A Note on the Pricing of Contingent Claims with a Mixture of Distributions in a Discrete-Time General Equilibrium Framework Luiz Vitiello and Ser-Huang Poon January 5, 200 Corresponding author. Ser-Huang Poon is at Manchester Business School, Crawford House, University of Manchester, Oxford Road, Manchester M3 9PL, UK. Tel: +44 6 275 043, Fax: +44 6 275 4023, email: ser-huang.poon@mbs.ac.uk. Luiz Vitiello is at London Metropolitan Business School, London Metropolitan University, 84 Moorgate, London EC2M 6SQ, UK, email: l.vitiello@londonmet.ac.uk.
A Note on the Pricing of Contingent Claims with a Mixture of Distributions in a Discrete-Time General Equilibrium Framework Abstract This paper presents a framework for the pricing of contingent claims based on the assumption that the underlying asset has a mixture of transformed normal distributions. Speci cally, the framework presented (i) can deliver risk neutral contingent claim pricing formulae, (ii) widens the set of distributions used in the mixture by assuming that the terminal price of the underlying security has a mixture of transformed-normal distributions, which contains the normal and lognormal distributions as special cases, and (iii) does not require the components of the mixture to have the same density as long as they belong to the family of transformed-normal distributions. An interesting aspect of mixtures of distributions, and in particular of the framework developed here, is that the actual and the risk neutral distributions may not have the same shape, which can lead to a non-monotonic pricing kernel. Keywords: Mixture of distributions, transformed-normal distribution, risk neutral valuation relationship, contingent claim pricing, discrete-time general equilibrium EFM classi cation: 40, 440, 30 2
A Note on the Pricing of Contingent Claims with a Mixture of Distributions in a Discrete-Time General Equilibrium Framework Introduction Mixtures of distributions have been widely applied to contingent claim as a way of extending the Black and Scholes (973) assumption of lognormally distributed asset prices, mainly because mixtures can cover a larger area in the skewness and kurtosis plane while the lognormal distribution is limited to a single line on this plane. In addition to pricing contingent claims, mixtures of distributions have also been used to estimate the implied risk neutral distribution of nancial securities (see for instance Bahra, 997; Söderlind and Svensson, 997; Jackwerth 999). The contingent claim pricing framework presented in this study is developed in a discrete time equilibrium economy, following Brennan (979) and Camara (2003). This framework (i) can deliver risk neutral contingent claim pricing formulae, (ii) widens the set of distributions used in the mixture by assuming that the terminal price of the underlying security has a mixture of transformed-normal distributions, which contains the normal and lognormal distribution as a special case, and (iii) does not require the components of the mixture to have the same density as long as they belong to the family of transformed-normal distributions. Due to their exibility, mixtures of distributions can produce densities with nonstandard shapes, making them an interesting tool to the pricing of contingent claims. Mixtures can also lead to a non-monotonic asset speci c pricing kernel as the actual and the risk neutral distributions may not have the same shape (see for instance Jackwerth and Rubinstein, 996; Ait-Sahalia and Lo, 998; Brown and Jackwerth, 2004). This paper is divided as follows: First we introduce the basic economy and the general form for the pricing kernel. Then, in section two, we provide the framework for the pricing of European contingent claims and illustrate its application through examples, which return new option pricing formulae. In section three we brie y discuss the implications of mixtures to contingent claim pricing and conclude. 2 The forward price equilibrium relationship Consider a risk-averse representative investor in ha complete market setting that maximises her expected utility of future wealth, max i. In equilibrium, the forward price of an underlying asset is given by 2 h i ~ () See Ritchey (990) and Melick and Thomas (997) and Vitiello and Poon (2008). 2 These results are obtained from the rst order condition for a maximum and from the application of conditional expectation proprieties. For a detailed derivation see Huang and Litzenberger (988). 3
where h 0 j ~ i h (2) i 0 is the asset speci c pricing kernel, the superscript of ( ) means the expectation is taken with respect to the actual probability, ~ is the random payo of the underlying asset, and 0 ( ) is the marginal utility function of the representative investor. It is assumed that ~ has a transformed normal distribution as speci ed in the following de nitions De nition (the transformed normal distribution) A random variable ~ has a transformed normal distribution if ~ ~ ~ + ~ (3) where ~ ~ is a strictly monotonic function, is a standard normal random variable, ~ 2 < and ~ 2 < + are the location and the scale parameter respectively. From De nition, if ~ has a normal distribution, then ~ has a transformed normal distribution. For instance, if ~ ln ~ then ~ has a lognormal distribution. De nition 2 (the density of ) ~ The random payo ~ has a transformed normal distribution according to De nition with terminal density X ~ (4) where is the weight on the component with P, 0 8 and " # ~ 0 p ~ exp ~ 2 2 2 ~ ~ 2 ~ (5) is the density function with location parameter ~ and scale parameter ~. The density in equation (5) follows directly from equation (3) and the density of a standard normal random variable. Given De nitions and 2, equation () can be written as Z ~ ~ Z Z Z ~X ~ ~ ~ ~ X ^~ ~ ~ ^ ~ ~ (6) 4
where ^ ~ is de ned as the risk adjusted density i.e. the product of the actual distribution and the asset speci c pricing kernel. The term risk adjusted comes from the fact that ^ ~ is adjusted for investor s risk aversion as it contains parameters that are related to the investor s preference. In order to obtain an applicable form for the pricing kernel and for the risk adjusted density further information about the utility function of the representative agent and the distribution of wealth is required, which leads to the following de nition De nition 3 (the marginal utility function) The representative investor s marginal utility function of wealth is 0 ~ exp h ~i (7) where the constant is the risk-aversion parameter, and ~ ~ ~ + ~ (8) has a transformed normal density according to De nition with location parameter ~ and scale parameter ~. As in Camara (2003) the above de nition does not require functions ~ ~ and ~ ~ to be the same i.e. the distribution of ~ and ~ can be di erent as long as they are in accordance with equation (3). Using de nitions to 3 it is possible to obtain a functional form for the asset speci c pricing kernel in equation (2), which is shown in the following proposition. Proposition 4 (the asset speci c pricing kernel) Given De nitions to 3, the asset speci c pricing kernel in equation (2) has the following form X where ~ ~ p 2 0 ~ ~ 2 ~ ( ~ ( ~ ) ( ~ + ~ ~ )) 2 (9) is the terminal density of ~ as in De nition 2, is the correlation coe cient of the density of ~ and ~, and P, 0 8. Proof. See appendix 5
Given De nition 2 and Proposition 4, the risk adjusted density is given by 3 ^ X 0 ~ ~ p 2 2 ( ( ~ ) ( ~ ~ + ~ ~)) 2 ~ (0) ~ 2 which is the product of the asset speci c pricing kernel and the physical density. The risk adjusted density has the same density and the same scale parameter as ~ ~, but with location parameter ~ + ~ ~. This result will be used in the next section to obtain the forward price in equation (6) and the contingent claim pricing framework. 2. The contingent claim pricing framework Let ~ be the contingent claim payo function. Then, using the same equilibrium arguments as in equation (6), the price of a contingent claim written on the forward price of ~ is given by h i h i Z ^ ~ () Since the density ^ ~ contains parameters related to investor preferences, if we wanted to price contingent claims using the risk adjusted density we would have to take into account these parameters, which would involve the estimation of several unobservable parameters. 4 One way of avoiding this problem is to work in a risk neutral setting, which means that investors would be insensitive to risk. 5 In order to do this one can try to replace these preference parameters by observable or marketable parameters, such as securities prices. This can be achieved by inverting 6 equation (6) and expressing the location parameter, ~ + ~ ~, as a function of (see Brennan, 979 and Camara, 2003 for instance). 3 This result is obtained by the multiplication of the pricing kernel in Proposition (4) and the density of ~ in De nition 2. Thus for ^ X h ~ + + i X ~ ~ + + ~ X ~ which leads to equation (0). 4 As Merton (973, p.6) points out, "... the expected return is not directly observable and estimates from past data are poor because of nonstationarity. It also implies that attempts to use the option price to estimate expected returns on the stock or risk-preferences of investors are doomed to failure". 5 Here, this would mean that the risk-aversion parameter is zero, 0. 6 Note however that it is not always possible to invert equation (6). 6
Nevertheless when the payo function is given by a mixture distribution there are densities and as a consequence location parameters (i.e. there is a set of parameters related to preference and wealth), which increases the complexity of the problem. If is the only price available, i.e. if there are not any other securities or derivative securities prices available that are related to, then further assumptions are needed. Here the location parameters of all densities are xed so as to be the same. That is, we assume that ~ + ~ ~ ^ 8 (2) which allows us to write ^ as a function of. Although this assumption restricts the range of skewness and kurtosis of the mixture, it still leaves the model with a great level of freedom to capture the terminal distribution of ~, mainly considering that the family of transformed normal distributions contains several high moment distributions. Thus, if it is possible to solve equation (6) for ^ and equation (2) holds, then equation () can be rewritten as h i h i where the superscript of ( ) means that the expectation is taken with respect to the risk neutral probability. In the following examples we show the application of the above framework to the pricing of European call options. It is assumed that the terminal value of ~ has a mixture of transformed normal distributions as in equation (3) and that equation (2) holds. Example 5 (The lognormal and mixture): Assume that ~ has a mixture of a lognormal and a densities 7 i.e. ~ ln ~ and ~2 ~ sinh ~ respectively. Then, the forward price in equation (6) is given by Z ~ h ~ ~ + 2 2 ~ 2 ~ i ~ Z ~ h ^ + 2 ^2 ~ i ~ (3) where ~ 2 ~ p ~ ~ 2 p ~2 ~ + p 2 2 2 ~ (ln( ~ ) ( ~ + ~ ~ )) 2 2 2 ~2 (sinh ( ~ ) ( ~ 2 + 2 ~ ~ 2)) 2 ^ ~ ^ 2 ~ p ~ 2 ~ (ln( 2 ) ( ~ ~ + ~ ~ )) 2 ~ (4) p p ~2 2 ~2 2 + (sinh ( ) ( ~ 2 ~ + 2 ~ 2)) 2 ~ 2 ~ (5) 7 For a detailed discussion of the distribution see Johsnon (949). 7
and P is the terminal density of ~ as in De nition 2. The value of is thus given by ~ + ~ ~ + 2 2 ~ + 2 2 2 2 ~ sinh ~2 + 2 ~ ~2 Since by assumption ~ + ~ ~ ~2 + 2 ~ ~2 ^, the above equation becomes ^+ 2 2 ~ + 2 2 2 2 2 ~ sinh(^) which allows us to solve for the variable ^, 8 0 q ^ ln@ + 2 + 2 2 2 2 + 2 2 2 ~ + 2 2 +2 2 A 2 2 2 + 2 2 2 2 Once we obtain an expression for ^ we can solve for the price of a contingent claim. For instance, let max ~ 0 be the payo of a call option with exercise price. From equation (), the price of this option is then h i h i Z max ~ 0 h ^ + 2 ^2 ~ i ~ which, after substituting for equations (4) and (5) and simplifying, yields the option pricing formula h i 2 2 ~ 2 2 2 ~ + 2 2 2 2 () (2) + 2 2 2 ~2 4 2 2 + 22 2 2 2 (3) 2 (4) 2 2 2 2 + 2 2 2 2 (2) 2 2 2 ~ (5) (6) where h i ln() ln 2 2 2 ~ + 2 2 2 2 ~ + ~ 3 h ln + p 2 + 8 The solution for^ has two roots but as to the log of a non-positive number. 2 ~ ln 4 3 ~2 2 2 2 + 2 2 2 2 5 4 2 ~ q + 2 + 2 2 2 2 + 2 2 2 ~ + 2 2 +2 2 r i ~2 + ~ 2 2 +2 2 2 2 + 2 2 ~ 2 + 2 2 +2 2, one of them leeds 8
Example 6 (The displaced lognormal and negative-skewed lognormal mixture): Assume that ~ has a mixture of a displaced lognormal distribution and a negative-skewed lognormal distribution, 9 i.e. i.e. ~ ln ~ and ~2 ~ ln 2 ~ respectively Then, as in example 5 Z ~ h ^ + 2 ^2 ~ i ~ (7) where ~ 2 ~ ~ ~ ~ p2 (ln( 2 2 ~ ) ( ~ + ~ )) 2 ~ ~ ~ ~ 2 ~ p2 2 2 (ln( 2 ) ( ~ 2 ~ + 2 ~ 2)) 2 ~ ~2 ^ ~ ^ 2 ~ p [ln( 2 ~ 2 ~ 2 ~ ) ( e + ~ ~ )] 2 ~ (8) ~2 p 2 2 ~ 2 2 [ln( 2 ) ( ~ e2 + 2 ~ ~2 )] 2 ~2 (9) P is the terminal density of ~ as in De nition 2 and ~ 2. The value of is given by ^+ 2 2 ~ + + 2 2 ^+ 2 2 2 ~ and solving for ^ ^ ln for exp 05 2 ~ 2 exp 05 2 ~2 " 2 2 2 2 ~ 2 2 2 2 ~ Considering the payo of a call option, becomes h i h i Z max # max ~ 0, equation () ~ 0 h ^ ~ + 2 ^2 ~ i ~ which, after substituting for equations (8) and (9) and simplifying the resulting equation, yields the option pricing formula h i 2 2 ~ () ( ) (2) + 2 ( 2 ) (3) 2 2 2 ~2 (4) (20) 9 See for instance Rubinstein (983) and Stapleton and Subrahmanyam (993) for the displaced lognormal and for the negative skewed lognormal respectively. 9
where 2 2 2 2 ~ 2 2 2 ~2 ln( ( )) ~ 2 ~ 3 ln(( 2 ) ) ~2 4 3 ~ 2 + ~ 3 Discussion and Final Remarks This paper extends the literature on the pricing of contingent claims with a mixture of distributions by allowing its components to have transformed-normal distributions. These components do not have to have the same density but must belong to the transformed-normal family. By introducing a restriction on the value of some distributional parameters, we show that it is possible to achieve a risk neutral valuation relationship. The framework presented here is particular interesting for pricing contingent claims on underlying securities that cannot be dynamically hedged. Also, given the exibility of mixtures of transformed normal distribution, the pricing models can be applied to a large range of securities, as they should be able to capture the distribution of their payo s. 0 Mixtures of (lognormal) distribution have also been used to estimate the implied distribution of nancial securities. Generally speaking, while mixture of lognormal distributions do not seem to perform neither better nor worse 2 than other methods, it guarantees a smooth behaviour of the tails and prevents negative probability (Campa, Chang and Reider, 998). In addition to this, the mixture of transformed-normal distributions introduced here shows that risk neutrality is achievable by construction and not by assumption, which leads to a clear link between actual and risk neutral distribution. Finally, when the underlying security price distribution is given by a mixture of distributions, actual and risk neutral distributions may not have the same shape. A direct consequence of this feature 3 is that the pricing kernel may be non-monotonic. 4 Figure shows a non-monotonic asset speci c pricing kernel obtained by the assumption 0 Vitiello and Poon (2008) for instance show several examples of densities and implied volatility smiles for a mixture of -distributions, a special case of the framework developed here. A comprehensive survey of methods for extracting risk neutral densities can be found in Bahra (997), Söderlind and Svensson (997) and Jackwerth (999). 2 Campa, Chang and Reider (998) and Coutant, Jondeau and Rockinger (999) found that mixture of lognormals produce similar results to the alternative methods. Bliss and Panigirtzoglou (2002) observed that a smoothed implied volatility method performs better than mixture. 3 That is, that the actual and the risk neutral densities may cross each other more than two times. 4 See for instance Brown and Jackwerth (2004) 0
2.8.6.4.2 m(s) 0.8 0.6 0.4 0.2 0 0.85 0.925.075.5.225.3.375.45 F(S) Figure : The asset speci c pricing kernel in equation (2) with 2 5, 4, 2, 35, 2 05, ~ ~ 28, and 2 ~ ~2 04. that the underlying security price has a mixture of two lognormal densities, where the asset speci c pricing kernel in Proposition (4) is given by X p ~ ~ ~ 2 2 2 ~ (ln( ~ ) ( ~ + ~ ~)) 2 (2) where P 2, is the lognormal density of, ~ 2 5, 4, 2, 35, 2 05, ~ ~ 28, and 2 ~ ~2 04.
Appendix Proof. (proposition 4 on the asset speci c pricing kernel) First we solve for the denominator of equation (2). Given equation (7) and considering that ~ ~ is normally distributed, we can use the de nition of the expected value of a lognormal random variable. Thus, h 0 ~ i exp ~ + 2 05 2 ~ (22) Second we solve for the numerator of equation (2). This requires additional steps. Let ~ j ~ be the conditional density of ~ given ~ and 2. Then h 0 ~ j ~ i Z 0 ~ ~ j ~ ~ Z ~ X 0 ~ ~ j ~ ~ ~ We let ~ j ~ ~ ~ P ~ 0 ~ ~ 0 ~ ~ p p ~ ~ ~ 2 2 " µ ~ ( ~ ) 2 µ ~ ~ ( ) 2( 2 ) + ~ ~ ~ ~ 2 2 ( ~ ( ~ ) ~ )( ~ ( ~ ) ~ ) ~ ~ where P, 0 8, ~ ~ is joint-density of ~ and, ~ is the density of ~ and P is the terminal density of ~ as in De nition 2. Using equation (7) the conditional expectation is h 0 ~ j ~ i X 0 ~ ~ 2 0 @ ~ (~ ) ~ ~ 2 A + ~ ~ (~ ) ~ + ~ ~ + 2( 2 ) 2 2 ~ Finally, substituting equations (22) and (23) into (2) yields equation (9). # (23) 2
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