Melino A. and S. Turnbull, 1990, "The Pricing of Foreign Currency Options with Stochastic Volatility", Journal of Econometrics, 45, 239-265.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....exotic options, however, do not always depend on the volatility in a monotone fashion, whence pricing consistently with the smile requires a more sophisticated model. Therefore, it is important to find e#cient ways to calculate exotic option values in exotic models. Melino and Turnbull showed in [11] that the assumption of stochastic volatility leads to a distribution of the underlying which is closer to empirical observations than the log normal distribution. Heston s stochastic volatility model [5] can explain the smile observed in foreign exchange vanilla options markets to some extend, ....

Melino, A. and Turnbull, S. M. (1990). Pricing Foreign Currency Options with Stochastic Volatility. Journal of Econometrics, 45, pp. 239-265.

....that the constant volatility BlackScholes (1973) model is subject to specification error. Various stochastic volatility option pricing models have been developed in the past decades to correct pricing bias caused by the constant volatility assumption. Heston (1993) Hull and White (1987) Melino and Turnbull (1990), Scott (1987) and Wiggins (1987) However, Fleming (1993) and Guo (1996) show that the implied variances extracted from the stochastic volatility model of Hull and White (1987) is a dominant, but still a biased estimator in terms of ex ante forecasting power in the stock index and foreign ....

....and White (1987) model still appears to be biased upward. The assumption of a zero correlation between innovations to volatility and innovations to aggregate consumption is often invoked to justify a zero risk premium for volatility risk (Hull and White (1987) This assumption is challenged by Melino and Turnbull (1990), who find that a stochastic volatility model with nonpositive price of volatility risk explains observed option prices better than the constant volatility models. Lamoureux and Lastrapes (1993) suggests that the market premium on variance risk is time varying, and that stochastic volatility ....

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Melino, A., and S. Turnbull, (1990),"Pricing Foreign Currency Options with Stochastic Volatility," Journal of Econometrics, 45, 7-39.

....model on which modern derivative pricing theory is based, and its properties are easy to derive. Estimation is very dicult because the likelihood is not tractable; implementation of stochastic volatility models are commonly through quasi maximum likelihood ( 64] and [43] and method of moments ([60]) Alternatively, people have used MCMC techniques: 46] and [79] implemented a single move Gibbs sampler; 82] improved it by sampling all the log volatilities in a group using a Metropolis Hastings algorithm; 48] used an o set mixture of normals approximation to log Chi square distribution ....

A Meliino and S M Turnbull. Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45:239-265, 1990.

....volatility models are motivated by clear evidence of predictable variation in conditional variance. This is apparent in the dynamics of implied volatilities (Table 2, for example) and in estimates of GARCH and related models. Notable applications to currency options include Bates (1996a) Melino and Turnbull (1990), and Taylor and Xu (1994) 15 The generic stochastic volatility model starts with log price changes of the form x t 1 = t 1 z 1=2 t t 1 ; 23) where t 1 is conditionally independent of z t and f t g is a sequence of iid draws with zero mean and unit variance. The new element, z t , is ....

Melino, Angelo, and Stuart Turnbull, 1990, "Pricing foreign currency options with stochastic volatility," Journal of Econometrics 45, 239-265.

....for hedging purposes. Amin and Jarrow (1991) realize the inconsistency of pricing cross currency derivatives without a proper model for the term structures, and they extend Heath, Jarrow and Morton s (1992) seminal work into an international setting. 3 Given an exogenously specified 1 See Melino and Turbull (1990, 1991) Bodurtha and Courtadon (1987) and Shastri and Tandon (1986) 2 See Cornell and Reinganum (1981) and Chang and Chang (1990) for discussions of why the mark tothe market effect in futures contracts can result in the divergence of futures and forward prices. 3 Amin and Bodurtha (1995) ....

Melino, A., and S. Turnbull, 1990, "Pricing Foreign Currency Options with Stochastic Volatility", Journal of Econometrics, 45, 239-265.

....allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile , allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino Turnbull (1990), our empirical findings strongly suggest the existence of a non zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information ....

....relative difference to measure option pricing errors. Our methodology is also different from other research based on observations of underlying state variables. First, different from the method of moments or GMM 4 used in Wiggins (1987) Scott (1987) Chesney Scott (1989) Jorion (1995) and Melino Turnbull (1990), the efficient method of moments (EMM) used in this paper yields efficient estimates of SV models as we shall see below, and the parameter estimates are not sensitive to the choice of particular moments. Second, our model allows for a richer structure for the state variable dynamics, for instance ....

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Melino, A. & Turnbull, S. M. (1990), `Pricing foreign currency options with stochastic volatility', Journal of Econometrics 45, 239--265.

....of predictable variation in conditional variance. This is apparent in the dynamics of implied volatilities #Table 2, for example#, in estimates of ARCH and related models, and in estimates of other models of stochastic volatility. Notable applications to currency options include Bates #1996a#, Melino and Turnbull #1990#, and Taylor and Xu #1994#. The generic stochastic volatility model starts with log price changes of x t 1 = # t 1 z 1=2 t t 1 ; #23# where t 1 is conditionally independentofz tand f t g is a sequence of iid draws with zero mean and unit variance. The new element, z t , is the ....

Melino, Angelo, and Stuart Turnbull, 1990, #Pricing foreign currency options with stochastic volatility," Journal of Econometrics 45, 239-265.

....the Barone Adesi Whaley approximation in the same way as for ARCH models. Examples of this practice are in Hans J. Knoch (1992) and Bates (1995) At present, the only other way to price American options under stochastic volatility is by solving a second order partial differential equation (Angelo Melino and Stuart Turnbull 1992), which is extremely computationally burdensome. Stochastic Volatility Option Models without Closed Form Solution. John C. Hull and Alan White (1987) Louis O. Scott (1987) and James B. Wiggins (1987) were among the first to develop option pricing models based on stochastic volatility. Hull and ....

....values under stochastic volatility were not significantly different from Black Scholes values, except for long maturity options. For equity options, Christopher Lamoureux and William Lastarapes (1993) offer evidence against the assumption of a zero volatility risk premium. For currency options, Melino and Turnbull (1992) found that a random volatility model yields option prices that are in closer agreement with the observed option prices than those of the Black Scholes model. While the numerical methods and computers currently available allow computation of these stochasticvolatility option prices, they are still ....

Melino, Angelo, and Stuart Turnbull. "The Pricing of Foreign Currency Options with Stochastic Volatility." Journal of Econometrics 45 (1992): 239-65.

.... Option Pricing Model Introduction Since Black and Scholes (1973) originally developed their option pricing formula, researchers have developed models that incorporate stochastic volatility (see Heston (1993) and references therein) These models have been successful in market option prices as in Melino and Turnbull (1990), Knoch (1992) Nandi (1996) Bakshi, Cao, and Chen (1997) Bates (1996 a,b) The two types of volatility models have been continuous time stochastic volatility models and discrete time GARCH models. Continuous time stochastic volatility models are effective for option pricing, but can be ....

Melino, Angelo and Stuart Turnbull, 1990, "The Pricing of Foreign Currency Options With Stochastic Volatility," Journal of Econometrics, 45, 239-265.

.... suggested by Engle (1982) and its various developments and extensions such as the Generalized ARCH (GARCH) model of Bollerslev (1986) and the Exponential GARCH (EGARCH) model of Nelson (1991) A di#erent approach consists of the Stochastic Volatility (SV) model as suggested by Taylor (1986) Melino and Turnbull (1990), and Harvey, Ruiz and Shephard (1992) A great deal of empirical and theoretical research has been done on these models during the last decade and they have provided an improved description of financial markets volatility. Consequently, they have been used with success in option pricing, ....

Melino, A. and Turnbull, S.M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239--265.

....in which volatility is treated as a latent process, are capable of demonstrating the volatility smile, but invite signi cant challenges in computation. Frequentist methods for SV models include generalized method of moments (GMM) and ecient methods of moments (EMM) Chesney and Scott (1989) Melino and Turn2 bull (1990), Gallant, Hsieh and Tauchen (1991) Anderson and Sorensen (1995) Gallant, Hsieh and Tauchen (1997) quasi maximum likelihood (QML) Harvey, Ruiz and Shephard (1994) On the other hand, the Bayesian approach is a natural alternative for studying hierarchical SV models with a latent volatility ....

Melino, A. and Turnbull, S.M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics. 45, 239-265.

....measure, one has to impose assumptions about the pricing of volatility. Many papers investigated option prices under stochastic volatility for the case that volatility has zero systematic risk (i.e. the volatility risk premium is zero, see e.g. Hull and White, 1987, and Renault and Touzi, 1996) Melino and Turnbull (1990) allowed for nonzero, constant and exogenous volatility risk premia. As the empirical results of Wiggins (1987) show, the non pricing of changes in volatility may not be justified. Duan identified an equivalent martingale measure Q by requiring that the conditional return distribution remains ....

Melino, A., Turnbull, S.M. (1990) Pricing Foreign Currency Options with Stochastic Volatility, Journal of Econometrics 45: 239--265.

....f Deltap t g series appear reasonable, with all parameters save ff 10 quite statistically significant. The finding that a two factor stochastic volatility diffusion model can adequately describe the marginal dynamics of a price movement series f Deltap t g alone is consistent with the findings of Melino and Turnbull (1990), Engle and Lee (1996) Gallant and Long (1997) among others. However, as seen from the middle and bottom portions of Table 1, the two factor model has considerable problems accounting for either the marginal dynamics of the range series fd t g or the joint dynamics f Deltap t ; d t g. Also, in ....

Melino, Angelo and Stuart M. Turnbull, "Pricing Foreign Currency Options with Stochastic Volatility," Journal of Econometrics 45, (1990) 239--266.

....approach, we include an example of an exotic American chooser option [16] written on a barrier put and call, which has a complex early exercise constraint. 2. Stochastic Volatility. Recently, there has been some interest in models where the volatility of the underlying asset is a random variable [17, 21, 15, 27]. Consider an option which is a function of the asset price s and the variance v, where s and v evolve according to [15] ds = sdt p vsdz 1 dv = Gamma v)dt oe p vdz 2 (1) where z 1 ; z 2 are Wiener processes [28] Note that p v is the instantaneous volatility of the asset price ....

A. Melino and S.M. Turnbull. Pricing foreign currency options with stochastic volatility. J. Econometrics, 45:239--265, 1990.

....assumption, which we will call the necessity assumption, is that the ARCH volatility has additional explanatory power given other relevant variables. This amounts to the assumption 1. Important competitors to the ARCH models include the discrete time stochastic volatility models discussed by [Melino and Turnbull 1990], Wiggins 1987] Harvey et al. 1994] Jacquier et al. 1994] and [Shephard et al. 1998] Due to the ease of implementing ARCH models, and their corresponding dominance among practitioners we focus attentions on ARCH in this study. 2 1. Nonparametric Testing of ARCH for Option Pricing that the ....

Melino, A. and S. Turnbull. 1990. Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45:239-265.

....to Dahlhaus (1997) we choose to follow his notation and use the dual indexed random variable, X t;T , to describe a locally stationary series. 3 Time Varying Stochastic Volatility In this section we introduce a time varying version of the stochastic volatility model developed by Wiggins (1987) Melino and Turnbull (1990), Harvey et al. 1994) and Taylor (1994) keeping it general enough so that it also extends the long memory stochastic volatility model proposed by Breidt et al. 1998) Define y t;T to be the time varying stochastic volatility model y t;T = expfH t;T =2gffl t (3) Phi(t=T ; B) 1 Gamma B) ....

Melino, A. and M. Turnbull (1990) "Pricing Foreign Currency Options with Stochastic 19 Volatility," Journal of Econometrics 45, 239-265.

....asset variance is constant over the life of the option. Substantial empirical evidence has been presented, see e.g. Bollerslev, Chou, and Kroner (1992) that stochastic variance characterizes many financial return time series. A number of option pricing models, such as Hull and White (1987b) Melino and Turnbull (1990, 1995) Amin and Ng (1993, 1994) and Duan (1995) have been developed that allow for stochastic volatility. However, each model provides a different specification for the variance process. The method for selecting the appropriate variance process and thus the correct model for option pricing and ....

Melino, A. and S. M. Turnbull, 1990, "Pricing Foreign Currency Options with Stochastic Volatility," Journal of Econometrics, 45, 239-265.

....the Black Scholes model. Therefore he proposed to build a composite model or to correlate the bias of the option prices to macroeconomic variables. A widely proposed approach to improve modelling of asset prices is the introduction of a stochastic volatility (see Hull, White (1987) Scott (1987) Melino, Turnbull (1990), Bates (1996) Heston (1993a) showed that the prices of these models may give the characteristic W shape and skewness in comparison to Black Scholes prices. However, these models do not explain the empirically observed magnitude of the smile effect (see Scott (1987) Wiggins (1987) Taylor, Xu ....

Melino, A. and Turnbull, S. M. 1990. Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45: 239-265.

....data can be used to estimate this premium 1 . For this reason, we shall attempt this with a model that is highly parametric, but complex enough to reflect an important number of observed volatility features: 1. volatility is positive; 2. volatility is rapidly mean reverting (see for example [10]) 3. volatility shocks are negatively correlated with asset price shocks. That is, when volatility goes up, stock prices tend to go down and vice versa. This is often referred to as leverage [4] and it at least partially accounts for a skewed distribution for the asset price that lognormal or ....

A. Melino and S.M. Turnbull. Pricing Foreign Currency Options with Stochastic Volatility. J. of Econometrics, 45:239--265, 1990.

....to the square root process of Cox, Ingersoll and Ross (1985) and oe can be interpreted as the radial distance from the origin of a multidimensional OU process. Two other models of note were proposed by Johnson and Shanno (1987) who modelled both the price and volatility as CEV processes, and Melino and Turnbull (1990) who took the price to be a CEV process and the logarithm of the volatility to be an OU process. 2.4 Transition densities Consider the model dP t P t = oe t dB t dt (13) doe t = fl(oe t )dW t (oe t )dt (14) where, for the moment, B and W are independent Brownian motions. Then oe and B ....

....consequence most ARV models lack one step transition densities for the process Y n . This means that it is frequently not possible to obtain maximum likelihood estimates for parameter values. Instead parameter values are frequently estimated using methods of moments techniques, see Taylor (1986) Melino and Turnbull (1990) and Duffie and Singleton (1993) Of particular interest is the autoregressive co efficient OE which governs the persistence of volatility shocks. According to Taylor (1994) most estimates of this parameter which are based on daily observations yield values greater than 0.95. Harvey, Ruiz and ....

MELINO, A., and TURNBULL, S.M. (1990): Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45, 239-265.

....or nonzero correlation, whereas in ARCH GARCH, there is none. In the original EGARCH paper [30] Nelson used maximum likelihood estimation (with a non Gaussian random variable replacing the second Brownian increment dZ) and subsequent studies, especially in the stochastic volatility literature [8, 27, 35, 39] use the Generalized Method of Moments (GMM) In using GMM, it is necessary to choose which moments to match and what weighting matrix to use. Indeed, there is a trade off between a large number of moments potentially better exploiting the data, but greatly reducing the accuracy with which the ....

....as the stochastic volatility model) gives some guidelines in this regard, but they strongly caution against using too many moments for high frequency data series such as ours. The empirical work mentioned so far all used daily data for example, Scott [35] used 4 moments, while Melino Turnbull [27] used 47. We are after specific groupings of the original parameters that the theory (Section 3) highlights as most important, so we do not undertake a global search for (ff; fi; m; ae) at one attempt. In addition, we have a very large dataset whose points are non evenly spaced this would make ....

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A. Melino and S. Turnbull. Pricing foreign currency options with stochastic volatility. J. Econometrics, 45:239--265, 1990.

....to irregularly spaced observations raises no new technical issues. There are now a considerable number of papers which focus on various econometric aspects of the SV model. Most deal with generalized method of moment (GMM) type estimators, examples being Taylor (1994) Duffie and Singleton (1993) Melino and Turnbull (1990), Andersen and Sorensen (1994) and Vetzal (1992) Unfortunately we know that GMM may be inefficient relative to likelihood based procedures. Indeed there is some evidence that even the suboptimal quasi likelihood procedure put forward by Harvey, Ruiz, and Shephard (1994) 1 may be more efficient ....

Melino, A. and S. M. Turnbull (1990). Pricing foreign currency options with stochastic volatility. J. Econometrics 45, 239--265.

....the estimated implied volatility from the class of Black Scholes (1973) constant volatility models is subject to model specification errors. Recently, various stochastic volatility option pricing models have been developed, examples include Scott (1987) Hull and White (1987,1988) Wiggins (1987) Melino and Turnbull (1990), Stein and Stein (1991) Heston (1993) These models make it theoretically attractive to adopt a logically consistent stochastic volatility model, and to explore the implication of the stochastic volatility model in the study of implied variance. The out of sample tests focus on examining the ....

....rate volatility suggests that the estimated implied variance from the constant volatility Black Scholes (1973) model is subject to specification error. Recently, various stochastic volatility models have been developed, for instance, by Scott (1987) Hull and White (1987,1988) Wiggins (1987) Melino and Turnbull (1990), and Heston (1993) These models make it theoretically attractive to recover the market expectation of future variance from a stochastic volatility model. However, no published article has extracted implied variance from a stochastic volatility model. 2 To meet this challenge, this paper seeks ....

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Melino, A. and S. Turnbull, 1990,"Pricing Foreign Currency Options with Stochastic Volatility, " Journal of Econometrics, 45, 7-39.

....that the coefficients in the moving average expansion (5) are not restricted to be positive. However, asymptotic results about the estimators have proved extremely hard to obtain, even when d = 0. 2. 3 LMSV A different approach, based on stochastic volatility (SV) models like those discussed by Melino and Turnbull (1990) and Harvey, Ruiz and Shephard (1994) has many appealing features as we now discuss. The Stochastic Volatility model is defined by y t = oe t t ; oe t = oe exp(v t =2) 6) where fv t g is independent of f t g, f t g is independent and identically distributed (iid) with mean zero and ....

....strategies will be considered here since the LMSV model is much more complicated than ARSV. Other methods for estimation from SV models have been proposed. A method of moments (MM) estimator, which avoids the problem of evaluating the likelihood function, was suggested by Taylor (1986) and Melino and Turnbull (1990). While easy to implement, MM estimators for parameters in the ARSV model have a num Long Memory in Stochastic Volatility 13 ber of disadvantages. The MM method seems relatively ineficient when some kind of persistence in the autocorrelations is present, as it is the case of nearly ....

Melino, A. and and Turnbull, S.M. (1990). Pricing foreign currency options with stochastic volatility. J. Econometrics 45, 239--265.

....distributed (iid) sequence with finite variance is both a martingale difference sequence and a white noise. In the iid case, the past of the series contains no information about the present or the future. It has been shown, however, e.g. Clark, 1973; Tauchen and Pitts, 1983; Nelson, 1988; Melino and Turnbull, 1990; Harvey, Ruiz and Shephard, 1994) that series arising in finance and econometrics cannot always be assumed to be iid. While the martingale difference property often appears plausible, the variance in a given realization seems to change over time. In fact, it is often the case that powers of the ....

....evaluate, since it is expressed as a T dimensional integral, where T is the number of observations. Several methods for estimation from SV models have been proposed. A method of moments (MM) estimator, which avoids the problem of evaluating the likelihood function, was suggested by Taylor (1986) Melino and Turnbull (1990), and most recently Vetzal (1992) While easy to implement, the MM estimator was shown to be inefficient and to perform poorly over repeated sampling (Jacquier, Polson and Rossi, 1994) Nelson (1988) Harvey and Shephard (1993) Ruiz (1994) and Harvey et al. 1994) after expressing the SV model ....

[Article contains additional citation context not shown here]

Melino, A. and and Turnbull, S.M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239--265.

....data can be used to estimate this premium 1 . For this reason, we shall attempt this with a model that is highly parametric, but complex enough to reflect an important number of observed volatility features: 1. volatility is positive; 2. volatility is rapidly mean reverting (see for example [10]) 3. volatility shocks are negatively correlated with asset price shocks. That is, when volatility goes up, stock prices tend to go down and vice versa. This is often referred to as leverage [4] and it at least partially accounts for a skewed distribution for the asset price that lognormal or ....

A. Melino and S.M. Turnbull. Pricing Foreign Currency Options with Stochastic Volatility. J. of Econometrics, 45:239--265, 1990.

....variables. An independent and identically distributed (iid) sequence with finite variance is both a MD sequence and a white noise, but series arising in finance and econometrics often cannot be assumed iid even when the MD property appears plausible (e.g. Clark, 1973; Tauchen and Pitts, 1983; Melino and Turnbull, 1990). Instead, the variance in a given realization seems to change smoothly over time, in the sense that large observations tend to be followed by large observations and small observations by small observations, as Mandelbrot (1963) noted. Two approaches have been proposed to model time dependent ....

Melino, A. and and Turnbull, S.M. (1990). Pricing foreign currency options with stochastic volatility. J. Econometrics 45, 239--265.

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Melino A. and S. Turnbull, 1990, "The Pricing of Foreign Currency Options with Stochastic Volatility", Journal of Econometrics, 45, 239-265.

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Melino A. and Turnbull S.M. (1990), Pricing foreign currency options with stochastic volatility, Journal of Econometrics, 45, pp. 239-265.

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Melino, A. and S.M. Turnbull (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239--265.

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Melino, A. and S.M. Turnbull (1990). Pricing foreign currency options with stochastic volatility . Journal of Econometrics 45, 239--265.

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Melino, A. and S.M. Turnbull (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239--265.

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Melino, A. and S.M. Turnbull, 1990, Pricing foreign currency options with stochastic volatility, Journal of Econometrics 45, 239-265.

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Melino, A. and S. Turnbull, 1990, Pricing Foreign Currency Options with Stochastic Volatility, Joural of Econometrics 45, 239-265.

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Melino, A. and S. Turnbull (1990), \Pricing Foreign Currency Options with Stochastic Volatility," Journal of Econometrics, 45, 239-265.

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Melino A. and S. Turnbull (1990), Pricing foreign currency options with stochastic volatility, Journal of Econometrics, 45, 239-265.

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Melino, A., and Turnbull, S. (1990), "Pricing Foreign Currency Options With Stochastic Volatility", Jounal of Econometrics 45, 7-39.

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Melino, Angelo and Stuart Turnbull (1990), "Pricing Foreign Currency Options with Stochastic Volatility," Journal of Econometrics, 45, 239265.

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Melino, A. and S. Turnbull ,1990,"Pricing Foreign Currency Options with Stochastic Volatility, " Journal of Econometrics , 45, 7-39.

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