SCOTT, L. O., "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Vol. 22, 1987, pp. 419-438.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as an integral of a lookback option over the variance of the log price. In this set up we can obtain prices analytically if we can find a way to describe the distribution of the variance. We price the passport option under the Hull and White model [14] and the Stein and Stein [23] and Scott [21]) models using power series methods. In this way we show that implementing stochastic volatility models for pricing passport options is quite straightforward. The results are compared to the prices obtained in the Black Scholes type model. We discover that the Black Scholes passport option price ....

....We introduce the passport option problem in 3 and in 4 prove the form of the optimal strategy for the class of models. The following section discusses pricing the passport option in our stochastic volatility framework. Then 6 prices the passport option under the Hull and White [14] Scott [21] and Stein and Stein [23] models and compares prices to the simple exponential Brownian motion model. Section 7 concludes, and an appendix contains proofs of results delayed from the main text. 2 Stochastic volatility models We will consider a broad class of models, which covers many of the ....

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Scott L.O.; Option pricing when the variance changes randomly: theory, es- timation and an application, Journal of Financial and Quantitiative Analysis, 22, p419-438, 1987. 19

....jumps to the volatility process. In response to these findings, Eraker, Johannes, and Polson (2000) henceforth EJP) use returns data to investigate the performance of models with jumps in volatil1 Stochastic volatility option pricing has been considered in Hull and White (1987) Wiggins (1987) Scott (1987), Chesney and Scott (1989) Melino and Thornbull (1990) Stein and Stein (1991) and Amin and Ng (1993) among others. 2 ity as well as prices using the class of jump in volatility models proposed by Du#e, Pan, and Singleton (2000) The DPS class of models generalizes the models in Merton (1976) ....

Scott, L. O. (1987). Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application. Journal of Financial and Quantitative Analysis, 22, 419--483.

....modeling. Meanwhile, it is evident that the relevant notion of daily volatility in this setting becomes , d . This 0 t 12 quantity is also of central importance for the pricing of derivative securities under stochastic volatility; see, e.g. Hull and White (1987) Melino (1994) Scott (1987) and Wiggins (1987) Equation (6) shows that r continues to provide an unbiased, albeit noisy, estimator of the relevant latent volatility factor for (1) t 1 2 daily returns, generalizing the results from the discrete time setting discussed earlier. 3.1 Continuous Time Modeling of Daily ....

Scott, L. (1987), "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, 22, 419-436.

....of alternative option pricing models, we use the 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 ....

....the same. This is a consequence of the fact that the individual t values are asymptotically equal with probability one in case of only one degree of freedom in the test. 9 Hull White (1987) Johnson Shanno (1987) Bailey Stulz (1989) Stein Stein (1991) and Heston (1993) 10 See e.g. Scott (1987), Wiggins (1987) Chesney Scott (1989) Melino Turnbull (1990) and Bakshi et al. 1997) 15 appropriate compensation for systematic asset, volatility, interest rate, or jump risks. Similarly, the no arbitrage models show that option prices are discounted future payoffs at the riskfree rate ....

Scott, L. O. (1987), `Option pricing when the variance changes randomly: Theory, estimators, and applications', Journal of Financial and Quantitative Analysis 22, 419--438.

....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 White as well as Scott made the questionable assumption that the risk premium of volatility is zero that is, the volatility risk is not priced in the options market and ....

Scott, Louis O. "Option Pricing When the Variance Changes Randomly: Theory, Estimation, and Application." Journal of Financial and Quantitative Analysis 22 (1987): 419-38.

.... late eighties and early nineties: V was modeled as a geometric Brownian motion (Hull White 1987) as a CoxIngersoll Ross process (Hull White 1988, Heston 1993) as the exponential of a Ornstein Uhlenbeck process (Wiggins 1987, Chesney Scott 1989) and as a squared Ornstein Uhlenbeck process (Scott 1987, Stein Stein 1991) The above papers all focus on pricing of a European call option written on a stock with price process P. Pricing is investigated for fixed value of the parameter q in the equation for V , and the majority of the papers pay no or little attention to estimation of q . Only ....

....Stein Stein 1991) The above papers all focus on pricing of a European call option written on a stock with price process P. Pricing is investigated for fixed value of the parameter q in the equation for V , and the majority of the papers pay no or little attention to estimation of q . Only Scott (1987) and Chesney Scott (1989) address the problem seriously and derive moment like estimators for the parameters. More recently, several estimation approaches have been suggested, some of which have earlier been applied for the discrete time versions of the models; see Shephard (1996) and Ghysels et ....

Scott, L. O. (1987), `Option pricing when the variance changes randomly: theory, estimation and an application', Journal of Financial and Quantitative analysis 22, 419--438.

.... late eighties and early nineties: V was modeled as a geometric Brownian motion (Hull White 1987) as a CoxIngersoll Ross process (Hull White 1988, Heston 1993) as the exponential of a Ornstein Uhlenbeck process (Wiggins 1987, Chesney Scott 1989) and as a squared Ornstein Uhlenbeck process (Scott 1987, Stein Stein 1991) All these papers focus on pricing of a European call option written on a stock with price process P = e X . Pricing is investigated for fixed value of the parameter q in the equation for V , and the majority of the papers pay no or little attention to estimation of q . ....

....Stein Stein 1991) All these papers focus on pricing of a European call option written on a stock with price process P = e X . Pricing is investigated for fixed value of the parameter q in the equation for V , and the majority of the papers pay no or little attention to estimation of q . Only Scott (1987) and Chesney Scott (1989) address the problem seriously and derive moment like estimators for the parameters. More recently, several estimation approaches have been suggested in the statistics literature. Below we briefly review the basic ideas; see Srensen (2000, Section 3.4) for a more ....

Scott, L. O. (1987), Option pricing when the variance changes randomly: theory, estimation and an application, J. Financial and Quantitative analysis 22, 419--438.

.... GARCH literature we refer to Bollerslev, Chou and Kroner (1992) Bera and Higgins (1993) Bollerslev, Engle and Nelson (1994) and Diebold and Lopez (1995) SV models are reviewed in, for example, Taylor (1994) Ghysels, Harvey and Renault (1996) and Shephard (1996) 2 See Hull and White (1987) Scott (1987) and Wiggins (1987) and Chesney and Scott (1989) 1 Although SV models are seen as a competitive alternative to GARCH models their empirical application has been limited. This can mainly be attributed to the difficulties that arise as a result of the intractability of the likelihood function ....

Scott, L.O. (1987), Option Pricing when the Variance Changes Randomly: Theory, Estimation and Application, Journal of Financial and Quantitative Analysis 22, 419-438.

.... literature we refer to Bollerslev, Chou and Kroner (1992) Bera and Higgins (1993) Bollerslev, Engle and Nelson (1994) and Diebold and Lopez (1995) SV models are reviewed in, for example, Taylor (1994) Ghysels, Harvey and Renault (1996) and Shephard (1996) 2 See Hull and White (1987) Scott (1987) and Wiggins (1987) and Chesney and Scott (1989) 1 Although SV models are seen as a competitive alternative to GARCH models their empirical application has been limited. This can mainly be attributed to the difficulties that arise as a result of the intractability of the likelihood function ....

Scott, L.O. (1987), Option Pricing when the Variance Changes Randomly: Theory, Estimation and Application, Journal of Financial and Quantitative Analysis 22, 419-438.

....of 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) ....

....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 (1993) Clewlow, Xu (1993) Bates (1997) Processes with jumps and mixed jump diffusion processes were introduced by Merton (1976) Naik, Lee (1990) Madan, Milne (1991) Heston (1993b) Bates (1991) especially to improve the tail behaviour of the stochastic ....

Scott, L. O. 1987. Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of Financial and Quantitative Analysis 22: 419-438.

....approach of [8] reviewed in [16, Section 1.2] or [3] in which volatility oe t is modeled as an Ito process driven by a Brownian motion that has a component independent of the Brownian motion W t driving the asset price. There has been a lot of analysis of specific Ito models in the literature [14, 17, 7] by numerical and analytical methods, many of which have ignored skew effects and or the volatility risk premium for tractability. Our goal [6] is to estimate these parameters from market data and to test their stability over time and thus the potential usefulness of stochastic volatility models ....

....(or, equivalently, log oe t is mean reverting OU) With a suitable initial distribution, the volatility process is stationary and ergodic which allows us to use averaging principles to approximate the option price, separating the minor and major fluctuations. This model has been considered in [14] and it is related to EGARCH models which, as shown in [11] are weak approximations to the continuous time diffusion. Another model that is stationary and can be similarly implemented and analyzed is when oe t is a mean reverting Feller (or Cox Ingersoll Ross) process [3] The final ingredient is ....

L. Scott. Option Pricing when the Variance changes randomly: Theory, Estimation, and an Application. J. Financial and Quantitative Analysis, 22(4):419--438, December 1987.

....of the 90 day historic volatility based on the above data. This limited evidence supports the contention that stock volatility is not constant and moreover that volatility shocks persist through time. This conclusion was reached by Mandelbrot (1963) Fama (1965) Blattberg and Gonedes (1974) and Scott (1987) amongst others. Stochastic volatility models are needed to describe and explain volatility patterns. 1.3 The Black Scholes paradigm and Option pricing One of the key contributions of mathematics to finance has been the development of formulae for the pricing of options and other derivative ....

....and more sophisticated models for the volatility are required which allow for further random changes in the level of volatility. In response to this need a series of models for asset price processes were proposed in the late 1980s which took volatility as an exogenous stochastic process. Scott (1987), Wiggins (1987) Hull and White (1987, 1988) Stein and Stein (1991) and Heston (1993) each proposed models of the form dP t P t = oe t dB t dt (8) where oe t , the stochastic volatility process, is itself the solution of a stochastic differential equation. Several candidate SDEs for the ....

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SCOTT, L.O. (1987): Option pricing when the variance changes randomly: theory, estimation and an application. Journal of Financial and Quantitative Analysis, 22, 419-438.

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SCOTT, L. O., "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Vol. 22, 1987, pp. 419-438.

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Scott, L.O. (1987). Option pricing when the variance changes randomly: Theory, estimation and an application. Journal of Financial and Quantitative Analysis 22, 419--439.

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Scott, L.O. (1987). Option pricing when the variance changes randomly: Theory, estimation and an application. Journal of Financial and Quantitative Analysis 22, 419--439.

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L. Scott. Option pricing when the variance changes randomly: theory, estimation and an application. Journal of Financial and Quantitative Analysis, 22:419--438, 1987.

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Scott, L., 1987, Option Pricing When the Variance Changes Randomly: Theory, Estimation and An Application, Jouvral of Financial and Quantitative Analysis 22, 419-438.

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Scott, L. (1987), \Option Pricing when the Variance Changes Randomly: Theory, Estimators and Applications," Journal of Financial and Quantitative Analysis, 22, 419-438.

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Scott, L. O. (1987), `Option pricing when the variance changes randomly: theory, estimation and an application', Journal of Financial and Quantitative analysis 22, 419--438.

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Scott L. (1987), Option Pricing When the Variance Changes Randomly: Theory, Estimation and An Application, Journal of Financial and Quantitative Analysis, vol. 22, 419-438.

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Scott L. (1987), Option Pricing When the Variance Changes Randomly: Theory, Estimation and An Application, Journal of Financial and Quantitative Analysis, vol. 22, 419-438.

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Scott L.O., Option pricing when the variance changes randomly: Theory, estimation, and application, Journal of Financial and Quantitative Analysis, 22, 419-438.

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Scott, L. [1987], Option Pricing when the variance changes randomly: Theory, Estimation, and an Application, Journal of Financial and Quantitative Analysis, Vol. 22, No. 4, pp. 419-438.

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Scott, L.O. (1987), "Option Pricing When the Variance Changes Randomly: Theory, Esimation, and an Application", Journal of Financial and Quantitative Analysis 22, 419-438.

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L.O. Scott- 1987- Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application- Journal of Financial and Quantitative Analysis- 22, 419-438.

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