Nonparametric Estimation of Risk-Neutral Distribution via the Empirical Esscher Transform/Estimacao Nao Parametrica da Distribuicao Neutra ao Risco atraves da Transformada de Esscher Empirica.
| Autor | Pereira, Manoel |
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Introduction
In most option pricing models, the fair price is determined from the expected value of its cash flow, under a risk-neutral probability (measure Q), and discounted by a risk-free rate. Under the assumption that the market is dynamically complete, it could be shown that every derivative security can be hedged and the measure Q is unique (Bingham and Kiesel, 2004). However, incomplete markets (1) exist for many reasons and, according to the second fundamental theorem of asset pricing, we have an infinite number of measures Q under which one can get prices of derivatives. Then, how to choose a measure Q from an infinite set of possible measures?
According to Danthine and Donaldson (2015), the literature highlights two approaches to this problem: models based on the general equilibrium (Arrow, 1964, Debreu, 1959, Lucas, 1978, Rubinstein, 1976, Brennan, 1979) and the models based on absence of arbitrage (BlackScholes, 1973, Cox and Ross, 1976, Harrison and Kreps, 1979, Harrison and Pliska, 1981). In the general equilibrium, the supply and demand interact in various markets affecting the prices of many goods simultaneously. The valuation of assets occurs when the markets are balanced, that is, when the supply equals the demand. Thus, from a theoretical connection between macroeconomics (aggregate consumption) and financial markets, the marginal rate of substitution is used to determine a measure Q by solving a utility maximization problem.
In absence of arbitrage, we are appealing to the law of one price. This states that the equilibrium prices of two separate units of what is essentially the same good should be identical. If this was not the case, a riskless and costless arbitrage opportunity would open up: buy extremely large amounts at the low price and sell them at the high price, forcing the two prices to converge. The first fundamental theory of asset pricing says that, if a market model has a measure Q, then it does not admit arbitrage. The conditions that the risk-neutral probability structure must satisfy are that the discounted price process has zero drift and it must also be equivalent to the original structure. Then, a class of pricing kernels, or Radon-Nikodym derivatives, can be specified and impose restrictions that ensure the existence of a risk-neutral measure. In this case, the measure Q can be obtained without completely characterizing equilibrium in the economy (Christoffersen, Elkamhi, Feunou, and Jacobs, 2010, Christoffersen, Jacobs and Ornthanalai, 2013).
In both cases, these approaches require the formulation of an explicit risk-neutral model and are restricted to a few probability distributions for the measure Q. First, because it is difficult to characterize the general equilibrium setup underlying a Risk-Neutral Valuation Relationship (RNVR), see for example Duan (1995, 1999). Second, it is possible to investigate option valuation for a large class of conditionally heteroskedastic processes (Gaussian or non-Gaussian), provided that the conditional moment generating function exists. Christoffersen, Jacobs and Wang (2004), cite that they help explain some stylized facts (smile effect, volatility variability over time and presence of clusters in certain periods) in a qualitative sense, but the magnitude of the effects is insufficient to completely solve the biases. The resulting pricing errors have the same sign as the Black-Scholes (1973) pricing errors, but are smaller in magnitude.
According to Haley and Walker (2010), the nonparametric option pricing techniques have expanded rapidly in recent years. It offers an alternative by avoiding possibly biased parametric restrictions and reducing the misspecification risk. As the change of measure does not involve the distribution of the model's innovations, this method of riskneutralization is applicable even when the moment generating function of the innovations' probability distribution does not exist. In these methods, the historical distribution of prices is used to predict the distribution of future asset prices. According to Stutzer (1996), by using past data to estimate the payoff distribution at expiration, it permits more accurate assessment of the likely pricing impact caused by investors' data-based beliefs about the future value distribution.
There are two ways to nonparametrically estimate risk-neutral probabilities implicit in financial instruments: the methods that seek to infer the empirical risk-neutral probability from the options market (2) (kernel, maximum entropy, and curve fitting (3)) and the methods that seek to infer the empirical risk-neutral probability from asset price (with or without option price), as canonical valuation developed by Stutzer (1996). In the case of canonical valuation, the maximum entropy principle is employed to transform the empirical distribution into its risk-neutral counterpart, by minimizing the Kullback-Leibler information criterion (KLIC).
Several papers have extended Stutzer's original work and demonstrated that the methodology is flexible and performs very well in the presence of realistic financial time series, see Gray and Newman (2005), Gray, Edwards, and Kalotay (2007), Alcock and Carmichael (2008), Haley et al. (2010) and Almeida and Azevedo (2014). Other researchers, as Haley et al. (2010) and Almeida et al. (2014), suggested the adoption of members of the Cressie-Read family of discrepancy functions as alternative ways of measuring distance in the space of probabilities.
This paper introduces an empirical version of the Esscher transform (1932) for nonparametric option pricing. We assume that the empirical pricing kernel (4) is known and given by an empirical version of the Esscher transform (1932). This assumption is reasonable, because it is well-known in the information theory (5) that a problem of maximum entropy has its solution in the form of the Esscher transform (Buchen and Kelly, 1996, Stutzer, 1996, Duan, 2002).
The empirical Esscher transform and the canonical valuation of Stutzer (1996) generate, in theory, the same risk-neutral measure: a measure that is exponential in the return of the underlying asset. The numerical differences that appear in this work are due to the different form in which the martingale condition is imposed. In the empirical Esscher transform, the martingale condition is imposed through the ratio of two empirical moment generating function's periods. In the Stutzer (1996) method, this condition is imposed via Euler's equation.
Duan (2002) also develops a nonparametric option pricing theory based on Esscher transform (1932). He uses a binary search to find the Esscher parameter and the measure Q is evaluated using the standard polynomial approximation formula. In our case, we use a consistent estimator for the moment generation function and we avoid the use of intensive computational methods. Hence, we obtain a method that does not require a set of restrictive assumptions for the formulation of a specific model; that provides a clear and easy way to obtain a risk-neutral distribution; is adaptable and flexible to respond to changes in the data generating process; and explores the whole cross-section information contained in the underlying asset's price.
The main contribution in this paper is testing the methodology proposed by Gerber and Shiu (1994) in a similar way to Stutzer (1996), Gray et al. (2005) and Haley et al. (2010). The test consists.of two parts: A Monte Carlo study that assesses the method's pricing ability in the Black and Scholes (1973) model and Heston (1993) model; and an empirical assessment of the method's pricing ability for European Call options on two Brazilian Companies (Vale and Petrobras). In empirical data, we propose a methodology to construct an unknown data generation process based on bootstrap with replacement on historical returns of the underlying asset. As in previous works, the pricing ability is measured using the Mean Absolute Percentage Error of an European Call.
The paper is organized as follows. Section 2 discusses the Esscher transform. In Section 3, we introduce the empirical Esscher transform. Section 4 presents the methodology we use to compare the different pricing methods, and the results are discussed in Section 5. Finally, Section 6 concludes.
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The Esscher transform
Let X be a random variable with probability density function fix) and let [theta] be a real number. Then, the Esscher transform (ET) of fix) with Esscher parameter [theta] is given by f (x;[theta]), defined as:
f (x; [theta]) = [e.sup.[theta]x]/[[[integral].sup.+[infinity].sub.-[infinity]] [e.sup.[theta]x] f(x)] f(x). (1)
Note that f(x; [theta]) is also a probability density function since it integrates one. Furthermore, the ET can be interpreted as a reweighted version of f(x), with reweighting function given by:
m(x:[theta]) = [e.sup.[theta]x]/[[[integral].sup.+[infinity].sub.-[infinity]] [e.sup.[theta]x] f(x)dx. (2)
The denominator of this expression represents the moment generating function (mgf) of f(x), denoted by:
M([theta]) = E[[e.sup.[theta]X] = [[integral].sup.+[infinity].sub.-[infinity]] [e.sup.[theta]x] f(x)dx. (3)
In this case, for the Esscher transform to exist, the mgf of X must exist, which precludes some well-known density functions, like the t-student. Hence, the ET of f(x) can be expressed as:
f(x;[theta]) = m(x;[theta])f(x) = [e.sup.[theta]x]/M([theta]). (4)
Consider now the ET of the density f([x.sub.T])of [X.sub.T], the log-return of an asset for a period T, given by:
[mathematical expression not reproducible]. (5)
Gerber and Shiu (1994) proposed to use the ET of [X.sub.T] as the risk-neutral distribution (RND) for the log-return of this asset. They call it Risk-Neutral Esscher Transform (RNET). In this context, f([x.sub.T]) is referred to as the physical probability measure P and f([x.sub.T]; [theta]), the ET of f([x.sub.T]), is identified as the risk-neutral measure Q or, still, the...
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