Framework
2.1. Option Model
For the empirical analysis, we first use an alternative to the prominent ad hoc Black-Scholes model of Dumas et al. (1998) provided by Derman (1999). (1) We allow each option to have its own Black-Scholes implied volatility depending on the exercise price K and time to maturity T. The sample of alternative values for time to maturity and the exercise price (and hence also moneyness) was split in [N.sub.T] time-to-maturity values and [N.sub.M] alternative moneyness values. The following functional form for the options' implied volatility will be applied in the remainder of the paper:
[IV.sub.ij] = [[omega].sub.0] + [[omega].sub.1][M.sub.i] + [[omega].sub.2][M.sub.i.sup.2] + [[omega].sub.3][T.sub.j] + [[omega].sub.4][T.sub.j.sup.2] + [[omega].sub.5][M.sub.i][T.sub.j] i= 1,...,[N.sub.M] j = 1,...,[N.sub.T], (1)
where [IV.sub.ij] denotes the implied volatility for a call option with moneyness [M.sub.i], and time to maturity [T.sub.j], For every exercise price and maturity we can compute the implied volatility and derive option prices using the Black-Scholes model (Black and Scholes 1973). Without loss of generality we focus on the price of a call option, as well as on the goal to arrive at a model that best forecasts out-of-sample call prices. For a call option with moneyness [M.sub.i], and time-to-maturity [T.sub.j], the price of a call option is defined as
[c.sub.ij] = BS([IV.sub.ij], [M.sub.i], [T.sub.j]; p) i = 1,...,[N.sub.M]j = 1,...,[N.sub.T], (2)
where BS([IV.sub.ij], [M.sub.i], [T.sub.j];p) denotes the theoretical call price according to the Black-Scholes formula and [c.sub.ij] is the observed call price. With p = ([omega].sub.0])...[[omega].sub.5])) we denote the vector of unknown parameters to be estimated.
2.2. Loss Functions
Empirical estimation of the model in Equations (1) and (2) requires the inclusion of an error term, or, stated otherwise, the formulation of a loss function. We propose and compare three alternative loss functions that are based on RMSE to estimate the parameters of the ad hoc Black-Scholes. The alternative loss functions read as follows:
1. The implied volatility error loss function: [L.sub.1] = (1/[N.sup.in] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with associated error term equal to [[eta].sub.ij.sup.1] = [
.IV.sub.ij], - [IV.sub.ij], i =...
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